Acoustical phenomena produce simple patterns that build on each other to produce complex musical patterns. The overtone series suggests scales which build into more and more complex chords. Musicians develop notational systems to represent these structures. The mathematics of combinatorics and software allows the presentations of large volumes of musical structures like scales, modes, and chords. These structures eventually evolve into complex mathematical patterns separated from their original context. Evolution of these patterns leads to representations of musical elements such as timbre, harmony, rhythm, melody, and form. Eventually, these patterns break down into music's true purpose: to express ephemeral human emotion and to aid in evolution.

Introduction

This document began as a simple introduction on how all of the complexities of music derive from variations, and how these patterns effect the main part of the Pattern Obsession guitar course. Eventually it grew organically into this quite complex manifesto, as pattern based mathematical functions tend to do.

Much of this text may be difficult for people without training in music theory. This is not a traditional theory text with "every good boy does find" or explaining half notes and whole notes. Instead, the text shows the underlying patterns that produce the music and then goes back later and explains the nomenclature as needed. Some theory concepts do not occur at all, as they are interspersed in the guitar method when necessary. The philosophy is to learn the music patterns first, then put the labels on it later.

Unfortunately, to write a_bout the increasing complexity if music, use of the jargon is necessary. The section of chord theory is particularly thick, and will lose even a lot of accomplished musicians. So, read through the text, and when your eyes start glazing over because of all the chord symbols or math, skip forward to a new section.

Nature of Sound

Consider a set with all possible musical combinations.

First start with the most basic sound possible. Sound occurs when objects vibrate in a medium, usually air, but possibly water, metal or any medium that can vibrate. Despite all the rumbling of ships passing by in Star Wars, no sound propagates in space without air. (The movie Alien got it right – "In space no one can hear you scream.")

To produce a sound, first disturb an object in space. For this theoretical example strike a perfectly flat surface that is infinitely rigid, like an impossibly stiff chalkboard suspended in the air in a frictionless manner. Striking the chalkboard causes it to move away from the listener slightly. When the surface moves away from the listener, the air molecules stretch apart slightly, a phenomenon known as rarefaction. Now consider this surface suspended by a perfectly efficient spring that wants the surface to return to its natural position. Striking the surface causes it to move away quickly at first, but as the spring begins to push back, the object slows down and eventually stops moving away. Once the energy of the initial disturbance dissipates, the surface moves back toward the listener, slowly at first and then gaining speed until it reaches its original position. The object moving back toward the listener eventually nullifies the rarefied air until the air pressure returns to whatever it began with.

But Newton proved that energy has to go somewhere, so the energy of the initial strike has produced a momentum of the moving surface. The surface does not stop, but continues toward the listener quickly at first, but eventually slowing down and stopping the same distance from the starting point as the initial movement away from the listener as the spring a_bsorbs the energy of the movement. As the surface moves toward the listener, the air molecules compress in a manner that parallels the movement of the surface.

But now the spring a_bsorbs the energy, and so wants to move the surface back to its original position, slowly at first, then increasing speed. Once back in the original position, the surface now has momentum the other direction and wants to repeat the cycle. Nature a_bhors a vacuum, so the air molecules next to the ones being compressed react by moving into the space with the lower air pressure. Then the compressed molecules that follow push the alternating compressions and rarefactions through the air until they eventually reach the ear drum. The mechanics of the ear translate these vibrations into what the brain perceives as sound.

Eventually the friction of the air would slow the object, but for this theoretical example, instead of air, imagine a perfectly consistent medium with no friction. Now, nothing would stop the cyclic movement of the surface moving away from the listener, back to its starting point, toward the listener the same distance, back to its starting point, and the repeating the cycle forever.

Sine Waves

Mathematically, the sine wave, y=sin(x) expressed mathematically, represents the compressions and rarefactions of the medium. Sine waves dominate nature from the waves on a pond, to the propagation of light, to the rolling of hills in geology.

The sine wave representing sound begins at the 0 point being the natural pressure of the medium at equilibrium, one standard atmosphere (or 14.7 pounds per square inch, 1,013.25 × 10 ³ dynes per square centimeter, 1,013.25 millibars, or 101.325 kilopascals using other measurement systems). The sine wave moving higher represents the air becoming more compressed, quickly at first, then tapering off at 90 degrees (π/2 in radians) returning to the equilibrium point at 180 degrees (π) reaching its most compressed state at 270 degrees (3π/2) and returning to the starting point at 360 degrees (2π). The number of times this cycle repeats in a second is called it frequency, measured in Hertz (Hz).

Many sounds wash over us as the brain develops, but certain sets of tones stick out. All sounds can be musical, but music tends to be categorized into sets of sounds. The set of sounds associated with melody and what the Western ear considers music is dominated by timbres with a musical pitch.

Pitch

Again, a set of all possible music would contain many notes we do not consider musical. Between the pitches that Western music considers acceptable notes are an infinite number of notes that sound out of tune to those reared listening to the standard scale. An Asian Indian folk singer might use these pitches, but mostly they are ignored as nonmusical.

Western music settles on a group of notes that approximate and subdivide the natural tones produced by a vibrating object. These notes sound most pleasing because they approximate mathematical patterns inherent in vibrating objects.

Overtones

A vibrating string moves back and forth at a certain frequency measured in cycles per second (Hz); a frequency which we perceive as the fundamental pitch of the note. But parts the string also vibrate naturally in other frequencies that mix to produce the timbre, or tone color, of the instrument.

A string tuned to A440, standard tuning pitch, vibrates at 440 times per second, but also has a second set of vibrations at double that frequency, 880 Hz, occurring halfway up the string. A guitar player can release this vibration by lightly touching the string half way, at the 12^th fret, producing a harmonic. (Think the opening notes of "Rounda_bout" by Yes). Since the original 440 pitch and the 880 pitch occur so often together, the human mind perceives the 880 frequency as actually the same pitch as 440, just higher. A note with a doubled frequency is an octave, or the interval (distance between two notes) of a Perfect 8th (P8) musically. (Why certain intervals are perfect and others are major, minor, diminished, or augmented is not important right now. Just observe the labels and their function will fall into place naturally and with further study.)

Even though the string is actually producing multiple sine waves at the same time with multiple frequencies, the brain tends to group these frequencies together rather than hearing them separately, just as different frequencies of light mix together to form colors. The string also naturally vibrates in thirds, at the seventh fret harmonic on a guitar. The frequency of this vibration is three times the fundamental, or 1320 Hz. (For a mix of seventh fret and 12^th fret harmonics, listen to the opening of "Red Barchetta" by Rush.) This vibration is so common that the mind perceives the distance between the notes as the most consonant interval, the perfect fifth (P5), or sol in solfege. (Solfeggio is do re mi f sol la ti do, listen to "Doe, a Deer" from The Sound of Music.)

The P5 interval is so universal that it occurs in many songs, think the opening notes to the Star Wars Theme. A chord based on the fifth scale degree is known as the dominant (V), which signals the return of the tonic (I), or root chord of a key (The chord built on the first note of the scale). This dominant-tonic relationship is ubiquitous in Western music, and the basis for many chord progressions.

The next interval in the overtone series is at four times the original frequency. Since four times the frequency is twice the frequency again doubled, the mind perceives this pitch as an octave above the octave at 880, or two octaves above the fundamental. The musical distance between the fifth and the octave above it is known as a perfect fourth (P4), the second most consonant interval. (Think the opening notes of "Here Come the Bride"). The chord built on the fourth degree of the scale is known as the subdominant (IV). Together the I, IV, and V chords are known as the primary chords. The subdominant-dominant-tonic relationship is the basis for cadence (ending a phrase) structures in Western music. So, see how the structures of western music all derive from a mathematical relationship based on the physics of sound.

Continuing up the overtone series, the next two naturally occurring intervals are the major third (M3) and the minor third (m3). Combinations of major thirds and minor thirds are the basis of the most common chord structures. A note plus a major third plus a minor third above it is known as a major chord, the most common chord. The minor chord is simply the two reversed (root, m3, M3.). Again, chord structure is built on naturally occurring mathematical relationships built of multiples of the fundamental frequency. Ancient music theorists did not consider the third to be a consonant interval, only fourths and fifths, but the modern ear hears the third as consonant because of its wide use in modern chord structure. Chords that sound good derive from natural mathematical sequences.

The next overtone is a minor seventh (m7) above the nearest octave of the fundamental. Chords built on the common triads are the next most commonly used chords, the seventh chord (7), which is a major chord plus m7 interval, and a minor seventh (m7), a minor chord plus a m7 interval ("m7" can designate the interval of the chord, depending on context).

The next two overtones are another octave, and then an interval a major second (M2) above that. The major second is the second scale degree, re, and also added to seventh chords to make ninth chords M9 and m9. In the scale, a ninth is essentially the same as the 2nd an octave up since the scale repeats after seven notes.

1 2 3 4 5 6 7 8 9

1 2 3 4 5 6 7 1 2

The overtones continue to eventually contain all notes, but they become functionally difficult to access and increasingly out of tune on any physical instrument. For some great harmonic work on the guitar, listen to "Top Jimmy" by Van Halen.

So, the intervals derived from the overtone series listed so far are

P1 P8 P5 P4 M3 m7 M2

(P1 designates a "unison" interval, or the fundamental frequency.)

Rearranged in the order from their nearest root note gives

P1 M2 M3 P4 P5 m7 P8

Starting arbitrarily on the note g is the easiest to express these intervals in musical notation. Musically, these intervals can translate to the notes

g a b c d f g

Again, why these notes have these labels is less important than knowing their function for now. The could just as easily be labeled with numbers or colors or animal names. Tradition dictates they be labeled with letters, lower case here to avoid confusion later when capital letters stand for something else.

Now we have six of the seven notes of a traditional grouping of notes, a scale. All the notes with consecutive letters are the same distance from the note preceding them with the exception of the interval from b to c. The distance for b to c is half the distance of the others, so this interval is a half-step or minor second (m2). The other more common distance is the whole step or major second (M2), which also derived from the overtone series.

Also troubling is the interval from d to f. The distance from d to f is three half steps, or minor third (m3), a larger interval than the other scale tones, and why the letter e is missing to leave room to put in another note. Pattern completion demands that interval be filled in. Without knowing from modern scales what note goes in the space, the e could be a half step or a whole step from the d. This ambiguity of the sixth scale degree can be exploited for variation later, but for now just define the sixth e as a whole step above the d just below it.

gabcdefg

Which is called the Mixolydian scale. Jazz players use Mixolydian applies to improvise over the dominant chord, so use this scale to improvise over the dominant chord (V) (More explanation on that later).

Modes

Do not be confused that this scale does not sound exactly like the "Doe, a Deer" song learned in elementary school. The modern major scale is not chiseled in stone, but derived from centuries of tradition. To explain how, consider the concept of modes.

All scales loop around on themselves naturally, so the scale from above becomes

g a b c d e f g a b c d e f g a …

Instead of starting on the first note as g, begin with c and work around the scale

cdefgabc

Now this scale sounds like the scale learned in elementary school.

To answer why the Mixolydian scale seems misplaced to modern ears, first explore all the modes. Ancient music often used the same combinations of scale relationships that modern music does, but not always with the same starting note.

Modes are scales begun on different beginning pitches, designated by the following labels:

Ionian Dorian Phrygian Lydian Mixolydian Aeolian Locrean

Using the notes from above, the modes would be:

Ionian	c	d	e	f	g	a	b
Dorian	d	e	f	g	a	b	c
Phrygian	e	f	g	a	b	c	d
Lydian	f	g	a	b	c	d	e
Mixolydian	g	a	b	c	d	e	f
Aeolian	a	b	c	d	e	f	g
Locrean	b	c	d	e	f	g	a

Descriptions of the modes

Ionian	Today's major scale
Dorian	Minor with a raised 6^th. Great for solos. Listen to "Greensleeves"
Phrygian	Rarely used, Eastern feel
Lydian	Nice raised 4^th alternate to major. Listen to Rush "Free Will" intro.
Mixolydian	The most mathematically natural. Solos over dominant V7 chord
Aeolian	Today's minor scale. Compare "What Child is This" to "Greensleeves"
Locrean	Very rarely used

Jazz players use these modes for solos that fit the chord progressions.

Leading Tones

Dorian Lydian, and Mixolydian scales were very common in ancient music, but Ionian became the standard for the modern ear. The reason Ionian sounds better than the Mixolydian scale that seems to derive from the overtone series proba_bly has to do with the concept of note "gravity."

Scales can contain and combinations of whole steps and half steps. When a note in a scale is a half-step away from another note, the "gravity" of the closer note pulls the melody toward that note.

For ease of demonstration, most of the following exercises are in the key (a key is a group of common notes) of C,

cdefgab

with no sharps or flats (a flat, b, is an accidental lowers a note in the same way a sharp raises it.) The following chart show the various classifications of the notes of the scale.

Note	c	d	e	f	g	a	b	c
Scale degree	1	2	3	4	5	6	7	8
Interval from tonic	P1	M2	M3	P4	P5	m6	M7	P8
Whole or Half Step		W	W	H	W	W	W	H
Solfege	Do	Re	Mi	Fa	Sol	La	Ti	Do

Pentatonic Scales

Pentatonic scales form an integral part of rock guitar playing. Five note pentatonic scales are the major or minor scales with two notes left out.

For major pentatonic, leave out the 4^th and 7^th scale degrees from the major scale

In C: c d e g a

Pentatonic scales are popular because all the notes in the scale sound good over most common musical changes.

Chords

Chords are groups of three or more notes played at the same time. Seeing a complicated chord like Em11b13b9 is intimidated to most musicians because of its complexity. Finding the chord in a reference table and then memorizing is one solution, but does not prepare the player for the next unfamiliar chord.

Once the player understands that most chords, even complicated jazz chords like Em11b13b9, derive from the patterns out of scales already known, the process leads to understanding of why these chords sound good and their functionality.

Major Triads

The patterns continue as the major scale stacks into chords. Begin with the standard major scale

Build a three-note chord (triad) by starting on C and skipping every other note.

c e g

These notes form is the C chord, the first and most common chord in the key of C.

The chord based on the first scale degree is called the Tonic, signified by the Roman numeral I.

Build the next chord starting on the fifth scale degree, G. Wrap the scale to supply all the notes

Start at g and skip every other note, yielding

gbd

These notes form the G chord, the second most common chord in C. The chord based on the fifth scale degree is called the dominant (V).

Continuing to build chords horizontally not efficient, and does not model chords in the real world since chords on the musical staff build vertically.

So, arrange the notes vertically with c at the bottom and the scale going up.

Build the next chord by starting on the 4^th scale degree, f and moving up, yielding

fac

These notes form the F chord, the next most common chord in C. The chord based on the fourth scale degree is called the sub-dominant (IV).

Together the I, IV, and V chords are known as the primary chords. All the primary chords are major chords. These three chords make up a surprising number of songs.

The next chord builds from the second scale degree. Rearrange the column to put the second degree, D, on the bottom, then skip every other note.

Yielding dfa, the D minor chord. The chord based on the second scale degree is called the super tonic (ii). Note that the Roman numeral representation is in small case for minor chords. The ii chord is a very common substitution for the IV chord in jazz. Instead of playing IV-V-I, a jazz player would substitute ii-V-I. This substitution works because the ii chord and the IV chord share two common notes. In C, ii is dfa and IV is fac.

Since table manipulation is so easy with today's software, for the next chord, use this table that has all the possible modes built vertically.

b	c	d	e	f	g	a
a	b	c	d	e	f	g
g	a	b	c	d	e	f
f	g	a	b	c	d	e
e	f	g	a	b	c	d
d	e	f	g	a	b	c
c	d	e	f	g	a	b

This table is from the modes section, just transposed into the key of C.

Next, find the triad built from the sixth scale degree, a, yielding

ace

the A minor chord. The chord based on the sixth scale degree is called the submediant (vi). Submediant chords can be used as a substitute for the tonic or as a pivot for a key change to the relative minor key.

This table is useful, but every other row is not being used, so why not just delete them?

b	c	d	e	f	g	a
g	a	b	c	d	e	f
e	f	g	a	b	c	d
c	d	e	f	g	a	b

Now all the chords lay out nicely. Use this table to find the chord based on the third scale degree

egb

The chord based on the third scale degree is called the mediant (vi). The mediant chord can act as a substitute for V or as a secondary dominant leading to vi, but is the least common of these chords.

The last triad is an unusual tonal color that adds functionality to chord progressions.

bdf

Based on the seventh scale degree is neither major nor minor since two minor triads stack on each other forming a diminished chord. The interval between the root and the fifth of this chord is a diminished fifth (d5). A diminished designation lowers a perfect interval a half step in the same manner a minor designation lowers a major interval.

The chord based on the seventh scale degree is known as the leading chord (vii_o) since the leading tone of B for the root gets sucked into the gravity of the C chord and wants to lead back to the tonic. vii_o can also be substitute for V or as a pivot chord to many key changes.

Root	Chord	Spelling	Roman	Function	Uses
c	C	ceg	I	Tonic	Establishes key
d	Dm	dfa	ii	Supertonic	Substitute for IV
e	Em	egb	iii	Mediant	Substitute for V
f	F	fac	IV	Subdominant	Precedes V
g	G	gbd	V	Dominant	Precedes I
a	Am	ace	vi	Sub Mediant	Substitute for I
b	Bo	bdf	vii_o	Leading	Substitute for V

Seventh Chords

The pattern continues with seventh chords. The techniques from the lesson on building triads grow directly into adding the next chord in the sequence, seventh chords. Seventh chords are simply the major triads with another note put on top. The new note is another third a_bout the last note in the triad.

Luckily, the work for completing a chart for building sevenths chords is already complete.

b	c	d	e	f	g	a
g	a	b	c	d	e	f
e	f	g	a	b	c	d
c	d	e	f	g	a	b

Simply read the columns bottom up, but add the fourth note.

So, the seventh chord built on the first scale degree is

cegb

A major chord with a major third added is a major 7^th (Maj7), or here CMaj7.

Roman numerals, Functions, and uses for seventh chords parallel those for major triads.

The seventh chord built on the fifth scale degree is

gbdf

A major chord with a minor third added is a 7^th chord (7), or here G7.

The seventh chord built on the second scale degree is

dfac

A minor chord with a minor third added is a minor 7^th chord (m7), or here Dm7.

The seventh chord built on the seventh scale degree is

bdfa

A diminished chord with a major third added is a half diminished 7^th chord (m7b5), or here Bmj7b5. The label Bo7 would seem to fit the pattern here, but that designation refers to a fully diminished seventh chord, which only occurs in minor keys naturally (explained fully later).

Full explanation of why chord symbols might have a flat or sharp designation will need to wait until after the explanation of key changes. Feel free to jump ahead to that section if not knowing what the chord symbols means bothers you, but for now, a better strategy for understanding is to just a_bsorb the chords and their functions, so that the understanding of the nomenclature will evolve naturally.

Root	Chord	Spelling
C	CMaj7	cegb
D	Dm7	dfac
E	Em7	egbd
F	F7	face
G	G7	gbdf
A	Am7	aceg
B	Bm7b5	bdfa

Ninth, Eleventh, and Thirteenth Chords

While ninth, eleventh, and thirteenth chords seem daunting, they are simply formed by adding more rows to the process already developed.

First, extend the chart to finish all possible notes in thirds

a	b	c	d	e	f	g
f	g	a	b	c	d	e
d	e	f	g	a	b	c
b	c	d	e	f	g	a
g	a	b	c	d	e	f
e	f	g	a	b	c	d
c	d	e	f	g	a	b

To find the ninth chord on the first scale degree (CMaj9), read up the first 5 rows

cegbd

To find the eleventh chord on the first scale degree (CMaj11), read up the first 6 rows

cegbdf

To find the thirteenth chord on the first scale degree (CMaj13), read all rows

cegbdfa

Root	9th Chord	Spelling	11^th Chord	Spelling	13^th Chord	Spelling
c	CMaj9	cegbd	CMaj11	cegbdf	CM13	cegbdfa
d	Dm9	dface	Dm11	dfaceg	Dm13	dfacegb
e	Em7b9	egbdf	Em11b9	egbdfa	Em11b13b9	egbdfac
f	FMaj9	faceg	FMaj9#11	facegb	FMaj13#11	facegbd
g	G9	gbdfa	G11	gbdfac	G13	gbdface
a	Am9	acegb	Am11	acegbd	Am13	acegbdf
b	Bm7b9b5	bdfac	Bm11b9b5	bdface	Bm11b13b9b5	bdfaceg

(Full explanation of the chord symbols comes later.)

Substitutions

Parsing the chord names is quite intimidation. But being able to break down Em11b13b9 is not as important as recognizing that this is a diatonic chord in the key of C. When reading music in the key of C, the appearance of this chord symbol should trigger that this is a 13^th chord based on the third scale degree. So, if the chord is unknown, just play an Em7, and the function is the same.

Careful reading of the patterns provides a more sophisticated way playing chords that function the same as the higher note chords. Playing a full thirteenth chord is impossible on the guitar because the six strings cannot play seven notes. So, the payer must choose notes to leave out.

The first note to leave out when making a substitution is the root note because that note is played by the bass payer anyway. In fact, leaving out the root on the lower pitched strings is a great jazz technique even on chords that are fully playable so as not to interfere with the bass player.

Consider the CMaj9 chord, cegbd. Removing the root note c yields

egbd

This is an Em7 chord, which is much more playable. Now look at the chart for seventh chords from above.

Root	Chord	Spelling
C	CMaj7	cegb
D	Dm7	dfac
E	Em7	egbd
F	FMaj7	face
G	G7	gbdf
A	Am7	aceg
B	Bm7b5	bdfa

The Em7 chord is the third row of this chart. Compare the seventh chart with the ninth chart, but move the Em7 chord to the first row, and all other chords up two rows, moving the first two rows to the bottom.

Root	9th Chord	Spelling	7th Chord	Substitution
C	CMaj9	cegbd	Em7	egbd
D	Dm9	dface	F7	face
E	Em7b9	egbdf	G7	gbdf
F	FM9	faceg	Am7	aceg
G	G9	gbdfa	Bm7b5	bdfa
A	Am9	acegb	CMaj7	cegb
B	Bm7b9b5	bdfac	Dm7	dfac

So, the table of seventh chords already mastered can substitute for the more complicated ninth chords without any loss of function, and actually sound better in jazz. Just be careful not to play guitar voicings with heavy emphasis on the lower pitched strings. An Em7 chord played with an open low E string will sound wrong because the low E will interfere with the bass player and also sound like the root of a chord. Simply mute the lover strings when doing substitutions.

Eleventh and Thirteenth Chord Substitutions

Since an eleventh chord is simply a ninth chord with another note on top, five-note ninth-chord substitutions for eleventh chords follow directly from the previous chart. Just use one of the pre_built 9^th chords based on the third scale degree.

11^th Chord	Spelling	9th Chord Substitution
CMaj11	cegbdf	Em7b9	egbdf
Dm11	dfaceg	FMaj9	faceg
Em11b9	egbdfa	G9	gbdfa
FM9#11	facegb	Am9	acegb
G11	gbdfac	Bm7b9b5	bdfac
Am11	acegbd	CMaj9	cegbd
Bm11b9b5	bdface	Dm9	dface

But, if the player is using substitutions for higher order chords because of lack of knowledge of how to play these chords, memorizing the difficult ninth chords may not be an option. The simplicity of the four-note substitution for the ninth chord is quite appealing since one would assume anyone attempting an eleventh chord already has a grasp of seventh chords.

A quick and dirty substitution for an eleventh chord might be to just eliminate the first two notes of the chord, building a seventh chord base on the fifth of the original chord.

11^th Chord	Spelling	7th Chord Substitution
CMaj11	cegbdf	G7	gbdf
Dm11	dfaceg	Am7	aceg
Em11b9	egbdfa	Bm7b5	bdfa
FMaj9#11	facegb	CMaj7	cegb
G11	gbdfac	Dm7	dfac
Am11	acegbd	Em7	egbd
Bm11b9b5	bdface	F7	face

However, this substitution, while simple, is hardly satisfying musically. With ninth chords, leaving out the root note is acceptable since the bass player supplies that note, or in solo work, the brain subconsciously imagines the note from the context of the progression. The third of the chord is actually a very important note because the third provides the information on whether the cord is major or minor.

A better substitution would be to leave out the root and the fifth of the chord. Since the fifth is the same for major and minor chords, the fifth is not essential for identifying the chord function.

Take the remaining notes and build a suspension based on the third above, as before. For example, the eleventh chord built on I in C is cegbdf. Leaving out the c gives egbdf. Also leaving out the fifth leaves ebdf. This could be an E chord with a suspended lowered second. For the 13^th chord, replace the f with a sus4.

11th		sus substitution		13th		sus substitution
I	Maj11	iii	7susb2	I	Maj13	iii	7sus4
ii	m11	IV	Maj7sus2	ii	m13	IV	Maj7sus#4
iii	m11b9	V	7sus2	iii	m11b9b13	V	7sus4
IV	Maj9#11	vi	7sus2	IV	Maj13#11	vi	7sus4
V	Maj7	vii	7b5susb2	V	13	vii	7b5sus4
vi	m11	I	Maj7sus2	vi	m11b13	I	Maj7sus4
vii	m11b5b9	ii	7sus2*	vii	m11b5b9b13	ii	7sus4*

*These substitutions are not optimal for diminished-based chords because they leave out the important b5. Learning the full chord may be necessary for advanced players.

The forms above are not the best for thirteenth chords because they leave out the seventh of the chord, which is an important note. A better substitution would include both the suspended fourth and the second.

13th			sus add substitution
I	Maj13	iii	7sus4add_b2
ii	m13	IV	Maj7sus#4add2
iii	m11b9b13	V	7sus4add2
IV	Maj13#11	vi	7sus4add2
V	13	vii	7b5sus4add_b2
vi	m11b13	I	Maj7sus4add2
vii	m11b5b9b13	ii	7sus4add2*

*Not optimal

Understanding the patterns can lead to other substitutions also.

Taking the Am11 chord acegbd and eliminating the root and fifth yields

cgbd

More than one chord contains these four notes. Rearranging the notes to

gbcd

Has the notation Gadd4, since the c is the fourth scale degree in the G scale.

Another notation with already learned patterns would be

G/C

The slash indicates a G chord with a c in the bass, although in this case the c need not be played as the lowest note.

11^th Chord	Spelling	Add 4 Sub.	/Chord	Spelling
CMaj11	cegbdf	Boadd4	B/E	ebdf
Dm11	dfaceg	Cadd4	C/F	fceg
Em11b9	egbdfa	Dmadd_b4	D/G	gdfa
FMaj9#11	facegb	Emadd4	E/A	aegb
G11	gbdfac	Fadd#4	F/B	bfac
Am11	acegbd	Gadd4	G/C	cgbd
Bm11b9b5	bdface	Amadd4	A/D	dace

Thirteenth chords are by their very nature complex, and the substitutions are also complex.

Six-note substitutions eliminating the bass are fairly rudimentary; just build an eleventh chord based on the third.

13^th Chord	Spelling	11^th Chord	Substitution
CMaj13	cegbdfa	Em11b9	egbdfa
Dm13	dfacegb	FMaj9#11	facegb
Em11b13b9	egbdfac	G11	gbdfac
FMaj13#11	facegbd	Am11	acegbd
G13	gbdface	Bm11b9b5	bdface
Am13	acegbdf	CMaj11	cegbdf
Bm11b13b9b5	bdfaceg	Dm11	dfaceg

An alternate five-note substitution is to change the add4 chords to sevenths with the same added 4^th.

13^th Chord	Spelling	Add 4 Sub.	/ Chord	Spelling
CMaj13	cegbdfa	Bm7b5add4	Bm7b5/E	ebdfa
Dm13	dfacegb	CMaj7add4	CMaj7/F	fcegb
Em11b13b9	egbdfac	Dm7add4	Dm7/G	gdfac
FMaj13#11	facegbd	Em7add4	Em7/A	aegbd
G13	gbdface	FMaj7add#4	FMaj7/B	bface
Am13	acegbdf	G7add4	G7/C	cgbdf
Bm11b13b9b5	bdfaceg	Am7add4	AMaj7/D	daceg

The quick and dirty method for four note substitutions builds seventh chords based on the seventh of the scale, but a better method is to eliminate the seventh, and build chords based on the 9^th with an added third,

Taking the CM13 chord cegbdfa and eliminating the root, fifth, and seventh yields

edfa

Rearranged to

defa

is a D minor add 2 (Dmadd2) since e is the second scale degree in D.

A slash notation alternative is Dm/E

13^th Chord	Spelling	Add 2 Sub.	/ Chord	Spelling
CMaj13	cegbdfa	Dmadd2	DMaj7/E	edfa
Dm13	dfacegb	Emadd2	EMaj7/F	fegb
Em11b13b9	egbdfac	Fadd2	FMaj7/G	gfac
FMaj13#11	facegbd	Gadd2	G7/A	agbd
G13	gbdface	Amadd2	AMaj7/B	bace
Am13	acegbdf	Boadd2	Bo/C	cbdf
Bm11b13b9b5	bdfaceg	Cadd2	CMaj7/D	dceg

An experienced jazz player might argue that eliminating the seventh of the chord is not acceptable because the difference between the major seventh and minor seventh is so important to the color of the chords. So, the five-note substitution for is more appropriate for the thirteenth chord. The player could eliminate either the ninth or eleventh (in addition to the root and fifth) and retain the seventh. However, chords based on these substitutions are more difficult to parse than the five note substitutions, so why use them?

Three note substitutions might have an application as substitutes for seventh chords.

Chord	Substitution
CMaj7	Em
Dm7	F
Em7	G
F7	Am
G7	Bo
Am7	C
Bm7b5	Dm

But are not satisfying for higher-level chords. A good use for these three note patterns is learning to parse eleventh chords

Consider the G11 chord gbdfac

Conceptualizing six notes is difficult, but the brain works well with two groups of three

gbd and fac

Since gbd is simply the G chord and fac is an F chord, the eleventh chord is just a G chord mixed with an F chord. So, to make an eleventh chord, take the root chord and add the triad based on the note one down the scale.

11^th Chord	Spelling	Chord 1	Chord 2
CMaj11	cegbdf	C	Bo
Dm11	dfacegb	Dm	C
Em11b9	egbdfac	Em	Dm
FMaj9#11	facegbd	F	Em
G11	gbdface	G	F
Am11	acegbdf	Am	G
Bm11b9b5	bdfaceg	Bo	Am

Thirteenth chords come from adding a seventh to the second column.

13^th Chord	Spelling	Chord 1	Chord 2
CMaj13	cegbdfa	C	BMaj7b5
Dm13	dfacegb	Dm	CMaj7
Em11b13b9	egbdfac	Em	DMaj7
FMaj13#11	facegbd	F	EMaj7
G13	gbdface	G	FMaj7
Am13	acegbdf	Am	G7
Bm11b13b9b5	bdfaceg	Bo	AMaj7

This process takes the quick and dirty method to a new level. Using these stacked chords is approachable in theory for even students with limited chord grasp. Am11 is difficult to understand, but seeing an A minor chord, going a step down to G and playing a G chord is easy to understand. Pattern recognition turns difficult chord into what they really are: clusters of simple chords.

The Learning Chords section of the guitar course has some more practical methods of using substitutions.

Harmonizing the Minor Scale

Consider again the ancient modes, this time centered on the C scale as Ionian (major).

Ionian	c	d	e	f	g	a	b
Dorian	d	e	f	g	a	b	c
Phrygian	e	f	g	a	b	c	d
Lydian	f	g	a	b	c	d	e
Mixolydian	g	a	b	c	d	e	f
Aeolian	a	b	c	d	e	f	g
Locrean	b	c	d	e	f	g	a

In some ancient cultures, Dorian was the preferred scale, but tradition dictates that the major mode dominates modern music. The second most prevalent mode in today's music uses the same notes as the Aeolian mode. Modern theorists refer to this scale minor, or the minor mode, or natural minor to separate this naturally occurring scale from variations of the minor scale.

The minor key with no flats or sharps is A minor

And for minor pentatonic scale, leave out the 2^nd and 6^th degrees from the minor scale.

In Am: a c d e g

The minor pentatonic scale is the most important scale for blues and rock music. Despite its simplicity, adapted with bends and other tricks, it forms the basis of a surprising number of rock and blues solos.

Even though the harmonizations previously discussed use C major as a home key, using A minor as a first chord makes more sense than C minor, because the C minor scale would have flatted notes in it.

So, Am becomes the tonic key, with the A minor chord spelled ace designated by the lower case Roman numeral (i). Natural minor chords need not be recalculated because they are simply the major diatonic chords rearranged and with the Roman numeral analysis adjusted to match the new pattern of majors and minors.

Root	Chord	Spelling	Roman
a	Am	ace	i
b	Bo	bdf	iio
c	C	ceg	III
d	Dm	dfa	iv
e	Em	egb	v
f	F	fac	VI
g	G	gbd	VII

The seventh chords follow the same rearranged pattern with another not a third above

Root	Chord	Spelling
a	AMaj7	aceg
b	Bmb5	bdfa
c	CMaj7	cegb
d	Dm7	dfac
e	Em7	egbd
f	F7	face
g	G7	gbdf

And continue to the higher note chords.

Root	9th Chord	Spelling	11^th Chord	Spelling	13^th Chord	Spelling
a	Am9	acegb	Am11	acegbd	Am13	acegbdf
b	Bmb9b5	bdfac	Bm11b9b5	bdface	Bm11b13b9b5	bdfaceg
c	CMaj9	cegbd	CMaj11	cegbdf	CMaj13	cegbdfa
d	Dm9	dface	Dm11	dfaceg	Dm13	dfacegb
e	Em7b9	egbdf	Em11b9	egbdfa	Em11b13b9	egbdfac
f	FMaj9	faceg	FMaj9#11	facegb	FMaj13#11	facegbd
g	G9	gbdfa	G11	gbdfac	G13	gbdface

Borrowed Chords

Returning to the G Mixolydian, scale

gabcdef

Because of note gravity, melodies that contain a f tend to sound more pleasing when followed by the e rather than moving up the scale to the next g.

So, a common substitution in Gregorian Chant was to substitute a note a half step higher than f, f#, with the # being called an accidental. The f# sounds better when moving from the seventh scale degree to the octave.

gabcdef#

This substitution is so pleasing to the ear that the pattern with the raised seventh, major scale, became ubiquitous.

In a G major scale, the f# is called a leading tone, because it leads back to the tonic, g.

Leading tones often infect other scales, such as the harmonic minor. Leading tones are also essential for secondary dominants, chromatic harmonies, and tritone substitutions. (Explanation of these terms come later.)

In the major scale, leading tone seventh degree pulls the root of the chord because of proximity and strong gravity of the tonic. This leading tone relationship is the reason the dominant seventh chord (V7) moves so nicely to the tonic I. In C, the b note in the gbd chord is pulled up c note and the f note is pulled down to e note leading to the tonic chord ceg. This dominant-tonic relationship (V-I) is essential to Western music.

In the minor more, the chord based on the fifth scale degree is minor, so the dominant-tonic relationship is weak. To reestablish the relationship, minor keys often borrow the dominant chord from the major key. So, in A minor the (v) chord egb becomes eg#b (V). A dominant chord substitute to signal an approaching chord a fourth above is called a secondary dominant.

The g# also works as a substitute in the chord based on the seventh scale degree. The gbd (VII) chord becomes g#bd (vii_o). This substitution reestablishes the leading tone relationship from major keys.

The substituting in the third triad containing g establishes a new chord. The (III) chord ceg becomes ceg# (III+). This is the first naturally occurring augmented chord, indicated in Roman numeral analysis with a + sign and chord chards as C+. Augmented chords are stacks of two major thirds, with the interval from the root to the fifth scale degree notated an augmented fifth (A5). An augmented interval raises a perfect interval a half step.

Root	Chord	Spelling	Roman
a	Am	ace	i
b	Bo	bdf	iio
c	C+	ceg#	III+
d	Dm	dfa	iv
e	E	egb	V
f	F	fac	VI
g	G#o	g#bd	Vii_o

The g# also infects the other possible chords. To find harmonizations, first consider a minor scale with the g replaced by g#

and harmonize this new scale in the same manner as the major scale harmonized previously.

To find the seventh chords, return to the stacked version of the triads, but with the note a moved to the bottom as the root and all the instances of g replaced by g#.

g#	a	b	c	d	e	f
e	f	g#	a	b	c	d
c	d	e	f	g#	a	b
a	b	c	d	e	f	g#

Observe the chords changed by the g#.

The chord based on e is a pattern from the major mode, eg#bd, or E7.

The other changes yield new chords.

The chord based on the root note a is a minor chord with a major seventh, notated as Am(Maj7).

The chord based on the third note c is an augmented chord with a major seventh, notated as CMaj7#5 (not C+7).

The chord based on the seventh degree is a g# diminished seventh chord, notated as g#o7. This chord is fully diminished, unlike the half-diminished chord from the major scale. A fully diminished chord has four notes all a minor third from each other. The seventh is an interval of a diminished seventh (d7) from the root. Diminished intervals lower perfect or minor intervals a half step.

Fully diminished chords are quite useful because they are symmetrical stacks of minor thirds, so the same diminished exists with different spellings in multiple keys.

The fully diminished scale is the first naturally occurring instance of a flatted note. A flat accidental lowers a tone a half step in the same manner a sharp raises it.

Only three diminished chords exist, g#bdf, ace_bg_b and a#c#eg, with all other diminished chords being different spellings of these three chords.

So, the chart of seventh chord harmonizations is

Root	Chord	Spelling
a	Am(Maj7)	aceg#
b	Bm7b5	bdfa
c	CMaj7#5	ceg#b
d	Dm7	dfac
e	E7	eg#bd
f	FMaj7	face
g	G#o7	g#bdf

Continue up the chart of notes.

f	g#	a	b	c	d	e
d	e	f	g#	a	b	c
b	c	d	e	f	g#	a
g#	a	b	c	d	e	f
e	f	g#	a	b	c	d
c	d	e	f	g#	a	b
a	b	c	d	e	f	g#

to produce ninth, eleventh, and thirteenth chords

Root	9th Chord	Spelling	11th	Spelling	13th	Spelling
a	Am(Maj9)	aceg#b	Am(Maj11)	aceg#bd	Am(Maj11)b13	aceg#bdf
b	Bm7b9b5	bdfac	Bm11b9b5	bdface	Bm11b13b9b5	bdfaceg#
c	CMaj9#5	ceg#b	CMaj11#5	ceg#bd	CMaj13#5	ceg#bdfa
d	Dm9	dface	Dm9#11	dfaceg	Dm13#11	dfaceg#b
e	E7b9	eg#bdf	E11b9	eg#bdfa	E11b13b9	eg#bdfac
f	FMaj7#9	faceg#	FMaj7#9#11	faceg#b	FMaj13#9#11	faceg#bd
g#	G#m7b9b5	g#bdfa	G#m11b9b5	g#bdfac	G#m13b9b5	g#bdface

These chord symbols are extremely difficult to parse, but become less important after realizing that these chords are just the simpler chords stacked on each other,

Melodic Minor

Raising the seventh scale degree in minor keys works well for chords, but leads to some challenges for melodies and solos based on these chords. The scale based on a minor pattern with a raised seventh is the harmonic minor scale, as from previous lessons

This is a strange sounding scale because of the distance from the f to the g# being three half steps. A three half-step interval is a minor third in most circumstances, but here f to g is not a third, but a second, so f to g# is an augmented second (A2). Augmented seconds are difficult to read and notate, and produce an eastern sounding feel which, although nice in certain situations, does not fit into the usual alternation of whole- and half-steps from other scales.

To alleviate the problem, the ascending melodic minor scale raises the sixth scale degree to

This pleasant-sounding scale has the minor character of the lowered third together with the functionality of the major scale. The raised sixth is also reminiscent of the Dorian mode.

However, the raised sixth and seventh lose some of the character of the minor mode. To restore the minor flavor, the melodic minor mode returns the sixth and seventh to their natural state when descending.

A harmonization of the descending melodic scale is the same as the natural melodic scale, but the addition of the raised 6^th (f#) adds some new wrinkles to harmonizing the ascending scale. Returning to the note stacking chart with f replaced by f# yields

f#	g#	a	b	c	d	e
d	e	f#	g#	a	b	c
b	c	d	e	f#	g#	a
g#	a	b	c	d	e	f#
e	f#	g#	a	b	c	d
c	d	e	f#	g#	a	b
a	b	c	d	e	f#	g#

Three triads affected are bdf# becoming Bm, df#a becoming D and f#ac becoming F#o

Root	Chord	Spelling
a	Am	ace
b	Bm	bdf#
c	C+	ceg#
d	D	df#a
e	E	egb
f#	F#o	f#ac
g#	G#o	g#bd

Continuing with seventh chords and up

Root		7^th Chord		Spelling
a		Am(Maj7)		aceg#
b		Bm7		bdf#a
c		CMaj7#5		ceg#b
d		D7		df#ac
e		E7		eg#bd
f#		F#m7b5		f#ace
g#		G#m7b5		g#bdf#

	9th Chord				11th		13th
a	Am(Maj9)		aceg#b		Am(Maj11)	aceg#bd	Am(Maj13)	aceg#bdf#
b	Bm7b9		bdf#ac		Bm11b9	bdf#ace	Bm13b9	bdf#aceg#
c	CMaj9#5		ceg#b		CMaj9#11#5	ceg#bd	CMaj13#11#5	ceg#bdf#a
d	D9		df#ace		D9#11	df#aceg	D13#11	df#aceg#b
e	E9		eg#bdf#		E11	eg#bdf#a	E11b13	eg#bdf#ac
f#	F#m9b5		f#aceg#		F#m11b5	f#aceg#b	F#m11b13b5	f#aceg#bd
g#	G#m7b57b9		g#bdf#a		G#m11b9b5	g#bdf#ac	G#m11b13b9b5	g#bdf#ace

Most of these chords are actually easier to parse than the harmonic version, although admittedly G#m11b13b9b5 is intimidating. But remember, these are just stacks of chords, so knowing all the nomenclature is not necessary.

All the note in the melodic minor scale reordered would be

a_bcdeff#gg#

Typically chords mixing the ascending and descending versions of the scale do not occur. But if they did, new charts would not be necessary.

a_bcdefg# is the harmonic minor, so use those charts.

a_bcdeff#gg is the Dorian mode, so use the chords from the major chart starting in the second row (or the 4^th row here). Or, take the melodic ascending chart and lower all the g# notes to g.

Comparison chart

Natural		Harmonic		Melodic Ascend
Am	ace	Am	ace	Am	ace
Bo	bdf	Bo	bdf	Bm	bdf#
C	ceg	C+	ceg#	C+	ceg#
Dm	dfa	Dm	dfa	D	df#a
Em	egb	E	egb	E	egb
F	fac	F	fac	F#o	f#ac
G	gbd	G#o	g#bd	G#o	g#bd
Am7	aceg	Am(Maj7)	aceg#	Am(Maj7)	aceg#
Bm7b5	bdfa	Bm7b5	bdfa	Bm7	bdf#a
CMaj7	cegb	CMaj7#5	ceg#b	CMaj7#5	ceg#b
Dm7	dfac	Dm7	dfac	D7	df#ac
Em7	egbd	E7	eg#bd	E7	eg#bd
F7	face	FMaj7	face	F#Maj7b5	f#ace
G7	gbdf	G#o7	g#bdf	G#Maj7b5	g#bdf#
Am9	acegb	Am(Maj9)	aceg#b	Am(Maj9)	aceg#b
Bm7b9b5	bdfac	Bm7b9b5	bdfac	Bm7b9	bdf#ac
CMaj9	cegbd	CMaj9#5	ceg#b	CMaj9#5	ceg#b
Dm9	dface	Dm9	dface	D9	df#ace
Em7b9	egbdf	E7b9	eg#bdf	E9	eg#bdf#
FMaj9	faceg	FMaj7#9	faceg#	F#m9b5	f#aceg#
G9	gbdfa	G#o7b9*	g#bdfa	G#m7b57b9	g#bdf#a
Am11	acegbd	Am(Maj11)	aceg#bd	Am(Maj11)	aceg#bd
Bm11b9b5	bdface	Bm11b9b5	bdface	Bm11b9	bdf#ace
CMaj11	cegbdf	CMaj11#5	ceg#bd	CMaj9#11#5	ceg#bd
Dm11	dfaceg	Dm9#11	dfaceg	D9#11	df#aceg
Em11b9	egbdfa	E11b9	eg#bdfa	E11	eg#bdf#a
FMaj9#11	facegb	FMaj7#9#11	faceg#b	F#m11b5	f#aceg#b
G11	gbdfac	G#o11b9*	g#bdfac	G#m11b9b5	g#bdf#ac
Am13	acegbdf	Am(Maj11)b13	aceg#bdf	Am(Maj13)	aceg#bdf#
Bm11b13b9b5	bdfaceg	Bm11b13b9b5	bdfaceg#	Bm13b9	bdf#aceg#
CMaj13	cegbdfa	CMaj13#5	ceg#bdfa	CMaj13#11#5	ceg#bdf#a
Dm13	dfacegb	Dm13#11	dfaceg#b	D13#11	df#aceg#b
Em11b13b9	egbdfac	E11b13b9	eg#bdfac	E11b13	eg#bdf#ac
FMaj13#11	facegbd	FMaj13#9#11	faceg#bd	F#m11b13b5	f#aceg#bd
G13	gbdface	G#o11b13b9*	g#bdface	G#m11b13b9b5	g#bdf#ace

*9^th, 11^th, and 13^th chords built on fully diminished chords are rare, so chord symbols representing these chords may vary.

Substitutions for these chords function just as in the major chords

For ninth chords, remove the root and play the seventh chord based on the third scale degree.

For eleventh chords, remove the root and play the ninth chord based on the third scale degree, or play the triad based on the seventh scale degree and add the third scale degree.

For thirteenth chords, remove the root and play the eleventh chord based on the third scale degree, or play the seventh chord based on the seventh scale degree and add the third scale degree.

Inversions

Often chords do not have the root in the bass note. The technical term for playing a chord note other than the root as the lowest note is an inversion.

A slash indicates and inversion, with the bass note after the slash.

Chord	Spelling	1st inv.	Spelling	2nd inv.	Spelling
C	ceg	C/E	egc	C/G	gce
Dm	dfa	Dm/F	fad	Dm/A	adf
Em	egb	Em/G	gbe	Em/B	beg
F	fac	F/A	acf	F/C	cfa
G	gbd	G/B	bdg	G/D	dg_b
Am	ace	Am/C	cea	Am/E	eac
Bo	bdf	Bo/D	dfb	Bo/F	fbd

Notes other than the bass note can be in any order.

Chords with more notes in the first and second inversions follow the patterns / the triads

G7/B, G7/D

Seventh chords could have an additional third inversion with the seventh in the bass.

G7/F

Ninth chords could have another additional fourth inversion /ninth in the bass.

G9/F, G9/A

Eleventh chords could have another additional fifth inversion /eleventh in the bass.

G11/F, G11/A, G11/C

Thirteenth chords could have an additional sixth inversion /thirteenth in the bass.

G13/F, G11/A, G11/C, G13/E

Added Notes

All the chords in the chords studied so far have had some kind of harmonic function; they are important to the chord structure. Sometimes notes can be added to a harmony for other reasons.

The substitutions section showed how an added 4 (add4) can substitute a ninth chord and an added second (add2) can substitute an eleventh chord. Possible diatonic add 2 chords are

major	Minor	Harmonic	Melodic
Cadd2	Amadd2	Amadd2	Amadd2
Dmadd2	Boadd_b2	Boadd_b2	Bmadd_b2
Emadd_b2	Cadd2	C+add2	C+add2
Fadd2	Dmadd2	Dmadd2	Dadd2
Gadd2	Emadd_b2	Eadd_b2	Eadd_b2
Amadd_b2	Fadd2	Fadd2	F#oadd2
Boadd2	Gadd2	G#oadd_b2	G#oadd_b2

Note the b2 on half step intervals starting on e, b, and g#.

Add2 intervals on seventh chords do not make sense since an added 2nd to a seventh chord just makes a ninth chord. Since ninth, eleventh, and thirteenth notes already have the note, adding a second here also does not apply.

Possible add4 chords include

major	Minor	Harmonic	Melodic
Cadd4	Amadd4	Amadd4	Amadd4
Dmadd4	Boadd4	Boadd4	Bmadd4
Emadd4	Cadd4	C+add4	C+add#4
Fadd#4	Dmadd4	Dmadd#4	Dadd4
Gadd4	Emadd4	Eadd4	Eadd4
Amadd4	Fadd#4	Fadd#4	F#oadd4
Boadd4	Gadd4	G#oadd4	G#oadd_b4
CMaj7add4	Am7add4	Am(Maj7)add4	Am(Maj7)add4
Dmadd4	Bm7b5add4	Bm7b5add4	Bm7add4
Em7add4	CMaj7add4	CMaj7#5add4	CMaj7#5add#4
F7add#4	Dm7add4	Dm7add#4	D7add4
G7add4	Em7add4	E7add4	E7add4
Am7add4	F7add#4	FMaj7add#4	F#Maj7b5add4
Bm7b5add4	G7add4	G#o7add4	G#m7b5add_b4

Add4 chords on ninths would be elevenths, and elevenths and thirteenths already have the fourth scale degree.

The other scale degree not yet discussed is the sixth scale degree.

Adding a sixth to a chord conjures up the feeling of a thirteenth chord because scale degrees 6 and 13 are just the same scale degree in a different octave. The designation (6) indicates an added sixth (not add6). A C6 chord is cega. Notice that cega is just aceg, the A minor seventh chord rearranged. Some chords have more than one possible name.

A 6^th can be added to higher note chords also, such as C67 C69 or any of the possibilities from the complicated chord charts (other than 13^th, which already have the 6^th). Add2 and add4 notes also appear in other chords not containing seconds and fourths.

major	Minor	Harmonic	Melodic
C6	Amadd_b6	Amb6	Am6
Dm6	Bob6	Bo6	Bm6
Emb6	C6	C6#5	C6#5
F6	Dm6	Dm6	D6
G6	Emadd_b6	Eb6	Eb6
Amb6	F6	F6	F#ob6
Boadd_b6	G6	G#oadd_b6	G#ob6
CMaj67	Am7add_b6	Am(Maj7)add_b6	Am(Maj7)6
Dm67	Bm7b5add_b6	Bm67b5	Bm67
EMaj7b6	CMaj67	CMaj7#56	CM67#5
F67	Dm67	Dm67	D67
G76	EMaj7b6	E7b6	E7b6
AMaj7add_b6	F67	FMaj76	F#m7b5add_b6
BMaj7b5add_b6	G67	G#o7b6	G#m7b5add_b6
CM69	Am9add_b6	Am(Maj9)b6	Am(Maj69)
Dm69	BMaj7b9b5b6	Bm9b5b6	Bm9b6
EMaj7b9add_b6	CM69	C(Maj9)6#5	C(Maj9)6#5
FM69	Dm69	Dm69	D69
G69	EMaj7b9b6	E7b9b6	E9b6
Am9b6	FM69	F(Maj9)b7	F#m9b5add_b6
Bm7b9b5add_b6	G69	G#o7b9add_b6	G#Maj7b9b5add_b6

Added notes on other scale degrees are just inversions, add3 would be first inversion, add5 would be second inversion, and add7 would be third inversion, so those add chords use the slash notation from the inversions section. A special case might be adding a note to a chord that did not have it, such as Boadd3, spelled bded#, but this would be rare indeed. Add notes can also be raised Fadd#4 or lowered Gadd_b2.

Multiple add not can appear on one chord. Triads could have two added notes

add2 and add4, Cadd2add4

add2 and add6, Cadd2add6

add4 and add6, Cadd4add6

or even all three Cadd2add4add6

Seventh chords could have an add4 and an add6, CMaj7add2add6.

Any other combination would be one of the higher order chords. One reason to use multiple add chords would be to leave out a note from a complicated chord. Cadd2add4add6 is just CMaj13 without the seventh degree.

Suspensions

Sometimes an added note replaces a chord tone rather than adds to the chord.

The most common of these suspensions replaces the third of the chord with a fourth. Replace the third of the C chord ceg with an f produces cfg, a Csus4 (or just Csus) chord. The gravity of the third e below the fourth f tends to drag the suspension down, so The suspended usually precedes the tonic chord or some other chord with an e. The classic suspension progression is G7-Csus-C, with the f in the G7 hanging into the Csus chord, thus delaying the resolution to the C chord in a pleasing matter. Suspended notes and be added to other chords also, such as C7sus4.

Suspensions follow the patterns of the add chords, with a few exceptions. The biggest difference is that since the suspension replaces the third scale degree, major and minor chords have the same suspension. Csus4 and Cmsus4 would be the same chord, so do not use Cmsus. The other difference is that chords with a diminished or augmented reference need to be rewritten, since the diminished implies a lowered third and augmented implies a raised third.

major	Minor	Harmonic	Melodic
Csus4	Asus4	Asus4	Asus4
Dsus4	Bosus4	Bosus4	Bsus4
Esus4	Csus4	C#5sus4	C#5sus#4
Fsus#4	Dsus4	Dsus#4	Dsus4
Gsus4	Esus4	Esus4	Esus4
Asus4	Fsus#4	Fsus#4	F#osus4
Bb5sus4	Gsus4	G#b5sus4	G#susb4b5
CMaj7sus4	A7sus4	AMaj7)us4	AMaj7sus4
D7sus4	B7b5sus4	B7b5sus4	B7sus4
E7sus4	CMaj7sus4	CMaj7#5sus4	CMaj7#5sus#4
F7sus#4	D7sus4	D7sus#4	D7sus4
G7sus4	E7sus4	E7sus4	E7sus4
A7sus4	F7sus#4	FMaj7sus#4	F#7b5sus4
B7b5sus4	G7sus4	G#b5sus4d7	G#7b5susb4

The next most common suspension is the 2^nd degree. A Csus2 (cdg) chord is similar to a Cadd2 chord, but the suspended note replaces the third. The second technically resolves upwards to the third, but in practice can move downward to the root by using Csus2 as a substitute for the more harmonically accurate Cadd2-C progression. Dsus4-D-Dsus2 is a common pattern on guitar because of its ease of play and usefulness for transitioning to the G and C chords.

major	Minor	Harmonic	Melodic
Csus2	Asus2	Asus2	Asus2
Dsus2	Bosusb2	Bosusb2	Bsusb2
Esusb2	Csus2	C#5sus2	C#5sus2
Fsus2	Dsus2	Dsus2	Dsus2
Gsus2	Esusb2	Esusb2	Esusb2
Asusb2	Fsus2	Fsus2	F#b5sus2
Bb5sus2	Gsus2	G#b5susb2	G#b5susb2

A variation of the suspended 2^nd that moves downward is called an appoggiatura. In a true appoggiatura would replace the root rather than the third, for example a four part harmony cegd moving to cegc. Chord notation does a poor job representing this, but Cadd9-C often approximates it.

A similar classical technique is to replace the upper root with the seventh cegb to cegc. The seventh resolves upward. In chord notation CMaj7-C can approximate this.

Another commonly used suspension is on the sixth scale degree. In Csus6 (cea), the 6^th scale degree replaces the fifth and resolves down to the fifth. The 6 of the sus6 chord could move upward to the seventh as a substitute for a passing chord Csus6-CMaj7-C, but this is not a true suspension, just using the Csus as a substitute for C6.

major	Minor	Harmonic	Melodic
Csus6	Amsusb6	Amsusb6	Amsus6
Dmsus6	Bmsusb6	Bmsus6	Bmsus6
Emsusb6	Csus6	C+sus6	C+sus6
Fsus6	Dmsus6	Dmsus6	Dsus6
Gsus6	Emsusb6	Esusb6	Esusb6
Amsusb6	Fsus6	Fsus6	F#msusb6
Bmsusb6	Gsus6	G#msusb6	G#msusb6
CMaj7sus6	Am7susb6	Am(Maj7)susb6	Am(Maj7)sus6
Dm7sus6	Bm7b5susb6	Bm7b5sus6	Bm7sus6
Em7susb6	CMaj7sus6	CMaj7#5sus6	CMaj7#5sus6
F7sus6	Dm7sus6	Dm7sus6	D7sus6
G7sus6	Em7susb6	E7susb6	E7susb6
AMaj7susb6	F7sus6	FMaj7sus6	F#Maj7b5susb6
BMaj7b5susb6	G7sus6	G#msusb6d7	G#Maj7b5susb6
CM9sus6	Am9susb6	Am(Maj9)susb6	Am(Maj9)sus6
Dm9sus6	BMaj7b9b5susb6	BMaj7b9b5sus6	BMaj7b9sus6
EMaj7b9susb6	CM9sus6	CMaj9#5sus6	CM9#5sus6
FM9sus6	Dm9sus6	Dm9sus6	D9sus6
G9sus6	EMaj7b9susb6	E7b9susb6	E9susb6
Am9susb6	FM9sus6	FMaj7#9sus6	F#m9b5susb6
BMaj7b9b5susb6	G9sus6	G#msusb6b7b9	G#Maj7b9b5susb6

Suspended chords have fewer combinations than add chords. Since sus2 and sus4 both replace the third, these would not appear together.

Either of these chords could appear with a sus6,

Csus4sus6

Csus2sus6

leading to a nice double resolution to C.

Consider the resolution CMaj7sus4sus6 to C. Try borrowing a trick from renaissance polyphonic music by resolving each of the suspensions separately

CMaj7sus4sus6- CMaj7sus- Csus4-C.

Passing Tones

Sometimes notes add to a chord to facilitate movement from one chord to another. A common guitar pattern when moving from G to C is

G-G/A-G/B-C

Analysis of the a in the G/A chord could lead to the (improper) conclusion that G/A is a Gsus2 chord, and music books often do notate the chord this way. The true function of this walking bass lick is to pass from the G to the C, hitting all the scale notes in between. Passing tones may appear notated as suspensions.

Passing tones are usually the 2^nd,4^th, and 6^th scale degrees moving between chord tones. In C, C/D, C/F, and C/A move between the chord inversions, but have no true functional harmony otherwise.

C, C/D, C/E. C/F, C/G, C/A, CMaj7/B, C

Listen to "Friend of the Devil" by the Grateful Dead for some nice passing tone work.

Chromatic Harmonies

Chromatic notes may also function as passing tones, but may also have a harmonic function known as chromatic harmony.

Consider the transition from F to C. The notes move from fac to ceg. Rearrange the notes on the C chord into first inversion to egc. Written here vertically:

c	c
a	g
f	e

This inversion highlights the natural gravity of the f to the e in a pleasing manner.

Now add an F minor (fa_be) chord between the two chords.

c	c	c
a	a_b	g
f	f	e

The transition from Fm to C now has an additional gravitational pull between the a_b and the G leading to a pleasing result. The Fm chord here would be an example of a chromatic harmony.

Any two chords with notes a step apart are candidates for chromatic harmonies. The most fruitful of these are ii-I, IV-V. Vi-V, since the chord roots are a step apart.

Consider the ii-I progression, dfa-ceg. The root, and fifth all have space for a chromatic note placed between them, with possible results being

a	a	g	a	a_b	g	a	a_b	g
f	f	e	f	f	e	f	f	e
d	d_b	c	d	d	c	d	d_b	c
ii	bii+	I	ii	iio	I	ii	bII	I
Dm	Db+	C	Dm	Do	C	Dm	D	C

Rearranging the ii chord to first inversion reveals more combinations

a	a_b	g
d	d#	e
f	f	c

So, new chords fad# and fa_bd# come out of the patterns. These chords would be difficult to analyze. Adding the seventh note c to the ii chord helps, yielding facd#. Since d# is enharmonic to Eb, the result is a misspelled F7 chord, which will suffice.

c	c		c	c		c	c
a	a	g	a	a_b	g	a	a_b	g
d	d#	e	d	d	e	d	d#	e
f	f	c	f	f	c	f	f	c
Dm7/F	F7	C	Dm7/F	DMaj7b5	C	Dm7/F	Fm7	C

Rearranging ii into third inversion yields another set of combinations.

c	c
f	f#	g
d	d#	e
a	a	c

c	c		c	c		c	c
f	f#	g	f	f	g	f	f#	g
d	d	e	d	d#	e	d	d#	e
a	a	c	a	a	c	a	a	c
Dm/A	D7	C	Dm/A	F7/A	C	Dm/A	A#o7	C

The A#o7 is proba_bly better written as F#o7/A since its function seems to be a secondary dominant of g. The series d-d#-e might interfere with the 4-3 (f-e) resolution. It could go into another octave leaving the f-e as a doubled note.

d	d#	e
c	c	g
f	f	e
a	a	c
Dm7/A	F7/A	C

Allowing for multiple octaves also allows combinations among the three groups. Now the d-d#-E pattern could with the d-d_b-c pattern in a different octave.

d	d#	e
a	a	g
f	f	e
d	d_b	c
Dm	Dbadd2	C

For the purposes of analysis, consider the D# an Eb to yield Dbadd2.

So, any combination of the chromatic harmonies is possible. As an alternate to doing all those combinations, make a list of all the chromatic lines

a	a_b	g
d	d_b	c
d	d#	e
f	f#	g

Now isolate the altered notes

a_b

d_b

Insert the altered noted into the chords one by one

c	c	c	c
a	a	a	a_b
f	f	f#	f
d_b	d#	d	d
DbMaj7#5	F7	D7	Dm7b5

Or in pairs, substituting enharmonics when necessary to aid analysis.

		e_b
c	c	c	c	c	c#	c
a	a_b	a	a	a_b	a	a_b
f#	f	f	f#	f	f	f#
d_b	d_b	d_b	d#	d#	d#	d
F#add#4	Fmb6/Db	F(b6)/Db	D#o7	Fmb7/D#	F#5b7	D7b5

Groups of three or four

	e_b
c	c	c	c#
a_b	a_b	a_b	a_b
f#	f	f#	f
d_b	d_b	d#	d#
DbMaj7sus4	Db9	D#add4d7(no 5th)	D#7add4(no 5th)

e_b
c	c#
a_b	a_b
f#	f
d_b	d#
Db11(no 3rd)	D#sus2

These last sets of notes seem to defy analysis. If anyone can come up with better chord symbols, please let me know. The chords notated here may not be the best versions of the groups of notes, depending on context. This is just a quick way to come up with all the possible chromatic harmonies. Adding more octaves could produce other chords, such as leaving d as the root and putting the d_b as an altered ninth, but these changes would just be variations of the chords here:

DbMaj7#5	F7	D7	DMaj7b5
F#add#4	Fmb6/Db	F(b6)/Db	D#o7
Fmb7/D#	F#5b7	D7b5
DbMaj7sus4	Db9	D#add4d7(no 5th)
D#7add4(no 5th)	Db11(no 3rd)	D#sus2

Moving from IV7-V7 yields three obvious chromatics

e		f
c	c#	d
a	a#	b
f	f#	g

With possible combinations

e		f	e		f	e		f
c	c	d	c	c	d	c	c#	d
a	a	b	a	a#	b	a	a	b
f	f#	g	f	f	g	f	f	g
FMaj7	F#o	G7	FMaj7	Fsus4	G7	FMaj7	F+	G7
e		f	e		f	e		f	e		f
c	c	d	c	c#	d	c	c#	d	c	c#	d
a	a#	b	a	a	b	a	a#	b	a	a#	b
f	f#	g	f	f#	g	f	f	g	f	f#	g
FMaj7	F#4b5	G7	FMaj7	F#m	G7	FMaj7	F#5sus4	G7	FMaj7	F#	G7

These combinations work without the seventh also. The e to f is a nice chromatic, so it adds to the process here.

Instead of working through all the inversions separately, use the techniques from the last section as shortcuts. First examine the chord tones of the final chord, where the chromatic harmony resolves, in this case, gbd. (Chromatic alterations could lead to other tones such as the seventh, ninth, eleventh, and thirteenth, but the strongest resolutions occur in order on the root, the fifth, and then the third.)

So, taking the notes g, b, and d, move out a whole step in either direction,

a-g-f

c#-b-a

e-d-c

Then look for notes in the first chord that are the same or enharmonic equivalents. Common notes include a, f, a again, e, d, and c. So, the possible places are

a-g

f-g

a-b

e-d

c-d

The note c# does not appear because it is not a chord tone. It might occur in a string of chromatics like d-c#-c-a, but again, save that for later.

Take the relationships and insert a chromatic note. If the direction of the line is downward, us a flat, and if it is upward use a sharp or the appropriate enharmonic equivalent.

a-a_b-g

f-f#-g

a-a#-b

e-e_b-d

c-c#-d

Make a table that replaces the notes of the original chord with the substitutions, doubling if necessary.

e_b

a# or a_b

Now begin substituting the altered tones for the original tones, first one by one.

e	e	e	e	e_b
c	c	c	c#	c
a	a_b	a#	a	a
f#	f	f	f	f
F#m7b5	Fm(Maj7)	FMaj7sus4	FMaj7#5	F7

Then add in pairs, using enharmonic when necessary.

e	e	e	e_b
c	c	c#	c
a_b	a#	a	a
f#	f#	f#	f#
Ab7#5/F#	F#7b5sus4	F#Maj7	F#o

	b_b
e	e	e_b
c#	c	c
a_b	a_b	a_b
f	f	f
Fm(Maj7)#5	Fm(Maj7)add4	FMaj7

	g#
e	e	e_b	e_b
c#	c	c	c#
a#	a#	a#	a
f	f	f	f
FMaj7#5sus4	FMaj#9sus4	F7sus4	F7#5

Then groups of three or more

b_b
e	e	e_b
c	c#	c
a_b	a_b	a_b
f#	f#	f#
AbMaj9#5/Gb	Ab7#5/Gb	Ab7/Gb

g#
e	e	e_b
c	c#	c
a#	a#	a#
f#	f#	f#
F#m9b5	F#7	F#Maj7b5

b_b		g#
e	e_b	e	e_b
c#	c#	c#	c#
a_b	a_b	a#	a#
f	f	f	f
Fm(Maj7)b5add4	Fm7#5	FMaj7#9#5sus4	F7#5sus4

b_b	g#
e_b	e_b	e_b	e_b
c#	c#	c#	c#
a	a	a#	a_b
f	f	f#	f#
F7#5add4	F7#9#5

b_b	g#	b_b	g#
e_b	e_b	e_b	e_b
c#	c#	c#	c#
a	a	a_b	a#
f#	f#	f	f#
F#m7add4	F#m7#9	Fm7b9#5	F9sus4

Summary of IV-V Chromatic Harmonies

F#Maj7b5	Fm(Maj7)	FMaj7sus	FMaj7#5
Ab7#5/F#	F#7b5sus4	F#Maj7	F#o
Fm(Maj7)#5	Fm(Maj7)add4	FMaj7	F7
FMaj7#5sus4	FMaj#9sus4	F7sus4	F7#5
AbMaj9#5/Gb	Ab7#5/Gb	Ab7/Gb	F7add4#5
F#m9b5	F#7	F#Maj7b5	F7#9#5
Fm(Maj7)b5add4	FMaj7#5	FMaj7#5sus4#9	F7#5sus4
F#m7add4	F#m7#9	FMaj7b9#5	F9sus4

Note that an of the chromatic harmonies work in both directions.

IVMaj7

IVsus4

Reversed to V7-IVsus4-IVMaj7 has the same notes, but the a# would be the enharmonic Bb to promote voice leading. In traditional notation, sharp accidentals look better moving up a-a#-b and flats moving down b-b_b-a.

Adding seconds, fourths, or sixths to these chords (when those degrees are not in the original chords) produces more complex chords with the same function. Defining those as ninth, eleventh, and thirteenth chords is difficult for many because of the manipulations necessary to force harmonic notation to its limits.

The chromatic harmonies based on other intervals have fewer chromatic harmonies, but still plenty. Consider V-I, the most important cadence.

Following the procedure, the possible places for chromatic harmonies leading to the tonic are

f-g-a

d-e-f#

b_b-c-d

so, the appropriate intervals to check are

f-g	f#
a-g	a_b
d-e	d#
f#-e	f

Looking at the notes to be replaced, gbd, the only appropriate replacement is changing the d to d#.

f	f	e
d	d#	e
b	b	g
g	g	c
V7	V7#5	I

The note f is already in the seventh chord, so does not need replacement.

Replacing the f with f# to lead to g actually weakens the cadence, as it removes the 4-3 resolution. The notes a_b and c# would be ninths and elevenths, which would be interesting, if not as strong a chromatic harmony as those based on the triad.

a	a_b	g
f	f#	g
d	d#	e
b	b	c
g	g	c
V9	VMaj7b9#5	I

This contains all the alterations. This chord contains a wealth of chromatics, and is easier to grasp than many of the awkward chords based on ii-I or IV-V.

Each chromatic taken one at a time yields

a	a	a_b
f	f#	f
d#	d	d
b	b	b
g	g	g
V9#5	VMaj9	Vb9

More than one

a	a_b	a_b	a_b
f#	f	f#	f#
d#	d#	d	d#
b	b	b	b
g	g	g	g
VMaj9#5	V9#5	VMaj7b9	VMaj7b9#5

And a few strange cords with f and f# (g_b) in different octaves.

a	a	a_b	a_b
f#	f#	f#	f#
f	f	f	f
d	d#	d	d#
b	b	b	b
g	g	g	g
VMaj9Add#6*	VMaj9#5Add#6	VMaj7b9Add#6	VMaj7b9#5Add#6

*A chord with both a minor seventh and a major seventh at the same time is difficult to analyze. The #6 is the minor seventh enharmonically, so Maj7add#6 is a chord with a major and a minor seventh at the same time. These chords would be rare, practically non-existent in actual music, but an interesting theory thought experiment.

With the exception of the last grouping, these chords are easier to grasp than previous examples, and additions of elevenths and thirteenths not as difficult.

GMaj7b9#5	G9#5	GMaj9	Gb9
GMaj9Add#6	GMaj9#5Add#6	GMaj7b9Add#6	GMaj7b9#5Add#6
GMaj11b9#5	G11#5	GMaj11	G11b9
GMaj11Add#6	GMaj11#5Add#6	GMaj11b9Add#6	GMaj11b9#5Add#6
GMaj13b9#5	G13#5	GMaj13	G13b9
GMaj13Add#6	GMaj13#5Add#6	GMaj13b9Add#6	GMaj13b9#5Add#6

Any chord change has possible chromatic harmonies. IV-I changes are similar to ii-I, so those serve as a good substitution since IV and ii contain overlapping notes. V-vi overlaps V-I for the same reason. Minor keys parallel the structure of major keys somewhat, with the obvious differences in thirds, and the addition of many more possibilities with raise sixths and sevenths. An exhaustive list of all the possibilities, such as the ones supplied here for the chosen intervals, would need to come in a separate document.

Two more sophisticated uses of chromatic harmonies secondary dominants and tritone substitution.

Secondary Dominants

The section on the harmonic minor scale showed how a manipulation of the (v) chord to (V) produced a more satisfying chord resolution to the tonic. This manipulation was simply a chromatic substitution of the third scale degree of the chord. The notes egb because eg#b.

A dominant-style chord substituting for another chord to highlight the chord after it is called a secondary dominant. Dominant style chords include major triads, in addition to seventh (7), ninth (9), eleventh (11), and thirteenth (13) chords. Other chords with a major third as the first interval can also serve the same function.

Consider the common IV-V-I cadence in the key of C.

c	d	g
a	b	e
f	g	c

Now, add a secondary dominant D major triad (even though he key suggests A minor) d f# a but spell it in first inversion f# to highlight the chromatic harmony.

c	d	d	g
a	a	b	e
f	f#	g	c

Notice how the gravity of the g in the bottom line pulls the f# up.

The lines are even cleaner when using a seventh chord.

	a
c	d	d	g
a	c	b	e
f	f#	g	c

Completing the four note chords with a V7 and rearranging provides lots of nice voice leading.

f	a	f	e
c	d	d	g
a	c	b	c
f	f#	g	low c

Now all four lines have a nice clean flow, and any of them could be a melody. Performing this exact pattern of a guitar would be difficult, so when recognizing chromatic harmonies, try to choose voicings that keep the lines together when possible.

Roman numeral notation would read

II7

The II7 notation for the secondary dominant is accurate, but does not really reflect the true function of the chord. The function of the D7 is to highlight the chord that comes after it, the G chord (V). D is the dominant chord in the key of G, so D is the V of G. Since G is the V of C, D is the V of [the V of C] (VofV). Substituting V7ofV for II7 yields.

V7ofV

If VofV is acceptable, then what a_bout secondary dominants of other notes? Perhaps to highlight an upcoming A minor (vi) chord, for a key change to the relative minor, a player would insert a secondary dominant just before it. To find the root of the proper secondary dominant, take the root note of the chord (in this case a) and move down a perfect fourth to e to form an E7 chord. The E7 chord could be notated as III7, but a better notation is Vofvi. Any chord can have a secondary dominant.

	In C		A Minor	Harmonic	Melodic
(VofI)	G	(Vofi)	E	E	E
Vofii	A	Vofiio	F#	F#	F#
Vofiii	B	VofIII	G	G	G
VofIV	C	Vofiv	A	A	A
VofV	D	Vofv	B	B	B
Vofvi	E	VofVI	C	C	C#
Vofvii_o	F#	VofVII	D	D#	D#

Any seventh (7) chord and its higher order permutations can be secondary dominant, so chords like V13ofIV are possible.

Chains of Secondary Dominants

Consider the cadence

D7	G7	C
V7ofV	V7	I

If any chord can have a secondary dominant associate with it, then what a_bout a secondary dominant for D7? Following the procedure, take d down a fourth to a then build a dominant A7, for example. A7 is certainly VI7 in the key of C, but that notation does not promote understanding. Follow this logic:

D7 is V7ofV

A7 is V of [D7] (Brackets added for clarity.]

So, A7 is V7of[VofV] or V7ofVofV. (Put the "7" on the first item only.)

Putting several secondary dominants together is called a chain of secondary dominants. The next in the chain is V7of{Vof[VofV]}, an E7 chord. Chaining secondary dominants is a great technique for folk music. Listen to "Her Majesty" by The Beatles for some great uses of secondary dominants and chromatic harmonies.

The full chain of secondary dominants would be

C7F7Bb7Eb7Ab7(Db7 or C#7)F#7B7E7A7D7G7C

Playing chords in series is a great exercise for learning the seventh chords, and is the basis for the chord weaving section of the guitar method.

Secondary Dominants Based on vii_o

Any chord with a dominant function can be a secondary dominant. The most common is the vii_o chord. A vii_o chord highlights the chord a half step above the root. So, vii_oofF is an Eo chord followed by and F chord.

	In C		A Minor	Harmonic	Melodic
(vii_oofI)	Bo	(vii_oofi)	G#o	G#o	G#o
vii_oofii	C#o	vii_oofiio	A#o	A#o	A#o
vii_oofiii	D#o	vii_oofIII	Bo	Bo	Bo
vii_oofIV	Eo	vii_oofiv	C#o	C#o	C#o
vii_oofV	F#o	vii_oofv	D#o	D#o	D#o
vii_oofvi	G#o	vii_oofVI	Eo	Eo	E#o
vii_oofvii_o	A#o	vii_oofVII	F#o	Fxo	Fxo

Half diminished seventh chords function the same way, as well as ninth, eleventh, and thirteenth chords based in the diminished triad. The vii_o7 chord can be extremely powerful as a pivot to a new key. Since vii_o7 is just a series of minor thirds, any note in the chord can be a secondary dominant leading to the chord just above it.

Tritone Substitutions

Jazz players use another type of chromatic harmony to pass from one chord to another. The tritone substitution uses a tritone (d5), three whole steps, as a passing tone between two chords.

Consider the very common ii-V-I jazz pattern in C.

	f
a	d	g
f	b	e
d	g	c
Dm	G7	C
ii	V7	I

One issue with this pattern is the big jump in the bass line d-g-c. So, instead of the g in the second chord, substitute the note that lies between the d in the first chord and the c in the third chord, d_b.

	f
a	g	g
f	b	e
d	d_b	c
Dm	G7b5/Db	C
ii	V7b5	I

The tritone here is from g, the expected root, to d_b, a tritone(d5) from g.

Now rearrange the last chord accentuation the 4-3 resolution.

	g	g
a	b	c
f	f	e
d	d_b	c
Dm	G7b5/Db	C
ii	V7b5	I

Tritone substitutions and other jazz tricks are an art form. Refer to a jazz text for more examples.

Chromatic harmonies may occur any place where a scale has a major second. The patterns 2-b2-1 (above), 2-#2-3, 4-#4-5, 6-b6-5, 6-b7-7-1 all lead to chord tones, so they make good chromatic harmonies. The ii-V7-I jazz pattern with these chromatic harmonies.

2-#2-3	Dm	G+7	C	d-d#-c
4-#4-5	Dm	GMaj7	C	f-f#-g
6-b6-5	Dm	G7b9	C	a-a_b-g
6-b7-7-1	Dm	G7#9-G7	C	a-a#-b-c

Misspelled Chords

Chromatic harmonization shows how notating an accidental as a flat suggests that the next note might be the pitch just below and notating with a sharp may signal an ascending line. The seventh degree of the F7 chord, spelled facEb, suggests the note should resolve downward to d, and that resolution is commonplace. But what if the same group of notes were spelled the enharmonic equivalent, facd#? Now the d# wants to resolve upward to an e.

Consider in the key of C an Ab7 chord. Written in the standard way, AbcEbGb, the chord seems like a VofDb secondary dominant, which would resolve to Db. But written AbcEbf#, now the f# suggest resolution to g, with e_b moving down to d and a_b down to g.

f#	g
e_b	d
c	b
a_b	g
Ger#6	V

Traditionally written with a I chord in second inversion delaying the resolutions.

f#	g	g
e_b	e_b	d
c	c	b
a_b	g	g
Ab	Cm/G	G

This is German sixth chord is the first of a number of altered chords widely used in classical music, the others being the French sixth

f#	g
d	d
c	b
a_b	g
Fr#6	V

Italian Sixth

f#	g
c	d
c	b
a_b	g
It#6	V

And English (or Swiss, or Alsatian, or double German)

f#	g	g
D#	e	d
c	c	b
a_b	g	g
En#6	C/G	V

This group, known as augmented sixth chords because of the distance between a_b an g#, usually precedes a V cord, but could theoretically resolve to other chords with notes a half step below the bass. Blues players use a chord based on b6 just before a dominant often.

A Neapolitan chord functions in a similar manner. N6 chords are major chords in first inversion built on the flat second scale degree. In C, the chord resolves down to I

d_b	c
a	g
f	e
Db/F	C/E

Traditionally delayed by the addition of a V chord.

		e
d_b	b	c
a	g	g
f	d	c
Db/F	G/D	C

The most famous misspelled chord is the Tristan chord b-d#-f-g#., from the opera Tristan und Isolda by Richard Wagner. Use of this chord signaled the move away from classical functional harmonies into more structurally built chords, and new tonalities based on chromaticism.

Interval Based Chords

So, far most of the chords listed are stacks of thirds on top of each other because that relationship follows from the physics of sound. But chords built on other intervals also have their place.

Quartal Chords

Chords based on fourths are called quartal chords.

Take the columns of notes from the original note stack, but extend to three octaves.

b	c	d	e	f	g	a
a	b	c	d	e	f	g
g	a	b	c	d	e	f
f	g	a	b	c	d	e
e	f	g	a	b	c	d
d	e	f	g	a	b	c
c	d	e	f	g	a	b
b	c	d	e	f	g	a
a	b	c	d	e	f	g
g	a	b	c	d	e	f
f	g	a	b	c	d	e
e	f	g	a	b	c	d
d	e	f	g	a	b	c
c	d	e	f	g	a	b
b	c	d	e	f	g	a
a	b	c	d	e	f	g
g	a	b	c	d	e	f
f	g	a	b	c	d	e
e	f	g	a	b	c	d
d	e	f	g	a	b	c
c	d	e	f	g	a	b

and choose every fourth note instead of every third, from c to f.

g	a	b	c	d	e	f
d	e	f	g	a	b	c
a	b	c	d	e	f	g
e	f	g	a	b	c	d
b	c	d	e	f	g	a
f	g	a	b	c	d	e
c	d	e	f	g	a	b

Any stacks of three or more notes make up a quartal chord.

Traditional analysis breaks down for quartal chords. Traditional analysis of quartal chords is possible but does not truly represent their function. For example, a suspender fourth would not usually resole down to a third as in traditional harmony. Here are some traditional forms forced onto the quartal chords:

13^th	CMaj13	Dm13	Em11b13b9	FMaj13#11	G13	Am11b13	Bm11b13b9b5
11^th	C11sus6	Dm11sus6	Em11b9susb6	FMaj9#11sus6	G11sus6	Am11susb6	Bm11b9susb6
5 note	CMaj7sus6add4	DMaj7sus6add4	EMaj7susb6ad4	FMaj7sus6add#4	G7sus6add4	AMaj7sus6badd4	BMaj7susb6ad4
7th	CMaj7add4(no5th)	DMaj7add4(no5)	EMaj7add4(no5)	FMaj7add#4(no5th)	G7add4(no5th)	AMaj7add4(no5)	BMaj7add4(no5)
triads	CMaj7sus4(no5th)	D7sus4(no5th)	E7sus4(no5th)	FMaj7sus#4(no 5th)	G7sus4(no5th)	A7sus4(no5th)	B7sus4(no5th)

Notice that no true ninth exists and the chords built on the seventh scale degree lack any diminished quality until the thirteenth chord.

Listen to music by De_bussy or the theme to the original Star Trek for uses of quartal chords.

Quartal chords based on other scales also exist. Many jazz players like to solo in fourths, so these patterns are useful even if not used in chords.

Quartal chords of the harmonic minor lead to some interesting substitutions, particularly on the seventh scale degree.

AmMaj11b13	Bm13b9b5	CMaj13#5	Dm13#11	E11b13b9	Fm6(Maj7)add#4	G#b5add#2add#6add_b9
AmMaj11susb6	Bm11b9sus6	C11#5sus6	Dm9#11sus6	E11b9susb6	FMaj7#9#11sus6	G#add#2add#6add_b9
AmMaj7susb6add4	Bm67add4	CMaj7#5sus6add4	DMaj7sus6add#4	E7susb6add4	FMaj7sus6add#4	G#add#2add#6
AmMaj7add4(no5th)	BMaj7add4(no5)	CMaj7add4(no5th)	DMaj7add#4(no5)	E7add4(no5th)	FMaj7add#4(no5th)	G#msusb7
AMaj7sus4(no5th)	B7sus4(no5th)	CMaj7sus4(no5th)	D7sus#4(no5th)	E7sus4(no5th)	FMaj7sus#4(no 5th)	G#o7(no5th)

On g# chords of 3 or more notes, the c is effectively the third, with the m3 treated as #2.

Quartal chords on melodic minor ascending

Am(Maj13)	Bm13b9	CMaj13#11#5	D13#11	E11b13b9	F#m6(Maj7)#11	G#7b13b9b5add#2
Am(Maj11)sus6	Bm11b9sus6	C9#11#5sus6	D9#11sus6	E11b9susb6	F#Maj7#9#11sus6	G#7b9add#2susb6
Am(Maj7)sus6add4	Bm67add4	CMaj7#5sus6add#4	D7sus6add#4	E7susb6add4	F#Maj7sus6add#4	G#7add#2susb6
Am(Maj7)add4(no5th)	BMaj7add4(no5)	CMaj7b5	D7b5	E7add4(no5th)	F#Maj7b5	G#7add#2(no5th)
AMaj7sus4(no5th)	B7sus4(no5th)	CMaj7sus#4(no5th)	D7sus#4(no5th)	E7sus4(no5th)	F#Maj7sus#4(no 5th)	G#7(no5th)

Quintal Chords

Quintal Chords stack in fifth instead of fourths.

f	g	a	b	c	d	e
b	c	d	e	f	g	a
e	f	g	a	b	c	d
a	b	c	d	e	f	g
d	e	f	g	a	b	c
g	a	b	c	d	e	f
c	d	e	f	g	a	b

Any stacks of three or more of these notes make up a quintal chord. Notice that quintal chord have no true seventh, ninths, or elevenths.

CMaj13	Dm13	Em11b13b9	FMaj13#11	G13	Am11b13	Bm11b13b9b5
C6Maj9	Dm69	Em9add_b6	F69	G69	Am9add_b6	BMaj7b9add_b6
C6add2	Dm6add2	Emadd2add_b6	F6add2	G6add2	Aadd2add_b6	Bmadd_b2add_b6
C6sus2	D6sus2	Esus2add_b6	F6sus2	G6sus2	Asus2add_b6	Bsusb2add_b6
Csus2	Dsus2	Esusb2	Fsus2	Gsus2	Asus2	Bsusb2

On the harmonic scale, again with some truly tortured chord analysis on g#.

Am11b13	Bm13b9b5	CMaj13#5	Dm13#11	E11b13b9	Fm6(Maj7)#11	G#b5add#2add#6add_b9
Am9add_b6	Bm67b9	C6Maj9#5	Dm69	E9add_b6	F67add#2	G#madd_b2add_b6add#6
Amadd2add_b6	Bm6add_b2	C6#5add2	Dm6add2	Eadd2add_b6	F6add#2	G#madd_b2add_b6
Asus2add_b6	B6susb2	C6#5sus2	D6sus2	Esus2add_b6	Fm6	G#susb2add_b
Asus2	Bsusb2	Csus2	Dsus2	Esus2	Fm	G#b5susb2

Melodic Minor Ascending

Am(Maj13)	B13b9	CMaj13#11#5	D13#11	E11b13b9	F#m6(Maj7)#11	G#7b5add#2add#6add_b9
Am(Maj11)	B67b9	C6Maj9#5	D69	E9add_b6	Fb77add#2	G#add2add_b6add#6
Amadd2add#4	B6add_b2	C6#5add2	D6add2	Eadd2add_b6	Fb7add#2	G#add2add_b6
Amsus2add#4	B6susb2	C6#5sus2	D6sus2	Esus2add_b6	Fb7sus#2	G#sus2add_b6
Asus2	Bsusb2	Csus2	Dsus2	Esusb2	F#sus#2	G#b5susb2

Since quintal chords are basically the circle of fifths, they are a good practice tool. Check out "Message in a Bottle" by The Police for use of quintal chords.

Chords based on sixths are theoretically possible, but since sixth are just groups of two thirds they tend to get washed into traditional harmony.

Chords based on sevenths are also theoretically possible, but the number of octaves necessary makes them unwieldy

d	e	f	g	a	b	c
e	f	g	a	b	c	d
f	g	a	b	c	d	e
g	a	b	c	d	e	f
a	b	c	d	e	f	g
b	c	d	e	f	g	a
c	d	e	f	g	a	b

CMaj13	Dm13	Em11b13b9	FMaj13#11	G13	Am11b13	Bm11b13b9b5
C6Maj7add4	D67add4	EMaj7add4add_b6	F6Maj7add#4	G67add4	AMaj7b6add4	BMaj7b6add4
C6Maj7sus4	D67sus4	E7add_b6sus4	F6Maj7sus#4	G67sus4	AMaj7b6sus4	BMaj7b6sus4
C6Maj7(no3rd)	D67(no3rd)	E7add_b6(no3rd)	F6Maj7(no3rd)	G67(no3rd)	Amzb6(no3rd)	BMaj7b6(no3rd)
CMaj7sus6(no3rd)	D7sus6(no3rd)	E7susb6(no3rd)	FMaj7sus6(no3rd)	G7sus6(no3rd)	AMaj7susb6(no3rd)	BMaj7susb6(no3rd)

Harmonic minor

Am(Maj11)b13	Bm13b9b5	CMaj13#5	Dm13#11	E11b13b9	Fm6(Maj7)#11	G#b5add#2add#6add_b9
Am(Maj7)add4add_b6	Bm67b5add4	C6Maj7#5add4	Dm67add#4	E7add4add_b6	F6Maj7add#4	G67add4add#2
Am(Maj7)add_b6sus4	Bm67b5sus4	C6Maj7#5sus4	D67sus#4	E7add_b6sus4	F6Maj7sus#4	G#7b5add_b6
AMaj7add_b6	B67sus#4	C6Maj7#5(no3rd)	D67(no3rd)	E7add_b6(no3rd)	F6Maj7(no3rd)	G#7b5add_b6(no3rd)
AMaj7susb6(no3rd)	B7sus6(no3rd)	CMaj7sus6(no3rd)	D7sus6(no3rd)	E7susb6(no3rd)	FMaj7sus6(no3rd)	G#7susb6(no3rd)

Melodic minor Ascending

Am(Maj11)b13	Bm13b9b5	CMaj13#11#5	D13#11	E11b13	F#m13	G#7b5add#2add#6add_b9
Am(Maj7)b6add4	Bm67add4	C6Maj7#5add#4	D67add#4	E7add4add_b6	F#m67add4	G67add4add#2
Am(Maj7)b6sus4	Bm67sus4	C6Maj7#5sus#4	D67sus#4	E7add_b6sus4	Fb77sus4	G#7b5add_b6
AMaj7add6(no3rd)	B67(no3rd)	C6Maj7#5(no3rd)	D67(no3rd)	E7add_b6(no3rd)	Fb77(no3rd)	G#7b5add_b6(no3rd)
AMaj7)us6(no3rd)	B7sus6(no3rd)	CMaj7sus6(no3rd)	D7sus6(no3rd)	E7susb6(no3rd)	F#7sus6(no3rd)	G#7susb6(no3rd)

Chords based on seconds are called clusters, discussed in the section on symmetrical chords.

The Circle of Fifths

The easiest way to order chords is chromatically, CC#DD#EFF#GG#AB.

However, in actual use chords tend form different group families. Remember from the beginning of the discussion on pitch, that the most consonant interval after the octave is the perfect fifth, defined initially as the note with three times the frequency of the fundamental frequency. Further development of the harmonies showed the importance of the tonic-dominant relationship. The tonic-dominant relationship puts chord with other chords a fifth apart. C and its G work together well, while G calls for its dominant D. In the key of G, the chords CDG produce the IV-V-I pattern essential to tonal music. So, when listing chords, an order that follows the pattern of dominants is more functional, developing from the very nature of the physics of music.

This pattern, the circle of fifths, is the first large pattern than many musicians see, and is integral to pattern obsession

Scales

Consider the C major scale

cdefgabc

The intervals between the notes follow the familiar pattern of whole steps (W) and half steps (H)

WWHWWWH

But starting at c is arbitrary, developed over years of tradition. Consider the Mixolydian mode starting on G, up a perfect fifth.

gabcdef

This scale sounds almost identical to the C scale moved, except for the last note. Changing the seventh of the mixolydian to mimic the sound of the leading tone is fundamental to the development of harmony, as seen before, so the f# here flows naturally.

gabcdef#

the G scale.

The same procedure works when moving to the dominant of G, which is d. Write out the notes of the previous chord, but start on d, the root of the dominant

def#gabc

then raise the seventh degree by adding a sharp to yield

def#gabc#

continue up a perfect fifth to A

a_bc#def#g#

Then up a perfect fifth E

ef#g#a_bc#d#

Notice the pattern emerging. The notes with the sharp added go in this order

fcgd

after the f, the sharps seem to add in the same order as the original. If this were true, the next sharp should be a# in the scale of B.

bc#d#ef#g#a#

and, indeed, it is. So, the order of sharps follows the same order as the circle if fifths, but lagging two behind.

Slow down at B to realize the next chord is based on F#, not F because the distance from b to f (d5) is smaller than the previous intervals. So, the next scale is the first one to begin on a sharp, F#, but still follows the pattern of adding sharps

f#g#a#bc#d#e#

one more to c#

c#d#e#f#g#a#b#

Everything is sharp now. Most theory textbooks bail out of the pattern by here and start using flats, but since this method is following patterns to their extreme, consider a scale built on G#. Following the pattern, take the scale from before, the

c#d#e#f#g#a#b#

rearrange it to start on the fifth

g#a#b#c#d#e#f#

and then raise the seventh by adding a sharp. Since f# already has a sharp, the seventh degree becomes f double sharp, fx.

g#a#b#c#d#e#fx

With that settled, jump back into the pattern, scale on d# with cx. The double sharps enter in the same pattern as the sharps did, fcgbdae_b

d#e#fxg#a#b#cx

Scale of A# with gx

a#b#cxd#e#fxgx

Scale of E# with dx

e#fxgxa#b#cxd#

Scale of B# with ax

b#cxd#e#fxgxax

At this point all twelve notes have a scale associated with them, so call it complete. Theoretically, the next scale would be on Fx, but B# is a good stopping point, and these last few scales are monstrosities anyway.

Circle of Fourths

Although b# is a valid note, most times the enharmonic equivalent of C makes a lot more sense. This beast

b#cxd#e#fxg#ax

is just a terribly spelled C scale.

cdefgab

So, outside of a theory textbook, almost no one would ever use the B# scale.

Start with the C scale and work backwards around the circle of fifths. Since a fifth going up is the same as a fourth going down, the two link together nicely.

Start with C and move up a fourth instead of a fifth to f. Now rearrange the notes of the C scale with the f first

fgabcde

This is the Lydian mode. To turn the Lydian into the major, just lower the fourth scale degree to b_b.

fgaBbcde

the next note a fourth up is that b_b just altered. Rearrange and lower the fourth (which just happened to be the last note of the previous scale)

BbcdEbfga

Continue the process to EB

EbfgAbBbcd

Then Ab

AbBbcDbEbfg

Notice the pattern of scales CFBbEbAb compared to the order of added flats BEAD. The order of added flats trails the key root pattern again by 2 places. Continue to Db

DbEbfGbAbBbc

GbAbBbCbDbEbf

and Cb

CbDbDbFbGbAbBb

At this point all notes are flat, but the pattern could continue for all enharmonic equivalents. The next scale would be Fb, with the lowered fourth going to b double flat (b_bb).

FbGbAb(Bb_b)CbDbEb

The double flats would enter in the same pattern the sharps did, beadgcf, the retrograde order of the sharps, fcgbdae_b. However, Fb completes the cycle, so no need to continue.

Summary chart from most flats to most sharps following the circle of fifths reading top to bottom.

fb	g_b	a_b	b_bb	c_b	d_b	e_b
c_b	d_b	e_b	fb	g_b	a_b	b_b
g_b	a_b	b_b	c_b	d_b	e_b	f
d_b	e_b	f	g_b	a_b	b_b	c
a_b	b_b	c	d_b	e_b	f	g
e_b	f	g	a_b	b_b	c	d
b_b	c	d	e_b	f	g	a
f	g	a	b_b	c	d	e
c	d	e	f	g	a	b
g	a	b	c	d	e	f#
d	e	f#	g	a	b	c#
a	b	c#	d	e	f#	g#
e	f#	g#	a	b	c#	d#
b	c#	d#	e	f#	g#	a#
f#	g#	a#	b	c#	d#	e#
c#	d#	e#	f#	g#	a#	b#
g#	a#	b#	c#	d#	e#	fx
d#	e#	fx	g#	a#	b#	cx
a#	b#	cx	d#	e#	fx	gx
e#	fx	gx	a#	b#	cx	d#
b#	cx	d#	e#	fx	gx	ax

So, the circle of fifths with all sharps is

CGDAEBF#C#G#D#A#E#

And the circle of fourths with all enharmonic equivalents written in flats is

CFBbEbABDbGbCbFb

In practice, the scales with double sharps or flats are usually ignored, leaving three sets of enharmonic chords in the traditional spelling of the circle of fifths

CGDAE(B/Cb)(F#/Gb)(C#/Db)AbEbBbF

And the circle of fourths

CFBbEbAb(Db/C#)(Gb/F#)(Cb/E)EADG

Any exploration of "all" keys usually contains these fifteen sets, with three keys being enharmonic equivalents.

Minor Scales

The natural minor scale is simply the major scale beginning on the sixth degree. Beginning the circle of fifths for minors starts not on C, but on its relative minor, Am because they both have the same number of sharps are flats, none.

The natural minor scales using the same patterns as the majors are

a_bcdefg

ef#gabcd

bc#def#ga

f#g#a_bc#de

c#d#ef#g#a_b

g#a#bc#d#ef#

d#e#f#g#a#bc#

a#b#c#d#e#f#g#

e#fxg#a#b#c#d#

b#cxd#e#fxg#a#

fxgxa#b#cxd#e#

cxd#e#fxgxa#b#

gxaxb#cxd#e#fx

a_bcdefg

defgab_bc

gab_bcde_bf

cde_bfga b_bb

fgab(b_b)cd_be_b

(b_b)cd_be_bfg_ba_b

e_bfg_ba_b(b_b)c_bd_b

a_b(b_b)c_bd_be_bfbg_b

d_be_bfbg_ba_b(b_bb)c_b

The minors with the double sharps as a root are not pleasant, but those keys are just theoretical anyway.

The harmonic minor scales raise the seventh of all

a_bcdefg#

ef#gabcd#

bc#def#ga#

f#g#a_bc#de#

c#d#ef#g#a_b#

g#a#bc#d#efx

d#e#f#g#a#bcx

a#b#c#d#e#f#gx

e#fxg#a#b#c#dx

b#cxd#e#fxg#ax

fxgxa#b#cxd#ex

cxd#e#fxgxa#bx

gxaxb#cxd#e#fx#*

a_bcdefg#

defgab_bc#

gab_bcde_bf#

cde_bfgab_b

fgab(b_b)cd_be

(b_b)cd_be_bfg_ba

e_bfg_ba_b(b_b)c_bd

a_b(b_b)c_bd_be_bfbg

d_be_bfbg_ba_b(b_bb)c

*Notice the extremely rare triple sharp in the gx minor scale.

The ascending melodic minor scale raises the sixth also

a_bcdef#g#

ef#gabc#d#

bc#def#g#a#

f#g#a_bc#d#e#

c#d#ef#g#a#b#

g#a#bc#d#e#fx

d#e#f#g#a#b#cx

a#b#c#d#e#fxgx

e#fxg#a#b#cxdx

b#cxd#e#fxgxax

fxgxa#b#cxdxex

cxd#e#fxgxaxbx

gxaxb#cxd#exfx#

a_bcdef#g#

defgabc#

gab_bcdef#

cde_bfgab

fgab(b_b)cde

(b_b)cd_be_bfga

e_bfg_ba_b(b_b)cd

a_b(b_b)c_bd_be_bfg

d_be_bfbg_ba_b(b_b)c

Key Signatures

The circle of fifths' most prevalent use is in establishing key signatures. Moving around the circle of fifths either adds one sharp or subtracts one flat from each key.

Key

B/Cb

F#/Gb

C#/Db

Number of sharps

Number of flats

added sharp

subtracted flat

Relative Minor

B/Cb

F#/Gb

C#/Db

Completing the theoretical keys:

C/Db_b

G/Ab_b

D/Eb_b

A/Bb_b

E/Fb

B/Cb

F#/Gb

C#/Db

G#/Ab

D#/Eb

A#/Bb

E#/F

B#/C

g_bb

d_bb

a_bb

e_bb

b_bb

A/Bb_b

E/Fb

B/Cb

F#/Gb

C#/Db

G#/Ab

D#/Eb

A#/Bb

E#/F

B#/C

C/Db_b

G/Ab_b

D/Eb_b

Key signatures placed on a traditional five-line staff also form a nice pattern, although the pattern needs to break to avoid putting the accidentals on ledger lines.

Chord Symbols

Some students wonder why the Pattern Obsession method uses terms such as complicated chord symbols without full explanation. This method teaches use of musical patterns in context rather than just a list of formulas. Traditional music theory classes might teach the chords by listing the types of chords--major, minor, diminished, and augmented--then listing the intervals in each chord, then having the student build chords built on the intervals. Teaching using brute force traditional drilling techniques may be a way to learn the forms, but without context they lose their meaning. So, only now at this point in the evolution of musical structure are all the tools necessary for an exhaustive lesson on how to use the chord symbols available.

To match a chord symbol to a series of notes, first compare the group of notes to a dominant chord based on the same note. Any deviations from the dominant would cause an alteration. So, in the key of C, the dominant builds on g.

The first part of the chord symbol is the letter corresponding to the root of the chord, not necessarily the lowest note or bass note, but the chord on which the analysis must be done.

In the triads, the dominant chord is a major triad. So, a symbol without any alterations is a major triad.

G is gbd

Or for F, build a dominant chord based on F, fac.

In analysis, first look at the third of the chord then the fifth.

If the third is major, and the fifth is unaltered, no alteration is necessary because it matches the dominant pattern.

If the third is minor and the fifth is unaltered, then add the m designation to indicate a minor third

Gm g_bbd Transposing to Fm yields fa_bc

If the third is major and the fifth is sharp, gbd#, then the chord is augmented, indicated by adding a +, or sometimes aug

G+ or Gaug

For F-fac#

The notation GMaj#5 also fits to indicate a major third with an augmented.

In the rare case with a major third and a minor 5^th, gbd_b, use of G(Maj)b5 might be OK, but do some further analysis. This is proba_bly some other chord disguised as a G.

If the third is minor, then check the fifth. If the fifth is unaltered, g_bbd for example, the chord symbol reflects the minor third by adding an m to the symbol, Gm in this case, or with an Fm being fa_bc as seen before. However if the third is lowered and the fifth is lowered, g_bbd_b, then the triad is diminished and notated

Go or sometimes Gdim.

Fo is fa_bc_b

If the third is minor and the fifth is augmented, g_bbd# notate as Gm#5

Fm#5 is fa_bc#. This chord would be rare.

Chords with an sixth in them but no fifth are suspended sixth chords since the sixth replaces the fifth. Use the sus6 tag to show that the fifth has been replaced by the 6^th.

g_be Gsus6 Fsus6 fad

Append the same tag to a minor chord

g_bbe Gmsus6 Fmsus6 fa_bd

and if the 6^th is lowered from the expected note, use susb6

g_be_b Gsusb6 Fsusb6 fad_b

g_bbe_b Gmsusb6 Fmsus6 fa_bd_b

If a third does not exist, or is neither major nor minor, the chord is proba_bly a suspension or an inversion replacing the third. Either a fourth or a second could replace the third, with the fourth being the most common. First, check for fourths. If a fourth exists and the fifth is unaltered, such as in the chord gcd, the c is replacing the third, so this is a Gsus4 (or Gsus) chord.

The transposition to Fsus4 has a b_b instead of a b, fb_bc, because the expected note in the scale always comes from the Mixolydian scale based on that root. So, with f as a root, the unaltered notes are fgab_bcde_b. Any note deviating from this group needs some king of marking to show the difference. The Mixolydian scale is just the major scale with a lowered seventh, if that helps.

Alternately, if the fifth is not a fifth at all but a sixth, gce, this might not be a G chord at all, but a C chord in second inversion C/G. For f the equivalent is Bb/F or f, b_b,d). Altered fifths or sixth could also lead to the following chords.

gcd# G(MAJ)#5sus4 F#5sus4 is fb_bc#

gcd_b G(Maj)b5sus4 Fb5sus4 is fb_bc_b (extremely rare)

A suspended fourth could also be in the same chord as a suspended sixth

gce Gsus4sus6 Fsus4sus6 fb_bd

This chord was also analyzed as C/G previously. Either could be correct. Many chords may have multiple symbols possible. The context of the music might indicate which is better. Other variations of sus4sus6 chords include

gce_b Gsus4susb6 Fsus4susb6 fb_bd_b

gc#e Gsus#4sus6 Fsus#4sus6 fbd

Note that in f, the #4 is not a b#, but a b natural. Since the expected tone is the b_b, the # designation cancels out the expected flat.

gc#e_b Gsus#4susb6 Fsus#4susb6 fb

Or possibly the g is the bass note of some inversion, depending on context.

gce_b Cm/G Bbm/F fb_bd_b

gce# Csus4/G Bbsus4/F fb_bd# (misspelled)

The suspension could also be altered

gc#d#

G(Maj)#5sus#4

F#5sus#4 is

fbc#

If the chord is neither major nor minor, and no fourth is apparent, check for suspended seconds replacing thirds.

gad Gsus2 Fsus2 fgc

gad_b Gb5sus2 Fb5sus2 fgc_b

gad# G(Maj)#5sus2 F#5sus2 fgc#

Or flatted seconds

gabd Gsusb2 Fsusb2 fg_bc

gabd_b Gb5susb2 Fb5susb2 fg_bc_b

gabd# G(Maj)#5susb2 Fsus2b#5 fg_bc#

If none of these patterns hold, try multiple suspensions or add notes.

Chords could be a suspended second together with a suspended 6^th

Gsus2sus6 gae Fsus2sus6 fgd

along with the permutations

Gsusb2sus6 gabe Fsusb2sus6 fgbd

Gsus2susb6 gae_b Fsus2susb6 fgd_bb

Gsusb2susb6 gabe_b Fsusb2susb6 fgbd_b

Note that none of these triads contains the seventh degree of the scale. Inclusion of the seventh degree of the scale strongly suggest that the apparent triad is actually a seventh chord with a missing note. The note combination g_bf is proba_bly an G7 chord (gbdf) without the fifth degree, G7(no 5^th). Since seventh chords are covered later, adding all the possible chords with missing notes here would lead to a lot of ambiguity and just muddy the waters.

Alternately, the g could be proba_bly a passing tone, third or higher inversion, or some kind of chromatic modulation. Finding a triad not fitting one of the patterns above is unlikely. Try analyzing the chord looking for its true root, then fit the bass note in as an inversion or passing tone. Check for misspelled notes. Remember that the same combination of notes can have multiple chord symbols depending on context.

Summary of possible triads with G as a bass note:

G	and	b	b_b	c	c#	a	a_b
with	Interval	M3	m3	P4	A4	M2	m2
d	P5	G	Gm	Gsus4	Gsus#4	Gsus2	Gsusb2
d_b	D5	G(Maj)b5	Go	G(Maj)b5sus4	G(Maj)b5sus#4	Gb5sus2	Gb5susb2
d#	A5	G+	Gm#5	G(MAJ)#5sus4	G(MAJ)#5sus#4	G(MAJ)#5sus2	G(MAJ)#5susb2
e	M6	Gsus6	Gmsus6	Gsus4sus6	Gsus#4sus6	Gsus2sus6	Gsus2susb6
e_b	m6	Gsusb6	Gmsusb6	Gsus4susb6	Gsus#4susb6	Gsus2susb6	Gsusb2susb6

Alternate analysis of the bottom left corner using inversions.

Em/G	Eo/G		C/G		C#o/G
Eb+/G		Eb/G		Cm/G

Seventh Chords

Luckily, most of the seventh chords are just a note put on top of the triads from the last step, so the grueling process of figuring out the base chords is complete. Three possible 7ths exist, f (7), f# (Maj7), and Fb (d7). Each seventh adds to the triads in its own chart. Just adding more lines to the existing chart would not work, as a complete chart would need to be three dimensional to cover all the possibilities.

Minor Sevenths

Due to a quirk in the notational system, the easiest seventh chord to notate is the minor seventh. Even though the interval of the seventh chord is a minor seventh, the chord notates with a 7 symbol with no m. The tradition happened because the seventh chord based on the dominant , gbdf, is by far the most widely used seventh chord. Because of the resolution back to the tonic, with f resolving to e to highlight the other natural chromatic resolution of the b leading to c, as discussed before the dominant seventh is widely used. Since this dominant seventh, sometimes referred to as a major-minor seventh in music theory textbooks, is so ubiquitous, the shorthand 7 became associated with it.

This tradition confuses some beginners since a major third in a chord symbol has no alteration, but a major seventh does need an alteration. For the same reason, this chart above centers around G rather than the central point of C in previous charts. The altered notation is most prevalent in the dominant chords, so chord symbols work on the assumption that the chord being modified is based on the fifth scale degree (Mixolydian) rather than the first.

So, the most prevalent seventh is the minor seventh (7). Adding a minor seventh to major and minor chords is easy, just append a 7 to the chord symbols.

G7 gbdf,

Gm7 g_bbdf

(All other chords in the following list are just the triads plus an f, so chord spellings will discontinue unless needed.)

The GMajb5 chord with an added seventh becomes G7b5 since the Maj that needed insertion to indicate the major third is now implied by the 7 designation.

For any chord with multiple alterations, list the highest degree unaltered symbol first and then the alterations in ascending order.

G7sus4

G7#5susb2 …

Two of the chord forms, the diminished and the augmented need changes. The diminished chord Go with a minor seventh, g_bbd_bf would not use the symbol Go7. That symbol is reserved for another chord to come later. Instead this chord, known as half diminished, treats the flat third as a minor and flat fifth of the diminished chords as an alteration, mb5 and then appends the seventh. Following the pattern of listing unaltered chords first, the correct order of placement is Gm7b5. Shorthand for half diminished chords appear in jazz charts appears as a circle with a line through it.

The correct notation for an augmented chord with a minor seventh attached is not G+7. That nomenclature is meaningless. Instead, the augmented fifth appears as a #5 in the chord G7#5.

Now the chart for minor sevenths is complete.

G7	Gm7	G7sus4	G7sus#4	G7sus2	G7susb2
G7b5	Gm7b5	G7b5sus4	G7b5sus#4	G7b5sus2	G7b5susb2
G7#5	Gm7#5	G7#5sus4	G7#5sus#4	G7#5sus2	G7#5susb2
G7sus6	Gm7sus6	G7sus4sus6	G7sus#4sus6	G7sus2sus6	G7sus2susb6
G7susb6	Gm7susb6	G7sus4susb6	G7sus#4susb6	G7sus2susb6	G7susb2susb6

One new group of inversion does occur, the chord where g is the minor seventh in third inversion, or

A7/G for major

AMaj7/G for minor

AMaj7b5/G for diminished and

A7#5/G for augmented, and all the alterations

A7/G	Am7/G	A7sus4/G	A7sus#4/G	A7sus2/G	A7susb2/G
A7b5/G	Am7b5/G	A7b5sus4/G	A7b5sus#4/G	A7b5sus2/G	A7b5susb2/G
A7#5/G	Am7#5/G	A7#5sus4/G	A7#5sus#4/G	A7#5sus2/G	A7#5susb2/G

Major Sevenths

The major seventh chord is the major seventh interval added to a chord, with the designation Maj7 (sometimes shortened to M7). Happily, the chord uses the same alterations as the 7 chord, so just replace all the 7s with Maj7. The only notational quirk is with minor chords plus major sevenths, where parentheses add clarity Gm(Maj7).

GMaj7	Gm(Maj7)	GMaj7sus4	GMaj7sus#4	GMaj7sus2	GMaj7susb2
GMaj7b5	Gm(Maj7)b5	GMaj7b5sus4	GMaj7b5sus#4	GMaj7b5sus2	GMaj7b5susb2
GMaj7#5	Gm(Maj7)#5	GMaj7#5sus4	GMaj7#5sus#4	GMaj7#5sus2	GMaj7#5susb2
GMaj7sus6	GMaj7sus6	GMaj7sus4sus6	GMaj7sus#4sus6	GMaj7sus2sus6	GMaj7sus2susb6
GMaj7susb6	GMaj7susb6	GMaj7sus4susb6	GMaj7sus#4susb6	GMaj7sus2susb6	GMaj7susb2susb6

The third inversion major seventh chords with G in the bass have a root of Ab.

AbMaj7/G	Abm(Maj7)/G	AbMaj7sus4/G	AbMaj7sus#4/G	AbMaj7sus2/G	AbMaj7susb2/G
AbMaj7b5/G	Abm(Maj7)b5/G	AbMaj7b5sus4/G	AbMaj7b5sus#4/G	AbMaj7b5sus2/G	AbMaj7b5susb2/G
AbMaj7#5/G	Abm(Maj7)#5/G	AbMaj7#5sus4/G	AbMaj7#5sus#4/G	AbMaj7#5sus2/G	AbMaj7#5susb2/G

Diminished Seventh

The diminished seventh interval (d7) is a special case because it only occurs on top of a diminished chord (o7) or (dim7). The diminished seventh chord, or fully diminished chord, is a symmetrical stack of four minor thirds, here g_bbd_bfb. Since the chords are symmetrical, any note of the chord could be the root, so Go, Eo, Bbo, and Dbo, along with their enharmonic equivalents, are all the same chord. Typically, a diminished seventh chord is not suspended because of the ambiguity of what the root actually is, although a chord like Go7sus2 might be theoretically possible.

Use of the diminished seventh outside the fully diminished chord would just be using the diminished seventh as a misspelled sixth, Dm6b5. These chords appear later in the list of sixth or thirteenth chords. See the section of chromatic harmonies for discussion of misspelled chords.

Summary of the seventh chords involving G

G7	Gm7	G7sus4	G7sus#4	G7sus2	G7susb2
G7b5	Gmj7b5	G7b5sus4	G7b5sus#4	G7b5sus2	G7b5susb2
G7#5	Gm7#5	G7#5sus4	G7#5sus#4	G7#5sus2	G7#5susb2
G7sus6	Gm7sus6	G7sus4sus6	G7sus#4sus6	G7sus2sus6	G7sus2susb6
G7susb6	Gm7susb6	G7sus4sus6	G7sus#4susb6	G7sus2susb6	G7susb2susb6
GMaj7	Gm(Maj7)	GMaj7sus4	GMaj7sus#4	GMaj7sus2	GMaj7susb2
GMaj7b5	Gm(Maj7)b5	GMaj7b5sus4	GMaj7b5sus#4	GMaj7b5sus2	GMaj7b5susb2
GMaj7#5	Gm(Maj7)#5	GMaj7#5sus4	GMaj7#5sus#4	GMaj7#5sus2	GMaj7#5susb2
GMaj7sus6	Gm(Maj7)sus6	GMaj7sus4sus6	GMaj7sus#4sus6	GMaj7sus2sus6	GMaj7sus2susb6
GMaj7susb6	Gm(Maj7)susb6	GMaj7sus4susb6	GMaj7sus#4susb6	GMaj7sus2susb6	GMaj7susb2susb6
Go7

Ninth Chords

Moving up the scale to the ninth adds two possibilities, the interval of the ninth (9) and the minor ninth (b9). An augmented ninth (#9) would be just a misspelling of the minor third, so it is omitted.

Adding an unaltered ninth to a chord with a minor seventh is straightforward since the 9^th is the expected tone in the Mixolydian scale. If an unaltered ninth adds to an unaltered seventh, the 9 replaces the 7, G9, Gm9 with any altered tones after remaining the same, Gm9sus#4 …

One quirk occurs when attempting to add a ninth to a chord with a second such as Gsus2. Since the ninth is enharmonic to the 2^nd, a chord like G9sus2 has the note a added twice, as the 2^nd and again as the ninth. Since this is redundant, those chords do not exist.

Adding an unaltered ninth to a chord with a suspended lowered ninth would be quite unusual, but theoretically possible. The charts retain these chords, but realize they are unlikely and many theorists would eliminate them completely.

G9	Gm9	G9sus4	G9sus#4	(redundant)	G9susb2
G9b5	Gm9b5	G9b5sus4	G9b5sus#4	(redundant)	G9b5susb2
G9#5	Gm9#5	G9#5sus4	G9#5sus#4	(redundant)	G9#5susb2
G9sus6	Gm9sus6	G9sus4sus6	G9sus#4sus6	(redundant)	G9sus2susb6
G9susb6	Gm9susb6	G9sus4susb6	G9sus#4susb6	(redundant)	G9susb2susb6

When adding a ninth to a chord with a major seventh, the designation Maj9 implies a major seventh, GMaj9. for chords with a minor third, Gm9(Maj7). Any further alternations follow in order, Gm(Maj9)b5

GMaj9	Gm(Maj9)	GMaj9sus4	GMaj9sus#4	(redundant)	GMaj9susb2
GMaj9b5	Gm(Maj9)b5	GMaj9b5sus4	GMaj9b5sus#4	(redundant)	GMaj9 b5susb2
GMaj9#5	Gm(Maj9)#5	GMaj9#5sus4	GMaj9#5sus#4	(redundant)	GMaj9#5susb2
GMaj9sus6	Gm(Maj9)sus6	GMaj9sus4sus6	GMaj9sus#4sus6	(redundant)	GMaj9sus2susb6
GMaj9susb6	Gm(Maj9)susb6	GMaj9sus4susb6	GMaj9sus#4susb6	(redundant)	GMaj9susb2susb6

Adding a ninth to a fully diminished seventh chord could be notated Go9, but this notation is not standard.. These chords are so rare that other analyses might be possible. Instead, the diminished seventh interval becomes the enharmonic sixth. Sixth chords will be covered later.

Altered Ninths

Adding a lowered ninth (b9) a chord is similar, but demands the retention of the seventh designation, G7b9. Notice that the 7 precedes the 9 because an unaltered interval always begins when possible. For minor chords, Gm7b9.

For chords with other alterations, the b9 appears first in the list after the unaltered 7, G7b9b5susb2, so the altered tones appear in descending numerical order. A b9 would be enharmonic to b2, so those chords are redundant.

G7b9	Gm7b9	G7b9sus4	G7b9sus#4	G7b9sus2	(redundant)
G7b9b5	Gm7b9b5	G7b9b5sus4	G7b9b5sus#4	G7b9b5ssus2	(redundant)
G7b9#5	Gm7b9#5	G7b9#5 us4	G7b9#5sus#4	G7b9#5sus2	(redundant)
G7b9sus6	Gm7b9sus6	G7b9sus4sus6	G7b9sus#4sus6	G7b9sus2sus6	(redundant)
G7b9susb6	Gm7b9susb6	G7b9sus4susb6	G7b9sus#4susb6	G7b9sus2susb6	(redundant)

For b9 combined with Maj7, the Maj7 replaces the 7, with parentheses for minor chords for clarity,. Note that even though the Maj7 is altered, it appears first for major chords, and just after the m for minor chords.

GMaj7b9	Gm(Maj7)b9	GMaj7b9sus4	GMaj7b9sus#4	GMaj7b9sus2	(x)
GMaj7b9b5	Gm(Maj7)b9b5	GMaj7b9b5sus4	GMaj7b9b5sus#4	GMaj7b9b5sus2	(x)
GMaj7b9#5	Gm(Maj7)b9#5	GMaj7b9#5sus4	GMaj7#5b9sus#4	GMaj7b9#5sus2	(x)
GMaj7b9sus6	Gm(Maj7)b9sus6	GMaj7b9sus4sus6	GMaj7b9sus#4sus6	GMaj7b9sus2sus6	(x)
GMaj7b9susb6	Gm(Maj7)b9susb6	GMaj7b9sus4susb6	GMaj7b9sus#4susb6	GMaj7b9sus2susb6	(x)

Summary of Ninth Chords

G9	Gm9	G9sus4	G9sus#4	G9susb2
G9b5	Gm9b5	G9b5sus4	G9b5sus#4	G9b5susb2
G9#5	Gm9#5	G9#5sus4	G9#5sus#4	G9#5susb2
G9sus6	Gm9sus6	G9sus4sus6	G9sus#4sus6	G9sus2susb6
G9susb6	Gm9susb6	G9sus4susb6	G9sus#4susb6	G9susb2susb6
GMaj9	Gm(Maj9)	GMaj9sus4	GMaj9sus#4	GMaj9susb2
GMaj9b5	Gm(Maj9)b5	GMaj9b5sus4	GMaj9b5sus#4	GMaj9 b5susb2
GMaj9#5	Gm(Maj9)#5	GMaj9#5sus4	GMaj9#5sus#4	GMaj9#5susb2
GMaj9sus6	Gm(Maj9)sus6	GMaj9sus4sus6	GMaj9sus#4sus6	GMaj9sus2susb6
GMaj9susb6	Gm(Maj9)susb6	GMaj9sus4susb6	GMaj9sus#4susb6	GMaj9susb2susb6
G7b9	Gm7b9	G7b9sus4	G7b9sus#4	G7b9sus2
G7b9b5	Gm7b9b5	G7b9b5sus4	G7b9b5sus#4	G7b9b5ssus2
G7b9#5	Gm7b9#5	G7b9#5 us4	G7b9#5sus#4	G7b9#5sus2
G7b9sus6	Gm7b9sus6	G7b9sus4sus6	G7b9sus#4sus6	G7b9sus2sus6
G7b9susb6	Gm7b9susb6	G7b9sus4susb6	G7b9sus#4susb6	G7b9sus2susb6
GMaj7b9	Gm(Maj7)b9	GMaj7b9sus4	GMaj7b9sus#4	GMaj7b9sus2
GMaj7b9b5	Gm(Maj7)b9b5	GMaj7b9b5sus4	GMaj7b9b5sus#4	GMaj7b9b5sus2
GMaj7b9#5	Gm(Maj7)b9#5	GMaj7b9#5sus4	GMaj7#5b9sus#4	GMaj7b9#5sus2
GMaj7b9sus6	Gm(Maj7)b9sus6	GMaj7b9sus4sus6	GMaj7b9sus#4sus6	GMaj7b9sus2sus6
GMaj7b9susb6	Gm(Maj7)b9susb6	GMaj7b9sus4susb6	GMaj7b9sus#4susb6	GMaj7b9sus2susb6

Eleventh Chords

Two possible eleventh additions include 11 and #11 (The b11 would be enharmonic with M3, so b11 is redundant). For chords with an unaltered 11^th, four possibilities exist.

Added to an unaltered ninth, G9 becomes G11 with the 9^th and 7th implied.

Added to a Maj9, GMaj9 becomes GMaj11, with 9^th and Maj7 implied.

Added to a G7b9 chord becomes G11b9, with an unaltered 7 implied

Added to a GMaj7b9 becomes GMaj11b9, with a Maj7 implied.

Since an 11^th is enharmonic with a 4^th, redundant chords such as G11sus4 are omitted.

G11	Gm11	(x)	G11sus#4	G11susb2
G11b5	Gm11b5	(x)	G11b5sus#4	G11b5susb2
G11#5	Gm11#5	(x)	G11#5sus#4	G11#5susb2
G11sus6	Gm11sus6	(x)	G11sus#4sus6	G11sus2susb6
G11susb6	Gm11susb6	(x)	G11sus#4susb6	G11susb2susb6
GMaj11	Gm(Maj11)	(x)	GMaj11sus#4	GMaj11susb2
GMaj11b5	Gm(Maj11)b5	(x)	GMaj11b5sus#4	GMaj11b5susb2
GMaj11#5	Gm(Maj11)#5	(x)	GMaj11#5sus#4	GMaj11#5susb2
GMaj11sus6	Gm(Maj11)sus6	(x)	GMaj11sus#4sus6	GMaj11sus2susb6
GMaj11susb6	Gm(Maj11)susb6	(x)	GMaj11sus#4susb6	GMaj11susb2susb6
G11b9	Gm11b9	(x)	G11b9sus#4	G11b9sus2
G11b9b5	Gm11b9b5	(x)	G11b9b5sus#4	G11b9b5sus2
G11b9#5	Gm11b9#5	(x)	G11b9#5sus#4	G11b9#5sus2
G11b9sus6	Gm11b9susb6	(x)	G11b9sus#4sus6	G11sus2b9sus6
G11b9susb6	Gm11b9susb6	(x)	G11b9sus#4susb6	G11sus2b9susb6
GMaj11b9	Gm(Maj11)b9	(x)	GMaj11b9sus#4	GMaj11b9sus2
GMaj11b9b5	Gm(Maj11)b9b5	(x)	GMaj11b9b5sus#4	GMaj11b9b5sus2
GMaj11b9#5	Gm(Maj11)b9#5	(x)	GMaj11b9#5sus#4	GMaj11b9#5sus2
GMaj11b9sus6	Gm(Maj11)b9sus6	(x)	GMaj11b9sus#4sus6	GMaj11b9sus2sus6
GMaj11b9susb6	Gm(Maj11)b9susb6	(x)	GMaj11b9sus#4susb6	GMaj11 b9sus2susb6

For altered 11^th, add the tag #11 to the ninth chord in all cases, omitting the sus#4 column. A #11b5 combination would be redundant.

G9#11	Gm9#11	G9#11sus4	(x)	G9#11susb2
(x)	(x)	(x)	(x)	(x)
G9#11#5	Gm9#11#5	G9#11#5sus4	(x)	G9#11#5susb2
G9#11sus6	Gm9#11sus6	G9#11sus4sus6	(x)	G9#11sus6susb2
G9#11susb6	Gm9#11susb6	G9#11sus4susb6	(x)	G9#11susb6susb2
GMaj9#11	Gm(Maj9)#11	GMaj9#11sus4	(x)	GMaj9#11susb2
(x)	(x)	(x)	(x)	(x)
GMaj9#11#5	Gm(Maj9)#11#5	GMaj9#11#5sus4	(x)	GMaj9#11#5susb2
GMaj9#11sus6	GMaj9m#11sus6	GMaj9#11sus4sus6	(x)	GMaj9#11sus6susb2
GMaj9#11susb6	GMaj9m#11susb6	GMaj9#11sus4susb6	(x)	GMaj9#11susb6susb2
G7#11b9	GMaj7#11b9	G7#11b9sus4	(x)	G7#11b9sus2
(x)	(x)	(x)	(x)	(x)
G7#11b9#5	GMaj7#11b9#5	G7#11b9#5sus4	(x)	G7#11b9#5sus2
G7#11b9sus6	GMaj7#11b9sus6	G7#11b9sus4sus6	(x)	G7#11b9sus6sus2
G7#11b9susb6	GMaj7#11b9susb6	G7#11b9sus4susb6	(x)	G7#11b9susb6sus2
GMaj7#11b9	Gm(Maj7)#11b9	GMaj7#11b9sus4	(x)	GMaj7#11b9sus2
(x)	(x)	(x)	(x)	(x)
GMaj7#11b9#5	Gm(Maj7)#11b9#5	GMaj7#11b9#5sus4	(x)	GMaj7#11b9#5sus2
GMaj7#11b9sus6	Gm(Maj7)#11b9sus6	GMaj7#11b9sus4sus6	(x)	GMaj7#11b9sus6sus2
GMaj7#11b9susb6	Gm(Maj7)#11b9susb6	GMaj7#11b9sus4susb6	(x)	GMaj7#11b9susb6sus2

Thirteenth Chords

Two possible additions of thirteenths include the 13 and the b13 (The #13 is enharmonic to m7).

When adding the unaltered 13 to a chord with an unaltered 11th, The thirteen replaces the 11 and the 11 is implied, along with any 9 or 7 that was implied buy the 11. G11 becomes G13 and Gm11 becomes Gm13. If the 11^th chord had any alterations following, they remain the same. G115susb2# becomes G13#5susb2.

Maj 11^th chords become Maj13, and imply the inherited Maj7.

Chords with altered #11 replace the unaltered number with the 13, so

G9#11 becomes G13#11, with the unaltered 9 and 7 implied.

G7#11b9 becomes G13#11b9 with an unaltered 7 implied.

Other major sevenths an ninths become 13^th with major seventh implied

GMaj9#11 becomes GMaj13#11 with an unaltered 9 and a major 7 implied.

GMaj7#11b9 becomes GMaj13#11b9.

G13	Gm13	G13sus#4	G13susb2
G13b5	Gm13b5	G13b5sus#4	G13b5susb2
G13#5	Gm13#5	G13#5sus#4	G13#5susb2
(x)	(x)	(x)	(x)
G13susb6	Gm13susb6	G13sus#4susb6	G13susb2susb6
GMaj13	Gm(Maj13)	GMaj13sus#4	GMaj13susb2
GMaj13b5	Gm(Maj13)b5	GMaj13b5sus#4	GMaj13 b5susb2
GMaj13#5	Gm(Maj13)#5	GMaj13#5sus#4	GMaj13#5susb2
(x)	(x)	(x)	(x)
GMaj13susb6	GmMaj13susb6	GMaj13sus#4susb6	GMaj13susb2susb6
G13b9	Gm13b9	G13b9sus#4	G13b9sus2
G13b9b5	Gm13b9b5	G13b9b5sus#4	G13 b9b5sus2
G13b9#5	Gm13b9#5	G13b9#5sus#4	G13 b9#5sus2
(x)	(x)	(x)	(x)
G13b9susb6	Gm13b9susb6	G13b9sus#4susb6	G13b9sus2susb6
GMaj13b9	Gm(Maj13)b9	GMaj13b9sus#4	GMaj13b9sus2
GMaj13b9b5	Gm(Maj13)b9b5	GMaj13b9b5sus#4	GMaj13b9b5sus2
GMaj13b9#5	Gm(Maj13)b9#5	GMaj13b9#5sus#4	GMaj13b9#5sus2
(x)	(x)	(x)	(x)
GMaj13b9susb6	Gm(Maj13)b9susb6	GMaj13b9sus#4susb6	GMaj13b9sus2susb6
G13#11	Gm13#11	G13#11sus4	G13#11susb2
G13#11#5	Gm13#11#5	G13#11#5sus4	G13#11#5susb2
G13#11sus6	Gm13#11sus6	G13#11sus4sus6	G13#11susb6sus2
G13#11susb6	Gm13#11susb6	G13#11sus4sus6	G13#11susb6susb2
GMaj13#11	Gm(Maj13)#11	GMaj13#11sus4	GMaj13#11susb2
(x)	(x)	(x)	(x)
GMaj13#11#5	Gm(Maj13)#11#5	GMaj13#11#5sus4	GMaj13#11#5susb2
(x)	(x)	(x)	(x)
GMaj13#11susb6	GMaj13m#11susb6	GMaj13#11sus4sus6	GMaj13#11susb6susb2
G13#11b9	Gm13#11b9	G13#11b9sus4	G13#11b9sus2
G13#11b9#5	Gm13#11b9#5	G13#11b9#5sus4	G13#11b9#5sus2
G13#11b9sus6	Gm13#11b9sus6	G13#11b9sus4sus6	G13#11b9sus2sus6
G13#11b9susb6	Gm13#11b9susb6	G13#11b9sus4sus6	G13#11b9sus2susb6
GMaj13#11b9	Gm(Maj13)#11b9	GMaj13#11b9sus4	GMaj13#11b9sus2
GMaj13#11b9b5	Gm(Maj13)#11b9b5	GMaj13#11b9b5sus4	GMaj13#11b9b5sus2
GMaj13#11b9#5	Gm(Maj13)#11b9#5	GMaj13#11b9#5sus4	GMaj13#11b9#5sus2
(x)	(x)	(x)	(x)
GMaj13#11b9susb6	Gm(Maj13)#11b9susb6	GMaj13#11b9sus4sus6	GMaj13#11b9sus2susb6

And lastly, adding the altered 13^th is simple. The b13 will never replace a 7, 9, or 11 since the altered tones always appear after the unaltered tones. And, since the altered tones go in order, b13 is always last. Omit rows with a susb6.

G11b13	Gm11b13	G11b13sus#4	G11b13susb2
G11b13b5	Gm11b13b5	G11b13b5sus#4	G11b13b5susb2
G11b13#5	Gm11b13#5	G11b13#5sus#4	G11b13#5susb2
G11b13sus6	Gm11b13sus6	G11b13sus#4sus6	G11b13susb6susb2
G11b13susb6	Gm11b13susb6	G11b13sus#4susb6	G11b13susb6susb2
GMaj11b13	Gm(Maj11)b13	GMaj11b13sus#4	GMaj11b13susb2
GMaj11b13b5	Gm(Maj11)b13b5	GMaj11b13b5sus#4	GMaj11b13b5susb2
GMaj11b13#5	Gm(Maj11)b13#5	GMaj11b13#5sus#4	GMaj11b13#5susb2
GMaj11b13sus6	Gm(Maj11)b13sus6	GMaj11b13sus#4sus6	GMaj11b13susb6sus2
(x)	(x)	(x)	(x)
G11b13b9	Gm11b13b9	G11b13b9sus#4	G11b13b9sus2
G11b13b9b5	Gm11b13b9b5	G11b13b9b5sus#4	G11b13b9b5sus2
G11b13b9#5	Gm11b13b9#5	G11b13b9#5sus#4	G11b13b9#5sus2
G11b13b9sus6	Gm11b13b9sus6	G11b13b9sus#4sus6	G11b13b9sus6sus2
G11b13b9susb6	Gm11b13b9susb6	G11b13b9sus#4susb6	G11b13b9susb6sus2
GMaj11b13b9	Gm(Maj11)b13b9	GMaj11b13b9sus#4	GMaj11b13b9sus2
GMaj11b13b9b5	Gm(Maj11)b13b9b5	GMaj11b13b9b5sus#4	GMaj11b13b9b5sus2
GMaj11b13b9#5	Gm(Maj11)b13b9#5	GMaj11b13b9#5sus#4	GMaj11b13susb6sus2
GMaj11b13b9sus6	Gm(Maj11)b13b9sus6	GMaj11b13b9sus#4sus6	GMaj11b13b9sus6 us2
(x)	(x)	(x)	(x)
G9b13#11	Gm9b13#11	G9b13#11sus4	G9b13#11susb2
G9b13#11#5	Gm9b13#11#5	G9b13#11#5sus4	G9b13#11#5susb2
G9b13#11sus6	Gm9b13#11sus6	G9b13#11sus4sus6	G9b13#11susb6susb2
G9b13#11susb6	Gm9b13#11susb6	G9b13#11sus4sus6	G9b13#11susb6susb2
GMaj9b13#11	Gm(Maj9)b13#11	GMaj9b13#11sus4	GMaj9b13#11susb2
GMaj9b13#11#5	Gm(Maj9)b13#11#5	GMaj9b13#11#5sus4	GMaj9b13#11#5susb2
GMaj9b13#11sus6	GMaj9mb13#11sus6	GMaj9b13#11sus4sus6	GMaj9b13#11susb6sus2
(x)	(x)	(x)	(x)
G7b13#11b9	GMaj7b13#11b9	G7b13#11b9sus4	G7b13#11b9sus2
G7b13#11b9#5	GMaj7b13#11b9#5	G7b13#11b9#5sus4	G7b13#11b9#5sus2
G7b13#11b9sus6	GMaj7b13#11b9sus6	G7b13#11b9sus4sus6	G7b13#11b9sus6sus2
G7b13#11b9susb6	GMaj7b13#11b9susb6	G7b13#11b9sus4sus6	G7b13#11b9susb6sus2
GMaj7b13#11b9	Gm(Maj7)b13#11b9	GMaj7b13#11b9sus4	GMaj7b13#11b9sus2
GMaj7b13#11b9#5	Gm(Maj7)b13#11b9#5	GMaj7b13#11b9#5sus4	(x)
GMaj7b13#11b9sus6	Gm(Maj7)b13#11b9sus6	GMaj7b13#11b9sus4sus6	G7b13#11b9sus2
(x)	(x)	(x)	(x)

Rules for Chord Symbols

So, now the pattern of notating chords is fully established.

(Bass note)(third)(seventh or higher)(alterations in reverse order)(suspensions in order)

Options:

Bass note: any of the 24 enharmonic pitches.

Third: major (blank), minor (m) diminished (o) or augmented (+).

The fifth takes its color from the minor column, fifth is perfect for major and minor, diminished for diminished, and augmented for augmented. Any notated alterations override these.

Seventh can be minor (7), major (Maj7) or diminished o7.

When notating al altered fifth in a seventh chord, add a separate #5 or b5 instead of using o or + (o7 is an exception)

Numbers above seven inherit the seventh from the 7 (9, 11, 13) or Maj7 (Maj9, Maj11, Maj13).

Any note not following these patterns appears as an alteration at the end of the list.

A # before an scale number raises the note. A natural note becomes sharp, a flat note becomes natural, and a sharp note becomes double sharp.

A b before an scale number lowers the note. A natural note becomes flat, a flat note becomes double flat, and a sharp note becomes natural.

The sus designation indicates the note replaces a chord tone. Sus4 an Sus2 both replace the third. Sus6 replaces the fifth.

Add Chords

So, far all the chords have been stacks of thirds or suspensions of those thirds. But a player might want to add a 9^th, 11^th, or 13^th equivalent note to chord without the seventh. To accomplish this, use an add chord. They typical add chords are add2, add4, and 6.

Add2

Addd2 chords use the same note as a ninth, but without the 7. So, a Gadd2 chord is gabd. The a could appear anywhere in the chord such as gbda.

Add2 cords can appear in any of the traids. The add designations appear to the end of the chord symbol not in the order of alterations, so an add 2 on a Gm#5 chord is Gm#5add2, sometimes with a space before the add, and sometimes with a capital A. Chords with a sus2 would not need to add a 2 since it is already there. A susb2 with and add2 is possible, but unlikely.

Gadd2	Gmadd2	Gsus4add2	Gsus#4add2	(x)	Gsusb2add2
G(Maj)b5add2	GMaj7b5add2	G(Maj)b5sus4add2	G(Maj)b5sus#4add2	(x)	Gb5susb2add2
G(Maj)#5add2	Gm#5add2	G(Maj)#5sus4add2	G(Maj)#5sus#4add2	(x)	G(MAJ)#5susb2add2
Gsus6add2	Gmsus6add2	Gsus4sus6add2	Gsus#4sus6add2	(x)	Gsus2sus6add2
Gsusb6add2	Gmsusb6add2	Gsus4sus6add2	Gsus#4susb6add2	(x)	Gsusb2susb6add2

The altered add_b2 follows the same pattern, with susb2add_b2 being redundant.

Gadd_b2	Gmadd_b2	Gsus4add_b2	Gsus#4add_b2	Gsus2add_b2	(x)
G(Maj)b5add_b2	GMaj7b5add_b2	G(Maj)b5sus4add_b2	G(Maj)b5sus#4add_b2	Gb5sus2add_b2	(x)
G(Maj)#5add_b2	Gm#5add_b2	G(MAJ)#5sus4add_b2	G(MAJ)#5sus#4add_b2	G(MAJ)#5sus2add_b2	(x)
Gsus6add_b2	Gmsus6add_b2	Gsus4sus6add_b2	Gsus#4sus6add_b2	Gsus2sus6add_b2	(x)
Gsusb6add_b2	Gmsusb6add_b2	Gsus4sus6add_b2	Gsus#4susb6add_b2	Gsus2susb6add_b2	(x)

Seventh chords would not have an add2 since a 2^nd added to a seventh chord would just make a ninth, which is already covered. An exception might be Go2add2 or gdin7add_b2, but theses table treat those chords as misspelled 6ths, covered later.

Ninth, eleventh, and thirteenth chords typically would not add a 2^nd because it would be redundant with the 9^th. An exception might be an altered b2 added to an unaltered ninth, G9add_b2. These mathematically possible but unlikely chords appear in a later table.

Add4 Chords

Add4 chords follow the patterns of eleventh chords with wither the seventh, the ninth, or both missing. And add4 would be redundant with a sus4.

Attached to triads:

Gadd4	Gmadd4	(x)	Gsus#4add4	Gsus2add4	Gsusb2add4
G(Maj)b5add4	Gmb5add4	(x)	G(Maj)b5sus#4add4	Gb5sus2add4	Gb5susb2add4
G+add4	Gm#5add4	(x)	G(MAJ)#5sus#4add4	G(MAJ)#5sus2add4	G(MAJ)#5susb2add4
Gsus6add4	Gmsus6add4	(x)	Gsus#4sus6add4	Gsus2sus6add4	Gsus2susb6add4

And with the altered #4, Note that b5#4 chords are redundant.

Gadd#4	Gmadd#4	Gsus4add#4	Gsus2add#4	Gsusb2add#4
(x)	(x)	(x)	(x)	(x)
G(Maj)#5add#4	Gm#5add#4	G(Maj)#5sus4add#4	G(Maj)#5sus2add#4	G(Maj)#5susb2add#4
Gsus6add#4	Gmsus6add#4	Gsus4sus6add#4	Gsus2sus6add#4	Gsus2susb6add#4

Add4 chords can appear on seventh chords, essentially an eleventh without the ninth, and inheriting the same redundancies from the triads. (Redundancies omitted from the chart to save space).

G7add4	GMaj7add4	G7sus#4add4	G7sus2add4	G7susb2add4
G7b5add4	GMaj7b5add4	G7b5sus#4add4	G7b5sus2add4	G7b5susb2add4
G7#5add4	GMaj7#5add4	G7#5sus#4add4	G7#5sus2add4	G7#5susb2add4
G7sus6add4	GMaj7sus6add4	G7sus#4sus6add4	G7sus2sus6add4	G7sus2susb6add4
G7susb6add4	GMaj7susb6add4	G7sus#4susb6add4	G7sus2susb6add4	G7susb2susb6add4
GMaj7add4	Gm(Maj7)add4	GMaj7sus#4add4	GMaj7sus2add4	GMaj7susb2add4
GMaj7#5add4	Gm(Maj7)#5add4	GMaj7#5sus#4add4	GMaj7#5sus2add4	GMaj7#5susb2add4
GMaj7sus6add4	GMaj7sus6add4	GMaj7sus#4sus6add4	GMaj7sus2sus6add4	GMaj7sus2susb6add4
GMaj7susb6add4	GMaj7susb6add4	GMaj7sus#4susb6add4	GMaj7sus2susb6add4	GMaj7susb2susb6add4
G7add#4	GMaj7add#4	G7sus4add#4	G7sus2add#4	G7susb2add#4
G7b5add#4	GMaj7b5add#4	G7b5sus4add#4	G7b5sus2add#4	G7b5susb2add#4
G7#5add#4	GMaj7#5add#4	G7#5sus4add#4	G7#5sus2add#4	G7#5susb2add#4
G7sus6add#4	GMaj7sus6add#4	G7sus4sus6add#4	G7sus2sus6add#4	G7sus2susb6add#4
G7susb6add#4	GMaj7susb6add#4	G7sus4sus6add#4	G7sus2susb6add#4	G7susb2susb6add#4
GMaj7add#4	Gm(Maj7)add#4	GMaj7sus4add#4	GMaj7sus2add#4	GMaj7susb2add#4
GMaj7#5add#4	Gm(Maj7)#5add#4	GMaj7#5sus4add#4	GMaj7#5sus2add#4	GMaj7#5susb2add#4
GMaj7sus6add#4	GMaj7sus6add#4	GMaj7sus4sus6add#4	GMaj7sus2sus6add#4	GMaj7sus2susb6add#4
GMaj7susb6add#4	GMaj7susb6add#4	GMaj7sus4sus6add#4	GMaj7sus2susb6add#4	GMaj7susb2susb6add#4

Add 4 chords could also add to the add2 chords (redundancies removed).

Gadd2add4	Gmadd2add4	Gsus#4add2add4	Gsusb2add2add4
G(Maj)b5add2add4	GMaj7b5add2add4	G(Maj)b5sus#4add2add4	Gb5susb2add2add4
G+add2add4	Gm#5add2add4	G(MAJ)#5sus#4add2add4	G(MAJ)#5susb2add2add4
Gsus6add2add4	Gmsus6add2add4	Gsus#4sus6add2add4	Gsus2susb6add2add4
Gsusb6add2add4	Gmsusb6add2add4	Gsus#4susb6add2add4	Gsusb2susb6add2add4
Gadd_b2add4	Gmadd_b2add4	Gsus#4add_b2add4	Gsus2add_b2add4
G(Maj)b5add_b2add4	GMaj7b5add_b2add4	G(Maj)b5sus#4ad_b2add4	Gb5sus2add_b2add4
G+add_b2add4	Gm#5add_b2add4	G(Maj)#5sus#4ad_b2add4	G(Maj)#5sus2add_b2add4
Gsus6add_b2add4	Gmsus6add_b2add4	Gsus#4sus6add_b2add4	Gsus2sus6add_b2add4
Gsusb6add_b2add4	Gmsusb6add_b2add4	Gsus#4susb6add_b2add4	Gsus2susb6add_b2add4
Gadd2add#4	Gmadd2add#4	Gsus4add2add#4	Gsusb2add2add#4
G+add2add#4	Gm#5add2add#4	G(Maj)#5sus4add2add#4	G(Maj)#5susb2ad2add#4
Gsus6add2add#4	Gmsus6add2add#4	Gsus4sus6add2add#4	Gsus2sus6add2add#4
Gsusb6add2add#4	Gmsusb6add2add#4	Gsus4susb6add2add#4	Gsusb2susb6add2add#4
Gadd_b2add#4	Gmadd_b2add#4	Gsus4add_b2add#4	Gsus2add_b2add#4
G+add_b2add#4	Gm#5add_b2add#4	G(Maj)#5sus4ad_b2add#4	G(Maj)#5sus2ad_b2add#4
Gsus6add_b2add#4	Gmsus6add_b2add#4	Gsus4sus6add_b2add#4	Gsus2sus6add_b2add#4
Gsusb6add_b2add#4	Gmsusb6add_b2add#4	Gsus4susb6add_b2add#4	Gsus2susb6add_b2add#4

6 Chords

Added sixth chords function like add2 and chords, but their chord symbols are different. The "add" is omitted, So, instead of Gadd6, use G6. The 6 goes after the m for minor chords, Gm6 and before any alterations, Gm6sus#4.

Diminished chords become Gm6b5, and augmented chords become G6#5, just as in seventh chords.

G6	Gm6	G6sus4	G6sus#4	G6sus2	G6susb2
G6b5	Gm6b5	G6b5sus4	G6b5sus#4	G6b5sus2	G6b5susb2
G6#5	Gm6#5	G6#5sus4	G6#5sus#4	G6#5sus2	G6#5susb2
(x)	(x)	(x)	(x)	(x)	(x)
G6susb6	Gm6susb6	G6sus4susb6	G6sus#4susb6	G6sus2susb6	G6susb2susb6

For the altered sixth, use add_b6 at the end Gadd_b6 and not G(b6). Note that #5 is redundant with b6, so that row is removed along with susbsadd_b6.

Gadd_b6	Gmadd_b6	Gsus4add_b6	Gsus#4add_b6	Gsus2add_b6	Gsusb2add_b6
G(M)b5add_b6	Gmb5add_b6	G(M)b5s4add_b6	G(M)b5s#4add_b6	Gb5sus2add_b6	Gb5susb2add_b6
Gsus6add_b6	Gmsus6add_b6	Gsus4sus6add_b6	Gsus#4sus6add_b6	Gsus2sus6add_b6	Gsus2susb6add_b6

For unaltered sixths added to seventh chords, the 6 precedes the 7, G67 or G6/7.

G67	Gm67	G67sus4	G67sus#4	G67sus2	G67susb2
G67b5	Gm67b5	G67b5sus4	G67b5sus#4	G67b5sus2	G67b5susb2
G67#5	Gm67#5	G67#5sus4	G67#5sus#4	G67#5sus2	G67#5susb2
(x)	(x)	(x)	(x)	(x)	(x)
G67susb6	Gsusb6	G67sus4sus6	G67sus#4susb6	G67sus2susb6	G67susb2susb6
G6Maj7	Gm6Maj7	G6Maj7sus4	G6Maj7sus#4	G6Maj7sus2	G6Maj7susb2
G6Maj7b5	Gm6Maj7b5	G6Maj7b5sus4	G6Maj7b5sus#4	G6Maj7b5sus2	G6Maj7b5susb2
G6Maj7#5	Gm6Maj7#5	G6Maj7#5sus4	G6Maj7#5sus#4	G6Maj7#5sus2	G6Maj7#5susb2
(x)	(x)	(x)	(x)	(x)	(x)
G6Maj7susb6	G6Maj7susb6	G6Maj7sus4susb6	G6Maj7sus#4susb6	G6Maj7sus2susb6	G6Maj7susb2susb6

For seventh chords with an altered sixth, just append the add_b6 tag to the end.

G7add_b6	GMaj7add_b6	G7sus4add_b6	G7sus#4add_b6	G7sus2add_b6	G7susb2add_b6
G7b5add_b6	GMaj7b5add_b6	G7b5sus4add_b6	G7b5sus#4add_b6	G7b5sus2add_b6	G7b5susb2add_b6
G7#5add_b6	GMaj7#5add_b6	G7#5sus4add_b6	G7#5sus#4add_b6	G7#5sus2add_b6	G7#5susb2add_b6
(x)	(x)	(x)	(x)	(x)	(x)
G7susb6add_b6	GMaj7susb6add_b6	G7sus4sus6add_b6	G7sus#4susb6add_b6	G7sus2susb6add_b6	G7susb2susb6add_b6
GMaj7add_b6	Gm(Maj7)add_b6	GMaj7sus4add_b6	GMaj7sus#4add_b6	GMaj7sus2add_b6	GMaj7susb2add_b6
GMaj7b5add_b6	Gm(Maj7)b5add_b6	GMaj7b5sus4add_b6	GMaj7b5sus#4add_b6	GMaj7b5sus2add_b6	GMaj7b5susb2add_b6
GMaj7#5add_b6	Gm(Maj7)#5add_b6	GMaj7#5sus4add_b6	GMaj7#5sus#4add_b6	GMaj7#5sus2add_b6	GMaj7#5susb2add_b6
GMaj7sus6add_b6	GMaj7sus6add_b6	GMaj7sus4sus6add_b6	GMaj7sus#4sus6add_b6	GMaj7sus2sus6add_b6	GMaj7sus2susb6add_b6
(x)	(x)	(x)	(x)	(x)	(x)

For ninth chords with unaltered ninths adding an unaltered 6, substitute 69 for 67.

G69	Gm69	G69sus4	G69sus#4	G69susb2
G69b5	Gm69b5	G69b5sus4	G69b5sus#4	G69b5susb2
G69#5	Gm69#5	G69#5sus4	G69#5sus#4	G69#5susb2
(x)	(x)	(x)	(x)	(x)
G69susb6	Gm69susb6	G69sus4susb6	G69sus#4susb6	G69susb2susb6
G6Maj9	Gm(Maj9)	G6Maj9sus4	G6Maj9sus#4	G6Maj9susb2
G6Maj9b5	Gm(Maj9)b5	G6Maj9b5sus4	G6Maj9b5sus#4	G6Maj9b5susb2
G6Maj9#5	Gm(Maj9)#5	G6Maj9#5sus4	G6Maj9#5sus#4	G6Maj9#5susb2
(x)	(x)	(x)	(x)	(x)
G6Maj9susb6	Gm6(Maj9)susb6	G6Maj9sus4sus6	G6Maj9sus#4susb6	G6Maj9susb2susb6
G67b9	GMaj7b9	G67b9sus4	G67b9sus#4	G67sus2b9
G67b9b5	GMaj7b9b5	G67b9b5sus4	G67b9b5sus#4	G67sus2b9b5
G67b9#5	GMaj7b9#5	G67b9#5sus4	G67b9#5sus#4	G67sus2b9#5
(x)	(x)	(x)	(x)	(x)
G67b9susb6	GMaj7b9susb6	G67b9sus4sus6	G67b9sus#4susb6	G67sus2b9susb6
G6Maj7b9	Gm6(Maj7)b9	G6Maj7b9sus4	G6Maj7b9sus#4	G6Maj7sus2b9
G6Maj7b9b5	Gm(6Maj7)b9b5	G6Maj7b9b5sus4	G6Maj7b9b5sus#4	G6Maj7sus2b9b5
G6Maj7b9#5	Gm6(Maj7)b9#5	G6Maj7b9#5sus4	G6Maj7b9#5sus#4	G6Maj7sus2b9#5
(x)	(x)	(x)	(x)	(x)
G6Maj7b9susb6	Gm6(Maj7)b9susb6	G6Maj7b9sus4susb6	G6Maj7b9sus#4susb6	G6Maj7sus2b9susb6

Altered sixths add_b6 symbols append to the end of ninth chords. (Redundancies removed)

G9add_b6	Gm9add_b6	G9sus4add_b6	G9sus#4add_b6	G9susb2add_b6
G9b5add_b6	Gm9b5add_b6	G9b5sus4add_b6	G9b5sus#4add_b6	G9b5susb2add_b6
G9sus6add_b6	Gm9sus6add_b6	G9sus4sus6add_b6	G9sus#4sus6add_b6	G9sus2susb6add_b6
GMaj9add_b6	Gm(Maj9)add_b6	GMaj9sus4add_b6	GMaj9sus#4add_b6	GMaj9susb2add_b6
GMaj9b5add_b6	Gm(Maj9)b5add_b6	GMaj9b5sus4add_b6	GMaj9b5sus#4add_b6	GMaj9b5susb2add_b6
GMaj9sus6add_b6	GmMaj9sus6add_b6	GMaj9sus4sus6add_b6	GMaj9sus#4sus6add_b6	GMaj9susb2susb6add_b6
G7b9add_b6	GMaj7b9add_b6	G7b9sus4add_b6	G7b9sus#4add_b6	G7sus2b9add_b6
G7b9b5add_b6	GMaj7b9b5add_b6	G7b9b5sus4add_b6	G7b9b5sus#4add_b6	G7sus2b9b5add_b6
G7b9sus6add_b6	GMaj7b9sus6add_b6	G7b9sus4sus6add_b6	G7b9sus#4sus6add_b6	G7sus2b9sus6add_b6
GMaj7b9add_b6	Gm(Maj7)b9add_b6	GMaj7b9sus4add_b6	GMaj7b9sus#4add_b6	GMaj7sus2b9add_b6
GMaj7b9b5add_b6	Gm(Maj7)b9b5add_b6	GMaj7b9b5sus4add_b6	GMaj7b9b5sus#4add_b6	GMaj7sus2b9b5add_b6
GMaj7b9sus6add_b6	Gm(Maj7)b9sus6add_b6	GMaj7b9sus4sus6add_b6	GMaj7b9sus#4sus6add_b6	GMaj7sus2b9sus6add_b6

A sixth would not add to an 11th chord because the result would just be a thirteenth chord. Also, a sixth would be redundant in a thirteenth chord. AN altered sixth might be added to an unaltered 13^th, but those chords are handled in the later tables on clusters.

The sixth chords could combine with added seconds.

G6add2	Gm6add2	G6sus4add2	G6sus#4add2	G6susb2add2
G6b5add2	Gm6b5add2	G6b5sus4add2	G6b5sus#4add2	G6b5susb2add2
G6#5add2	Gm6#5add2	G6#5sus4add2	G6#5sus#4add2	G6#5susb2add2
G6susb6add2	Gm6susb6add2	G6sus4susb6add2	G6sus#4susb6add2	G6susb2susb6add2
Gadd2add_b6	Gmadd2add_b6	Gsus4add2add_b6	Gsus#4add2add_b6	Gsusb2add2add_b6
G(Maj)b5add2add_b6	Gmb5add2add_b6	G(Maj)b5sus4add2add_b6	G(Maj)b5sus#4add2add_b6	Gb5susb2add2add_b6
Gsus6add2add_b6	Gmsus6add2add_b6	Gsus4sus6add2add_b6	Gsus#4sus6add2add_b6	(x)
G6add_b2	Gm6add_b2	G6sus4add_b2	G6sus#4add_b2	G6sus2add_b2
G6b5add_b2	Gm6b5add_b2	G6b5sus4add_b2	G6b5sus#4add_b2	G6b5sus2add_b2
G6#5add_b2	Gm6#5add_b2	G6#5sus4add_b2	G6#5sus#4add_b2	G6#5sus2add_b2
G6susb6add_b2	Gm6susb6add_b2	G6sus4susb6add_b2	G6sus#4susb6add_b2	G6sus2susb6add_b2
Gadd_b2add_b6	Gmadd_b2add_b6	Gsus4add_b2add_b6	Gsus#4add_b2add_b6	Gsus2add_b2add_b6
G(M)b5ad_b2add_b6	Gmb5add_b2add_b6	G(Maj)b5sus4add_b2add_b6	G(Maj)b5sus#4add_b2add_b6	Gb5sus2add_b2add_b6
Gsus6add_b2add_b6	Gmsus6ad_b2add_b6	Gsus4sus6add_b2add_b6	Gsus#4sus6add_b2add_b6	Gsus2sus6add_b2add_b6

Sixth chords may also combine with add4 chords.

G6add4	Gm6add4	G6sus#4add4	G6sus2add4
G6b5add4	Gm6b5add4	G6b5sus#4add4	G6b5sus2add4
G6#5add4	Gm6#5add4	G6#5sus#4add4	G6#5sus2add4
G6susb6add4	Gm6susb6add4	G6sus#4susb6add4	G6sus2susb6add4
Gadd4add_b6	Gmadd4add_b6	Gsus#4add4add_b6	Gsus2add4add_b6
G(Maj)b5ad4add_b6	Gmb5add4add_b6	G(M)b5sus#4ad4add_b6	Gb5sus2add4add_b6
Gsus6add4add_b6	Gmsus6add4add_b6	Gsus#4sus6add4add_b6	Gsus2sus6add4add_b6
G67add4	Gm67add4	G67sus#4add4	G67sus2add4
G67b5add4	Gm67b5add4	G67b5sus#4add4	G67b5sus2add4
G67#5add4	Gm67#5add4	G67#5sus#4add4	G67#5sus2add4
G67susb6add4	Gsusb6add4	G67sus#4susb6add4	G67sus2susb6add4
G6Maj7add4	Gm6Maj7add4	G6Maj7sus#4add4	G6Maj7sus2add4
G6Maj7b5add4	Gm6Maj7b5add4	G6Maj7b5sus#4add4	G6Maj7b5sus2add4
G6Maj7#5add4	Gm6Maj7#5add4	G6Maj7#5sus#4add4	G6Maj7#5sus2add4
Gm6add#4	G6sus4add#4	G6sus2add#4	G6susb2add#4
Gm6b5add#4	G6b5sus4add#4	G6b5sus2add#4	G6b5susb2add#4
Gm6#5add#4	G6#5sus4add#4	G6#5sus2add#4	G6#5susb2add#4
Gm6susb6add#4	G6sus4susb6add#4	G6sus2susb6add#4	G6susb2susb6add#4
Gmadd#4add_b6	Gsus4add#4add_b6	Gsus2add#4add_b6	Gsusb2add#4add_b6
Gmb5add#4add_b6	G(Maj)b5sus4add#4add_b6	Gb5sus2add#4add_b6	Gb5susb2add#4add_b6
Gmsus6add#4add_b6	Gsus4sus6add#4add_b6	Gsus2sus6add#4add_b6	Gsusb2susb6add#4add_b6
Gm67add#4	G67sus4add#4	G67sus2add#4	G67susb2add#4
Gm67b5add#4	G67b5sus4add#4	G67b5sus2add#4	G67b5susb2add#4
Gm67#5add#4	G67#5sus4add#4	G67#5sus2add#4	G67#5susb2add#4
Gsusb6add#4	G67sus4sus6add#4	G67sus2susb6add#4	G67susb2susb6add#4
Gm6Maj7add#4	G6Maj7sus4add#4	G6Maj7sus2add#4	G6Maj7susb2add#4
Gm6Maj7b5add#4	G6Maj7b5sus4add#4	G6Maj7b5sus2add#4	G6Maj7b5susb2add#4
Gm6Maj7#5add#4	G6Maj7#5sus4add#4	G6Maj7#5sus2add#4	G6Maj7#5susb2add#4

Finally, sixth chords could combine with both added seconds and added fourths, essentially creating a thirteenth chord with no seventh.

G6add2add4	Gm6add2add4	G6sus#4add2add4	G6sus2add2add4
G6b5add2add4	Gm6b5add2add4	G6b5sus#4add2add4	G6b5sus2add2add4
G6#5add2add4	Gm6#5add2add4	G6#5sus#4add2add4	G6#5sus2add2add4
G6susb6add2add4	Gm6susb6add2add4	G6sus#4susb6add2add4	G6sus2susb6add2add4
Gadd2add4add_b6	Gmadd2add4add_b6	Gsus#4add2add4add_b6	Gsus2add2add4add_b6
G(Maj)b5add2add4add_b6	Gmb5add2add4add_b6	G(Maj)b5sus#4add2add4add_b6	Gb5sus2add2add4add_b6
Gsus6add2add4add_b6	Gmsus6add2add4add_b6	Gsus#4sus6add2add4add_b6	Gsus2sus6add2add4add_b6
Gm6add2add#4	G6sus4add2add#4	G6sus2add2add#4	G6susb2add2add#4
Gm6b5add2add#4	G6b5sus4add2add#4	G6b5sus2add2add#4	G6b5susb2add2add#4
Gm6#5add2add#4	G6#5sus4add2add#4	G6#5sus2add2add#4	G6#5susb2add2add#4
Gm6susb6add2add#4	G6sus4susb6add2add#4	G6sus2susb6add2add#4	G6susb2susb6add2add#4
Gmadd2add#4add_b6	Gsus4add2add#4add_b6	Gsus2add2add#4add_b6	Gsusb2add2add#4add_b6
Gmb5add2add#4add_b6	G(Maj)b5sus4add2add#4add_b6	Gb5sus2add2add#4add_b6	Gb5susb2add2add#4add_b6
Gmsus6add2add#4add_b6	Gsus4sus6add2add#4add_b6	Gsus2sus6add2add#4add_b6	Gsusb2susb6add2add#4add_b6
G6add_b2add4	Gm6add_b2add4	G6sus#4add_b2add4	G6sus2add_b2add4
G6b5add_b2add4	Gm6b5add_b2add4	G6b5sus#4add_b2add4	G6b5sus2add_b2add4
G6#5add_b2add4	Gm6#5add_b2add4	G6#5sus#4add_b2add4	G6#5sus2add_b2add4
G6susb6add_b2add4	Gm6susb6add_b2add4	G6sus#4susb6add_b2add4	G6sus2susb6add_b2add4
Gadd_b2add4add_b6	Gmadd_b2add4add_b6	Gsus#4add_b2add4add_b6	Gsus2add_b2add4add_b6
G(b5)add_b2add4add_b6	Gmb5add_b2add4add_b6	G(M)b5sus#4ad_b2add4add_b6	Gb5sus2add_b2add4add_b6
Gsus6add_b2add4add_b6	Gmsus6add_b2add4add_b6	Gsus#4sus6add_b2add4add_b6	Gsus2sus6add_b2add4add_b6
Gm6add_b2add#4	G6sus4add_b2add#4	G6sus2add_b2add#4	G6susb2add_b2add#4
Gm6b5add_b2add#4	G6b5sus4add_b2add#4	G6b5sus2add_b2add#4	G6b5susb2add_b2add#4
Gm6#5add_b2add#4	G6#5sus4add_b2add#4	G6#5sus2add_b2add#4	G6#5susb2add_b2add#4
Gm6susb6add_b2add#4	G6sus4susb6add_b2add#4	G6sus2susb6add_b2add#4	G6susb2susb6add_b2add#4
Gmadd_b2add#4add_b6	Gsus4add_b2add#4add_b6	Gsus2add_b2add#4add_b6	Gsusb2add_b2add#4add_b6
Gmb5add_b2add#4add_b6	G(M)b5sus4add_b2add#4add_b6	Gb5sus2add_b2add#4add_b6	Gb5susb2add_b2add#4add_b6
Gmsus6ad_b2add#4add_b6	Gsus4sus6add_b2add#4add_b6	Gsus2sus6add_b2add#4add_b6	Gsusb2susb6add_b2add#4add_b6

Cluster Chords

At this point, most theory textbooks would state that all the chords have been represented. In fact, some of the chords listed here would not be considered legitimate musically. However, certain note combinations still do not have a chord symbol the corresponds. For example, what is the symbol for the entire chromatic scale played all at once in a cluster?

Using the cords from the traditional stacks of thirds and add notes, the chord with the most notes have nine notes, such as G13add_b2add#4, gbdfacea_bc#. With the alterations used already, an add_b6 e_b could produce the ten note chord G13add_b2add#4add_b6, gbdfacea_bc#e_b. Rearranged, gaba_bcc#de_bef, so the notes b_b and f# do not occur.

Atypical Added Notes

The note Bb is usually the minor third in G, but using the typical m tag would eliminate the b natural implied in the 13^th chord, so using m is not possible. The tag add_b3, is possible, but using the enharmonic equivalent add#2 is more in line with the pattern of seconds, thirds, an fourths being added tones. So, the symbol G13add_b2add#2add#4add_b6, gbdfaceAbc#e_ba#, would contain eleven of twelve notes. That leaves the f#.

The f# in G usually associates with chords such as Maj7, Maj9, Maj11, and Maj13. So, a first guess might be to use G13Maj7add_b2add#2add#4add_b6 as a way to add the f#. But this is really too close to GMaj13, and would be confusing. Instead, use the Maj13 to add the b, and add back the minor seventh as add#6, the enharmonic equivalent.

So, the tools add#2 and Maj7add#6 will be necessary to fill in all the clusters of notes possible, with the unused add_b4 and the d7 tag from the diminished chord available if necessary.

Add#2 Chords

Several clusters already happen in previous analysis, for example the G9add_b2 chord has g, Ab, and a all together in the same chord. But none of the previous chords had the major third and the minor third in the same chord. Consider the cluster g(Bb)bd. This is a major triad with a minor third added. For the sake on analysis, consider the Bb an a#, and the chord becomes Gadd#2.

For a minor chord the notes are the same, so although an analysis of Gmadd_b4 is possible, this chord is redundant and therefore skipped. Suspension Sus4 and Sus2 imply no third, so an add#2 on top of a suspension would not be appropriate since add#2 is just a purposely misspelled m3.

Diminished triads become G(Maj)b5add#2.

G+ becomes G(Maj)#5add#2

The suspended chords sixth just add the add#2 tag at the end

Gadd#2	(x)	(x)	(x)	(x)	(x)
G(Maj)b5add#2	(x)	(x)	(x)	(x)	(x)
GMaj#5add#2	(x)	(x)	(x)	(x)	(x)
Gsus6add#2	(x)	(x)	(x)	(x)	(x)
Gsusb6add#2	(x)	(x)	(x)	(x)	(x)

So, only the first column of each table needs consideration since all chords build on the triads

G7add#2	G9add#2	G11add#2	G13add#2	G11b13add#2
G7b5add#2	G9b5add#2	G11b5add#2	G13b5add#2	G11b13b5add#2
G7#5add#2	G9#5add#2	G11#5add#2	G13#5add#2	G11b13#5add#2
G7sus6add#2	G9sus6add#2	G11sus6add#2	(x)	G11b13sus6add#2
G7susb6add#2	G9susb6add#2	G11susb6add#2	G13susb6add#2	G11b13susb6add#2
GMaj7add#2	GMaj9add#2	GMaj11add#2	GMaj13add#2	GMaj11b13add#2
GMaj7b5add#2	GMaj9b5add#2	GMaj11b5add#2	GMaj13b5add#2	GMaj11b13b5add#2
GMaj7#5add#2	GMaj9#5add#2	GMaj11#5add#2	GMaj13#5add#2	GMaj11b13#5add#2
GMaj7sus6add#2	GMaj9sus6add#2	GMaj11sus6add#2	(x)	GMaj11b13sus6add#2
GMaj7susb6add#2	GMaj9susb6add#2	GMaj11susb6add#2	GMaj13susb6add#2	(x)
	G7b9add#2	G11b9add#2	G13b9add#2	G11b13b9add#2
	G7b9b5add#2	G11b9b5add#2	G13b9b5add#2	G11b13b9b5add#2
	G7b9#5add#2	G11b9#5add#2	G13b9#5add#2	G11b13b9#5add#2
	G7b9sus6add#2	G11b9sus6add#2	(x)	G11b13b9sus6add#2
	G7b9susb6add#2	G11b9susb6add#2	G13b9susb6add#2	G11b13b9susb6add#2
	GMaj7b9add#2	GMaj11b9add#2	GMaj13b9add#2	GMaj11b13b9add#2
	GMaj7b9b5add#2	GMaj11b9b5add#2	GMaj13b9b5add#2	GMaj11b13b9b5add#2
	GMaj7b9#5add#2	GMaj11b9#5add#2	GMaj13b9#5add#2	GMaj11b13b9#5add#2
	GMaj7b9sus6add#2	GMaj11b9sus6add#2	(x)	GMaj11b13b9sus6add#2
	GMaj7b9susb6add#2	GMaj11b9susb6add#2	GMaj13b9susb6add#2	(x)
		G9#11add#2	G13#11add#2	G9b13#11add#2
		G9#11b5add#2	G13#11b5add#2	G9b13#11b5add#2
		G9#11#5add#2	G13#11#5add#2	G9b13#11#5add#2
		G9#11sus6add#2	G13#11sus6add#2	G9b13#11sus6add#2
		G9#11susb6add#2	G13#11susb6add#2	G9b13#11susb6add#2
		GMaj9#11add#2	GMaj13#11add#2	GMaj9b13#11add#2
		GMaj9#11b5add#2	GMaj13#11b5add#2	GMaj9b13#11b5add#2
		GMaj9#11#5add#2	GMaj13#11#5add#2	GMaj9b13#11#5add#2
		GMaj9#11sus6add#2	(x)	GMaj9b13#11sus6add#2
		GMaj9#11susb6add#2	GMaj13#11susb6add#2	(x)
		G7#11b9add#2	G13#11b9add#2	G7b13#11b9add#2
		G7#11b9b5add#2	G13#11b9b5add#2	G7b13#11b9b5add#2
		G7#11b9#5add#2	G13#11b9#5add#2	G7b13#11b9#5add#2
		G7#11b9sus6add#2	G13#11b9sus6add#2	G7b13#11b9sus6add#2
		G7#11b9susb6add#2	G13#11b9susb6add#2	G7b13#11b9susb6add#2
		GMaj7#11b9add#2	GMaj13#11b9add#2	GMaj7b13#11b9add#2
		GMaj7#11b9b5add#2	GMaj13#11b9b5add#2	GMaj7b13#11b9b5add#2
		GMaj7#11b9#5add#2	GMaj13#11b9#5add#2	GMaj7b13#11b9#5add#2
		GMaj7#11b9sus6add#2	(x)	GMaj7b13#11b9sus6add#2
		GMaj7#11b9susb6add#2	GMaj13#11b9susb6add#2	(x)

And the add#2 appends to all the other add chords, with only the first column being of interest.

Gadd2add#2	G7add#2add4	Gadd2add#2add4
G(Maj)b5add2add#2	G7b5add#2add4	G(Maj)b5add2add#2add4
G(Maj)#5add2add#2	G7#5add#2add4	G+add2add#2add4
Gsus6add2add#2	G7sus6add#2add4	Gsus6add2add#2add4
Gsusb6add2add#2	G7susb6add#2add4	Gsusb6add2add#2add4
Gadd_b2add#2	GMaj7add#2add4	Gadd_b2add#2add4
G(Maj)b5add_b2add#2	GMaj7#5add#2add4	G(Maj)b5add_b2add#2add4
G(Maj)#5add_b2add#2	GMaj7sus6add#2add4	G+add_b2add#2add4
Gsus6add_b2add#2	GMaj7susb6add#2add4	Gsus6add_b2add#2add4
Gsusb6add_b2add#2	G7add#2add#4	Gsusb6add_b2add#2add4
Gadd#2add4	G7b5add#2add#4	Gadd2add#2add#4
G(Maj)b5add#2add4	G7#5add#2add#4	G+add2add#2add#4
G+add#2add4	G7sus6add#2add#4	Gsus6add2add#2add#4
Gsus6add#2add4	G7susb6add#2add#4	Gsusb6add2add#2add#4
Gadd#2add#4	GMaj7add#2add#4	Gadd_b2add#2add#4
(x)	GMaj7#5add#2add#4	G(Maj)#5add_b2add#2add#4
G+add#2add#4	GMaj7sus6add#2add#4	Gsus6add_b2add#2add#4
Gsus6add#2add#4	GMaj7susb6add#2add#4	Gsusb6add_b2add#2add#4

G6add#2		G9add#2add_b6	G6b5add#2add4
G6b5add#2		G9b5add#2add_b6	G6#5add#2add4
G6#5add#2		G9sus6add#2add_b6	G6susb6add#2add4
(x)		GMaj9add#2add_b6	Gadd#2add4add_b6
G6susb6add#2		GMaj9b5add#2add_b6	G(Maj)b5add#2add4add_b6
G67add#2	G69add#2	GMaj9sus6add#2add_b6	Gsus6add#2add4add_b6
G67b5add#2	G69b5add#2	G7b9add#2add_b6	G67add#2add4
G67#5add#2	G69#5add#2	G7b9b5add#2add_b6	G67b5add#2add4
(x)	(x)	G7b9sus6add#2add_b6	G67#5add#2add4
G67susb6add#2	G69susb6add#2	GMaj7b9add#2add_b6	G67susb6add#2add4
G6Maj7add#2	G6Maj9add#2	GMaj7b9b5add#2add_b6	G6Maj7add#2add4
G6Maj7b5add#2	G6Maj9b5add#2	GMaj7b9sus6add#2add_b6	G6Maj7b5add#2add4
G6Maj7#5add#2	G6Maj9#5add#2	G6add2add#2	G6Maj7#5add#2add4
(x)	(x)	G6b5add2add#2	G6add2add#2add4
G6Maj7susb6add#2	G6Maj9susb6add#2	G6#5add2add#2	G6b5add2add#2add4
G7add#2add_b6	G67b9add#2	G6susb6add2add#2	G6#5add2add#2add4
G7b5add#2add_b6	G67b9b5add#2	Gadd2add#2add_b6	G6susb6add2add#2add4
G7#5add#2add_b6	G67b9#5add#2	G(Maj)b5add2add#2add_b6	Gadd2add#2add4add_b6
(x)	(x)	Gsus6add2add#2add_b6	G(Maj)b5add2add#2add4add_b6
G7susb6add#2add_b6	G67b9susb6add#2	G6add_b2add#2	Gsus6add2add#2add4add_b6
GMaj7add#2add_b6	G6Maj7b9add#2	G6b5add_b2add#2	G6add_b2add#2add4
GMaj7b5add#2add_b6	G6Maj7b9b5add#2	G6#5add_b2add#2	G6b5add_b2add#2add4
GMaj7#5add#2add_b6	G6Maj7b9#5add#2	G6susb6add_b2add#2	G6#5add_b2add#2add4
GMaj7sus6add#2add_b6	(x)	Gadd_b2add#2add_b6	G6susb6add_b2add#2add4
	G6Maj7b9susb6add#2	G(Maj)b5add_b2add#2add_b6	Gadd_b2add#2add4add_b6
		Gsus6add_b2add#2add_b6	G(b5)add_b2add#2add4add_b6
		G6add#2add4	Gsus6add_b2add#2add4add_b6

At this point the permutations are becoming unwieldy because of all the possibilities, but if each add chord appends to all previous chords systematically, then all possibilities exist.

Add#6 Chords

An added augmented sixth allows for the seventh and flat seventh scale degrees to exist in the same chord. Do not confuse these chords with the augmented sixth chords used for voice leading, such as the German+6, although some overlap in function exists.

An add#6 chord would be redundant for triads because the b7 is enharmonic to the minor 7 in the (7) chord (These are the German, French, and Italian sixth, purposely misspelled for voice leading purposes).

The GMaj7add#6 would be gbdf#f. A seventh (7) chord with the same notes could have an analysis of G7add_b8, but this would be redundant with the simpler GMaj7add#6. So, add#6 type chords based on the (7) chord are redundant.

Appending the add#6 tag to the chords without other added notes is straightforward.

Sevenths

GMaj7add#6	Gm(Maj7)add#6	GMaj7sus4add#6	GMaj7sus#4add#6	GMaj7sus2add#6	GMaj7susb2add#6
GMaj7b5add#6	Gm(Maj7)b5add#6	GMaj7b5sus4add#6	GMaj7b5sus#4add#6	GMaj7b5sus2add#6	GMaj7b5susb2add#6
GMaj7#5add#6	Gm(Maj7)#5add#6	GMaj7#5sus4add#6	GMaj7#5sus#4add#6	GMaj7#5sus2add#6	GMaj7#5susb2add#6
GMaj7sus6add#6	Gm(Maj7)sus6add#6	GMaj7sus4sus6add#6	GMaj7sus#4sus6add#6	GMaj7sus2sus6add#6	GMaj7sus2susb6add#6
GMaj7susb6add#6	Gm(Maj7)susb6add#6	GMaj7sus4susb6add#6	GMaj7sus#4susb6add#6	GMaj7sus2susb6add#6	GMaj7susb2susb6add#6

Ninths

GMaj9add#6	Gm(Maj9)add#6	GMaj9sus4add#6	GMaj9sus#4add#6	GMaj9susb2add#6
GMaj9b5add#6	Gm(Maj9)b5add#6	GMaj9b5sus4add#6	GMaj9b5sus#4add#6	GMaj9b5susb2add#6
GMaj9#5add#6	Gm(Maj9)#5add#6	GMaj9#5sus4add#6	GMaj9#5sus#4add#6	GMaj9#5susb2add#6
GMaj9sus6add#6	Gm(Maj9)msus6add#6	GMaj9sus4sus6add#6	GMaj9sus#4sus6add#6	GMaj9sus2susb6add#6
GMaj9susb6add#6	Gm(Maj9)msusb6add#6	GMaj9sus4susb6add#6	GMaj9sus#4susb6add#6	GMaj9susb2susb6add#6
GMaj7b9add#6	Gm(Maj7)b9add#6	GMaj7b9sus4add#6	GMaj7b9sus#4add#6	GMaj7sus2b9add#6
GMaj7b9b5add#6	Gm(Maj7)b9b5add#6	GMaj7b9b5sus4add#6	GMaj7b9b5sus#4add#6	GMaj7sus2b9b5add#6
GMaj7b9#5add#6	Gm(Maj7)b9#5add#6	GMaj7b9#5sus4add#6	GMaj7b9#5sus#4add#6	GMaj7sus2b9#5add#6
GMaj7b9sus6add#6	Gm(Maj7)b9sus6add#6	GMaj7b9sus4sus6add#6	GMaj7b9sus#4sus6add#6	GMaj7sus2b9sus6add#6
GMaj7b9susb6add#6	Gm(Maj7)b9susb6add#6	GMaj7b9sus4susb6add#6	GMaj7b9sus#4susb6add#6	GMaj7sus2b9susb6add#6

Elevenths and Thirteenths

GMaj11add#6	Gm(Maj11)add#6	GMaj11sus#4add#6	GMaj11susb2add#6
GMaj11b5add#6	Gm(Maj11)b5add#6	GMaj11b5sus#4add#6	GMaj11b5susb2add#6
GMaj11#5add#6	Gm(Maj11)#5add#6	GMaj11#5sus#4add#6	GMaj11#5susb2add#6
GMaj11sus6add#6	Gm(Maj11)sus6add#6	GMaj11sus#4sus6add#6	GMaj11sus2susb6add#6
GMaj11susb6add#6	Gm(Maj11)msusb6add#6	GMaj11sus#4susb6add#6	GMaj11susb2susb6add#6
GMaj11b9add#6	Gm(Maj11)b9add#6	GMaj11b9sus#4add#6	GMaj11sus2b9add#6
GMaj11b9b5add#6	Gm(Maj11)b9b5add#6	GMaj11b9b5sus#4add#6	GMaj11sus2b9b5add#6
GMaj11b9#5add#6	Gm(Maj11)b9#5add#6	GMaj11b9#5sus#4add#6	GMaj11sus2b9#5add#6
GMaj11b9sus6add#6	Gm(Maj11)b9sus6add#6	GMaj11b9sus#4sus6add#6	GMaj11sus2b9sus6add#6
GMaj11b9susb6add#6	Gm(Maj11)b9susb6add#6	GMaj11b9sus#4susb6add#6	GMaj11sus2b9susb6add#6
GMaj9#11add#6	Gm(Maj9)#11add#6	GMaj9#11sus4add#6	GMaj9susb2#11add#6
GMaj9#11b5add#6	Gm(Maj9)#11b5add#6	GMaj9#11b5sus4add#6	GMaj9susb2#11b5add#6
GMaj9#11#5add#6	Gm(Maj9)#11#5add#6	GMaj9#11#5sus4add#6	GMaj9susb2#11#5add#6
GMaj9#11sus6add#6	GMaj9m#11sus6add#6	GMaj9#11sus4sus6add#6	GMaj9sus2#11susb6add#6
GMaj9#11susb6add#6	GMaj9m#11susb6add#6	GMaj9#11sus4sus6add#6	GMaj9susb2#11susb6add#6
GMaj7#11b9add#6	Gm(Maj7)#11b9add#6	GMaj7#11b9sus4add#6	GMaj7sus2#11b9add#6
GMaj7#11b9b5add#6	Gm(Maj7)#11b9b5add#6	GMaj7#11b9b5sus4add#6	GMaj7sus2#11b9b5add#6
GMaj7#11b9#5add#6	Gm(Maj7)#11b9#5add#6	GMaj7#11b9#5sus4add#6	GMaj7sus2#11b9#5add#6
GMaj7#11b9sus6add#6	Gm(Maj7)#11b9sus6add#6	GMaj7#11b9sus4sus6add#6	GMaj7sus2#11b9sus6add#6
GMaj7#11b9susb6add#6	Gm(Maj7)#11b9susb6add#6	GMaj7#11b9sus4sus6add#6	GMaj7sus2#11b9susb6add#6
GMaj13add#6	Gm(Maj13)add#6	GMaj13sus#4add#6	GMaj13susb2add#6
GMaj13b5add#6	Gm(Maj13)b5add#6	GMaj13b5sus#4add#6	GMaj13b5susb2add#6
GMaj13#5add#6	Gm(Maj13)#5add#6	GMaj13#5sus#4add#6	GMaj13#5susb2add#6
GMaj13susb6add#6	GmMaj13susb6add#6	GMaj13sus#4susb6add#6	GMaj13susb2susb6add#6
GMaj13b9add#6	Gm(Maj13)b9add#6	GMaj13b9sus#4add#6	GMaj13sus2b9add#6
GMaj13b9b5add#6	Gm(Maj13)b9b5add#6	GMaj13b9b5sus#4add#6	GMaj13sus2b9b5add#6
GMaj13b9#5add#6	Gm(Maj13)b9#5add#6	GMaj13b9#5sus#4add#6	GMaj13sus2b9#5add#6
GMaj13b9susb6add#6	Gm(Maj13)b9susb6add#6	GMaj13b9sus#4susb6add#6	GMaj13sus2b9susb6add#6
GMaj13#11add#6	Gm(Maj13)#11add#6	GMaj13#11sus4add#6	GMaj13susb2#11add#6
GMaj13#11b5add#6	Gm(Maj13)#11b5add#6	GMaj13#11b5sus4add#6	GMaj13susb2#11b5add#6
GMaj13#11#5add#6	Gm(Maj13)#11#5add#6	GMaj13#11#5sus4add#6	GMaj13susb2#11#5add#6
GMaj13#11susb6add#6	GMaj13m#11susb6add#6	GMaj13#11sus4sus6add#6	GMaj13susb2#11susb6add#6
GMaj13#11b9add#6	Gm(Maj13)#11b9add#6	GMaj13#11b9sus4add#6	GMaj13sus2#11b9add#6
GMaj13#11b9b5add#6	Gm(Maj13)#11b9b5add#6	GMaj13#11b9b5sus4add#6	GMaj13sus2#11b9b5add#6
GMaj13#11b9#5add#6	Gm(Maj13)#11b9#5add#6	GMaj13#11b9#5sus4add#6	GMaj13sus2#11b9#5add#6
GMaj13#11b9susb6add#6	Gm(Maj13)#11b9susb6add#6	GMaj13#11b9sus4sus6add#6	GMaj13sus2#11b9susb6add#6
GMaj11b13add#6	Gm(Maj11)b13add#6	GMaj11b13sus#4add#6	GMaj11susb2b13add#6
GMaj11b13b5add#6	Gm(Maj11)b13b5add#6	GMaj11b13b5sus#4add#6	GMaj11susb2b13b5add#6
GMaj11b13#5add#6	Gm(Maj11)b13#5add#6	GMaj11b13#5sus#4add#6	GMaj11susb2b13#5add#6
GMaj11b13sus6add#6	Gm(Maj11)b13sus6add#6	GMaj11b13sus#4sus6add#6	GMaj11sus2b13susb6add#6
GMaj11b13b9add#6	Gm(Maj11)b13b9add#6	GMaj11b13b9sus#4add#6	GMaj11sus2b13b9add#6
GMaj11b13b9b5add#6	Gm(Maj11)b13b9b5add#6	GMaj11b13b9b5sus#4add#6	GMaj11sus2b13b9b5add#6
GMaj11b13b9#5add#6	Gm(Maj11)b13b9#5add#6	GMaj11b13b9#5sus#4add#6	GMaj11sus2b13b9#5add#6
GMaj11b13b9sus6add#6	Gm(Maj11)b13b9sus6add#6	GMaj11b13b9sus#4sus6add#6	GMaj11sus2b13b9sus6add#6
GMaj9b13#11add#6	Gm(Maj9)b13#11add#6	GMaj9b13#11sus4add#6	GMaj9susb2b13#11add#6
GMaj9b13#11b5add#6	Gm(Maj9)b13#11b5add#6	GMaj9b13#11b5sus4add#6	GMaj9susb2b13#11b5add#6
GMaj9b13#11#5add#6	Gm(Maj9)b13#11#5add#6	GMaj9b13#11#5sus4add#6	GMaj9susb2b13#11#5add#6
GMaj9b13#11sus6add#6	GMaj9mb13#11sus6add#6	GMaj9b13#11sus4sus6add#6	GMaj9sus2b13#11susb6add#6
GMaj7b13#11b9add#6	Gm(Maj7)b13#11b9add#6	GMaj7b13#11b9sus4add#6	GMaj7sus2b13#11b9add#6
GMaj7b13#11b9b5add#6	Gm(Maj7)b13#11b9b5add#6	GMaj7b13#11b9b5sus4add#6	GMaj7sus2b13#11b9b5add#6
GMaj7b13#11b9#5add#6	Gm(Maj7)b13#11b9#5add#6	GMaj7b13#11b9#5sus4add#6	GMaj7sus2b13#11b9#5add#6
GMaj7b13#11b9sus6add#6	Gm(Maj7)b13#11b9sus6add#6	GMaj7b13#11b9sus4sus6add#6	GMaj7sus2b13#11b9sus6add#6

Add_b2 Clusters

The Add_b2 (Alternately add_b9) chord appeared previously, but in order to keep the flow of the tables, not all instances made it to the tables. The Add9 include g, a_b, and a all together. Take all the chords with 9ths or add2 and append an add_b2, (610 occurrences.)

G9add_b2	Gm9add_b2	G9sus4add_b2	G9sus#4add_b2
G9b5add_b2	Gm9b5add_b2	G9b5sus4add_b2	G9b5sus#4add_b2
G9#5add_b2	Gm9#5add_b2	G9#5sus4add_b2	G9#5sus#4add_b2
G9sus6add_b2	Gm9sus6add_b2	G9sus4sus6add_b2	G9sus#4sus6add_b2
G9susb6add_b2	Gm9susb6add_b2	G9sus4susb6add_b2	G9sus#4susb6add_b2
GMaj9add_b2	Gm(Maj9)add_b2	GMaj9sus4add_b2	GMaj9sus#4add_b2
GMaj9b5add_b2	Gm(Maj9)b5add_b2	GMaj9b5sus4add_b2	GMaj9b5sus#4add_b2
GMaj9#5add_b2	Gm(Maj9)#5add_b2	GMaj9#5sus4add_b2	GMaj9#5sus#4add_b2
GMaj9sus6add_b2	Gm(Maj9)msus6add_b2	GMaj9sus4sus6add_b2	GMaj9sus#4sus6add_b2
GMaj9susb6add_b2	Gm(Maj9)msusb6add_b2	GMaj9sus4susb6add_b2	GMaj9sus#4susb6add_b2
G11add_b2	Gm11add_b2	G11sus#4add_b2
G11b5add_b2	Gm11b5add_b2	G11b5sus#4add_b2
G11#5add_b2	Gm11#5add_b2	G11#5sus#4add_b2
G11sus6add_b2	Gm11sus6add_b2	G11sus#4sus6add_b2
G11susb6add_b2	Gm11susb6add_b2	G11sus#4susb6add_b2
GMaj11add_b2	Gm(Maj11)add_b2	GMaj11sus#4add_b2
GMaj11b5add_b2	Gm(Maj11)b5add_b2	GMaj11b5sus#4add_b2
GMaj11#5add_b2	Gm(Maj11)#5add_b2	GMaj11#5sus#4add_b2
GMaj11sus6add_b2	Gm(Maj11)sus6add_b2	GMaj11sus#4sus6add_b2
GMaj11susb6add_b2	Gm(Maj11)msusb6add_b2	GMaj11sus#4susb6add_b2
G9#11add_b2	Gm9#11add_b2	G9#11sus4add_b2
G9#11b5add_b2	Gm9#11b5add_b2	G9#11b5sus4add_b2
G9#11#5add_b2	Gm9#11#5add_b2	G9#11#5sus4add_b2
G9#11sus6add_b2	Gm9#11sus6add_b2	G9#11sus4sus6add_b2
G9#11susb6add_b2	Gm9#11susb6add_b2	G9#11sus4sus6add_b2
GMaj9#11add_b2	Gm(Maj9)#11add_b2	GMaj9#11sus4add_b2
GMaj9#11b5add_b2	Gm(Maj9)#11b5add_b2	GMaj9#11b5sus4add_b2
GMaj9#11#5add_b2	Gm(Maj9)#11#5add_b2	GMaj9#11#5sus4add_b2
GMaj9#11sus6add_b2	GMaj9m#11sus6add_b2	GMaj9#11sus4sus6add_b2
GMaj9#11susb6add_b2	GMaj9m#11susb6add_b2	GMaj9#11sus4sus6add_b2
G13add_b2	Gm13add_b2	G13sus#4add_b2
G13b5add_b2	Gm13b5add_b2	G13b5sus#4add_b2
G13#5add_b2	Gm13#5add_b2	G13#5sus#4add_b2
G13susb6add_b2	Gm13susb6add_b2	G13sus#4susb6add_b2
GMaj13add_b2	Gm(Maj13)add_b2	GMaj13sus#4add_b2
GMaj13b5add_b2	Gm(Maj13)b5add_b2	GMaj13b5sus#4add_b2
GMaj13#5add_b2	Gm(Maj13)#5add_b2	GMaj13#5sus#4add_b2
GMaj13susb6add_b2	GmMaj13susb6add_b2	GMaj13sus#4susb6add_b2
G13#11add_b2	Gm13#11add_b2	G13#11sus4add_b2
G13#11b5add_b2	Gm13#11b5add_b2	G13#11b5sus4add_b2
G13#11#5add_b2	Gm13#11#5add_b2	G13#11#5sus4add_b2
G13#11sus6add_b2	Gm13#11sus6add_b2	G13#11sus4sus6add_b2
G13#11susb6add_b2	Gm13#11susb6add_b2	G13#11sus4sus6add_b2
GMaj13#11add_b2	Gm(Maj13)#11add_b2	GMaj13#11sus4add_b2
GMaj13#11b5add_b2	Gm(Maj13)#11b5add_b2	GMaj13#11b5sus4add_b2
GMaj13#11#5add_b2	Gm(Maj13)#11#5add_b2	GMaj13#11#5sus4add_b2
GMaj13#11susb6add_b2	GMaj13m#11susb6add_b2	GMaj13#11sus4sus6add_b2
G11b13add_b2	Gm11b13add_b2	G11b13sus#4add_b2
G11b13b5add_b2	Gm11b13b5add_b2	G11b13b5sus#4add_b2
G11b13#5add_b2	Gm11b13#5add_b2	G11b13#5sus#4add_b2
G11b13sus6add_b2	Gm11b13sus6add_b2	G11b13sus#4sus6add_b2
G11b13susb6add_b2	Gm11b13susb6add_b2	G11b13sus#4susb6add_b2
GMaj11b13add_b2	Gm(Maj11)b13add_b2	GMaj11b13sus#4add_b2
GMaj11b13b5add_b2	Gm(Maj11)b13b5add_b2	GMaj11b13b5sus#4add_b2
GMaj11b13#5add_b2	Gm(Maj11)b13#5add_b2	GMaj11b13#5sus#4add_b2
GMaj11b13sus6add_b2	Gm(Maj11)b13sus6add_b2	GMaj11b13sus#4sus6add_b2
G9b13#11add_b2	Gm9b13#11add_b2	G9b13#11sus4add_b2
G9b13#11b5add_b2	Gm9b13#11b5add_b2	G9b13#11b5sus4add_b2
G9b13#11#5add_b2	Gm9b13#11#5add_b2	G9b13#11#5sus4add_b2
G9b13#11sus6add_b2	Gm9b13#11sus6add_b2	G9b13#11sus4sus6add_b2
G9b13#11susb6add_b2	Gm9b13#11susb6add_b2	G9b13#11sus4sus6add_b2
GMaj9b13#11add_b2	Gm(Maj9)b13#11add_b2	GMaj9b13#11sus4add_b2
GMaj9b13#11b5add_b2	Gm(Maj9)b13#11b5add_b2	GMaj9b13#11b5sus4add_b2
GMaj9b13#11#5add_b2	Gm(Maj9)b13#11#5add_b2	GMaj9b13#11#5sus4add_b2
GMaj9b13#11sus6add_b2	GMaj9mb13#11sus6add_b2	GMaj9b13#11sus4sus6add_b2
Gadd2add_b2	Gmadd2add_b2	Gsus4add2add_b2	Gsus#4add2add_b2
G(Maj)b5add2add_b2	Goadd2add_b2	G(Maj)b5sus4add2add_b2	G(Maj)b5sus#4add2add_b2
G(Maj)#5add2add_b2	Gm#5add2add_b2	G(MAJ)#5sus4add2add_b2	G(MAJ)#5sus#4add2add_b2
Gsus6add2add_b2	Gmsus6add2add_b2	Gsus4sus6add2add_b2	Gsus#4sus6add2add_b2
Gsusb6add2add_b2	Gmsusb6add2add_b2	Gsus4sus6add2add_b2	Gsus#4susb6add2add_b2
Gadd2add4add_b2	Gmadd2add4add_b2	Gsus#4add2add4add_b2
G(Maj)b5add2add4add_b2	Goadd2add4add_b2	G(Maj)b5sus#4add2add4add_b2
G+add2add4add_b2	Gm#5add2add4add_b2	G(MAJ)#5sus#4add2add4add_b2
Gsus6add2add4add_b2	Gmsus6add2add4add_b2	Gsus#4sus6add2add4add_b2
Gsusb6add2add4add_b2	Gmsusb6add2add4add_b2	Gsus#4susb6add2add4add_b2
Gadd2add#4add_b2	Gmadd2add#4add_b2	Gsus4add2add#4add_b2
G+add2add#4add_b2	Gm#5add2add#4add_b2	G(MAJ)#5sus4add2add#4add_b2
Gsus6add2add#4add_b2	Gmsus6add2add#4add_b2	Gsus4sus6add2add#4add_b2
Gsusb6add2add#4add_b2	Gmsusb6add2add#4add_b2	Gsus4sus6add2add#4add_b2
G69add_b2	Gm69add_b2	G69sus4add_b2	G69sus#4add_b2
G69b5add_b2	Gm69b5add_b2	G69b5sus4add_b2	G69b5sus#4add_b2
G69#5add_b2	Gm69#5add_b2	G69#5sus4add_b2	G69#5sus#4add_b2
G69susb6add_b2	Gm69susb6add_b2	G69sus4susb6add_b2	G69sus#4susb6add_b2
G6Maj9add_b2	Gm(Maj9)add_b2	G6Maj9sus4add_b2	G6Maj9sus#4add_b2
G6Maj9b5add_b2	Gm(Maj9)b5add_b2	G6Maj9b5sus4add_b2	G6Maj9b5sus#4add_b2
G6Maj9#5add_b2	Gm(Maj9)#5add_b2	G6Maj9#5sus4add_b2	G6Maj9#5sus#4add_b2
G6Maj9susb6add_b2	Gm6(Maj9)susb6add_b2	G6Maj9sus4sus6add_b2	G6Maj9sus#4susb6add_b2
G9add_b2add_b6	Gm9add_b2add_b6	G9sus4add_b2add_b6	G9sus#4add_b2add_b6
G9b5add_b2add_b6	Gm9b5add_b2add_b6	G9b5sus4add_b2add_b6	G9b5sus#4add_b2add_b6
G9sus6add_b2add_b6	Gm9sus6add_b2add_b6	G9sus4sus6add_b2add_b6	G9sus#4sus6add_b2add_b6
GMaj9add_b2add_b6	Gm(Maj9)add_b2add_b6	GMaj9sus4add_b2add_b6	GMaj9sus#4add_b2add_b6
GMaj9b5add_b2add_b6	Gm(Maj9)b5add_b2add_b6	GMaj9b5sus4add_b2add_b6	GMaj9b5sus#4add_b2add_b6
GMaj9sus6add_b2add_b6	GmMaj9sus6add_b2add_b6	GMaj9sus4sus6add_b2add_b6	GMaj9sus#4sus6add_b2add_b6
G6add2add_b2	Gm6add2add_b2	G6sus4add2add_b2	G6sus#4add2add_b2
G6b5add2add_b2	Gm6b5add2add_b2	G6b5sus4add2add_b2	G6b5sus#4add2add_b2
G6#5add2add_b2	Gm6#5add2add_b2	G6#5sus4add2add_b2	G6#5sus#4add2add_b2
G6susb6add2add_b2	Gm6susb6add2add_b2	G6sus4susb6add2add_b2	G6sus#4susb6add2add_b2
Gadd2add_b2add_b6	Gmadd2add_b2add_b6	Gsus4add2add_b2add_b6	Gsus#4add2add_b2add_b6
G(Maj)b5add2add_b2add_b6	Gmb5add2add_b2add_b6	G(Maj)b5sus4add2add_b2add_b6	G(Maj)b5sus#4add2add_b2add_b6
Gsus6add2add_b2add_b6	Gmsus6add2add_b2add_b6	Gsus4sus6add2add_b2add_b6	Gsus#4sus6add2add_b2add_b6
G6add2add4add_b2	Gm6add2add4add_b2	G6sus#4add2add4add_b2	G6sus2add2add4add_b2
G6b5add2add4add_b2	Gm6b5add2add4add_b2	G6b5sus#4add2add4add_b2	G6b5sus2add2add4add_b2
G6#5add2add4add_b2	Gm6#5add2add4add_b2	G6#5sus#4add2add4add_b2	G6#5sus2add2add4add_b2
G6susb6add2add4add_b2	Gm6susb6add2add4add_b2	G6sus#4susb6add2add4add_b2	G6sus2susb6add2add4add_b2
Gadd2add4add_b6add_b2	Gmadd2add4add_b6add_b2	Gsus#4add2add4add_b6add_b2	Gsus2add2add4add_b6add_b2
G(Maj)b5add2add4add_b6add_b2	Gmb5add2add4add_b6add_b2	G(Maj)b5sus#4add2add4add_b6add_b2	Gb5sus2add2add4add_b6add_b2
Gsus6add2add4add_b6add_b2	Gmsus6add2add4add_b6add_b2	Gsus#4sus6add2add4add_b6add_b2	Gsus2sus6add2add4add_b6add_b2
Gm6add2add#4add_b2	G6sus4add2add#4add_b2	G6sus2add2add#4add_b2
Gm6b5add2add#4add_b2	G6b5sus4add2add#4add_b2	G6b5sus2add2add#4add_b2
Gm6#5add2add#4add_b2	G6#5sus4add2add#4add_b2	G6#5sus2add2add#4add_b2
Gm6susb6add2add#4add_b2	G6sus4susb6add2add#4add_b2	G6sus2susb6add2add#4add_b2
Gmadd2add#4add_b6add_b2	Gsus4add2add#4add_b6add_b2	Gsus2add2add#4add_b6add_b2
Gmb5add2add#4add_b6add_b2	G(Maj)b5sus4add2add#4add_b6add_b2	Gb5sus2add2add#4add_b6add_b2
Gmsus6add2add#4add_b6add_b2	Gsus4sus6add2add#4add_b6add_b2	Gsus2sus6add2add#4add_b6add_b2
G9add#2add_b2	G11add#2add_b2	G13add#2add_b2	G11b13add#2add_b2
G9b5add#2add_b2	G11b5add#2add_b2	G13b5add#2add_b2	G11b13b5add#2add_b2
G9#5add#2add_b2	G11#5add#2add_b2	G13#5add#2add_b2	G11b13#5add#2add_b2
G9sus6add#2add_b2	G11sus6add#2add_b2		G11b13sus6add#2add_b2
G9susb6add#2add_b2	G11susb6add#2add_b2	G13susb6add#2add_b2	G11b13susb6add#2add_b2
GMaj9add#2add_b2	GMaj11add#2add_b2	GMaj13add#2add_b2	GMaj11b13add#2add_b2
GMaj9b5add#2add_b2	GMaj11b5add#2add_b2	GMaj13b5add#2add_b2	GMaj11b13b5add#2add_b2
GMaj9#5add#2add_b2	GMaj11#5add#2add_b2	GMaj13#5add#2add_b2	GMaj11b13#5add#2add_b2
GMaj9sus6add#2add_b2	GMaj11sus6add#2add_b2		GMaj11b13sus6add#2add_b2
GMaj9susb6add#2add_b2	GMaj11susb6add#2add_b2	GMaj13susb6add#2add_b2
	G9#11add#2add_b2	G13#11add#2add_b2	G9b13#11add#2add_b2
	G9#11b5add#2add_b2	G13#11b5add#2add_b2	G9b13#11b5add#2add_b2
	G9#11#5add#2add_b2	G13#11#5add#2add_b2	G9b13#11#5add#2add_b2
	G9#11sus6add#2add_b2	G13#11sus6add#2add_b2	G9b13#11sus6add#2add_b2
	G9#11susb6add#2add_b2	G13#11susb6add#2add_b2	G9b13#11susb6add#2add_b2
	GMaj9#11add#2add_b2	GMaj13#11add#2add_b2	GMaj9b13#11add#2add_b2
	GMaj9#11b5add#2add_b2	GMaj13#11b5add#2add_b2	GMaj9b13#11b5add#2add_b2
	GMaj9#11#5add#2add_b2	GMaj13#11#5add#2add_b2	GMaj9b13#11#5add#2add_b2
	GMaj9#11sus6add#2add_b2		GMaj9b13#11sus6add#2add_b2
	GMaj9#11susb6add#2add_b2	GMaj13#11susb6add#2add_b2
Gadd2add#2add_b2	G7add#2add4add_b2	Gadd2add#2add4add_b2
G(Maj)b5add2add#2add_b2	G7b5add#2add4add_b2	G(Maj)b5add2add#2add4add_b2
G(Maj)#5add2add#2add_b2	G7#5add#2add4add_b2	G+add2add#2add4add_b2
Gsus6add2add#2add_b2	G7sus6add#2add4add_b2	Gsus6add2add#2add4add_b2
Gsusb6add2add#2add_b2	G7susb6add#2add4add_b2	Gsusb6add2add#2add4add_b2
Gadd#2add4add_b2	G7b5add#2add#4add_b2	Gadd2add#2add#4add_b2
G(Maj)b5add#2add4add_b2	G7#5add#2add#4add_b2	G+add2add#2add#4add_b2
G+add#2add4add_b2	G7sus6add#2add#4add_b2	Gsus6add2add#2add#4add_b2
Gsus6add#2add4add_b2	G7susb6add#2add#4add_b2	Gsusb6add2add#2add#4add_b2
Gadd#2add#4add_b2	GMaj7add#2add#4add_b2
(x)	GMaj7#5add#2add#4add_b2
G+add#2add#4add_b2	GMaj7sus6add#2add#4add_b2
Gsus6add#2add#4add_b2	GMaj7susb6add#2add#4add_b2
G9add#2add_b2add_b6	G69add#2add_b2
G9b5add#2add_b2add_b6	G69b5add#2add_b2
G9sus6add#2add_b2add_b6	G69#5add#2add_b2
GMaj9add#2add_b2add_b6	G69susb6add#2add_b2
GMaj9b5add#2add_b2add_b6	G6Maj9add#2add_b2
GMaj9sus6add#2add_b2add_b6	G6Maj9b5add#2add_b2
G6add2add#2add_b2	G6Maj9#5add#2add_b2
G6b5add2add#2add_b2	G6Maj9susb6add#2add_b2
G6#5add2add#2add_b2
G6susb6add2add#2add_b2	GMaj7sus2add#6add_b2
Gadd2add#2add_b6add_b2	GMaj7b5sus2add#6add_b2
G(Maj)b5add2add#2add_b6add_b2	GMaj7#5sus2add#6add_b2
	GMaj7sus2sus6add#6add_b2
	GMaj7sus2susb6add#6add_b2
GMaj9add#6add_b2	Gm(Maj9)add#6add_b2	GMaj9sus4add#6add_b2	GMaj9sus#4add#6add_b2
GMaj9b5add#6add_b2	Gm(Maj9)b5add#6add_b2	GMaj9b5sus4add#6add_b2	GMaj9b5sus#4add#6add_b2
GMaj9#5add#6add_b2	Gm(Maj9)#5add#6add_b2	GMaj9#5sus4add#6add_b2	GMaj9#5sus#4add#6add_b2
GMaj9sus6add#6add_b2	Gm(Maj9)msus6add#6add_b2	GMaj9sus4sus6add#6add_b2	GMaj9sus#4sus6add#6add_b2
GMaj9susb6add#6add_b2	Gm(Maj9)msusb6add#6add_b2	GMaj9sus4susb6add#6add_b2	GMaj9sus#4susb6add#6add_b2
GMaj11add#6add_b2	Gm(Maj11)add#6add_b2	GMaj11sus#4add#6add_b2
GMaj11b5add#6add_b2	Gm(Maj11)b5add#6add_b2	GMaj11b5sus#4add#6add_b2
GMaj11#5add#6add_b2	Gm(Maj11)#5add#6add_b2	GMaj11#5sus#4add#6add_b2
GMaj11sus6add#6add_b2	Gm(Maj11)sus6add#6add_b2	GMaj11sus#4sus6add#6add_b2
GMaj11susb6add#6add_b2	Gm(Maj11)msusb6add#6add_b2	GMaj11sus#4susb6add#6add_b2
GMaj9#11add#6add_b2	Gm(Maj9)#11add#6add_b2	GMaj9#11sus4add#6add_b2
GMaj9#11b5add#6add_b2	Gm(Maj9)#11b5add#6add_b2	GMaj9#11b5sus4add#6add_b2
GMaj9#11#5add#6add_b2	Gm(Maj9)#11#5add#6add_b2	GMaj9#11#5sus4add#6add_b2
GMaj9#11sus6add#6add_b2	GMaj9m#11sus6add#6add_b2	GMaj9#11sus4sus6add#6add_b2
GMaj13add#6add_b2	Gm(Maj13)add#6add_b2	GMaj13sus#4add#6add_b2
GMaj13b5add#6add_b2	Gm(Maj13)b5add#6add_b2	GMaj13b5sus#4add#6add_b2
GMaj13#5add#6add_b2	Gm(Maj13)#5add#6add_b2	GMaj13#5sus#4add#6add_b2
GMaj13susb6add#6add_b2	GmMaj13susb6add#6add_b2	GMaj13sus#4susb6add#6add_b2
GMaj13#11add#6add_b2	Gm(Maj13)#11add#6add_b2	GMaj13#11sus4add#6add_b2
GMaj13#11b5add#6add_b2	Gm(Maj13)#11b5add#6add_b2	GMaj13#11b5sus4add#6add_b2
GMaj13#11#5add#6add_b2	Gm(Maj13)#11#5add#6add_b2	GMaj13#11#5sus4add#6add_b2
GMaj13#11susb6add#6add_b2	GMaj13m#11susb6add#6add_b2	GMaj13#11sus4sus6add#6add_b2
GMaj11b13add#6add_b2	Gm(Maj11)b13add#6add_b2	GMaj11b13sus#4add#6add_b2
GMaj11b13b5add#6add_b2	Gm(Maj11)b13b5add#6add_b2	GMaj11b13b5sus#4add#6add_b2
GMaj11b13#5add#6add_b2	Gm(Maj11)b13#5add#6add_b2	GMaj11b13#5sus#4add#6add_b2
GMaj11b13sus6add#6add_b2	Gm(Maj11)b13sus6add#6add_b2	GMaj11b13sus#4sus6add#6add_b2
GMaj9b13#11add#6add_b2	Gm(Maj9)b13#11add#6add_b2	GMaj9b13#11sus4add#6add_b2
GMaj9b13#11b5add#6add_b2	Gm(Maj9)b13#11b5add#6add_b2	GMaj9b13#11b5sus4add#6add_b2
GMaj9b13#11#5add#6add_b2	Gm(Maj9)b13#11#5add#6add_b2	GMaj9b13#11#5sus4add#6add_b2
GMaj9b13#11sus6add#6add_b2	GMaj9mb13#11sus6add#6add_b2	GMaj9b13#11sus4sus6add#6add_b2

Add#4 Clusters

The next cluster of notes to add would be the add#4 along with 4ths found in sus4 chords and 11^th chords. Appending a sus#4 to any chord with an unaltered sus4 or 11the fills in the gaps. Adding a sus#4 to a diminished style chord with a b5 would be redundant. The altered 11ths also need to add to 13^th and Maj13 style chords since they have an implied 11^th.

Gsus4add#4	GMaj7sus4add#4	Gsus4add2add#4
G(MAJ)#5sus4add#4	GMaj7#5sus4add#4	G(MAJ)#5sus4add2add#4
Gsus4sus6add#4	GMaj7sus4sus6add#4	Gsus4sus6add2add#4
Gsus4susb6add#4	GMaj7sus4susb6add#4	Gsus4sus6add2add#4
G7sus4add#4	G9sus4add#4	Gsus4add_b2add#4
G7#5sus4add#4	G9#5sus4add#4	G(MAJ)#5sus4add_b2add#4
G7sus4sus6add#4	G9sus4sus6add#4	Gsus4sus6add_b2add#4
G7sus4susb6add#4	G9sus4susb6add#4	Gsus4sus6add_b2add#4
GMaj7sus4add#4	GMaj9sus4add#4	Gmadd4add#4
GMaj7#5sus4add#4	GMaj9#5sus4add#4	Gm#5add4add#4
GMaj7sus4sus6add#4	GMaj9sus4sus6add#4	Gmsus6add4add#4
GMaj7sus4susb6add#4	GMaj9sus4susb6add#4
G7sus4add#4	G7b9sus4add#4	GMaj7b9sus4add#4
G7#5sus4add#4	G7b9#5sus4add#4	GMaj7b9#5sus4add#4
G7sus4sus6add#4	G7b9sus4sus6add#4	GMaj7b9sus4sus6add#4
G7sus4sus6add#4	G7b9sus4susb6add#4	GMaj7b9sus4susb6add#4
G11add#4	Gm11add#4	G11susb2add#4
G11#5add#4	Gm11#5add#4	G11#5susb2add#4
G11sus6add#4	Gm11sus6add#4	G11sus2susb6add#4
G11susb6add#4	Gm11susb6add#4	G11susb2susb6add#4
GMaj11add#4	Gm(Maj11)add#4	GMaj11susb2add#4
GMaj11#5add#4	Gm(Maj11)#5add#4	GMaj11#5susb2add#4
GMaj11sus6add#4	Gm(Maj11)sus6add#4	GMaj11sus2susb6add#4
GMaj11susb6add#4	Gm(Maj11)msusb6add#4	GMaj11susb2susb6add#4
G11b9add#4	Gm11b9add#4	G11sus2b9add#4
G11b9#5add#4	Gm11b9#5add#4	G11sus2b9#5add#4
G11b9sus6add#4	Gm11b9susb6add#4	G11sus2b9sus6add#4
G11b9susb6add#4	Gm11b9susb6add#4	G11sus2b9susb6add#4
GMaj11b9add#4	Gm(Maj11)b9add#4	GMaj11sus2b9add#4
GMaj11b9#5add#4	Gm(Maj11)b9#5add#4	GMaj11sus2b9#5add#4
GMaj11b9sus6add#4	Gm(Maj11)b9sus6add#4	GMaj11sus2b9sus6add#4
GMaj11b9susb6add#4	Gm(Maj11)b9susb6add#4	GMaj11sus2b9susb6add#4
G13add#4	Gm13add#4	G13susb2add#4
G13#5add#4	Gm13#5add#4	G13#5susb2add#4
G13susb6add#4	Gm13susb6add#4	G13susb2susb6add#4
GMaj13add#4	Gm(Maj13)add#4	GMaj13susb2add#4
GMaj13#5add#4	Gm(Maj13)#5add#4	GMaj13#5susb2add#4
GMaj13susb6add#4	GmMaj13susb6add#4	GMaj13susb2susb6add#4
G13b9add#4	Gm13b9add#4	G13sus2b9add#4
G13b9#5add#4	Gm13b9#5add#4	G13sus2b9#5add#4
G13b9susb6add#4	Gm13b9susb6add#4	G13sus2b9susb6add#4
GMaj13b9add#4	Gm(Maj13)b9add#4	GMaj13sus2b9add#4
GMaj13b9#5add#4	Gm(Maj13)b9#5add#4	GMaj13sus2b9#5add#4
GMaj13b9susb6add#4	Gm(Maj13)b9susb6add#4	GMaj13sus2b9susb6add#4
G7add4add#4	GMaj7add4add#4	G7sus2add4add#4
G7#5add4add#4	GMaj7#5add4add#4	G7#5sus2add4add#4
G7sus6add4add#4	GMaj7sus6add4add#4	G7sus2sus6add4add#4
G7susb6add4add#4	GMaj7susb6add4add#4	G7sus2susb6add4add#4
GMaj7add4add#4	Gm(Maj7)add4add#4	GMaj7sus2add4add#4
GMaj7#5add4add#4	Gm(Maj7)#5add4add#4	GMaj7#5sus2add4add#4
GMaj7sus6add4add#4	GMaj7sus6add4add#4	GMaj7sus2sus6add4add#4
GMaj7susb6add4add#4	GMaj7susb6add4add#4	GMaj7sus2susb6add4add#4
Gadd2add4add#4	Gmadd2add4add#4	Gsusb2add2add4add#4
G+add2add4add#4	Gm#5add2add4add#4	G(MAJ)#5susb2add2add4add#4
Gsus6add2add4add#4	Gmsus6add2add4add#4	Gsus2susb6add2add4add#4
Gsusb6add2add4add#4	Gmsusb6add2add4add#4	Gsusb2susb6add2add4add#4
Gadd_b2add4add#4	Gmadd_b2add4add#4	Gsus2add_b2add4add#4
G+add_b2add4add#4	Gm#5add_b2add4add#4	G(MAJ)#5sus2add_b2add4add#4
Gsus6add_b2add4add#4	Gmsus6add_b2add4add#4	Gsus2sus6add_b2add4add#4
Gsusb6add_b2add4add#4	Gmsusb6add_b2add4add#4	Gsus2susb6add_b2add4add#4
G6sus4add#4	G69sus4add#4	G9sus4add#4add_b6
G6#5sus4add#4	G69#5sus4add#4	G9sus4sus6add#4add_b6
G6sus4susb6add#4	G69sus4susb6add#4	GMaj9sus4add#4add_b6
Gsus4add#4add_b6	G6Maj9sus4add#4	GMaj9sus4sus6add#4add_b6
Gsus4sus6add#4add_b6	G6Maj9#5sus4add#4	G7b9sus4add#4add_b6
G67sus4add#4	G6Maj9sus4sus6add#4	G7b9sus4sus6add#4add_b6
G67#5sus4add#4	G67b9sus4add#4	GMaj7b9sus4add#4add_b6
G67sus4sus6add#4	G67b9#5sus4add#4	GMaj7b9sus4sus6add#4add_b6
G6Maj7sus4add#4	G67b9sus4sus6add#4	G6sus4add2add#4
G6Maj7#5sus4add#4	G6Maj7b9sus4add#4	G6#5sus4add2add#4
	G6Maj7b9#5sus4add#4	G6sus4susb6add2add#4
G6Maj7sus4susb6add#4	G6Maj7b9sus4susb6add#4	Gsus4add2add#4add_b6
G7sus4add#4add_b6		Gsus4sus6add2add#4add_b6
G7#5sus4add#4add_b6		G6sus4add_b2add#4
G7sus4sus6add#4add_b6		G6#5sus4add_b2add#4
GMaj7sus4add#4add_b6		G6sus4susb6add_b2add#4
GMaj7#5sus4add#4add_b6		Gsus4add_b2add#4add_b6
GMaj7sus4sus6add#4add_b6		Gsus4sus6add_b2add#4add_b6
		G6add4add#4
G6#5add4add#4	Gm6add4add#4	G6sus2add4add#4
G6susb6add4add#4	Gm6#5add4add#4	G6#5sus2add4add#4
Gadd4add#4add_b6	Gm6susb6add4add#4	G6sus2susb6add4add#4
Gsus6add4add#4add_b6	Gmadd4add#4add_b6	Gsus2add4add#4add_b6
G67add4add#4	Gmsus6add4add#4add_b6	Gsus2sus6add4add#4add_b6
G67#5add4add#4	Gm67add4add#4	G67sus2add4add#4
G67susb6add4add#4	Gm67#5add4add#4	G67#5sus2add4add#4
G6Maj7add4add#4	Gsusb6add4add#4	G67sus2susb6add4add#4
G6Maj7#5add4add#4	Gm6Maj7add4add#4	G6Maj7sus2add4add#4
	Gm6Maj7#5add4add#4	G6Maj7#5sus2add4add#4
G6add2add4add#4	Gm6add2add4add#4	G6sus2add2add4add#4
G6#5add2add4add#4	Gm6#5add2add4add#4	G6#5sus2add2add4add#4
G6susb6add2add4add#4	Gm6susb6add2add4add#4	G6sus2susb6add2add4add#4
Gadd2add4add#4add_b6	Gmadd2add4add#4add_b6	Gsus2add2add4add#4add_b6
Gsus6add2add4add#4add_b6	Gmsus6add2add4add#4add_b6	Gsus2sus6add2add4add#4add_b6
G6add_b2add4add#4	Gm6add_b2add4add#4	G6sus2add_b2add4add#4
G6#5add_b2add4add#4	Gm6#5add_b2add4add#4	G6#5sus2add_b2add4add#4
G6susb6add_b2add4add#4	Gm6susb6add_b2add4add#4	G6sus2susb6add_b2add4add#4
Gadd_b2add4add#4add_b6	Gmadd_b2add4add#4add_b6	Gsus2add_b2add4add#4add_b6
Gsus6add_b2add4add#4add_b6	Gmsus6add_b2add4add#4add_b6	Gsus2sus6add_b2add4add#4add_b6
G11add#2add#4	G7add#2add4add#4	Gadd2add#2add4add#4
G11#5add#2add#4	G7#5add#2add4add#4	GM#5add2add#2add4add#4
G11sus6add#2add#4	G7sus6add#2add4add#4	Gsus6add2add#2add4add#4
G11susb6add#2add#4	G7susb6add#2add4add#4	Gsusb6add2add#2add4add#4
GMaj11add#2add#4	GMaj7add#2add4add#4	Gadd_b2add#2add4add#4
GMaj11#5add#2add#4	GMaj7#5add#2add4add#4
GMaj11sus6add#2add#4	GMaj7sus6add#2add4add#4	GM#5add_b2add#2add4add#4
GMaj11susb6add#2add#4	GMaj7susb6add#2add4add#4	Gsus6add_b2add#2add4add#4
G11b9add#2add#4		Gsusb6add_b2add#2add4add#4
G11b9#5add#2add#4	G13add#2add#4
G11b9sus6add#2add#4	G13#5add#2add#4	G7susb2add4add#4
G11b9susb6add#2add#4	G13susb6add#2add#4	G7#5susb2add4add#4
GMaj11b9add#2add#4	GMaj13add#2add#4	G7sus2susb6add4add#4
GMaj11b9#5add#2add#4	GMaj13#5add#2add#4	G7susb2susb6add4add#4
GMaj11b9sus6add#2add#4	GMaj13susb6add#2add#4	GMaj7susb2add4add#4
GMaj11b9susb6add#2add#4	G13b9add#2add#4	GMaj7#5susb2add4add#4
G11b13add#2add#4	G13b9#5add#2add#4	GMaj7sus2susb6add4add#4
G11b13#5add#2add#4	G13b9susb6add#2add#4	GMaj7susb2susb6add4add#4
G11b13sus6add#2add#4	GMaj13b9add#2add#4
G11b13susb6add#2add#4	GMaj13b9#5add#2add#4
GMaj11b13add#2add#4	GMaj13b9susb6add#2add#4
GMaj11b13#5add#2add#4	G13#11add#2add#4	G13#11b9#5add#2add#4
GMaj11b13sus6add#2add#4	G13#11#5add#2add#4	G13#11b9sus6add#2add#4
G11b13b9add#2add#4	G13#11sus6add#2add#4	G13#11b9susb6add#2add#4
G11b13b9#5add#2add#4	G13#11susb6add#2add#4	GMaj13#11b9add#2add#4
G11b13b9sus6add#2add#4	GMaj13#11add#2add#4	GMaj13#11b9#5add#2add#4
G11b13b9susb6add#2add#4	GMaj13#11#5add#2add#4	GMaj13#11b9susb6add#2add#4
GMaj11b13b9add#2add#4	GMaj13#11susb6add#2add#4
GMaj11b13b9#5add#2add#4	G13#11b9add#2add#4

Add_b6 Clusters

The final cluster group is the add_b6 cluster, which contains both an added sixth tone and an add_b6. Append add_b6 this cluster on any previous chord with a 6 or 13, eliminating as redundant those already with b6 or #5

Gsus6add_b6	Gmsus6add_b6	Gsus4sus6add_b6	Gsus#4sus6add_b6	Gsus2sus6add_b6
G7sus6add_b6	GMaj7sus6add_b6	G7sus4sus6add_b6	G7sus#4sus6add_b6	G7sus2sus6add_b6
GMaj7sus6add_b6	Gm(Maj7)sus6add_b6	GMaj7sus4sus6add_b6	GMaj7sus#4sus6add_b6	GMaj7sus2sus6add_b6
G9sus6add_b6	Gm9sus6add_b6	G9sus4sus6add_b6	G9sus#4sus6add_b6
GMaj9sus6add_b6	Gm(Maj9)msus6add_b6	GMaj9sus4sus6add_b6	GMaj9sus#4sus6add_b6
G7b9sus6add_b6	GMaj7b9sus6add_b6	G7b9sus4sus6add_b6	G7b9sus#4sus6add_b6	G7sus2b9sus6add_b6
GMaj7b9sus6add_b6	Gm(Maj7)b9sus6add_b6	GMaj7b9sus4sus6add_b6	GMaj7b9sus#4sus6add_b6	GMaj7sus2b9sus6add_b6
G11sus6add_b6	Gm11sus6add_b6	G11sus#4sus6add_b6
GMaj11sus6add_b6	Gm(Maj11)sus6add_b6	GMaj11sus#4sus6add_b6
G9#11sus6add_b6	Gm9#11sus6add_b6	G9#11sus4sus6add_b6	G9sus2#11sus6add_b6
GMaj9#11sus6add_b6	GMaj9m#11sus6add_b6	GMaj9#11sus4sus6add_b6	GMaj9sus2#11sus6add_b6
G7#11b9sus6add_b6	GMaj7#11b9sus6add_b6	G7#11b9sus4sus6add_b6	G7sus2#11b9sus6add_b6
GMaj7#11b9sus6add_b6	Gm(Maj7)#11b9sus6add_b6	GMaj7#11b9sus4sus6add_b6	GMaj7sus2#11b9sus6add_b6
G13add_b6	Gm13add_b6	G13sus#4add_b6	G13susb2add_b6
G13b5add_b6	Gm13b5add_b6	G13b5sus#4add_b6	G13b5susb2add_b6
GMaj13add_b6	Gm(Maj13)add_b6	GMaj13sus#4add_b6	GMaj13susb2add_b6
GMaj13b5add_b6	Gm(Maj13)b5add_b6	GMaj13b5sus#4add_b6	GMaj13b5susb2add_b6
G13b9add_b6	Gm13b9add_b6	G13b9sus#4add_b6	G13sus2b9add_b6
G13b9b5add_b6	Gm13b9b5add_b6	G13b9b5sus#4add_b6	G13sus2b9b5add_b6
G13#11add_b6	Gm13#11add_b6	G13#11sus4add_b6	G13susb2#11add_b6
G13#11b5add_b6	Gm13#11b5add_b6	G13#11b5sus4add_b6	G13susb2#11b5add_b6
G13#11sus6add_b6	Gm13#11sus6add_b6	G13#11sus4sus6add_b6
GMaj13#11add_b6	Gm(Maj13)#11add_b6	GMaj13#11sus4add_b6	GMaj13susb2#11add_b6
GMaj13#11b5add_b6	Gm(Maj13)#11b5add_b6	GMaj13#11b5sus4add_b6	GMaj13susb2#11b5add_b6
G13#11b9add_b6	Gm13#11b9add_b6	G13#11b9sus4add_b6	G13sus2#11b9add_b6
G13#11b9b5add_b6	Gm13#11b9b5add_b6	G13#11b9b5sus4add_b6	G13sus2#11b9b5add_b6
G13#11b9sus6add_b6	Gm13#11b9sus6add_b6	G13#11b9sus4sus6add_b6	G13sus2#11b9sus6add_b6
GMaj13#11b9add_b6	Gm(Maj13)#11b9add_b6	GMaj13#11b9sus4add_b6	GMaj13sus2#11b9add_b6
GMaj13#11b9b5add_b6	Gm(Maj13)#11b9b5add_b6	GMaj13#11b9b5sus4add_b6	GMaj13sus2#11b9b5add_b6
Gsus6add2add_b6	Gmsus6add2add_b6	Gsus4sus6add2add_b6	Gsus#4sus6add2add_b6	Gsus2sus6add2add_b6
Gsus6add_b2add_b6	Gmsus6add_b2add_b6	Gsus4sus6add_b2add_b6	Gsus#4sus6add_b2add_b6	Gsus2sus6add_b2add_b6
Gsus6add4add_b6	Gmsus6add4add_b6		Gsus#4sus6add4add_b6	Gsus2sus6add4add_b6
Gsus6add#4add_b6	Gmsus6add#4add_b6	Gsus4sus6add#4add_b6		Gsus2sus6add#4add_b6
G7sus6add4add_b6	GMaj7sus6add4add_b6	G7sus#4sus6add4add_b6	G7sus2sus6add4add_b6
GMaj7sus6add4add_b6	GMaj7sus6add4add_b6	GMaj7sus#4sus6add4add_b6	GMaj7sus2sus6add4add_b6
G7sus6add#4add_b6	GMaj7sus6add#4add_b6	G7sus4sus6add#4add_b6	G7sus2sus6add#4add_b6	G6susb2add_b6
GMaj7sus6add#4add_b6	GMaj7sus6add#4add_b6	GMaj7sus4sus6add#4add_b6	GMaj7sus2sus6add#4add_b6	G6b5susb2add_b6
Gsus6add2add4add_b6	Gmsus6add2add4add_b6	Gsus#4sus6add2add4add_b6		G67susb2add_b6
Gsus6add_b2add4add_b6	Gmsus6add_b2add4add_b6	Gsus#4sus6add_b2add4add_b6	Gsus2sus6add_b2add4add_b6	G67b5susb2add_b6
Gsus6add2add#4add_b6	Gmsus6add2add#4add_b6	Gsus4sus6add2add#4add_b6	Gsus2sus6add2add#4add_b6	G6Maj7susb2add_b6
Gsus6add_b2add#4add_b6	Gmsus6add_b2add#4add_b6	Gsus4sus6add_b2add#4add_b6	Gsus2sus6add_b2add#4add_b6	G6Maj7b5susb2add_b6
G6add_b6	Gm6add_b6	G6sus4add_b6	G6sus#4add_b6	G6sus2add_b6
G6b5add_b6	Gm6b5add_b6	G6b5sus4add_b6	G6b5sus#4add_b6	G6b5sus2add_b6
G67add_b6	Gm67add_b6	G67sus4add_b6	G67sus#4add_b6	G67sus2add_b6
G67b5add_b6	Gm67b5add_b6	G67b5sus4add_b6	G67b5sus#4add_b6	G67b5sus2add_b6
G6Maj7add_b6	Gm6Maj7add_b6	G6Maj7sus4add_b6	G6Maj7sus#4add_b6	G6Maj7sus2add_b6
G6Maj7b5add_b6	Gm6Maj7b5add_b6	G6Maj7b5sus4add_b6	G6Maj7b5sus#4add_b6	G6Maj7b5sus2add_b6
G69add_b6	Gm69add_b6	G69sus4add_b6	G69sus#4add_b6	G69susb2add_b6
G69b5add_b6	Gm69b5add_b6	G69b5sus4add_b6	G69b5sus#4add_b6	G69b5susb2add_b6
G6Maj9add_b6	Gm(Maj9)add_b6	G6Maj9sus4add_b6	G6Maj9sus#4add_b6	G6Maj9susb2add_b6
G6Maj9b5add_b6	Gm(Maj9)b5add_b6	G6Maj9b5sus4add_b6	G6Maj9b5sus#4add_b6	G6Maj9b5susb2add_b6
G67b9add_b6	GMaj7b9add_b6	G67b9sus4add_b6	G67b9sus#4add_b6	G67sus2b9add_b6
G67b9b5add_b6	GMaj7b9b5add_b6	G67b9b5sus4add_b6	G67b9b5sus#4add_b6	G67sus2b9b5add_b6
G6Maj7b9add_b6	Gm6(Maj7)b9add_b6	G6Maj7b9sus4add_b6	G6Maj7b9sus#4add_b6	G6Maj7sus2b9add_b6
G6Maj7b9b5add_b6	Gm(6Maj7)b9b5add_b6	G6Maj7b9b5sus4add_b6	G6Maj7b9b5sus#4add_b6	G6Maj7sus2b9b5add_b6
G6add2add_b6	Gm6add2add_b6	G6sus4add2add_b6	G6sus#4add2add_b6	G6susb2add2add_b6
G6b5add2add_b6	Gm6b5add2add_b6	G6b5sus4add2add_b6	G6b5sus#4add2add_b6	G6b5susb2add2add_b6
G6add_b2add_b6	Gm6add_b2add_b6	G6sus4add_b2add_b6	G6sus#4add_b2add_b6	G6sus2add_b2add_b6
G6b5add_b2add_b6	Gm6b5add_b2add_b6	G6b5sus4add_b2add_b6	G6b5sus#4add_b2add_b6	G6b5sus2add_b2add_b6
G6add4add_b6	Gm6add4add_b6	G6sus#4add4add_b6	G6sus2add4add_b6
G6b5add4add_b6	Gm6b5add4add_b6	G6b5sus#4add4add_b6	G6b5sus2add4add_b6
G67add4add_b6	Gm67add4add_b6	G67sus#4add4add_b6	G67sus2add4add_b6
G67b5add4add_b6	Gm67b5add4add_b6	G67b5sus#4add4add_b6	G67b5sus2add4add_b6
G6Maj7add4add_b6	Gm6Maj7add4add_b6	G6Maj7sus#4add4add_b6	G6Maj7sus2add4add_b6
G6Maj7b5add4add_b6	Gm6Maj7b5add4add_b6	G6Maj7b5sus#4add4add_b6	G6Maj7b5sus2add4add_b6
Gm6add#4add_b6	G6sus4add#4add_b6	G6sus2add#4add_b6	G6susb2add#4add_b6
Gm6b5add#4add_b6	G6b5sus4add#4add_b6	G6b5sus2add#4add_b6	G6b5susb2add#4add_b6
Gm67add#4add_b6	G67sus4add#4add_b6	G67sus2add#4add_b6	G67susb2add#4add_b6
Gm67b5add#4add_b6	G67b5sus4add#4add_b6	G67b5sus2add#4add_b6	G67b5susb2add#4add_b6
Gm6Maj7add#4add_b6	G6Maj7sus4add#4add_b6	G6Maj7sus2add#4add_b6	G6Maj7susb2add#4add_b6
Gm6Maj7b5add#4add_b6	G6Maj7b5sus4add#4add_b6	G6Maj7b5sus2add#4add_b6	G6Maj7b5susb2add#4add_b6
G6add2add4add_b6	Gm6add2add4add_b6	G6sus#4add2add4add_b6	G6sus2add2add4add_b6
G6b5add2add4add_b6	Gm6b5add2add4add_b6	G6b5sus#4add2add4add_b6	G6b5sus2add2add4add_b6
Gm6add2add#4add_b6	G6sus4add2add#4add_b6	G6sus2add2add#4add_b6	G6susb2add2add#4add_b6
Gm6b5add2add#4add_b6	G6b5sus4add2add#4add_b6	G6b5sus2add2add#4add_b6	G6b5susb2add2add#4add_b6
G6add_b2add4add_b6	Gm6add_b2add4add_b6	G6sus#4add_b2add4add_b6	G6sus2add_b2add4add_b6
G6b5add_b2add4add_b6	Gm6b5add_b2add4add_b6	G6b5sus#4add_b2add4add_b6	G6b5sus2add_b2add4add_b6
Gsus6add#2add_b6	Gsus6add2add#2add_b6	G7sus6add#2add4add_b6	Gsus6add2add#2add4add_b6
G13add#2add_b6	Gsus6add_b2add#2add_b6		Gsus6add_b2add#2add4add_b6
G13b5add#2add_b6	Gsus6add#2add4add_b6	G67b9add#2add_b6	Gsus6add#2add#4add_b6
GMaj13add#2add_b6	G6add#2add_b6	G67b9b5add#2add_b6	G6b5add#2add4add_b6
GMaj13b5add#2add_b6	G6b5add#2add_b6
G13b9add#2add_b6	G67add#2add_b6	G69add#2add_b6
G13b9b5add#2add_b6	G67b5add#2add_b6	G69b5add#2add_b6	G67add#2add4add_b6
GMaj13b9add#2add_b6	G6Maj7add#2add_b6	G6Maj9add#2add_b6	G6Maj7add#2add4add_b6
GMaj13b9b5add#2add_b6	G6Maj7b5add#2add_b6	G6Maj9b5add#2add_b6	G6Maj7b5add#2add4add_b6
G13#11add#2add_b6	G13#11b9add#2add_b6	G6b5add2add#2add_b6	G6add2add#2add4add_b6
G13#11b5add#2add_b6	G13#11b9b5add#2add_b6		G6b5add2add#2add4add_b6
G13#11sus6add#2add_b6	G13#11b9sus6add#2add_b6
GMaj13#11add#2add_b6	GMaj13#11b9add#2add_b6
GMaj13#11b5add#2add_b6	GMaj13#11b9b5add#2add_b6		GMaj7sus2sus6add#6add_b6
GMaj7sus6add#6add_b6	Gm(Maj7)sus6add#6add_b6	GMaj7sus4sus6add#6add_b6	GMaj7sus#4sus6add#6add_b6
GMaj9sus6add#6add_b6	Gm(Maj9)msus6add#6add_b6	GMaj9sus4sus6add#6add_b6	GMaj9sus#4sus6add#6add_b6
GMaj7b9sus6add#6add_b6	Gm(Maj7)b9sus6add#6add_b6	GMaj7b9sus4sus6add#6add_b6	GMaj7b9sus#4sus6add#6add_b6
GMaj11b9sus6add#6add_b6	Gm(Maj11)b9sus6add#6add_b6	GMaj11b9sus#4sus6add#6add_b6	GMaj11sus2b9sus6add#6add_b6
GMaj9#11sus6add#6add_b6	GMaj9m#11sus6add#6add_b6	GMaj9#11sus4sus6add#6add_b6
GMaj7#11b9sus6add#6add_b6	Gm(Maj7)#11b9sus6add#6add_b6	GMaj7#11b9sus4sus6add#6add_b6	GMaj7sus2#11b9sus6add#6add_b6
GMaj13add#6add_b6	Gm(Maj13)add#6add_b6	GMaj13sus#4add#6add_b6	GMaj13susb2add#6add_b6
GMaj13b5add#6add_b6	Gm(Maj13)b5add#6add_b6	GMaj13b5sus#4add#6add_b6	GMaj13b5susb2add#6add_b6
GMaj13b9add#6add_b6	Gm(Maj13)b9add#6add_b6	GMaj13b9sus#4add#6add_b6	GMaj13sus2b9add#6add_b6
GMaj13b9b5add#6add_b6	Gm(Maj13)b9b5add#6add_b6	GMaj13b9b5sus#4add#6add_b6	GMaj13sus2b9b5add#6add_b6
GMaj13#11add#6add_b6	Gm(Maj13)#11add#6add_b6	GMaj13#11sus4add#6add_b6	GMaj13susb2#11add#6add_b6
GMaj13#11b5add#6add_b6	Gm(Maj13)#11b5add#6add_b6	GMaj13#11b5sus4add#6add_b6	GMaj13susb2#11b5add#6add_b6
GMaj13#11b9add#6add_b6	Gm(Maj13)#11b9add#6add_b6	GMaj13#11b9sus4add#6add_b6	GMaj13sus2#11b9add#6add_b6
GMa13#11b9b5add#6add_b6	Gm(Maj13)#11b9b5add#6add_b6	GMaj13#11b9b5sus4add#6add_b6	GMaj13sus2#11b9b5add#6add_b6
GMaj9sus6add_b2add_b6	Gm(Maj9)msus6add_b2add_b6	GMaj9sus4sus6add_b2add_b6	GMaj9sus#4sus6add_b2add_b6
G11sus6add_b2add_b6	Gm11sus6add_b2add_b6	G11sus#4sus6add_b2add_b6
GMaj11sus6add_b2add_b6	Gm(Maj11)sus6add_b2add_b6	GMaj11sus#4sus6add_b2add_b6
G9#11sus6add_b2add_b6	Gm9#11sus6add_b2add_b6	G9#11sus4sus6add_b2add_b6
G13add_b2add_b6	Gm13add_b2add_b6	G13sus#4add_b2add_b6
G13b5add_b2add_b6	Gm13b5add_b2add_b6	G13b5sus#4add_b2add_b6
GMaj13add_b2add_b6	Gm(Maj13)add_b2add_b6	GMaj13sus#4add_b2add_b6
GMaj13b5add_b2add_b6	Gm(Maj13)b5add_b2add_b6	GMaj13b5sus#4add_b2add_b6
G13#11add_b2add_b6	Gm13#11add_b2add_b6	G13#11sus4add_b2add_b6
G13#11b5add_b2add_b6	Gm13#11b5add_b2add_b6	G13#11b5sus4add_b2add_b6
G13#11sus6add_b2add_b6	Gm13#11sus6add_b2add_b6	G13#11sus4sus6add_b2add_b6
GMaj13#11add_b2add_b6	Gm(Maj13)#11add_b2add_b6	GMaj13#11sus4add_b2add_b6
GMaj13#11b5add_b2add_b6	Gm(Maj13)#11b5add_b2add_b6	GMaj13#11b5sus4add_b2add_b6
Gsus6add2add_b2add_b6	Gmsus6add2add_b2add_b6	Gsus4sus6add2add_b2add_b6	Gsus#4sus6add2add_b2add_b6
Gsus6add2add4add_b2add_b6	Gmsus6add2add4add_b2add_b6	Gsus#4sus6add2add4add_b2add_b6
Gsus6add2ad#4add_b2add_b6	Gmsus6add2add#4add_b2add_b6	Gsus4sus6add2add#4add_b2add_b6
G69add_b2add_b6	Gm69add_b2add_b6	G69sus4add_b2add_b6	G69sus#4add_b2add_b6
G69b5add_b2add_b6	Gm69b5add_b2add_b6	G69b5sus4add_b2add_b6	G69b5sus#4add_b2add_b6
G6Maj9add_b2add_b6	Gm(Maj9)add_b2add_b6	G6Maj9sus4add_b2add_b6	G6Maj9sus#4add_b2add_b6
G6Maj9b5add_b2add_b6	Gm(Maj9)b5add_b2add_b6	G6Maj9b5sus4add_b2add_b6	G6Maj9b5sus#4add_b2add_b6
G6add2add_b2add_b6	Gm6add2add_b2add_b6	G6sus4add2add_b2add_b6	G6sus#4add2add_b2add_b6
G6b5add2add_b2add_b6	Gm6b5add2add_b2add_b6	G6b5sus4add2add_b2add_b6	G6b5sus#4add2add_b2add_b6
G6add2add4add_b2add_b6	Gm6add2add4add_b2add_b6	G6sus#4add2add4add_b2add_b6	G6sus2add2add4add_b2add_b6
G6b5add2add4add_b2add_b6	Gm6b5add2add4add_b2add_b6	G6b5sus#4add2add4add_b2add_b6	G6b5sus2add2add4add_b2add_b6
Gm6add2add#4add_b2add_b6	G6sus4add2add#4add_b2add_b6	G6sus2add2add#4add_b2add_b6	GMaj13#11b5add#2add_b2add_b6
Gm6b5add2ad#4add_b2ad_b6	G6b5sus4add2add#4add_b2add_b6	G6b5sus2add2add#4add_b2add_b6	G11sus6add#2add_b2add_b6
G9sus6add#2add_b2add_b6	G11sus6add#2add_b2add_b6		G69add#2add_b2add_b6
G13add#2add_b2add_b6	G9sus6add#2add_b2add_b6	G11sus6add#2add_b2add_b6	G69b5add#2add_b2add_b6
G13b5add#2add_b2add_b6	GMaj9sus6add#2add_b2add_b6	GMaj11sus6add#2add_b2add_b6	G6Maj9add#2add_b2add_b6
GMaj13add#2add_b2add_b6	G9#11sus6add#2add_b2add_b6		GMa7sus2sus6add#6add_b2add_b6
GMa13b5add#2add_b2add_b6	GMaj9#11sus6add#2add_b2add_b6
G13#11add#2add_b2add_b6	Gsus6add2add#2add_b2add_b6	G7sus6add#2add4add_b2add_b6	Gsus6add2ad#2add4add_b2add_b6
GM13#11add#2add_b2add_b6	Gsus6add#2add4add_b2add_b6	G7sus6add#2add#4add_b2add_b6	Gsus6ad2ad#2add#4add_b2add_b6
GM9sus6add#6add_b2add_b6	Gm(Maj9)sus6add#6add_b2add_b6	GMaj9sus4sus6add#6add_b2add_b6	GM9sus#4sus6add#6add_b2add_b6
GM11sus6ad_b7add_b2add_b6	Gm(M11)sus6add#6add_b2add_b6	GM11sus#4sus6ad_b7add_b2add_b6
GMaj13add#6add_b2add_b6	Gm(Maj13)add#6add_b2add_b6	GMaj13sus#4add#6add_b2add_b6
GM13b5add#6add_b2add_b6	Gm(Maj13)b5add#6add_b2add_b6	GMa13b5sus#4add#6add_b2add_b6
GM13#11add#6add_b2add_b6	Gm(Ma13)#11add#6add_b2add_b6	GMa13#11sus4add#6add_b2add_b6	GM13#11sus4sus6ad_b7ad_b2ad_b6
Gsus4sus6add#4add_b6	GMaj7sus4sus6add#4add_b6	Gsus4sus6add2add#4add_b6
G7sus4sus6add#4add_b6	G9sus4sus6add#4add_b6	Gsus4sus6add_b2add#4add_b6
GMaj7sus4sus6add#4add_b6	GMaj9sus4sus6add#4add_b6	Gmsus6add4add#4add_b6
G7sus4sus6add#4add_b6	G7b9sus4sus6add#4add_b6	GMaj7b9sus4sus6add#4add_b6
G11sus6add#4add_b6	Gm11sus6add#4add_b6
G13add#4add_b6	Gm13add#4add_b6	G13susb2add#4add_b6
GMaj13add#4add_b6	Gm(Maj13)add#4add_b6	GMaj13susb2add#4add_b6
G13b9add#4add_b6	Gm13b9add#4add_b6	G13sus2b9add#4add_b6
GMaj13b9add#4add_b6	Gm(Maj13)b9add#4add_b6	GMaj13sus2b9add#4add_b6
G7sus6add4add#4add_b6	GMaj7sus6add4add#4add_b6	G7sus2sus6add4add#4add_b6
GMaj7sus6add4add#4add_b6	GMaj7sus6add4add#4add_b6	GMaj7sus2sus6add4add#4add_b6
Gsus6add2add4add#4add_b6	Gmsus6add2add4add#4add_b6
Gsus6add_b2ad4add#4add_b6	Gmsus6add_b2add4add#4add_b6	Gsus2sus6add_b2add4add#4add_b6
G6sus4add#4add_b6	G69sus4add#4add_b6
G6Maj9sus4add#4add_b6	Gsus6ad_b2ad#2add4add#4add_b6	G6Maj7add4add#4add_b6
G6M9sus4sus6add#4add_b6	G13add#2add#4add_b6	G13#11sus6add#2add#4add_b6
G67b9sus4add#4add_b6	GMaj13add#2add#4add_b6	GMaj13#11add#2add#4add_b6
G67b9sus4sus6add#4add_b6	G13b9add#2add#4add_b6	G13#11b9add#2add#4add_b6
G6Maj7b9sus4add#4add_b6	GMaj13b9add#2add#4add_b6
G6sus4add_b2add#4add_b6	G13#11add#2add#4add_b6	G13#11b9sus6add#2add#4add_b6
G6add4add#4add_b6	Gm6add4add#4add_b6	G6sus2add4add#4add_b6
G6add2add4add#4add_b6	Gm6add2add4add#4add_b6	G6sus2add2add4add#4add_b6
G6add_b2add4add#4add_b6	Gm6add_b2add4add#4add_b6	G6sus2add_b2add4add#4add_b6
G11sus6add#2add#4add_b6	G7sus6add#2add4add#4add_b6	Gsus6add2add#2add4add#4add_b6
GM11sus6add#2add#4add_b6		GMaj7sus6add#2add4add#4add_b6

No 3rd Chords

Consider the chord built on the notes ge_bef. This chord is proba_bly a G67add_b6 with the third scale degree removed, notated G67add_b6(no3rd). In previous chords suspensions removed thirds and replaced them with other tones, but now the only way to represent all of the add_b6 and add#6 chords is by using the (no3rd) tag. Minor, diminished, augmented, and suspended 2nds or 4ths designations with no3rd would be redundant.

Gsus6add_b6(no3rd)	G6add_b6(no3rd)
G7sus6add_b6(no3rd)	G6b5add_b6(no3rd)
GMaj7sus6add_b6(no3rd)	G67add_b6(no3rd)
G9sus6add_b6(no3rd)	G67b5add_b6(no3rd)
GMaj9sus6add_b6(no3rd)	G6Maj7add_b6(no3rd)
G7b9sus6add_b6(no3rd)	G6Maj7b5add_b6(no3rd)
GMaj7b9sus6add_b6(no3rd)	G69add_b6(no3rd)
G11sus6add_b6(no3rd)	G69b5add_b6(no3rd)
GMaj11sus6add_b6(no3rd)	G6Maj9add_b6(no3rd)
G9#11sus6add_b6(no3rd)	G6Maj9b5add_b6(no3rd)
GMaj9#11sus6add_b6(no3rd)	G67b9add_b6(no3rd)
G7#11b9sus6add_b6(no3rd)	G67b9b5add_b6(no3rd)
GMaj7#11b9sus6add_b6(no3rd)	G6Maj7b9add_b6(no3rd)
G13add_b6(no3rd)	G6Maj7b9b5add_b6(no3rd)
G13b5add_b6(no3rd)	GMaj7sus6add#6add_b6(no3rd)
GMaj13add_b6(no3rd)	GMaj9sus6add#6add_b6(no3rd)
GMaj13b5add_b6(no3rd)	GMaj7b9sus6add#6add_b6(no3rd)
G13b9add_b6(no3rd)	GMaj11b9sus6add#6add_b6(no3rd)
G13b9b5add_b6(no3rd)	GMaj9#11sus6add#6add_b6(no3rd)
G13#11add_b6(no3rd)	GMaj7#11b9sus6add#6add_b6(no3rd)
G13#11b5add_b6(no3rd)	GMaj13add#6add_b6(no3rd)
G13#11sus6add_b6(no3rd)	GMaj13b5add#6add_b6(no3rd)
GMaj13#11add_b6(no3rd)	GMaj13b9add#6add_b6(no3rd)
GMaj13#11b5add_b6(no3rd)	GMaj13b9b5add#6add_b6(no3rd)
G13#11b9add_b6(no3rd)	GMaj13#11add#6add_b6(no3rd)
G13#11b9b5add_b6(no3rd)	GMaj13#11b5add#6add_b6(no3rd)
G13#11b9sus6add_b6(no3rd)	GMaj13#11b9add#6add_b6(no3rd)
GMaj13#11b9add_b6(no3rd)	GMa13#11b9b5add#6add_b6(no3rd)
GMaj13#11b9b5add_b6(no3rd)

At this point all possible chords appear in the chart, without inversions, transpositions, or enharmonically spelled chords. A summary for all the chords so far is on the we_bsite http://patternobsession.richardrepp.com

All the Chords

"Check out Guitar George/He knows all the chords." ("Sultans of Swing" by Dire Straits). By "all the cords" Mark Knopfler proba_bly means all the chords traditionally used in jazz, which as shown before, is a great deal. Now twelve tone techniques combined with modern technology for data analysis, a list of literally all the chords based on the chromatic scale is possible.

Using statistical techniques, the number of possible 12 tone combinations without repeated notes computes to 12! (12*11*10*…*2*1) or 479,001,600. For any given tone row, eleven transpositions exist, so divide 479,001,600 by twelve to get the number of unique rows, 39,916,800.

Notes	Combinations	Enharmonics	Total unique
3	220	4	55
4	495	3	165
5	792	2 groups of 5	330
6	924	2	462
7	792	1	462
8	495	1	330
9	220	1	220
10	66	1	55
11	12	2	11
12	1	1	1
			&nbnbsp;

Total 4017 chord combinations, with 2091 being unique, plus the diminished seventh.

Chord Spelling

The list started with chord spellings, but did not continue all the way to the end to save space. Instead of going through and spelling each chord one by one, technology can aid in finding all the combinations.

Step 1. First list all the combinations of notes taken three at a time.

g, g#, a	g, g#, a, a#	g, g#, a, a#, b	g, g#, a, a#, b, c	…
g, g#, a#	g, g#, a, b	g, g#, a, a#, c	g, g#, a, a#, b, c#
g, g#, b	g, g#, a, c	g, g#, a, a#, c#	g, g#, a, a#, b, d
g, g#, c	g, g#, a, c#	g, g#, a, a#, d	g, g#, a, a#, b, d#
g, g#, c#	g, g#, a, d	g, g#, a, a#, d#	g, g#, a, a#, b, e
g, g#, d	g, g#, a, d#	g, g#, a, a#, e	g, g#, a, a#, b, f
g, g#, d#	g, g#, a, e	g, g#, a, a#, f	g, g#, a, a#, b, f#
g, g#, e	g, g#, a, f	g, g#, a, a#, f#	g, g#, a, a#, c, c#
g, g#, f	g, g#, a, f#	g, g#, a, b, c	g, g#, a, a#, c, d
g, g#, f#	g, g#, a#, b	g, g#, a, b, c#	g, g#, a, a#, c, d#
g, a, a#	g, g#, a#, c	g, g#, a, b, d	g, g#, a, a#, c, e
g, a, b	g, g#, a#, c#	g, g#, a, b, d#	g, g#, a, a#, c, f
g, a, c	g, g#, a#, d	g, g#, a, b, e	g, g#, a, a#, c, f#
g, a, c#	g, g#, a#, d#	g, g#, a, b, f	g, g#, a, a#, c#, d
		continues	…

This list gives all the combinations possible. Several online sites will combine items automatically, so coding is not necessary. The above sort is for the group [gg#aa#bcc#dd#eff#]. These combinations are possible, but not very useful musically. Instead using the set [gbdface Bb Db F# Ab Eb] produces a much more musical result.

The groups are much easier to sort than the previous chord list also. If the last sorts by size of cell, all the four note chord group together, and the same for the higher chords. A few tweaks made the final sort more musically appropriate on the first column of notes based on the major triad. The list sort does well for major chords, but the minor chords are scattered around the combinations. The minors need to be resorted after the chord symbols go in.

Adding the chord symbols takes time, but since the patterns grow sequentially, the process is much faster than if the chords were not in order. The final list is a much more organized listing of chords, with fewer mistakes, and this list has the spellings also.

Available on http://patternobsession.richardrepp.com

Inversions

So, far most of the chords have been in root position with G as the root note and also the bass note (lowest note in the chord). Chords lay also be inverted, with the bass note being different than the root and the root somewhere else in the chord. Inversions may happen on any chord tone from third to thirteenth, including suspensions treated as chord tones. For example, in the G13 chord the first inversion begins on the third chord tone, b, the second inversion on the next chord tone.

Bass note	g	b	d	f	a	c	e
Inversion		1st	2nd	3rd	4th	5th	6th
Symbol	G13	G13/B	G13/D	G13/F	G13/A	G13/C	G13/E

Chords with altered notes follow the same pattern, so GMaj7#11b5 would be

Bass	g	b_b	d_b	f	a	c#	e
Inv.		1st	2nd	3rd	4th	5th	6th
Symbol	GMaj7#11b5	GMaj7#11b5/Bb	GMaj7#11b5/Db	GMaj7#11b5/F	GMaj7#11b5/A	GMaj7#11b5/C#	G13/G

In a suspended chord, the suspension takes the place of the note being replaced, so sus2 would be

Bass note	g	c	d	f	a	c	e
Inversion		1st	2nd	3rd	4th	5th	6th
Symbol	G13sus2	G13sus2/C	G13sus2/D	G13sus2/F	G13sus2/A	G13sus2/C	G13sus2/E

For add notes, the chord is technically not an inversion, but a passing tone in the bass. However, for notational purposes, they seem the same. So, the chord Gadd2/A would have the A in the bass and the other tones stacked above in any order, agbd, for example. Note that agbd is also A7sus4add2, so the inversions have chord tones with the same spelling. Whether to use Gadd2/A or A7sus4add2 would depend on context.

G Bass Inversions

Calculating all the inversions one by one would be prohibitive, even with the aid of technology. If the goal is producing all possible suspensions, then a better method is to find all the suspensions with g as the bass (not root). The result will be all the /G chords, which then can easily transpose into any key.

So, in first inversion working with the G chord, the put the G as the bass note, and then figure out what chord has G as the third, ?/G. In this case, the chord with G as its major third is Eb, so the result is Eb/G.

The beauty of this method is that any chord with a major third follows the same pattern. So, a G7#11b5 chord has a corresponding chord of Eb7#11b5/G. Remember that Eb7#11b5/G is not the first inversion of G7#11b5, but just a transitional step, a chord with the same color so that all possibilities compute mathematically for easy transposition later.

Continuing for the 5^th, find a chord whose fifth is g, the C chord. Use the chord C/G as a working bass. Now any chord with an unaltered fifth corresponds to the C/G chord with the same altered notes. For example, GMaj7#11b5 has a corresponding substitute of CMaj7#11b5/G.

For the fourth inversion, find the chord where g is the seventh, A7/G.

Completing the pattern:

Inversion		1st	2nd	3rd	4th	5th	6th
Substitute	(G)	Eb/G	C/G	A7/G	F9/G	D11/G	Bb13/G

(Side note: The pattern of substitutions follows a reversed BbMaj13 chord, which could be an alternate substitute for the final chord.)

First inversions appear on Sus2 and Sus4 chords, with second inversions on Sus6.

Inversion	1st	1st	2nd
Substitute	Asus2/G	Dsus4/G	Bbsus6/G

Altered chord tone follow a similar pattern. With the m3, G is the minor third of Em, Em/G. G is the b5 of C#, C#o/G and so on.

1st	2nd	3rd	4th	5th	6th
Em/G	C#o/G	AbMaj7/G	F#7b9/G	C#11b9/G	B11b13/G
m3	d5	Maj7	b9	#11	b13

1st	1st	2nd
F#susb2/G	Dbsus#4/G	Bbsusb6/G
susb2	sus#4	susb6

Now use software to substitute the correct corresponding chords to their counterparts. An excel function that searches for each possibility of first inversion chords (here starting in the cell A1) would be

=IF(ISERROR(FIND("sus#4",A1)=TRUE),

IF(ISERROR(FIND("susb2",A1)=TRUE),

IF(ISERROR(FIND("sus2",A1)=TRUE),

IF(ISERROR(FIND("sus4",A1)=TRUE),

IF(ISERROR(FIND("m",A1)=TRUE),

CONCATENATE(SUBSTITUTE(A1,"G","Eb"),"/G"),

CONCATENATE(SUBSTITUTE(A1,"G","E"),"/G")),

CONCATENATE(SUBSTITUTE(A1,"G","D"),"/G")),

CONCATENATE(SUBSTITUTE(A1,"G","A"),"/G")),

CONCATENATE(SUBSTITUTE(A1,"G","F#"),"/G")),

CONCATENATE(SUBSTITUTE(A1,"G","Db"),"/G"))

Because of the nature of nested if statements in Excel, the first substitution for sus#4 actually happens on the last line, then the others follow in reverse order. If no alterations appear, the substitution reverts to a default of major.

Once completed the formula applies to all 3600+ root chords, and then is adaptable to the other inversions. (Note that No3rd chords do not have a true first inversion, so do not appear.)

The list of inversions has too many rows for this document, so is in a separate table on http://patternobsession.richardrepp.com

Second Inversion Chords

In the second inversion, the following roots are possible, including the previously unanalyzed b5 and #5.

5	b5	#5	sus6	susb6
C/G	G#Maj(b5)	GbMaj#5/G	Bbsus6/G	Bsusb6/G

Leading to the substitutions

=IF(ISERROR(FIND("b5",A1)=TRUE),

IF(ISERROR(FIND("b7",A1)=TRUE),

IF(ISERROR(FIND("sus6",A1)=TRUE),

IF(ISERROR(FIND("susb6",A1)=TRUE),

CONCATENATE(SUBSTITUTE(A1,"G","C"),"/G"),

CONCATENATE(SUBSTITUTE(A1,"G","B"),"/G")),

CONCATENATE(SUBSTITUTE(A1,"G","Bb"),"/G")),

CONCATENATE(SUBSTITUTE(A1,"G","Gb"),"/G")),

CONCATENATE(SUBSTITUTE(A1,"G","G#"),"/G"))

No5th chords do not have a second inversion.

Third Inversion Chords

When programming for third inversion chords, three possibilities are, 7, Maj7, and no seventh (o7 is in the next section). Sevenths exist in 7^th, 9^th, 11^th, and 13^th chords. The following script checks for 7, 9, 11, 1nd 13, and f none of those numbers exist, then returns a blank cell. If any of those chord tines do exist, the script checks if they are Maj, and if so substitutes AbMaj7/G, and if not, A7/G. (A more efficient program could avoid the repeated lines.)

=IF(ISERROR(FIND("7",A1)=TRUE),

IF(ISERROR(FIND("9",A1)=TRUE),

IF(ISERROR(FIND("11",A1)=TRUE),

IF(ISERROR(FIND("13",A1)=TRUE),"",

IF(ISERROR(FIND("Maj",A1)=TRUE), CONCATENATE(SUBSTITUTE(A1,"G","A"),"/G"), CONCATENATE(SUBSTITUTE(A1,"G","Ab"),"/G"))),

IF(ISERROR(FIND("Maj",A1)=TRUE), CONCATENATE(SUBSTITUTE(A1,"G","A"),"/G"), CONCATENATE(SUBSTITUTE(A1,"G","Ab"),"/G")))

Diminished Seventh

As always, the diminished seventh chord is a special case as its symmetrical nature renders inversions moot. A Go7 chord, GBbDbFb in first inversion is jus BbDbFbG, which is just another diminished seventh chord with the same notes, Bbo7/G. Similarly, the other inversions are Dbo7/G and Fbo7/G, along with any alternate spellings, but these are all essentially the same chord.

Higher Inversions

Coding for the higher inversions becomes simpler each time, with one less chord family to check.

9th, 11th, 13th	9	b9	A9/G	F# 7b9/G
11th, 13th	11	#11	Cb 9#11	D11/G
13	13	b13	Bb 13/G	B11b13/G

(The coding is essentially the same as above with one less line each time.)

Bass Passing Tones

Add notes as passing tones in the bass follow the same patterns as the altered chord tones, however they can occur with any chord, so numbering the inversions would not apply. A Gadd2 chord is still based in F just as the G9 chord was, Fadd2/G but now the F is no longer required to apply only to ninth chords and higher.

The additional atypical added tones of add#2 and add#6 yield chords Fbadd#2/G and Bb_badd#6/G. The Bb_badd#6/G chord would more likely appear as enharmonic A7/G.

F#add_b2/G	Fadd2/G	Fbadd#2/G	Cadd4/G	Cbadd#4/G	Badd_b6/G	Bb6/G	Bb_badd#6/G
add_b2	add2	add#2	add4	add#4	add_b6	6	add#6

At this point, all chords with bass note G including inversions appear in the chart. This is essentially a list of every possible chord type.

See the we_b site http://patternobsession.richardrepp.com for a full list.

Mapping to G

Using G as a root note helped avoid calculating each inversion separately, but the result is unappealing musically because the chords with different root notes are difficult to compare. The next step is mapping all the chords to a G root. Here are the present chords compared to a more musically desirable set with the same symbols:

Root	b2	2	#2	M3	m3	4	#4
G	F#add2b2/G	Fadd2/G	Fbadd2/G	Eb/G	Em/G	Dadd4/G	Dbadd#4/G
G	Gadd_b2/Ab	Gadd2/A	Gadd2/A#	G/B	Gm/Bb	Gadd4/C	Gadd#4/C#

5	b5	#5	6	b6	b7	7	Maj7
C/G	C#o/G	Cb+/G	Bb6/G	Badd_b6/G	Bb_badd#6	A7/G	AbMaj7/G
G/D	Go/Db	G+/D#	G6/E	Gadd_b6/Eb	Gadd#6/E#	G7/F	GMaj7/F#

For the purposes of the translation, all that needs to change are the root notes and the bass note, so suspensions and other chords have equivalent translations. Since none of the root notes repeat, just search for each root note, substitute the appropriate new bass note, and change the root to G. Code:

=IF(A1="","",

IF(ISERROR(FIND("F#",A1)=TRUE),

IF(ISERROR(FIND("Fb",A1)=TRUE),

IF(ISERROR(FIND("F",A1)=TRUE),

IF(ISERROR(FIND("Eb",A1)=TRUE),

IF(ISERROR(FIND("E",A1)=TRUE),

IF(ISERROR(FIND("Db",A1)=TRUE),

IF(ISERROR(FIND("D",A1)=TRUE),

IF(ISERROR(FIND("C#",A1)=TRUE),

IF(ISERROR(FIND("Cb",A1)=TRUE),

IF(ISERROR(FIND("C",A1)=TRUE),

IF(ISERROR(FIND("Bb_b",A1)=TRUE),

IF(ISERROR(FIND("Bb",A1)=TRUE),

IF(ISERROR(FIND("B",A1)=TRUE),

IF(ISERROR(FIND("Ab",A1)=TRUE),

IF(ISERROR(FIND("A",A1)=TRUE),

A1,

SUBSTITUTE(SUBSTITUTE(A1,"A","G"),"/G","/F")),

SUBSTITUTE(SUBSTITUTE(A1,"Ab","G"),"/G","/F#")),

SUBSTITUTE(SUBSTITUTE(A1,"B","G"),"/G","/Eb")),

SUBSTITUTE(SUBSTITUTE(A1,"Bb","G"),"/G","/E")),

SUBSTITUTE(SUBSTITUTE(A1,"Bb_b","G"),"/G","/E#")),

SUBSTITUTE(SUBSTITUTE(A1,"C","G"),"/G","/D")),

SUBSTITUTE(SUBSTITUTE(A1,"Cb","G"),"/G","/D#")),

SUBSTITUTE(SUBSTITUTE(A1,"C#","G"),"/G","/Db")),

SUBSTITUTE(SUBSTITUTE(A1,"D","G"),"/G","/C")),

SUBSTITUTE(SUBSTITUTE(A1,"Db","G"),"/G","/C#")),

SUBSTITUTE(SUBSTITUTE(A1,"E","G"),"/G","/Bb")),

SUBSTITUTE(SUBSTITUTE(A1,"Eb","G"),"/G","/B")),

SUBSTITUTE(SUBSTITUTE(A1,"F","G"),"/G","/A")),

SUBSTITUTE(SUBSTITUTE(A1,"Fb","G"),"/G","/A#")),

SUBSTITUTE(SUBSTITUTE(A1,"F#","G"),"/G","/Ab")))

Sorting

This formula outputs all the chords with root G, but the order is alpha_betical rather than musical. Sort the list according to musical preference, typically similar to the way this method introduced the chords. Now the list is quite useful, proba_bly the most representative of all the lists for comparing the various chords and inversions.

Complete list on http://patternobsession.richardrepp.com

Transpositions

Now that the list of all chords with a single root is finished, the last step is to transpose that list into all keys. As learned in the section on key signatures, the term "all keys" phrase is misleading, because theoretically an infinite number of keys exist if multiple sharps and flat occur. The chart of extended key signatures is quite useful here in automating the key changes using patterns.

fb	g_b	a_b	b_bb	c_b	d_b	e_b
c_b	d_b	e_b	fb	g_b	a_b	b_b
g_b	a_b	b_b	c_b	d_b	e_b	f
d_b	e_b	f	g_b	a_b	b_b	c
a_b	b_b	c	d_b	e_b	f	g
e_b	f	g	a_b	b_b	c	d
b_b	c	d	e_b	f	g	a
f	g	a	b_b	c	d	e
c	d	e	f	g	a	b
g	a	b	c	d	e	f#
d	e	f#	g	a	b	c#
a	b	c#	d	e	f#	g#
e	f#	g#	a	b	c#	d#
b	c#	d#	e	f#	g#	a#
f#	g#	a#	b	c#	d#	e#
c#	d#	e#	f#	g#	a#	b#
g#	a#	b#	c#	d#	e#	fx
d#	e#	fx	g#	a#	b#	cx
a#	b#	cx	d#	e#	fx	gx
e#	fx	gx	a#	b#	cx	d#
b#	cx	d#	e#	fx	gx	ax

The list shows 21 possible keys. To find a matrix to calculate all possible chords in each of the keys, first compare to the tendencies of the G chords, as all will be the same.

Maj7

The chord tones for the transpositions will match the chord tones from the scales already calculated, so just drop those into the correct places.

Maj7

g_b

a_b

b_bb

c_b

d_b

e_b

c_b

d_b

e_b

g_b

a_b

b_b

g_b

a_b

b_b

c_b

d_b

e_b

d_b

e_b

g_b

a_b

b_b

a_b

b_b

d_b

e_b

a_b

b_b

e_b

b_b

Note that the seventh scale degree belongs in the Maj7 slot, not 7.

Filling in the altered tones is a bit more difficult, but using the table lessens the need for manual transposition. First, find the comparison row g and fill in the appropriate notes.

a_b

d_b

The take the altered notes one by one and fill in their circles of fifth so that the columns and the rows align.

g_b

a_b

b_bb

c_b

d_b

e_b

c_b

d_b

e_b

g_b

a_b

b_bb

b_b

g_b

a_b

b_b

c_b

d_b

e_b

d_b

e_b

g_b

a_b

b_b

c_b

a_b

b_b

d_b

e_b

g_b

e_b

a_b

b_b

d_b

b_b

e_b

a_b

b_b

e_b

b_b

This process almost works, but leave a space at the top and adds an extra row at the bottom. The row at the bottom would be the Maj7 of a key not in the range (Fx), so it can be deleted. The empty space in the first row is filled by the note a P4 above the note just below it, or Eb_b. F is the easiest column to compute, so with the other columns, use an extension of the circle of fifths.

Take the circle of fifths and extend it another octave above and below. Calculation of each cell is not necessary, as the pattern continues the same as the notes below, just with triple flats and sharps when necessary. So, the first column extends to

fb_b

c_bb

g_bb

d_bb

a_bb

e_bb

b_bb

c_b

(…)

With all the extensions being

f_bb	g_bb	a_bb	b_bbb	c_bb	d_bb	e_bb
c_bb	d_bb	e_bb	f_bb	g_bb	a_bb	b_bb
g_bb	a_bb	b_bb	c_bb	d_bb	e_bb	fb
d_bb	e_bb	fb	g_bb	a_bb	b_bb	c_b
a_bb	b_bb	c_b	d_bb	e_bb	fb	g_b
e_bb	fb	g_b	a_bb	b_bb	c_b	d_b
b_bb	c_b	d_b	e_bb	fb	g_b	a_b
fb	g_b	a_b	b_bb	c_b	d_b	e_b
c_b	d_b	e_b	fb	g_b	a_b	b_b
g_b	a_b	b_b	c_b	d_b	e_b	f
d_b	e_b	f	g_b	a_b	b_b	c
a_b	b_b	c	d_b	e_b	f	g
e_b	f	g	a_b	b_b	c	d
b_b	c	d	e_b	f	g	a
f	g	a	b_b	c	d	e
c	d	e	f	g	a	b
g	a	b	c	d	e	f#
d	e	f#	g	a	b	c#
a	b	c#	d	e	f#	g#
e	f#	g#	a	b	c#	d#
b	c#	d#	e	f#	g#	a#
f#	g#	a#	b	c#	d#	e#
c#	d#	e#	f#	g#	a#	b#
g#	a#	b#	c#	d#	e#	fx
d#	e#	fx	g#	a#	b#	cx
a#	b#	cx	d#	e#	fx	gx
e#	fx	gx	a#	b#	cx	dx
b#	cx	d#	e#	fx	gx	ax
fx	gx	d#	e#	cx	dx	ex
cx	dx	a#	b#	gx	ax	bx
gx	ax	e#	fx	dx	ex	fx#
dx	ex	b#	cx	ax	bx	cx#
ax	bx	fx	gx	ex	fx#	gx#
ex	fx#	cx	dx	bx	cx#	dx#
bx	cx#	gx	ax	fx#	gx#	ax#

Now the columns have enough space for all the columns of the transposition map. (Hint: Read the map from the middle rows and work out rather than looking at all the b_bs in the first row.)

Maj7

g_bb

g_b

a_b

a_bb

b_bb

b_b

c_b

c_bb

d_b

d_bb

e_bb

e_b

c_b

d_bb

d_b

e_b

e_bb

g_b

g_bb

a_b

d_bb

b_bb

b_b

g_b

a_bb

a_b

b_b

b_bb

c_b

d_b

d_bb

e_b

a_bb

d_b

e_bb

e_b

g_b

a_b

a_bb

b_b

e_bb

c_b

a_b

b_bb

b_b

c_b

d_b

e_b

e_bb

b_bb

g_b

e_b

g_b

a_b

b_b

b_bb

d_b

b_b

c_b

d_b

e_b

c_b

a_b

g_b

a_b

b_b

c_b

g_b

e_b

d_b

e_b

g_b

d_b

b_b

a_b

b_b

d_b

a_b

e_b

a_b

e_b

b_b

e_b

b_b

fx#

cx#

fx#

Once the map is complete, use these arrays to substitute the transposed column for the reference column. (The coding would not make sense out of context because it refers to variable rows.)

The complete list of 490,959 chords.

Parallel Modes

Previous discussion of modes showed them in their context within their relative modes, C Ionian with D Dorian with E Phrygian and so on. Jazz players use this context for soloing. In a I-IV-V-I chord progression in C, the player uses the modes C Ionian-F Lydian-G Mixolydian-C Ionian. All the Greek makes the process sound much more difficult that the practice, which is just using the same seven notes but highlighting different tones to match the chords underneath. In Ionian, c,e, and g would be the anchor notes to begin and end phrases with, in Lydian, f, a, c, ad Mixolydian, b, d, and g.

Transposing the modes to the same root note allows comparison of the different tones.

The relative modes in C are

C Ionian	c	d	e	f	g	a	b
D Dorian	d	e	f	g	a	b	c
E Phrygian	e	f	g	a	b	c	d
F Lydian	f	g	a	b	c	d	e
G Mixolydian	g	a	b	c	d	e	f
A Aeolian	a	b	c	d	e	f	g
B Locrean	b	c	d	e	f	g	a

C Ionian is the major scale, which remains constant.

D Dorian transposed down a step becomes C Dorian

e_b

b_b

Which is a C Minor scale with a raised sixth since a_b is the expected note in C minor.

E Phrygian transposed down two steps yields C Phrygian

d_b

e_b

a_b

b_b

Which is the C minor scale with a lowered second.

F Lydian transposed down a perfect fourth becomes C Lydian

The major scale with a raised fourth degree.

G Mixolydian down a perfect fifth becomes

b_b

the major scale with a lowered seventh.

A Aeolian down a major sixth is C Aeolian, or C minor

e_b

a_b

b_b

and, finally B Locrean becomes C Locrean

d_b

e_b

g_b

a_b

b_b

Which resembles a minor scale with a lowered second and a lowered (diminished) fifth.

Harmonic Minor Mode Equivalents

In the a harmonic minor scale, the seventh of the scale degree raised to g# yielding

This scale transposed to C Harmonic Minor yields

e_b

a_b

Harmonic minor is not a traditional modal scale, but treat the patterns as if they were modal yields some interesting scales. For example, taking the second degree of harmonic minor and filling in the scale yields this pattern

Which is similar to the Locrean mode with a raised sixth . This scale transposed into C is

d_b

e_b

g_b

b_b

Beginning the A Harmonic minor scale on the third tone yields

The major scale with an augmented fifth. This scale is already in C so needs no transposition for comparison. Continuing the scale on the fourth, fifth, and sixth degrees yields

d		e	f			g#	a		b	c
e	f			g#	a		b	c		d
f			g#	a		b	c		d		e

Respectively

D Dorian Raised Fourth

E Phrygian Raised Third

F Lydian Raised Second

And then transposed into the C parallel versions,

C Dorian #4

e_b

b_b

C Phrygian ♮3

d_b

a_b

b_b

C Lydian #2

(Note that the ♮3 designation on the Phrygian mode raises the usually lowered third to the natural position. The symbol #3 would be interpreted as an augmented third e#.)

The final scales break the modal pattern comparisons down. Following the previous pattern, an altered Mixolydian chord should come next, but that chord would have a raised root. The notes

Do not resemble a Mixolydian chord is any way, so referring to this scale as an altered Mixolydian is not satisfying. Functionally, the scale relates to a natural minor scale the way a Locrean scale relates to a major scale, so a borrowed Locrean designation with a lowered fourth and a lowered seventh might make sense. However, this designation is confusing since it breaks the pattern. Instead observe the seventh chord built on this scale,

g# b d f

This is a fully diminished seventh chord, so this text refers to the scale as the Diminished mode. Transposed into C yields

C Diminished Mode

d_b

e_b

g_b

a_b

b_bb

Melodic Minor Mode Ascending Equivalents

In the a melodic minor scale ascending, the seventh of the scale degree raised to g# yielding

Transposing to C yields C Melodic Minor

e_b

The scale built on the second degree of this scale and then transposed into C would be

d_b

e_b

b_b

This scale could be a Dorian scale with a lowered second, but that analysis breaks the pattern of Locrean scales usually holding that position. A more consistent analysis would be as a Locrean scale with a raised sixth and a raised fifth.

The same technique continuing up the next three scale degrees yields

Ionian #5 #4

Dorian #4 ♮3

b_b

Phrygian ♮3 #2

a_b

b_b

The scales built on the fourth and fifth degrees have the issue of the raised root breaking the pattern of the traditional mode equivalent analysis.

c		d	e_b		f	g_b		a_b		b_b
c	d_b		e_b	fb		g_b		a_b		b_b

For the harmonic minor, the term Diminished Mode substituted for Mixolydian because that scale contained the fully diminished chord. These new scales contain the half diminished chord c e_b g_b b_b, so they could have an analysis of half diminished modes. To be consistent with the previous analysis, the chord in the typical Mixolydian position is the Half Diminished made and the chord in the typical Lydian position is half diminished with a natural 4^th. (The natural raises the lowered fourth from the half diminished mode.)

The following chart shows all the parallel modes compared, with common enharmonic tones aligned regardless of scale degree.

Ionian

Dorian

e_b

b_b

Phrygian

d_b

e_b

a_b

b_b

Lydian

Mixolydian

b_b

Aeolian

e_b

a_b

b_b

Locrean

d_b

e_b

g_b

a_b

b_b

Altered for Harmonic

Ionian #5

Dorian #4

e_b

b_b

Phrygian ♮3

d_b

a_b

b_b

Lydian #2

Diminished

d_b

e_b

g_b

a_b

b_bb

Harmonic Minor

e_b

a_b

Locrean b7

d_b

e_b

g_b

b_b

Altered Twice for Melodic

Ionian #5 #4

Dorian #4 ♮3

b_b

Phrygian ♮3 #2

a_b

b_b

Half Diminished ♮4

e_b

g_b

a_b

b_b

Half Diminished

d_b

e_b

g_b

a_b

b_b

Melodic Minor

e_b

Locrean b7 #5

d_b

e_b

b_b

Parallel Mode Chords

Previous analysis showed the modal chords based on the major scale to be

C	CMaj7	CM9	CM11	CM13
Dm	Dm7	Dm9	Dm11	Dm13
Em	Em7	Em7b9	Em11b9	Em11b13b9
F	F7	FM9	FM9#11	FM13#11
G	G7	G9	G11	G13
Am	Am7	Am9	Am11	Am13
Bo	Bm7b5	Bm7b9b5	Bm11b9b5	Bm11b13b9b5

The analysis of the modal equivalent chords for A Harmonic Minor (rearranged to put C at the top)

C+	CMaj7#5	CMaj9#5	CMaj11#5	CMaj13#5
Dm	Dm7	Dm9	Dm9#11	Dm13#11
E	E7	E7b9	E11b9	E11b13b9
F	FMaj7	FMaj7#9	FMaj7#9#11	FMaj13#9#11
G#o	G#o7	G#Maj7b9b5	G#m11b9b5	G#m13b9b5
Am	Am(Maj7)	Am(Maj9)	Am(Maj11)	Am(Maj11)b13
Bo	Bm7b5	Bm7b9b5	Bm11b9b5	Bm11b13b9b5

And A Melodic Minor rearranged

C+	CMaj7#5	CM9#5	CM9#11#5	CM13#11#5
D	D7	D9	D9#11	D13#11
E	E7	E9	E11	E11b13
F#o	F#m7b5	F#m9b5	F#m11b5	F#m11b13b5
G#o	G#m7b5	G#m7b57b9	G#m11b9b5	G#m11b13b9b5
Am	Am(Maj7)	Am(Maj9)	Am(Maj11)	Am(Maj13)
Bm	Bm7	Bm7b9	Bm11b9	Bm13b9

Finding the relative modal equivalents for these new scales is simple. To find the relative modal equivalent based on the second scale degree, just take the top row and move it to the bottom.

Dm	Dm7	Dm9	Dm11	Dm13
Em	EMaj7	EMaj7b9	Em11b9	Em11b13b9
F	F7	FM9	FM9#11	FM13#11
G	G7	G9	G11	G13
Am	Am7	Am9	Am11	Am13
Bo	Bm7b5	Bm7b9b5	Bm11b9b5	Bm11b13b9b5
C	CMaj7	CM9	CM11	CM13

This group transposed down to C for comparison purposes is

Cm	Cm7	Cm9	Cm11	Cm13
Dm	Dm7	DMaj7b9	Dm11b9	Dm11b13b9
Eb	Eb7	EbMaj9	EbMaj9#11	EbMaj13#11
F	F7	F9	F11	F13
Gm	Gm7	Gm9	Gm11	Gm13
Ao	Am7b5	A7b9b5	Am11b9b5	Am11b13b9b5
Bb	BbMaj7	BbMaj9	BbMaj11	BbMaj13

This group comes from the Dorian patterns, but using the Greek modes would be confusing and inaccurate, so label this group as the second scale degree base.

Then to convert to the parallel version, just treat the D Minor as like an Ionian mode, the E minor like Dorian, et cetera. The Greek mode names no longer apply; instead label the group as having a second-degree base.

The third-degree base, starting on E minor from the Phrygian position then transposed to C, would be

Cm	Cm7	Cm7b9	Cm11b9	Cm11b13b9
Db	DbMaj7	DbMaj9	DbMaj9#11	DbMaj13#11
Eb	Eb7	Eb9	Eb11	Eb13
Fm	Fm7	Fm9	Fm11	Fm13
Go	Gm7b5	Gm7b9b5	Gm11b9b5	Gm11b13b9b5
Ab	AbMaj7	AbMaj9	AbMaj11	AbMaj13
Bbm	Bbm7	Bbm9	Bbm11	Bbm13

For the Harmonic minor equivalent shown above, the G# changes the chords. Rearranging this group leads to some interesting chords, particularly in the modes based on the fifth scale degree G#.

G#o	G#o7	G#m7b9b5	G#m11b9b5	G#m13b9b5
Am	Am(Maj7)	Am(Maj9)	Am(Maj11)	Am(Maj11)b13
Bo	BMaj7b5	BMaj7b9b5	Bm11b9b5	Bm11b13b9b5
C+	CMaj7#5	CMaj9#5	CMaj11#5	CMaj13#5
Dm	Dm7	Dm9	Dm9#11	Dm13#11
E	E7	E7b9	E11b9	E11b13b9
F	FMaj7	FMaj7#9	FMaj7#9#11	FMaj13#9#11

Transposed to C for comparison.

Co	Co7	Cm7b9b5	Cm11b9b5	Cm13b9b5
Dbm	Dbm(Maj7)	Dbm(Maj9)	Dbm(Maj11)	Dbm(Maj11)b13
Ebo	Ebm7b5	Ebm7b9b5	Ebm11b9b5	Ebm11b13b9b5
E+	EMaj7#5	EMaj9#5	EMaj11#5	EMaj13#5
F#m	F#m7	F#m9	F#m9#11	F#m13#11
Ab	Ab7	Ab7b9	Ab11b9	Ab11b13b9
Bb_b	Bb_bMaj7	Bb_bMaj7#9	Bb_bMaj7#9#11	Bb_bMaj13#9#11

The Melodic minor modal equivalents also produce new chords. The Melodic fourth scale degree modal equivalents

F#o	F#m7b5	F#m9b5	F#m11b5	F#m11b13b5
G#o	G#m7b5	G#m7b57b9	G#m11b9b5	G#m11b13b9b5
Am	Am(Maj7)	Am(Maj9)	Am(Maj11)	Am(Maj13)
Bm	Bm7	Bm7b9	Bm11b9	Bm13b9
C+	CMaj7#5	CM9#5	CM9#11#5	CM13#11#5
D	D7	D9	D9#11	D13#11
E	E7	E9	E11	E11b13

Translate to

Co	Cm7b5	Cm9b5	Cm11b5	Cm11b13b5
Do	Dm7b5	Dm7b57b9	Dm11b9b5	Dm11b13b9b5
Ebm	Ebm(Maj7)	Ebm(Maj9)	Ebm(Maj11)	Ebm(Maj13)
Fm	Fm7	Fm7b9	Fm11b9	Fm13b9
Gb+	GbMaj7#5	GbMaj9#5	GbMaj9#11#5	GbMaj13#11#5
Ab	Ab7	Ab9	A#11b9	Ab13#11
Bb	Bb7	Bb9	Bb11	Bb11b13

The complete set of chords is in the document Parallel Chords based on All Modes and Alterations."

These chords families are useful, but for comparison purposes, extracting the notes with the same scale degree yields

Scale Degree Base
First	C	CMaj7	CMaj9	CMaj11	CMaj13
Second	Cm	Cm7	Cm9	Cm11	Cm13
Third	Cm	Cm7	Cm7b9	Cm11b9	Cm11b13b9
Fourth	C	C7	CMaj9	CMaj9#11	CMaj13#11
Fifth	C	C7	C9	C11	C13
Sixth	Cm	Cm7	Cm9	Cm11	Cm13
Seventh	Co	Cm7b5	Cm7b9b5	Cm11b9b5	Cm11b13b9b5
First Harmonic	C+	CMaj7#5	CMaj9#5	CMaj11#5	CMaj13#5
Second Harmonic	Cm	Cm7	Cm9	Cm9#11	Cm13#11
Third Harmonic	C	C7	C7b9	C11b9	C11b13b9
Fourth Harmonic	C	CMaj7	CMaj7#9	CMaj7#9#11	CMaj13#9#11
Fifth Harmonic	Co	Co7	Cm7b9b5	Cm11b9b5	Cm13b9b5
Sixth Harmonic	Cm	Cm(Maj7)	Cm(Maj9)	Cm(Maj11)	Cm(Maj11)b13
Seventh Harmonic	Co	Cm7b5	Cm7b9b5	Cm11b9b5	Cm11b13b9b5
First Melodic	C+	CMaj7#5	CMaj9#5	CMaj9#11#5	CMaj13#11#5
Second Melodic	C	C7	C9	C9#11	C13#11
Third Melodic	C	C7	C9	C11	C11b13
Fourth Melodic	Co	Cm7b5	Cm9b5	Cm11b5	Cm11b13b5
Fifth Melodic	Co	Cm7b5	Cm7b9b5	Cm7b11b5	Cm11b13b9b5
Sixth Melodic	Cm	Cm(Maj7)	Cm(Maj9)	Cm(Maj11)	Cm(Maj13)
Seventh Melodic	Cm	Cm7	Cm7b9	Cm11b9	Cm13b9

The differences are even more pronounced on other scale degrees, particularly those with an altered note such as the fifth scale degree

First	G	G7	G9	G11	G13
Second	Gm	Gm7	Gm9	Gm11	Gm13
Third	Go	Gm7b5	Gmb9b5	Gm11b9b5	Gm11b13b9b5
Fourth	G	GMaj7	GMaj9	GMaj11	GMaj13
Fifth	Gm	Gm7	Gm9	Gm11	Gm13
Sixth	Gm	GMaj7	GMaj7b9	Gm11b9	Gm11b13b9
Seventh	Gb	Gb7	GbMaj9	GbMaj9#11	GbMaj13#11
First Harmonic	G#o	G#o7	G#Maj7b9b5	G#m11b9b5	G#m13b9b5
Second Harmonic	Gm	Gm(Maj7)	Gm(Maj9)	Gm(Maj11)	Gm(Maj11)b13
Third Harmonic	Go	GMaj7b5	GMaj7b9b5	Gm11b9b5	Gm11b13b9b5
Fourth Harmonic	G+	GMaj7#5	GMaj9#5	GMaj11#5	GMaj13#5
Fifth Harmonic	Gbm	Gbm7	Gbm9	Gbm9#11	Gbm13#11
Sixth Harmonic	G	G7	G7b9	G11b9	G11b13b9
Seventh Harmonic	Gb	GbMaj7	GbMaj7#9	GbMaj7#9#11	GbMaj13#9#11
First Melodic	G#o	G#m7b5	G#m7b9b5	G#m11b9b5	G#m11b13b9b5
Second Melodic	Gm	Gm(Maj7)	Gm(Maj9)	Gm(Maj11)	Gm(Maj13)
Third Melodic	Gm	GMaj7	GMaj7b9	Gm11b9	Gm13b9
Fourth Melodic	Gb+	GbMaj7#5	GbMaj9#5	GbMaj9#11#5	GbMaj13#11#5
Fifth Melodic	Gb	Gb7	Gb9	G#11b9	Gb13#11
Sixth Melodic	G	G7	G9	G11	G11b13

Note how this group has the notes G, Gb, and G# as possible roots and also contains major, minor, diminished, and augmented triads as the building blocks for more complex chords.

The complete chart is in a separate document, "Parallel Chords Fully Interleaved And Arranged."

This chart is useful, but removing the context of the scale degrees and just listing the different chord forms in their positions makes for a better comparison. The chords groups for I, IV, and V based variations are

C	C7	C7b9	C11	C11b13
C+	Cm(Maj7)	C9	C11b9	C11b9#13
Cm	Cm7	Cm(Maj9)	C9#11	C13
Co	CMaj7b5	CMaj7b9	Cm(Maj11)	C13#11
	CMaj7	Cm7b9b5	Cm11	Cm(Maj11)b13
	CMaj7#5	Cm9	Cm11b5	Cm(Maj13)
	Co7	Cm9b5	Cm11b9	Cm11b13b5
		CMaj7#9	Cm11b9b5	Cm11b13b9b5
		CMaj9	Cm11b9b5	Cm11b13b9
		CMaj9#5	Cm9#11	Cm13
			Cm9b5	Cm13#11
			CMaj11	Cm13b5
			CMaj11#5	Cm13b9
			CMaj7#9#11	Cm13b9b5
			CMaj9#11	Cm11b13b9b5
			CMaj9#11#5	CMaj13
				CMaj13#11
				CMaj13#11#5
				CMaj13#5
				CMaj13#9#11

F	F#m7b5	F#m7b9b5	F#m11b5	F#m11b13b5
F#o	F#o7	F#Maj7b9b5	F#m11b9b5	F#m11b13b9b5
Fb+	F7	F#m9b5	F#m11b9b5	F#m13b9b5
Fb+	FbMaj7#5	F7b9	F11	F#m11b13b9b5
Fm	Fm(Maj7)	F9	F11b9	F11b13
	FMaj7	FbMaj9#5	F9#11	F11b9#13
		Fm(Maj9)	FbMaj11#5	F13
		FMaj7b9	FbMaj9#11#5	F13#11
		Fm9	Fm(Maj11)	FbMaj13#11#5
		FMaj7#9	Fm11	FbMaj13#5
		FMaj9	Fm11b9	Fm(Maj11)b13
			Fm9#11	Fm(Maj13)
			Fm9b5	Fm11b13b9
			FMaj11	Fm13
			FMaj7#9#11	Fm13#11
			FMaj9#11	Fm13b5
				Fm13b9
				FMaj13
				FMaj13#11
				FMaj13#9#11

G	G#Maj7b5	G#m7b9b5	G#m11b9b5	G#m13b9b5
G#o	G#o7	G#Maj7b9b5	G#m11b9b5	G#m11b9b5b13
G+	G7	G7b9	G11	G11b13
Gb	Gb7	G9	G11b9	G11b9#13
Gb+	GbMaj7	Gb9	G#11b9	G13
Gbm	Gbm7	Gbm9	Gbm9#11	Gb13#11
Gm	GbMaj7#5	GbMaj7#9	GbMaj7#9#11	Gbm13#11
Go	Gm(Maj7)	GbMaj9	GbMaj9#11	GbMaj13#11
	Gm(Maj7)	GbMaj9#5	GbMaj9#11#5	GbMaj13#11#5
	GMaj7	Gm(Maj9)	Gm(Maj11)	GbMaj13#9#11
	GMaj7b5	GMaj7b9b5	Gm11	Gm(Maj11)b13
	GMaj7	GMaj7b9	Gm11b5	Gm(Maj13)
	GMaj7#5	Gm9	Gm11b9b5	Gm11b13b5
		Gm9b5	Gm11b9	Gm11b13b9b5
		GMaj9	Gm9b5	Gm11b13b9b5
		GMaj9#5	GMaj11	Gm11b13b9
			GMaj11#5	Gm13
				Gm13b5
				Gm13b9
				GMaj13
				GMaj13#5

These groupings are the most useful of all since they show all the possible substitutions for each scale degree. These substitution could be for pivot chords to a key change, voice leading, or simply to add color to chord progressions by using borrowed chords from other modes.

The complete list is in the file, "Parallel Chords Unique.xlsx."

Distilling this group even further by removing the root context yields the naturally occurring chord forms

(major)	7	9	11	13
				13#11
				11b13
		7b9	11b9	11b13b9
	Maj7	Maj9	Maj11	Maj13
			Maj9#11	Maj13#11
		m7#9	m7#11#9	m13#11#9
m	m7	m9	m11	m13
			m9#11	m13#11
		m7b9	m11b9	m13b9
				m11b13b9
	m(Maj7)	m(Maj9)	m(Maj11)	m(Maj13)
				m(Maj11)b13
+	Maj7#5	Maj9#5	Maj11#5	Maj13#5
			Maj9#11#5	Maj13#11#5
o	m7b5	m9b5	m11b5	m11b13b5
		Maj7b9b5	m11b9b5	m11b13b9b5
				m13b9b5
				Maj7b13b9b5
	o7

The entire list or 2000+ chord forms from previous pattern exploration was too expansive for anyone to learn, so this group is a more approachable goal. This is still an intimidating list for a beginner, though, so try learning the chords in groups.

Group 1 contains the most common chords and the majority of popular songs just contain these

(major)

Group 2 adds enough chords to get through most songs

Maj7

Group 3 adds some more interesting chords to the mix

m7b5

Group 4 chords finish out the naturally occurring seventh chords, but are actually more rare than some of the group 5 chords

m(Maj7)

Maj7#5

Group 5 chords add the upper notes for jazz flavorings

Maj9

Maj11

Maj13

m11

m13

Group 6 chords finish up the ninth chords and their 11^th and 13^th counterparts

m(Maj9)

m(Maj11)

m(Maj13)

Maj9#5

Maj11#5

Maj13#5

m9b5

m11b5

m11b13b5

7b9

11b9

11b13b9

Maj7b9

m11b9

m13b9

Maj7#9

Maj7#11#9

Maj13#11#9

m7b9b5

m11b9b5

m11b13b9b5

And finally group 7 chords only exist naturally as 11ths or 13ths. These are for the true jazz masters.

Maj9#11

Maj13#11

m9#11

m13#11

Maj9#11#5

Maj13#11#5

13#11

11b13

m(Maj11)b13

m13b9b5

m11b13b9

m7b13b9b5

Of course, many other chords exist, but to find a context for them, look other places than the traditional modal scales and alterations. See the All Spelled Chords table for the list of all chords.

Symmetrical Patterns

So, far, the patterns have evolved naturally from the overtone series into music that is naturally pleasing to the ear. But pattern obsession eventually evolves into combinations of notes that are not based on harmonics, but rather based on mathematical relationships among notes.

Symmetrical Scales

Many scales do not fit into the modes listed previously. Selected scales that use different patterns are known as symmetrical, including chromatic, whole tone, and octatonic scales.

Chromatic Scale

The major scale is satisfying, but pattern obsession requires that the gaps in between the notes must be filled. The scale contains whole steps (W) and half steps (h). Every place in the major scale pattern (WWHWWWH) with a whole step can have a note placed between them, yielding a scale with all half steps, or the 12-note chromatic scale.

cc#dd#eff#gg#aa#b

Or alternately,

cd_bde_befg_bga_bab_bb

In function, only one chromatic scale exists since the scale is has the same pattern no matter which note begins.

This scale is useful for technique development and to provide passing tones for improvised melodies.

Whole Tone Scales

Whole scales are similar to chromatic in that they all have a set interval between the notes. In the whole tone scale, all intervals are a whole step. To find the 6-note whole tone scale, begin with a note of the chromatic scale and skip every other note

cdef#g#a#

In function, only two whole tone scales exist. The other is

d_be_bfgab

Since starting on any note in the scale just produces the same scale again, they are functionally the same.

The scale a floating feel because it is not necessarily anchored to any key, major or minor. A good example is Stevie Wonder's "You are the Sunshine of my Life."

The symmetrical scales and chords provide a notational challenge because music notation derived from major scales. The scale wraps around to C again

(cdef#g#a#cdef#g#a#) since the distance between a# and c is two half steps, the same as a whole step even though the interval a# to c looks strange because it is a diminished second (d2), a rarely notated interval. The enharmonic spelling cdef#g#b# could be used with b# replacing c enharmonically, but this would be problematic to continue us the scale without using double sharps.

Octatonic Scales

The rarely used octatonic scale is appropriate for solos over a fully diminished seventh chord. The 8-note octatonic scale alternates half steps and whole steps.

	H	W	H	W	H	W	H
c	d_b	e_b	e	f#	g	a	b_b

This scale also functionally has two other possibilities, the others being

a_b

b_b

c_b

and

e_b

g_b

a_b

The scales may appear with the whole steps followed by the half steps.

c	d	e_b	f	g_b	a_b	a	b
d_b	e_b	e	f#	g	a	b_b	c
d	e	f	g	a_b	b_b	c_b	d_b

Symmetrical Chords

Just as intervals diatonic scales stack to become chords, the intervals in the symmetrical scales mix to become chord that are different from the those bund to the harmonic structure of the modes.

Tone Clusters

The chromatic scale is the set of all twelve notes. A theorist could make an argument that all chords are simply stacks of chromatic notes separate by intervals.

Consecutive chromatic tones played closely together produce clusters of dissonant notes more reminiscent of percussion than harmony. A piano player can easily bang a bunch of notes together to form a chromatic cluster, but the distance between chromatic notes on different strings on a guitar keep clusters out of reach. of course players can use small dissonant clusters, such as in the theme from South Park.

Clusters mad from major 2nds are more interesting clusters.

Clusters based on the whole tone scale

cdef#g#a#

can have an ethereal sound since the notes do not imply any harmonic structure with the a_bsence of thirds. Possible intervals include M2, M3, A4, A5, and m7, and these intervals are the same ascending or descending and any interval is an equal distance from any pitch.

Augmented Chords

The obvious chord derived from the whole tone scale

cdef#g#a#

is the augmented chord.

Only four augmented chords exist

ceg#

df#a#

d_bfa

e_bgb

With any other augmented chords being just different spellings of these four. Any of the notes of the augmented chord can serve as the root, so these chords do seem unanchored. Augmented chords can serve a dominant function in tonal harmony, or as pivot to a key change. A true augmented seventh does not make sense since skipping another tone simple brings the chord back to its tonic. (ceg#c, for example). Augmented chords can borrow sevenths from other chords, though, as seen in the III+7 harmonization in the harmonic minor chord.

Allowing for enharmonic spellings, chords derived from the whole tone scale could harmonize as

C+, C+(7), C+(9), C+(11), C(11b13)

Since the scales are symmetrical, chords built on other scale tones would follow the same pattern

D+, D+(7), D+(9), …

And since only two truly unique whole tone scales exist, patterns on C# complete the group of possible chords.

C#+, C#+(7), C#+(9), C#+(11), C#(11b13)

Diminished Chords

Unlike augmented chords, diminished chords, based on stacks of minor thirds, are an important part of tonal harmony. The diminished seventh chord is a special case because it need not be tethered to a particular harmonic structure. Only four truly unique diminished seventh chords exist

c e_b g_b a

d_b e g b_b

d f a_b c_b

with all other chords being different spellings of the same notes. Because of the overlapping nature of these spellings, the same diminished seventh chord can coexist in several keys at once, so they make great transition chords for key changes. The distinctive sound is reminiscent woman tied to the train tracks in a old silent movie.

The octatonic scale

d_b/c#

e_b/d#

efb

f#/g_b

b_b/a#

is not usually harmonized, but does contain a surprising number of chords.

Based on C as a root, the scale yields major, minor, and diminished triads, each of which can use b_b as the 7^th, d_b as b9, f# as #11, and a as 13 or d7 depending on context.

C	C7	C7b9	C7#11b9	C13#11b9
Cm	CMaj7	CMaj7b9	CMaj7#11b9	Cm13#11b9
Co	CMaj7b5	CMaj7b9b5	CMaj7#11b9b5	Cm13#11b9b5
(Co)	Co7	Co7b9	Co7#11b9	Co13#11b9

(Co) indicates that the chord repeats, but holds its place in the table for clarity.

The tonic harmonies did not produce some of these chords, so the octatonic scale yields new, interesting sounds.

C# as a root yields either minor or diminished chords, with b_b as a diminished 7^th (b7), d# as 9, f# as 11, and a as b13.

C#m	C#mb7	C#m9b7	C#m11b7	C#m11b13b7
C#o	C#o7	C#9b5b7	C#o11	C#11b7b5

For chords based on C, the d_b serves as an add_b9 or susb2, while chords based on C# use the D# as a 9 or sus2.

Cadd_b9	C7add_b9	Csusb2	C7susb2
Cmadd_b9	CMaj7add_b9	(Csusb2)	(C7susb2)
Coadd_b9	CMaj7b5add_b9	Cb5susb2	C7b5susb2
(Coadd_b9)	Co7add_b9	(Cosusb2)	C7b5susb2
C#add2	C#badd2	C#sus2	C#b7sus2
C#oadd2	C#o7add2	C#b5sus2	C#o7b5sus2

9^th, 11^th, and 13^th chords already have a ninth, so adding a ninth to a higher note chord such as C7b9add_b9 is redundant. If the following tables seem to have empty areas, that is proba_bly where a redundant chord would go.

The f# serves as an add#4 or a sus#4 in appropriate chords cased in C, or as add4 or sus4 in chords based in C#.

Cadd#4	C7add#4	Csus#4	C7sus#4	C7b9sus#4
Cmadd#4	CMaj7add#4	(Csus#4)	(C7sus#4)	(C7b9sus#4)
Coadd#4	CMaj7b5add#4	Cb5sus#4	(C7b5sus#4)	C7b9b5sus#4
(Coadd#4)	Co7add#4	(Cb5sus#4)	Cb5sus#4d7	Cb5sus#4d7b9
C#madd4	C#mb7add4	C#sus4	C#(b7sus4)	C#(9b7sus4)
C#oadd4	C#o7add4	C#b5sus4	C#b5sus4d7	C#(9b5sus4d7)

Using an fb as the 4^th for chords based on C yields changes to the added cords that do not have a major 3^rd. Suspended b4 does no make sense since b4 and M3 are enharmonic.

Cmadd_b4	CMaj7add_b4
Coadd_b4	CMaj7b5add_b4
(Coadd_b4)	Co7add_b4

And the fb also changes the elevenths, forcing the use of add#2

CMaj7b9add#2	CMaj13b9add#2
(x)	CMaj13b9b5add#2
CMaj7b9b11b5add#2	CMaj13b9b5add#2

An fb in chords based on C# is theoretically possible, but would be a doubly diminished 4th (b_b4). This substitution does not seem necessary, as beginning to use double flats opens up too many worm cans, like c=d_bb

Using the note a as (6) for chords based in C and (b6) for chords based in C# yields

C6	C67	C67b9	Csus6	C7sus6	C7b9sus6
Cm6	Cm67	Cm67b9	Cmsus6	CMaj7sus6	CMaj7b9sus6
Co6	Cm67b5	Cm67b9b5	(Cmsus6)	(CMaj7sus6)	(CMaj7b9sus6)
(Co6)	Co67	Co67b9	(Cosus6)	Cmsus6d7	Cmsus6d7b9
C#mb6	C#mb6b7	C#m9b6b7	C#msusb6	C#msusb6b7	C#m9susb6b7

Using a# instead of a yields (6) for C# chords and b7 for fully diminished chords without b7

C#m6	C#m6b7	C#m69b7	C#msus6	C#msus6b7	C#m9sus6b7
C#o(6)	C#o7(6)	C#o7(69)	C#m(sus6)b5	C#(sus6)b5d7	C#m9b5sus6d7
Co7b7	C#9b7b5b7	C#11b7b5	C#11b7b5b13

And the thirteenth chords equivalents based on fully diminished chords

CMaj7b9b5add#2add#6 and C#m13b5

So, just the first two notes as roots combine to 147 unique chord combinations. of course, the add- sus- and 6 chords could combine into many other combinations, such as Csus4 add2.

Since the chords are symmetrical, chords with roots on other notes mirror the same patterns as the ones based in C. the notes d# and e harmonize to D#, D#m, D#o, Em, and Eo, with all the other variations following the same pattern. Four possible groupings exist, beginning on every other note in the scale.

So, 159 chords x 4 possible scale positions x 3 possible octatonic scales equals 1920 unique chords, not including combinations of add- and sus- chords, and not including enharmonic spellings. Have fun playing them all.

Symmetrical Quartal Chords

Stacks of half steps produce clusters, alternating whole steps produce whole tone clusters, alternating whole tones produce augmented chords, stacks of minor thirds produce diminished seventh chords. Stacks of major thirds are just alternating whole tones, so they produce the augmented chords.

Continuing the process, stacking perfect fourths would produce a stack of notes following the circle of fourths.

The following stack of notes, written enharmonically since traditional scale notation has long since broken down.

c_b/b

g_b/f#

dd/c#

a_b/g#

d#/e_b

b_b/a#

f/e#

c/b#

Now rearrange into something approximating traditional scale order

c_b/b

b_b/a#

a_b/g#

g_b/f#

f/e#

e_b/d#

d_b/c#

c/b#

Possible triads built on the note c include

g_b	g#	g#	g_b
e_b	e#	e_b	e#
c	c	c	c
Co	CMaj#5sus4	Cm#5	CMajb5sus4

Since a #3rd is just a 4th, rewrite as

Csus4

Cm#5

Cb5sus4

Using a suspended chord as a triad is new, but remember that this notation has its basis in a totally different type / harmony, so the analysis is imperfect.

Possible sevenths include b (Maj7) or Bb (7)

CMaj7b5

C7sus4

CMaj7#5

C7b5sus4

CoMaj7

CMaj7sus4

Cm(Maj7)#5

C(Maj7)b5sus4

Possible added notes include ninth d# (#9) 11^th f (b11)or f# (#11)and 13^th Ab (b13). Since b11 is enharmonic to M3 when octaves are ignored, that chord be skipped for chords based on a major third as the first interval. B11 is not a viable chord form, so use add#2 to simulate. Similarly, #11 is enharmonic to b5 in the same octave, and #9 is enharmonic to b3, and b13 is enharmonic to #5.

#9	b11	#11	b13
(repeat)	CMaj7b5add#2	(repeat)	CMaj7b13b5
C7#9sus4	(repeat)	C7#11sus4	C7b13sus4
(repeat)	CMaj7b#5add#2	CMaj7#11#5	(repeat)
C7b5sus4#9	(repeat)	(repeat)	C7b13b5sus4
(repeat)	(x)	CoMaj7#11	CoMaj7b13
CMaj7#9sus4	(repeat)	CMaj7#11sus4	CMaj7b13sus4
(repeat)	C(Maj7)#5add#2	Cm(Maj7)#11#5	(repeat)
C(Maj7)b5sus4	C(Maj7)b5sus4add#2	(repeat)	C(Maj7)b13b5sus4

The #9 could also be an add#2 to appropriate chords, with other add chords such as add_b4 or add#4 or add_b6 when the base chords do not have an enharmonic note. Refer to the section on octatonic harmonization for strategies.

Suspended chords, however, do not make sense in this context because no true third exists. Suspensions could replace other chord tones. Adding a sus#2 to the CMaj7sus4 to make CMaj7sus4sus#2 removes an e and replaces it with D#, but this has little to do with a true suspended function from tonal harmony.

Because the scale is symmetrical, any note in the scale can serve a first note (the term tonic no longer truly applies). Substituting the next note, f, in the original scale yields

Fsus4

Fm#5

Fb5sus4

Tonal chord representations can only go so far with chords based on mathematical relationships. With the harmonization of the octatonic scale, chord symbols can be forced upon the symmetrical chords. But with the sets of notes like Circle of Fourth chords, the system breaks down and some chords cannot be analyzed in any meaningful way using tonic harmonies. To form chords based on math rather than the physics of sound, use set theory, discussed in the next major section.

Higher Interval Chords

Continuing the pattern of building chords based on larger and larger intervals, the next interval would be a tritone (d5). Since a tritone is just two minor thirds stacked, any chords based on the tritone are simply repeats of the diminished harmonies.

The tritone is also the center of a twelve-tone scale, and since these chords are symmetrical, they reflect around the tritone. The next interval above a tritone is a perfect fifth (P5). With octaves ignored, a perfect fifth up would be the same note as a perfect fourth down.

A scale based on perfect fifths cgdaebf#c#ge_bb_bf is the same scale as the scale of fourths once rearranged and respelled. So, the chords based on these spellings

Since the chords are symmetrical, moving down the scale has the same intervallic relationship as moving up, so inversions of the chords are also from the same chord set, although analysis may provide different spellings.

The pattern continues for intervals greater than the fifth. A minor sixth is a major third below the octave, and so inherits the chord structures from the augmented chords. major sixths match with minor third patterns, Minor seventh with whole tone scales, and major sevenths produce a chromatic scale. Octaves are a special case because they produce no harmonies. Intervals above the octave simply repeat the pattern.

Twelve Tone Music

Wagner's Tristan chord led to expansion of harmonies in the Romantic era, followed by impressionist music using pattern chords, including a lot of fourths and fifths. The next major evolution in music was to use all twelve notes of the chromatic scale in patterns defined mathematically. Arnold Schoenberg began a style of music known as twelve-tone music, or serial music.

In twelve tone music, all twelve notes of the chromatic scale fill their own place in a tone row. Take all the pitches and place them in a row without repetition. For example,

a_b	b	g	f#	e_b	e	a	d	f	b_b	d_b	c
1	2	3	4	5	6	7	8	9	10	11	12

(Note: Traditional systems number from 0-11 instead of 1-12.)

This tone row allows two possibilities, played forward, or retrograde from 12-1

c	d_b	b_b	f	d	a	e	e_b	f#	g	b	a_b
12	11	10	9	8	7	6	5	4	3	2	1

Then add another row above with all the pitches transposed up a half step.

a	c	a_b	g	e	f	b_b	e_b	f#	b	d	d_b
a_b	b	g	f#	e_b	e	a	d	f	b_b	d_b	c
1	2	3	4	5	6	7	8	9	10	11	12

Continue adding rows until all 12 possibilities fill in

P₁₂

b_b

e_b

a_b

d_b

P₁₁

d_b

e_b

a_b

b_b

P₁₀

a_b

e_b

d_b

b_b

P₉

e_b

b_b

d_b

a_b

P₈

e_b

d_b

b_b

a_b

P₇

d_b

b_b

e_b

a_b

P₆

d_b

a_b

b_b

e_b

P₅

e_b

b_b

a_b

d_b

P₄

b_b

a_b

d_b

e_b

P₃

b_b

d_b

a_b

e_b

P₂

a_b

b_b

e_b

d_b

P₁

a_b

e_b

b_b

d_b

Now calculate the inversions order.

Take the original tone row and calculate the interval up from one tone to the next, substituting enharmonic intervals for clarity.

	m3	m6	Maj7	M6	m2	P4	P4	m3	P4	m3	Maj7
a_b	b	g	f#	e_b	e	a	d	f	b_b	d_b	c
1	2	3	4	5	6	7	8	9	10	11	12

Now start with those intervals and move down the interval from the note before it. Write as a column instead of a row.

a_b	Down
f	m3
a	m6
b_b	Maj7
d_b	M6
c	m2
g	P4
d	P4
b	m3
f#	P4
e_b	m3
e	Maj7

Now rearrange the table to match this order

P₁

a_b

e_b

b_b

d_b

P₁₀

a_b

e_b

d_b

b_b

P₂

a_b

b_b

e_b

d_b

P₃

b_b

d_b

a_b

e_b

P₆

d_b

a_b

b_b

e_b

P₅

e_b

b_b

a_b

d_b

P₁₂

b_b

e_b

a_b

d_b

P₇

d_b

b_b

e_b

a_b

P₄

b_b

a_b

d_b

e_b

P₁₁

d_b

e_b

a_b

b_b

P₈

e_b

d_b

b_b

a_b

P₉

e_b

b_b

d_b

a_b

The table rows reads left to right (Primary), or right to left (Retrograde) as before, but with more starting places for transpositions.

Columns read top to bottom are Inversions, while read bottom to top are Retrograde Inversions.

Tone rows began as serial music, all notes played in order either as melodies or stacked into chords. Much of the early attempts were difficult to listen to, but later composers such as Alban Berg tried to make more approachable music. Listen to Berg's Violin Concerto for music based in serial patterns that has tonal elements.

The original song "Free Fall" uses the tone rows more loosely than the strict serialist music, and incorporates more approachable harmonies.

Fretboard Patterns

At this point music has moved away from functional harmony and just become a series of patterns translated into notes. Patterns can come from other places besides mathematical set theory. A guitar player can just find fingerings that work and move them around from string to string, ignoring the musical context.

Consider the fret pattern 0-2-3 on the guitar

e---0-2-3

moved from string to string.

e---0-2-3------------------------------

b---------0-2-3------------------------

d---------------0-2-3------------------

g---------------------0-2-3------------

a---------------------------0-2-3------

e---------------------------------0-2-3

This pattern does not fit any standard, but it sounds good and is easy to play. Eddie Van Halen is a master of the Fretboard pattern, giving him a unique sound. Check out "Hot for Teacher" by Van Halen for this technique.

Stochastic Processes

So, far, the patterns all come from evolution of natural chord tendencies, patterns based on scales, patterns based on math, and patterns based on instrumentation. Pattern can also come from random sources. To form a tone row based on randomness, get a twelve sided Dungeons and Dragons die and roll it. Write down the note corresponding to the number. Then roll again for the next note, ignoring repeats for true serial music. After twelve rolls, the row is complete.

Using randomness in music is stochastic composition. Occasionally random chance can reveal patterns that not might reveal themselves through study of traditional techniques. So, the most highly structured music now takes on its next logical evolution, breaking down all structure and moving toward chaos.

All Possible Scales

A set of all possible scales or tone rows is easy to complete mathematically. Defining a scale as a group of seven notes, the permutations begin thusly

c d e f g a b

c d e f g a b_b

c d e_b e f g a

c d e f g a_b a

et cetera, until all 792 combinations are complete. See the document "All Possible Scales" for a complete list. But scales can contain more or less than seven notes, so the document contains lists of scales up to twelve notes (the chromatic scale). Defining a scale as having at least five notes (anything less would be an arpeggio or interval), the All Possible Scales contains 3302 scales mathematically produced, without context.

Technically, this set would not contain all possible scales, since some scales can have different notes in different octaves, for example major in the first octave then minor in the second. A calculation of all these scales would be functionally infinite. Calculations in the guitar section show a possibility of 119 Trillion scales just on the guitar and more for instruments with a greater range. Another set of scales containing all the microtones would be literally infinite.

Improvisation

Improvisation is a type of controlled randomness based on instinct. Since the patterns break down more and more, relying on instinct is the next logical step. Jazz improvisation is what people most think of, but everything can be improvised once the pattern start breaking down, not just melody.

The Grateful Dead were known for improvising everything. They did not come to concerts with a fixed set list, they just worked out on stage what songs to play. Then, the songs differed each time they played them. Verses and choruses roughly stayed the same, but the solo sections veered off into unusual directions, and occasionally they ended with a different song than they started with. Totally improvised section of their concert known as "Space" gave up fixed rhythms and just worked the tone colors in a psychedelic manner. Twentieth century composer wrote pieces that did not have notation, just gave a general description of what mood to evoke.

Microtones

Once the traditional structures break down, all traditions are on the chopping block. All of the notation so far has come out of the chromatic scale, which evolved from the natural vibrations of objects. But an infinite amount of other pitches occur between the traditional pitches.

Musicians use these extra pitches frequently for effect. The seventh of a major scale sounds a bit better played a little sharp because playing sharp enhances the leading tone effect of resolving to the tonic. Asian Indian music uses lots of microtones in its melodies. Vibrato, alternation of small pitch intervals, is common for voice and many other instruments, and is essential for most fretted string instruments. A singer can scoop up to a pitch for effect, and guitar bends from one note to another are a staple for rock guitar playing.. Listen to music by David Gilmour for excellent examples of pitch modulation using bends, finger tremolo, whammy bar tremolo, and bends both upward and downward using the whammy bar.

Some instruments, like the trombone, have sliding pitch possibilities and can use microtones for effect. Listen to "Lassus Trombones." A pedal steel guitar, or even a standard guitar using a slide, can slide between entire chords for a twangy, country effect.

Temperament

All those examples are passing uses of pitch as an effect, retuning to traditional notes. Pitch change can also effect notes in a more permanent manner.

Traditional music divides the twelve traditional pitches into equally spaced intervals. But the natural overtone series does not follow that pattern exactly. Consider the tuning note, A440, vibrating at exactly 440 cycles per second (hz), and call this note A4 because it is on the fourth octave of the piano. Octaves are perfectly in tune on many instruments. Octaves are perfectly in tune on many instruments. The section on development of pitch showed how the interval of an octave plus a perfect fifth is three times the frequency of the fundamental. So, A5, the octave above A4 should have a frequency of 880 Hz, and G6, G just above A5 should have a frequency of 1760 Hz.

But equally tempered instruments like the piano do not tune to this relationship. The way to tune an instrument equally might be to have an equal number of frequencies between each note. But the problem with trying to calculate equal intervals is that just dividing by 12 will not work.

Consider the octave from 440 Hz to 880 Hz. This octave has a range of 440 Hz from the lower to the higher note. Now consider the octave from 880 Hz to 1760 Hz. This octave has a range of 880 Hz. So, the range of 440 Hz in the first octave sounds like the same distance of 880 Hz in the second octave even though it seems twice as large by looking at the number of Hz between the notes.

Frequency differences work logarithmically rather than arithmetically, meaning that the distances multiply powers of two rather than adding. Octaves from 55-110-220-440-880-1760 all sound the same. Each octave has twice the frequency range.

Now consider the octave from 880 Hz to 1760 Hz. This octave has a range of 880 Hz and a hallway point of 1320 Hz. 1320 is also 440*3, which is also consistent with the octave plus perfect fifth being three times the frequency, the fundamental relationship the development of tonal harmony.

The differences in measuring arithmetically and geometrically also show when comparing octaves. The perfect fifth is halfway arithmetically

The octave 440-880 has a halfway point E at 660, 220 Hz away.

The octave 880 -1760 has a halfway point at 1320 Hz, 440 Hz away.

So, stating that a perfect fifth has a distance of 220 Hz is only true for one octave of one scale.

So, the math seems to suggest that a perfect fifth is halfway between the two octaves, and observed arithmetically, it is. But consider the set of twelve notes

a-a#-b-c-c#-d-d#-e-f-f#-g-g#-a

Start with the lower note a and count up six half steps to d#.

Start with the upper note a and count down six half steps to d#.

So, musically halfway up the scale is not the perfect fifth, but the tritone (the Devil's interval, forbidden in the Middle Ages in some cultures).

To find the frequency difference between the true halfway point of the scale, use multiplication. Some number multiplied by the fundamental 440 must equal the frequency of the center tone d# then the same number multiplied by the frequency of the d# must equal 880, the octave. Mathematically using f(d#) to represent the

frequency of d#, using k to represent the unknown multiplier. (Here f is not the musical note f, but the function, like f(x).)

440*k=f(d#) and f(d#)*k=880

In the second equation, divide both sides by k

f(d#)=880/k

But we know f(d#)=440x from the first equation, so

440x=880/k

Multiply both sides by k

440k^2=880

k^2=2

x=sqrt(2) The square root of 2, a_bout 1.414.

So, multiply the frequency of a pitch by 1.414 to get the frequency of the tritone above it.

440*1.414 = 622.254 Hz

Then to get to the next octave, multiply again

622.254*1.414 = 880. Hz.

(If you got a slightly lower number it is because of rounding error. Try 1.414213562 or use a calculator and do the actual square root of two.)

Using the same logic to calculate half steps is a little more complicated. Looking for a number x the multiplied by the fundamental yields the note a half step above it

Define a function f(x) that returns the frequency a half step above it.

f(x)=k*x

In this case x=440, but writing the general function allows its use later for other frequencies.

To find the next note up, a whole step, take the plug the frequency of the half step, kx, back into the function recursively.

f(kx)=k*(kx)=kkx or k^2x

To find the next note, a minor third, plug k^2x back in recursively,

f(k^2x)=k(k^2)x=k^3x

Continue up the octave

k^4x	c#
k^5x	d
k^6x	d#
k^7x	e
k^8x	f
k^9x	f#
k^10x	g
k^11x	g#
k^12x	a

The result k^12x is the octave. The frequency of the octave is also twice the original frequency, or 2x.

k^12x=2x

Dividing both sides by x yields

k^12=2 (or k*k*k*k*k*k*k*k*k*k*k*k=2)

So, k is the twelfth root of 2, or approximately 1.059463094

So, starting with A440, the pitches of the chromatic scale are (calculations shown)

440	a
466.1637615	a#	440*1.059463094
493.8833013	b	466.16376151809*1.059463095
523.2511306	c	493.883301256124*1.059463096
554.365262	c#	523.251130601197*1.059463097
587.3295358	d	554.365261953744*1.059463098
622.2539674	d#	587.329535834815*1.059463099
659.2551138	e	622.253967444162*1.059463100
698.4564629	f	659.25511382574*1.059463101
739.9888454	f#	698.456462866008*1.059463102
783.990872	g	739.988845423269*1.059463103
830.6093952	g#	783.990871963499*1.059463104
880	a	830.60939515989*1.059463105

This set of frequencies is nice to have around when coding. It shows all notes of the scale equally set apart, or equal temperament, the most common modern tuning system.

The problem with equal temperament is that it does not line up exactly with the natural resonances. The analysis of the interval P5 dating all the way back to the ancient Greeks (Pythagorean tuning) shows that the P5 note e should have a frequency of 660, not 659.2551138. This time the difference is not due to rounding error, but inherent in the system.

Since the modern ear is so accustomed to equally tuned scales, the note played sounds fine. But a fifth played slightly sharp from equal tuning sounds a bit more natural, so instrumentation not bound to equal temperament, such as a cappella vocal music, often contains performance practice that gravitates toward a slightly sharp fifth, a pure fifth.

Now consider the third. The overtone of A110 (two octaves down from A440) is a c# in the octave being discussed. This c# vibrates naturally at a frequency of 550 Hz. But the equally tempered version of the third is at 554.37 Hz. While the difference between the pure fifth and equally tempered fifth is small, the difference between the pure major third and the equally tempered major third is noticeable. The third of the chord is the pitch that tends to be the most difficult to tune for this reason. A third played slightly lower would be more acceptable. Blues players take advantage of the ambiguity of the third by playing the third of the chord somewhere between a minor and major third.

So, why not just tune the instrument to the natural pitches? Tuning the note e a little sharp and the c# a bit more flat would be easy on the piano, and the result would be nice in the key of a. But what if the piece modulated to F major? The note a is now the third in f, so not only does it have the natural tendency to be sharp, now the piano is tuned even more sharp on that note and the result is a chord that sounds out of tune.

So, modern tuning opts for the compromise of equal temperament, with each scale slightly out of tune with pure intervals, but no jarring keys caused by trying to adhere to a strictly mathematical scale. Classical players of the past experimented with different temperaments, such as just temperament, but since the advent of chromatic tuners and songs with lots of modulations, equal temperament dominates. Synthesizers often have settings for temperament buried in all their settings.

Sometimes pianos tune to a stretch tuning, with every octave a bit sharper than the one below it to take advantage of pure fifths. Stretch tuned pianos can be almost a half step out of tune from the bottom keys to the top keys.

Alternate Pitch Sets

But the microtones discussed in the last section are still simply variations on tonal harmony. What a_bout a different number of notes in the scale? Composers have experimented with quarter tones, placing notes exactly halfway between the notes of the chromatic scale, yielding a 24-note set of pitches in each octave. This arrangement retains the pitch and voice leading possibilities of 12 tone music, and in fact, doubles the number of places where they can occur. To calculate the intervals in this scale, multiply each frequency by the 24^th root of 2, 1.0293. Music using these scales just sound out of tune at first, but careful listening can pick out the patterns eventually.

But if the goal is to break down traditional patterns, then why simply adjust the twelve-tone scale. Why not discard it completely and derive new patterns. Now that a mathematical formula exists. To divide an octave into n parts, multiply the fundamental by the nth root of 2. For math geeks-[2^(1/n)]. Or even more chaotic, choose random or intuited places to put notes all along the infinite spectrum of musical frequencies.

Placing notes anywhere along the musical spectrum and adding them together is indeed the next logical step, but it stretches the definition of harmony to its breaking point. The dictionary definition of harmony is, "the combination of simultaneously sounded musical notes to produce chords and chord progressions having a pleasing effect." (Google Dictionary). But the idea of combinations of notes being pleasing ended long ago when the combinations of notes started being based on patterns rather than musical function. Further combinations of notes are not true harmonies, but are simply timbre clusters, groups of tones that have a discordant effect rather than pleasing.

Timbre

Timbre is "color" the musical sounds. Descriptions of timbre such as "warm" tend to be nonscientific and difficult to quantize, but these terms belie the natural mathematic functions that underlay all timbres.

White Noise

Continuing the pattern of adding more and more chaos to the musical structure to its extreme, the most chaotic combination of tones possible would be all frequencies played simultaneously. White noise is the hiss sound an old school television makes when not tuned to any station. Since "all" frequencies would be an infinite amount, and would be impossible to produce since many would add destructively with others, to calculate white noise, use a random number generator to produce a mix of a large number of frequencies with an equal chance of any frequency appearing at any time. The more frequencies in the mix, the closer it will approximate true theoretical white noise.

Pink Noise

But the ear perceives frequency logarithmically, not arithmetically, as shown in the discussion of temperament. So, adding frequencies together with combinations in white noise produces a texture the overemphasizes higher frequencies. In pink noise, frequencies continue to add randomly, but with an equal number of frequencies in each octave.

Pink noise is softer sound and more useful to engineers. An engineer will project pink noise onto a performance space to measure the acoustical properties of the space. By monitoring the way the pink noise interacts with the space using a microphone, the engineer then adjusts the equalization of the mix to counteract any anomalies cause by irregularities in the performance space.

Subtractive Synthesis

At this point, no more frequencies are available to add to the series of tones. The progression of patterns moved naturally from the most basic tone, the sine wave, to vibrations of a string showing the naturally occurring scales, to chords built on those scales, to chords built by filling in the missing intervals on those scales, to mathematical permutations of those notes, to the addition of microtones, and finally to total chaos, white noise. Now the only way to continue discovering patterns is to begin to take away sounds from white noise, or subtractive synthesis. When taking away frequencies, the concept of amplitude now becomes important.

Amplitude

Amplitude is the technical term for volume. The unit of measurement for amplitude is the Decibel, dB, measuring Sound Pressure Level, dB SPL. Often the SPL tag does not appear, but to be technically accurate, dB SPL is most correct since decibels are just a mathematical relationship and not a true quantity. Decibels might also come into play in a home stereo system, measuring the voltage inside the circuits, bBV, or in electronic equipment measuring power, dBm.

Decibels show the difference between a measurement of a quantity compared to a reference value. For example, in dB SPL, the change in air pressure compares the softest sound half the population can hear, 20 uP (micro Pascals-a unit of pressure change).

The ears are amazing in the range of sound they can perceive. The pressure changes in a very loud sound are over a million times those of a softer sound. And the perception of the amplitudes does not spread evenly either; just as in frequency bands, more differences in volume happen at higher air pressure levels. The reason is that the ear hears logarithmically, with the higher amplitude bands being more compressed than the lower bands.

The Decibel measurement compares the pressure change (P) of the sound to a standard pressure (P₀). Since the relationship is logarithmic, adding or subtracting from the reference vale does not make sense. A difference between 20 uP and 120 uP is relatively large, but a difference between 1,000,000 uP and 1,000,100 would not even be noticeable.

So, difference measure by dividing the pressure change by the reference value. P/P₀. Taking the logarithm of a divided relationship converts a logarithmic relationship into a linear relationship, and humans are much better at understanding linear relationships. The formula color for bB SPL is

SPL = 20Log(P/P₀) dB

The math is interesting in several respects. Substituting the softest sound for P yields

20Log(P₀/P₀) =

20Log1=

20*0

=0 dB

So, 0 d_b is the softest sound half the population can hear. Notice that 0 dB is not a_bsolute silence. Theoretical a_bsolute silence would be an pressure of 0.

20Log(0/P₀) = color

20Log(0) =

UNDEFINED.

(Taking the log of 0 is akin to dividing by 0; the result is not defined.)

But taking logs very close to 0 does yield a result. As numbers get smaller and smaller approaching 0, their logs get more and more negative. So, as P approaches 0, the limit of 20Log(P/P₀) approaches negative infinity.

The loudest sound has no mathematical limit, but the limitations of the ear give a realistic maximum volume of 120 dB, commonly known as the threshold of pain. Ears can sense amplitudes of some frequencies up to 160 dB, known as the threshold of feeling.

Mathematically, a difference of 3 dB SPL is a doubling of the amount to raw pressure change. But humans perceive a 3 dB SPL changes as barely noticeable. The perception of acoustics puts a 10 bB SPL change as a doubling of perceived volume.

Refer to OSHA guidelines for the maximum amount of time to spend at any volume level.

Filters

So, subtractive synthesis uses filters to remove frequency bands from complex sounds. The simplest of these filters are the EQ knobs on a home stereo system. Many stereo systems have knobs labeled treble or bass. Shelf filters like these have a cutoff value predetermined by the manufacturer, and any change to the knob lowers or raises the frequencies above that cutoff for treble and below the cutoff for bass.

An exact cutoff frequency is not perfect in the real world, so a graph of the frequencies around the cutoff have a slope. A mathematically theoretically perfect cutoff is known as a brick wall filter.

The useful high pass filter allows frequencies above the cutoff to pass through. With a cutoff typically around 80Hz, the high pass cuts out lower sounds on tracks that have no low tones, avoiding rumbling, muddy sounds interfering with a mix. The term low cut is proba_bly more intuitive, but high pass is the term more often found.

For frequency bands in the middle range, two cutoff frequencies are necessary, an upper and a lower. Often these types of filters occur in multiple groups, with a common setup being the 31-band equalizer with 3 separate sliders than can raise or lower the frequency band.

If the frequency band is adjustable, then the filter is quasi-parametric, with an adjustable center frequency and amplitude. These filters usually use knobs instead of sliders. Often, they appear in groups of two to four on mixing boards.

If a filter has control over amplitude, the center frequency, and the width of the frequency band, the filter is parametric, or fully parametric. The measurement for the width of the frequency band is Q (Quality), with a smaller Q being a larger frequency band. A filter with a large Q is a notch filter, designed to remove unwanted frequencies such as electrical hum.

All filters introduce phasing problems and artificial colors, so use sparingly. Filters work better to cut unwanted bands away and sound less natural adding. Appropriate use of filtering includes adjusting for room imbalances, problems with equipment, making instruments stand out in the mix by giving them their own frequency mix, and as a special effect like a telephone voice. Avoid using EQ to make tones "sound good." Instead, work on microphone choice or placement for a more natural sound.

Formants

The discussion of vibration of a string showed how musical tones built from the overtone series, with a musical note being a combination of sine waves with frequencies that are multiples of the fundamental. But the amplitude of each of these frequencies is not equal. Typically, each overtone is softer than the overtone below it, each with its own amplitude. The relative amplitude of each of these frequencies gives the sound its timbre, or tone color. Fast Fourier Transforms FFTs are the mathematical method for calculating thee frequency relationships.

Natural phenomena filter out groups of these frequencies in bands called formants. The most common use of formants is in analyzing the voice. Vocal tone produced at the larynx has a buzzy sound later filtered by the tongue and jaw placement. If the tongue is back in the mouth, lower frequencies filter out leaving an oo sound for a closed jaw. If the tongue is further forward, then more high frequencies pass leaving an ee sound. A closed jaw filters out the lower frequencies, while an open jaw allows lower frequency bands through, allowing an aah sound to pass. These groups of frequency bands are the first and second formants of the voice. Another format occurs when the sound vibrates in the sinus area and surrounding bones. Frequencies produced in area, known as the singer's formant, allow a singer to project over an orchestra since few orchestral sounds lie is the frequency band.

Compression/Limiting

Amplitude can also function as its own effect over the entire audio band. Compression is an effect that reduces the dynamic range of sounds. Compression flattens out higher amplitudes to allow softer sounds through. A compressor sets an threshold amplitude, and then reduces all sound louder than that amplitude by a ratio, typically written like 4:1. The compression has a makeup gain that raises the volume of all sounds.

A compressor with a very high ratio is a limiter. Limiters set an upper level on how loud a sound can be. Limiter are useful as a protection device to make sure that the volume does not get too loud and blow a speaker. Modern music typically overuses compressing effects to make the sound louder. Unfortunately, the music becomes flat dynamically.

Expander/Gate

An expander takes all sounds below a threshold and reduces them by a ratio. An expander with a high ratio is a gate. A noise gate is useful in filtering out guitar him or other unwanted ambient sounds.

Distortion

Pushing the amplitude of a signal beyond the limits of the circuitry or speakers causes distortion. Distortion is unwanted in mixing, but guitar players have used distortions such a tube screamers to produce interesting timbres.

Envelopes

Compressors and limiters have one more set of settings typically. To understand these setting, time as a parameter for acoustic effects is necessary. Time came into play when discussing frequencies (cycles per second), but so far, all the discussion has been on static tones. Time will turn out to be the foundation for most of musical effects, processing, and composition.

The missing compressor settings are attack time, the amount of time before a compressor starts, and decay time, the amount of time the compression lasts. Attack time allows a sharp attack like to pass through without unwantedly overly compressing sounds with a sharp beginning. Decay time controls how long until the sound returns to normal dynamic levels. Improper use of decay leads to a mix that "pumps" or "breathes" unnaturally.

The attack and decay are part of a sound's envelope. A piano has a quick attack and a long decay, while a voice singing a vowel has a smoother attack. A snare drum has a sharp attack and a small decay.

Tremolo

Envelope changes can occur in the middle of notes also. An effect with repeated periodic changes in amplitude is called tremolo, popular in surf music. Tremolo measures in Hz (cycles per second) just like frequency.

Amplitude Modulation

When the frequency of a tremolo passes the frequency of the lowest frequency the ear can perceive, 20 Hz, the tremolo becomes a pitch unto itself, separate from the original pitch. These pitches add together in unusual ways, producing three tones, the fundamental original tone, a second band above the fundamental as a factor of the amplitude modulation, and a third frequency below the fundamental the same amount as the second band was above.

Frequency Modulation

Vibrato, alternating pitch rather than amplitude, at a frequency above 20 Hz has a similar but even more profound effect on the tone. Vibrato produces even more frequency bands at evenly spaced intervals around the fundamental. Mathematically, the resultant frequency is the two sine waves multiplied together.

Because the resultant waveform derives from mathematical relationships rather than the overtone series, the resultant sound is unnatural and unusual. 19080s music used synthesizer like the Yamaha DX7 to exploit these new tones. Other musical timbres that do not follow the overtone series are bell tones and percussive sounds, which tend to have randomness associated with their overtones.

Autotune

Another modulation of frequency in popular music is pitch shifting. Before computers musicians could raise or lower the pitch of a sound by increasing the tape speed or turntable speed, but then the tempo of the song also increases, and the result was a chipmunk sounding voice. Now the software can change the pitch without changing tempo.

Pitch shifting is useful for transposition, and widely used in autotuning. Autotune is a revolution for music, but the original overly autotune like "Believe" by Cher has gone out of style and been replaced by more subtle tuning that is often imperceptible.

Pitch shifting also allows automatic harmonization. Software can analyze the chords of a song and provide automatic harmonies. To make the harmonies more realistic and avoid the chipmunk sound, the formats do not move, just the fundamental pitch.

Quantizing

Error correction like autotune applied to timing is called quantization. Quantization moves sound that are not exactly on time to the nearest correct place. Doing the opposite, taking sounds that are too precise and unnatural such as computer-generated tones, and adding some randomness is humanization.

Phase

Unfortunately, when the technology starts moving things around or changing the pitch, the notes do not line up the way then should or out of phase. To understand phase as an introduction to time based effects, consider the equation for the sine wave, y=sin(x). Instead of the parent function, consider a function with all possible transformations

f(x)=a*sin(b(x-c)) + d

The added letters represent ways to transform the simple sine wave into any sine wave.

The letter d is the offset, which signifies where the sine wave starts. This is usually set to one atmospheric pressure, but could change with altitude adjustment or temperature changes. Since sound is usually relative to the sounds around it, the d is usually ignored or set to 0 with the understanding of constant air pressure.

The a variable represents the amplitude of the wave. A higher a corresponds to a louder sound.

The term b represents the frequency of the sound. This term works inversely to what might be expects, in that a higher value of b corresponds to a lower pitch and vice versa.

The c represents the phase of the wave a sine wave begins at 0 and starts growing, but another wave plotted on the same axis may not start exactly at the same place. The c represents the shift horizontally.

In amplitude modulation, the a is another sine wave that affects the carrier wave. The math would be

f(x,a)=sin(a)sin(b(x-c))

And in frequency modulation, b is variable.

f(x,b)=a*sin[sin(b)(x-c)]

Phase is important because two or more sounds interact together in the air. If one wave is compressing the air molecules while the other wave tries to rarefy the air, the two waves are out of phase and destructive interference occurs. Conversely, if the two waves are in sync with each other, constructive interference produces a louder sound at that frequency. Phase is an important part of mixing music that is difficult to her for an untrained listener but can have a profound effect masking or enhancing frequencies in the mix.

Changing phase can also have beneficial effects. A signal copied and then delayed slightly produces an effect called a phase shifter as the two waves interact with each other. A slightly longer phase shift is a flanger, coined from when engineers would put their finger slightly on the flange of the tape reel to slow it down slightly.

More complex phase shifting becomes a chorus sound, similar to the sound of multiple people singing with slight variations in pitch. A basic phase shift formula would be

f(x,c) = a*sin(b(x-sin(c))

The previous formulas showed only one additional variable, but any combination is possible, leading to

f(x,a,b,c) = sin(a)(sin[sin(b)(x-sin(c)])

And then each of the sine waves could have their own amplitude, frequency, and phase shifts

f(x,a,b,c)=a₁sin(b₁(a-c₁))(sin(a₂sin(b₂(b-c₂))(x-sin(a₃(sin(b₃(c-c₃)))))

(Note that the subscripted values a₁, b₁, et cetera, are usually Greek characters in FFTs, simplified here. FFT math is extremely complicated, and out of the scope of this document.)

And the variables a, b, or c need not be sine waves, but can be any function. This formula

F(x,b)=sin(tan(b)(x)) would sweep through all frequencies from high to low repeatedly.

Larger phase differences stop sounding like a chorus effect and begin to sound like two different sounds. Doubling is effective as an effect or it can simulate stereo by delaying between speakers. Slightly longer differences enhance the doubling to a slapback effect.

Echo

As the difference between the waves lengthens, phase is no longer a consideration and now an echo effect known as a delay occurs. Echo is a natural phenomenon occurring when a sound wave strikes a surface. Depending on the material struck, frequencies may be a_bsorbed, reflected, refracted, or transmitted.

Absorbed sounds filter out, with higher frequencies more likely a_bsorbed. Material such as fiberglass a_bsorbs most sound. Refracted sounds strike a rough surface and dissipate in different directions in the air. Some materials transmit the sound into the next space. Metal transmits sound waves well. Glass allows low frequencies to pass through while reflecting higher frequencies. The reflected sounds echo back, and delay mimics this echo.

This reflection of sound inverts the phase of the sound wave. Phase also comes into play in listening environments. In a room with parallel walls, a sound can bounce back and forth between the walls causing constructive or destructive interference, depending on frequency.

Sound has a velocity traveling through different materials. The speed of sound through air is 1130 feet per second at 70 degrees Fahrenheit, then adjusting 1.1 feet per second for every degree of temperature difference. Since sound is measured in Hz, or cycle per second, calculation the wavelength of sound is simple, just divide the speed of sound by the frequency (feet per second/cycles per second=feet per cycle). So, a sound of 1130 Hz would have a wavelength of 1 foot.

Phasing problems occur when the wavelength matches the distance between parallel surfaces. Lower frequencies have larger wavelengths. A frequency of 113Hz, around the note a on the electric bass, has a wavelength of 10 feet. So, in a room with 10 foot parallel walls, the sound would bounce off, inverting its phase, and the inverted sound would mix with the original sound casing destructive interference. The overtones in the sound also reflect, with some overtones adding together destructively and others constructively, changing the nature of the timbre.

To minimize the effect of phasing, architects design spaces without parallel surfaces. But sound interaction is complex as multiple echoes bounce from wall to wall, and some phasing occurs in any room. To adjust for room differences, an engineer projects pink noise into the space, then analyses the resultant sound mix with a microphone. Then, equalizer counteract any frequencies changed by the room anomalies.

Delay

Delay is prevalent in music, and mathematical patterns help set delays. Delay settings, mimic the natural pattern of echo, for example delays may contain filters which mimic the a_bsorption of material. The most important setting on a delay is delay time, often measure in milliseconds.

An important use of delay times is in setting remote speakers in a large space. Since sound has a noticeable speed, electronically delivered sound, which travels at the speed of light, reaches the speakers before the sound from the stage. The listener perceives the first sound heard as the source, and the second as an echo, so the brain places the source of the sound to the speaker, not the stage, causing an unnerving effect.

To mitigate this problem, set a delay of the speakers to match their distance from the stage. A distance of 113 feet divided by 1130 feet per second gives 0.1 seconds, or 100 ms delay time to match, adjusting accordingly for temperature.

Similar mathematics is useful to have the delay match the beat of the song, enhancing the rhythmic effect of the delay. Music with a tempo of 60 beats per minute (bpm) would have a natural delay time of 1 second, or 1000ms. To calculate delays to match the beat of the song,

Delay time =6000ms per minute/bpm of the music.

Multiples of the resultant delay time also have musical effect. To have the beat occur on half beats, just multiply by two, or every other beat divide by 2.

Reverberation

Multiple delays may occur on the same effect, causing multiple echoes. The echoes usually have a decay envelope so that each successive echo has a lower amplitude that the one before it. Each echo can have an echo of its own, called feed_back. Use of high levels of feed_back causes the echoes to mix together into a mix so that individual echoes are no longer discernable, but interpreted by the brain as reverberation.

Reverb mimics the groups of reflections occurring in a natural space. The first sound reaching the ears is on a direct line from the sound source, and the brain localizes that sound as the source. Next, sounds that reflects off near surfaces (first order reflections) reach the ear, slightly softer and filtered by the a_bsorption of the reflecting material. Then, some sound bounces off multiple surfaces before reaching the ear (second order reflections), again becoming softer, more filtered, and effected by constructive and destructive interference. The brain uses this complex information to decode the acoustics of the space.

Reverb comes in different types, mimicking the types of spaces that produce reverb. Room reverbs tend to be shorter, mimicking recording in a small room. Hall reverbs are larger, with variations such as cathedral for very large spaces. Music composed for these large spaces such as renaissance polyphony takes advantage of the natural reverberation for effect.

Some reverbs are contrived. A plate reverb mimics the old school process of projecting the sound into a large metal plate for a brighter reverb. Spring reverb comes out of old amplifiers that contained springs, popular is surf music.

Sampling

Very long delay times without any decay became the basis for musical sampling. Sampling is taking a musical phrase and repeating it as a basis for a backing track. Digital sampling techniques revolutionized music production and are the basis for much rap and hip-hop music.

Taking short samples allowed conversion from the analog domain to the digital domain. Sound breaks down into very short units, often 44,100 samples per second. The 44,100 number for CD quality (Red Book) sound came from the frequency limits of the human ear. The human ear hears up to 20,000 Hz. To take an accurate sample, a minimum of two samples per cycles are necessary, leading to 40,000 samples per second. The extra space is headroom to account for filter slopes. Today engineers commonly sample at higher rates as high as 192 kHz to decrease the errors in processing and allow for phenomena such as localization, which may have influences faster than 20 kHz.

The samples also measure the amplitude changes in discreet unit since computers do not do well with smooth curves. Typical sampling breaks down the amplitude into bits (computer 1s and 0s). Red book audio standard is 16 bits, or 65,536 steps of amplitude breakdown, although many engineers prefer 24 bits for higher quality.

Rhythm

Of course, musicians were manipulating time long before the advent of technology. The patterns presented here have grown from single tones to stacks of tones to repeated notes, and now have come full circle to the fundamental aspect of music, repeated tones forming rhythm. Rhythm is arguable the most important aspect of music.

In the broadest sense, music is simply sound organized in time, and does not require a set rhythm. Musical elements can be placed at non-repetitive places in time, as found in many styles like ambient music or slow introductions to songs (think early Pink Floyd). In fact, a set of all possible music would have more music without a set rhythm than with a steady rhythm.

But humans find music with a set rhythm satisfying. Even before the ears develop in the womb, the developing human feels repeated sounds such as the mother's heartbeat, breathing rhythm, and the ba_by's heartbeat later. In life, rhythms surround us, footsteps, waves on the beach, or a noisy dryer. Music mimics these rhythms. And this pulse represents well through mathematical patterns.

Some music, such as some African folk drumming, uses a steady rhythm without a set grouping, but most music breaks the rhythms down into repeating groups. The most common grouping is four pules per group (measure) repeated. Most Western music chooses a set group, and repeats this group throughout the song, notating the group with a time signature.

The basic representation of the pulse if often a quarter note. The quarter note is a whole note divided into two half notes then divided again, so mathematical property of a geometric series comes into play. This division continues to eighth notes, 16^th, 32^nd, 64^th, and theoretically on to infinitely small division based on dividing each pervious division by two.

Pulses may be in groups other than 4. Keeping the pulse the same, groups could include two quarter notes per grouping (measure) indicated by 2/4, three per measure, 3/4, a waltz feel, or any number 5/4, 6/4, et cetera. And other divisions could arbitrarily receive the pulse, so 2/2 time is essentially the same as 4/4, but the groupings suggest playing twice as fast, or "cut" time.

In a time signature of 3/8, three eighth notes per measure is essentially the same division notated differently. But in practice, 3/8 tends to suggest a faster division of the pulse into three with one pulse per measure. The more common 6/8 time traditionally has two beats per measure each with three divisions. These compound time signatures establish a different rhythmic feel than the time signatures based on divisions of 2.

Duple time signature simulates the division of the compound time signatures by using a triplet, placing three notes in the space of two. Advanced music can put any group of notes into a beat, so complex music such as some songs by Steve Vai place 5, 7, 9, or any number of divisions per beat.

Polyrhythms

These divisions placed against each other form interlocking patterns. Three against two would align

2--|x--x--|x--x--|

3--|x-x-x-|x-x-x-|

three against 4

3--|x---x---x---|x---x---x---|

4--|x--x--x--x--|x--x--x--x--|

Traditional African music utilizes these polyrhythms to great effect, and some modern drummers also use polyrhythms.

Time signatures do not need to constant, either. A measure may butt up against another measure with any time signature. Progressive bands like Dream Theatre use multiple rhythms to great effect (Listen to "Dance of Eternity"). The music can be disconcerting, though, since rhythm is reminiscent of the heartbeat, and an irregular heartbeat makes us uncomfortable.

Music then weaves together different musical elements for interest. A standard rock rhythm alternates a kick drum and a snare drum, weaving in high hats and cymbals.

Melody

Weaving together pitches with rhythms leads to melodies. Gregorian chant, firm which all Western music evolves, began as a simple line chanted by monks. Since the church had the funds and education to write down the melodies sung by different monasteries, musicians began to use these basic patterns to build more complex music.

Texture

The texture of a single line is known as monophonic texture. Combining two melodies on top of each other would be biophonic music, and more than two would be polyphonic. Polyphonic music of the middle ages, renaissance, and into the baroque era is often contrapuntal, with different lines counterbalancing each other in interesting combinations. This counterpoint is why voice leading like the chromatic harmonies exists even now.

Even though several musical melodies occurred at once without necessarily being broken down into set measures, the arrangements tended to end on what the modern ear would hear as a chord, or at least a perfect fifth. Eventually these chord structures evolved into common cadences, or end points to phrases.

As composers started to develop an ear for chords working together, homorhythmic music developed, with all lines moving in synch in a modern style chord progression (think the introduction to "Bohemian Rhapsody" by Queen). Most modern music is a combination of homorhythmic and monophonic texture known as homophonic texture, melody plus accompaniment. One more texture, heterophonic, is more than one variation on a melody at the same time, as might be found in some jazz music and twentieth century composition.

Cadences

Patterns also appear on larger scales in music. Again, considering the set of all possible music statistically, musical phrases could begin or end on any chord. However, humans prefer music that mimics language patterns. Sentences end with punctuation, while phrases end with a cadence.

The most common final cadence is V-I, discussed in detail previously. Alternate variations on V-I include vii_o-I, V-I, V7-I, along with all the variations of V7 chords and substitutes for V. A pre-dominant phase based on subdominant variations IV-V-I, ii-V-I, et cetera often precedes the dominant. This authentic cadence denotes finality to a phrase.

Cadences that also end V-I with the root note as the highest and lowest tones are perfect authentic cadences, often found as the very last chords in a music from the classical era. In a minor key, the borrowed I chord often substitutes for the naturally occurring I chord, iv-V-I replacing iv-V-I for a more optimistic ending to a sad sounding piece. The altered note is a Pickard third.

A softer but still prevalent cadence is IV-I, known as a plagal cadence. The plagal cadence is prevalent in church music, and is known as the amen cadence.

A cadence that mimics a question mark is the half cadence ending on V. A half cadence could be part of a call and response pattern. The composer plays a phrase ending on the half cadence, then repeats the phrase with variation, ending on a final cadence.

A deceptive cadence mimics an authentic cadence, but ends on the relative minor vi instead of I, IV-V7-vi. This cadence is often a transition to the minor key.

Jazz music uses turnarounds at the ends of parses to move to a section or return to a repeated section. Typical turnarounds according to Wikipedia include

I–vi–ii–V

I-VI-ii-V

I–VI–II–V

I–♭iii^o–ii⁷–V⁷

I–vi–♭VI⁷^♯¹¹–V

V–IV–I

I–♭III–♭VI–♭II⁷

iii-VI-ii-V

Form

Music again mimics languages in larger scales. Just as a cadence implies a sentence structure, a section in music mimics a paragraph. Sections in music are represented by capital letters, not to be confused with chord symbols.

The simplest form, binary form, has two contrasting sections, AB, often repeated AABB. A variation popular in jazz is the ternary form, AABA. The B section is the bridge, or middle eight, while the A section is the head.

Classical music developed into more complicated forms. The rondo form ABACABA introduces the concept of a developmental section, C, that can vary from the two main forms. Development is most prevalent in the sonata form.

Sonata forms have many variations, but tend to have similar elements. They may begin with an introductory section, but the main part of the form is called the exposition. The exposition can have two contrasting themes. After the themes are introduced and concluded, a development section explores the possible relationships among the thematic material. The final main section is the recapitulation, with the themes restated in the tonic key as a resolution. An optional coda section as an ending may follow.

Folk music often uses strophic form, ABABAB with alternating sections. The A section is the verse and the B section is the chorus. The A section varies from verse to verse, but the B section tends to remain the same each time.

Typical song form is

(Intro)-Verse1-Chorus-Verse2-Chorus-Bridge-(Verse3)-Chorus-(Outro).

Optional phrases show in parentheses, with the third verse often a repeat of Verse1 or a_bsent altogether. The bridge section is contrasting material, or possible a solo over the chords of the song, or a rap. The Outro (or coda) may contain repeated choruses often with a fade out.

Music may also be through composed, or chain form. An alternate is theme and variation, with each section being based on the same initial theme.

Groups

Patterns continue to develop as different musical pieces group together. Baroque music grouped songs into the groups sonata di camera or sonata di chiesa (church), often four to six songs together. This evolved into the classical grouping of three movements

Allegro (a fast song), a slower song, and a finale.

Which developed into the classical symphony

Allegro in sonata form

Slow movement

Dance movement, often in 3/4

Finale

Songs also combine for dramatic purposes. An opera or musical groups songs to tell a story. Other classical groups of music include cycles of songs.

Popular music uses albums as grouping. Albums today are often just collections of singles, with the hits at the beginning, but some styles of music have albums that were a single unit, expected to be played from beginning to end. The album Thick as a Brick by Jethro Tull is one long 45-minute song.

Styles

Larger scale groupings are known as styles. Music from a certain era with similar characteristic patterns groups together naturally, and as an aid to finding music in the record store. The evolution of style is the most fascinating example of how simple musical concepts grow into complex forms.

Although a few notations from ancient music exists, deciphering them would be quite difficult without any supporting recordings. Greek music has enough descriptions to piece together the use of modes and mathematical relationships in music. These modes are apparent in Gregorian chant, among the first notated music that is decipherable today.

Since Pope Gregory I (ca. 504-670) ordered monastic chants be transcribed, musicians could compare and evolve the musical patterns. Renaissance music used melodies to create polyphonic music. Later the music became more choral and regulated. Baroque music ornamented these chords to extreme levels. Classical music formalized everything, creating music with set forms and strict rules. Romantic music began breaking and evolving the rules into new soundscapes. Twentieth century classical music shattered all the rules and began composing based on patterns.

Popular music included influences from classical music and relied heavily on folk traditions. Slaves composed soulful melodies that reflected their horrible living conditions. Theses melodies became spirituals. Thee spirituals affected early jazz music in producing blues and later race music in a divided nation.

Appalachian folk traditions became American country music. Styles form church music influenced folk music, and vice versa. Operas became musicals, with Gilbert and Sullivan and later into staged productions with more popular styling's.

The influence of recording technology had a huge impact on music. Now music from different cultures could influence musicians not able to her live music. Jazz became swing with a more popular appeal. The youth culture began to have more prominence.

Race music, country, and popular music all joined together into an art form known as rock and roll. Now white audiences heard music influenced by segregated society, first mimicked by white artists, and the slowly accepting music from all cultures into the mainstream. British artists combined rock and roll with more blues influence to create rock music.

Then The Beatles and other British invasion artists experimented with more avant garde techniques, influence by the drug culture and the social upheaval of the sixties. The seventies produced more polished arena rock and rock became big business. Country music became more influenced by rock music. Disco dance music became popular. Punk rock reacted to the commercialization of music into a rawer form. Heavy metal music began with Black Sabbath and grew into its own genre.

Technology continued to influence music. Recording techniques improved and influenced music. Synthesizers and sequencers became commonplace. In the 1980s MTv made music into more of a visual medium than before. Heavy metal became a popular art form and more stylized with band more known for their hair than their music.

Early rap artists revolutionized music by removing melodies and replacing them with rhythmic speech patterns. Sampling also became common as artists mixed together a pastiche of samples of other music. The rap music became popular with young white audiences because of its authenticity and novelty.

In the 1990s the Seattle based movement known as grunge revolutionized rock. 80s hair bands with their elaborate costuming and virtuosic style became passé. The 90s saw rock slowly dying a the predominant popular music form, to be replaced by rap and its offshoot, hip-hop. Napster revolutionized music distribution so now purchasing music was not necessary. Loss of revenue hurt many artists.

The 2000s saw the popularity of more women artists, from divas such as Mariah Carey to the folksier Lilith Fair artists. Rock became nurock, and slowly lost popularity. Country music is more heavily influenced by pop to the point where they are indistinguishable. And the technology continues to evolve, with autotuning, auto-harmonization, quantizing, and looping becoming more commonplace.

Each of these steps builds on the influences of previous music, either adapting to the old styles or rejecting them. The more one studies the history of music, the more amazing details of how musical styles grow organically are apparent.

Psychoacoustics

A discussion of musical pattern would not be complete without thinking a_bout how the human body interprets sound. The way humans perceive sound is far different than the mathematical analysis of acoustics implies. Much of the interpretation of sound becoming music comes from biology.

The first part of the body affecting sound is the ear. The ear not only funnels sound into the body, but also filters and amplifies frequencies because of the shape of the ear. Sound from behind the head tends to have fewer high frequencies, for example. The shape of the ear is how humans can perceive if a sound is directly above them, an important evolutionary skill if one is trying to avoid being eaten by tigers.

Since humans have two ears, they perceive sound in stereo. Sound form the side reaches one ear slightly before the other, and this timing difference aids in localization. The sound reaching the second ear is also a bit softer and has fewer high frequencies. Reflected reverberation also helps humans pinpoint a sound source quite accurately as more reverberation comes from the direction of the source.

The shape of the ear canal also affects the sound. Since the ear canal is a tube, a natural resonant frequency occurs, just as in any flute-like instrument. The natural resonance of the ear canal is a_bout 6 kHz, so tones with that frequency are a bit louder than others, Humans find this frequency pleasant because of it familiarity, so engineers often boost the frequency for effect.

The eardrum itself is not a perfect microphone and colors the sound. Eardrums may be damaged by loud sounds or illness. The bones in the inner ear are not perfect transmitters of sound either.

Much of the physical limitations of hearing occur at the cochlea. The cochlea's hair-like structures of differing length vibrate with the sound coming in. Sorter hairs vibrate in tune with higher frequencies. Loud noises may damage these hairs lading to hearing loss. As humans age, these hairs die, with the shorter hairs tending to die first. So, older people have a narrower frequency range than the textbook 20-20 kHz range. Student who know this set their cell phone ring tones to pitches higher than the teacher can hear to avoid detection. Shop keepers may project high frequencies to drive away pesky teenagers and not affect their older preferred clients.

The inner era also has a built-in compression mechanism which constricts the inner ear for protection. Sort, loud sounds such as a gunshot damage the ear because they happen too quickly for the ear to adjust. Humans have such a large dynamic range, for 0 dB SPL for some people all the way up to 160 dB SPL (not recommended) for some frequencies in the middle range. But after a while the hairs in the cochlea just cannot vibrate any more, and thus the limit to hearing.

The hairs also mask the vibration of other hairs nearby. Loud sounds limit the vibration of the cochlea even in frequencies not heard at the time. Frequencies near other frequencies tend to get masked, as the hairs are all too close together to vibrate at the same time.

But the organ that distorts sound the most is the human brain. The brain is proficient at filling in the blanks of a perceived pattern, which is why jazz substitutions wok. The brain of an experiences jazz listener hears the notes being skipped even though they are not there. The brain also imagines low frequency notes played on earbuds that are too small to produce those frequencies. This interpolation of sound based on previous experiences is shy people can enjoy even poorly recorded music; they just imagine what the music should be.

The brain often hears musical elements not suggested by the acoustic phenomena. Pervious discussions how show a combination of sound waves mixes together interpreted in the brain as a musical note. These acoustical phenomena process in the brain to create the music. The way technical measurements interpret sound differs from the musical interpretation is shown in the following chart:

Comparison chart technical vs. musical terms

Frequency	Amplitude	Overtone	Beats per	Minutes
440 Hz	85 dB	Series	Minute	Seconds
Pitch	Dynamics	Timbre	Rhythm	Measures
The note a	Forte	Tone color	Allegro	Beats

Aesthetics

Possibly, music contains no patterns at all inherently, the brain simply projects its need to find patterns onto sound, and music without pattern is just not recognized as music. The old philosophical trope, "if a tree falls in the forest and no one is there to hear it, does it make a sound?" comes to mind. This point could be argued, but certainly the sound that the tree might make would not be music. A definition of music must include the human mind's interpretation of sound to be valid. The sound of a falling tree could be musical in certain contexts. But calling all sound music is too broad since if all sound were music, then the term "music" would be unnecessary.

Music plays an important part in the human psyche because of its attachments to emotions and meaning. Other physical phenomena follow patterns just as complex as music, but something a_bout musical patterns touches that ephemeral part of what "being human" might be. So, what is the purpose of music?

A formalist would say that music is an art form within itself, separated from the listener. Many twentieth century composers took the attitude that music "sounding good" or pleasing an audience was not as important as its form being well thought out. Elliot Carter had the idea to compose without worrying a_bout the listener, being more concerned with the piece and the performer. Bill Nye once joked that, "the peculiar characteristic of classical music is that it is really so much better than it sounds."

Others see a more functional purpose for music. Patriotic songs are meant for inspiration and supporting a political ideology than their form. Elevator music an phone hold music also has a specific function, to cover dead air just as wallpaper covers bare walls. Music at a party also has the function of covering awkward silences, and too complex music at a social situation might distract from the social purpose. Some groups define certain styles as "our music" and are protective of musical genres for political, ethnic, or social reasons.

But the true purpose of music must have to do with its connection to the human emotions, or the aesthetic. Music is the least inherently representational of the art forms. A painting often looks like an a_bstraction of something from the physical world, but music only refers to other musical patterns and traditions. People associate musical patterns with emotions—minor keys are "sad" because of tradition. Other cultures did not hear the minor modes as sad, so Israeli music seems sad to Western ears, but was not sad in its original context.

This easy attachment to emotions is what makes music ubiquitous and powerful. So, all the patterns of music developed here previously are less important that the effect of those patterns on the human psyche. Previous exploration of patterns showed how the more organized music got, such as in twelve tone music, the more random it actually sounded. We also saw that the extreme limit of following a pattern is to break the pattern down completely, rejecting all patterns. So, now the exploration of music, which began with purely scientific explanation of sound waves that built into musical elements, has now come to its conclusion that music really is a model of ephemeral human emotions.

But music is more than just a mental concept; music enhances all three parts of the human, the body, mind, and spirit. Physically music is difficult, and requires training and coordination of the muscles. Intellectually, the study of the musical patterns stimulates the mind. Metaphysically, music represents the intangible elements of human experience that are difficult to represent within the limits of language. Music can represent those aspects of being human that we know exist, but have a hard time explaining, such as emotions.

Evolution

Music exists in every culture known, even in cultures that attempt to suppress it. So, the existence of music must be inherent to our beings, or why would it survive? Spiritual people have an easy answer to why music exists, a divine being blessing humans with a gift, but from an evolutionary standpoint the question of why music evolved is more complicated.

To understand the evolutionary music, consider the first rock band. No, way before Bill Hayley or Chuck Berry, like when a caveman started banging two literal rocks together and liked the sound. The banging of the rocks had an effect on the caveman and the culture.

An immediate effect might have been that the other cavemen were frightened by the noise and ran away. This process would help establish dominance in a primate culture. Primates exhibit this pattern of noise making as intimidation today.

Other cavemen would observe the behavior and copy to establish their own dominance. The cavemen who make the music moved up the dominance hierarchy, and also began to develop increased coordination and other health benefits. The musicians in the group now have increased dexterity and are good with a rock, which helps when flinging rocks at an antelope for food and other necessities.

The female cavemen noticed this increase in a_bility and began to see the musicians as potential mates. So, the musicians were more likely to have offspring, and these offspring would inherit the music tendencies. Musicians even today seem to have an advantage in potential sexual encounters.

As the musical patterns evolved, not only physical dexterity improved, but also mental acuity. Playing music at a young age improves several types of IQ, including spatial-temporal reasoning. One type of reasoning intertwined with music is language.

We have seen how the music people prefer tends to mimic the patterns of language. So, is music mimicking the language, or did the music come first and then language adapted to that? Imagine the musical wailings of a pre-human communicating an emotion and then those wailings developing into early forms of words. Languages today still have pitch inflections to represent meaning; a rising pitch indicates a question in Western languages. Some Eastern languages are even more dependent on pitch. In Chinese, a word can have different meanings depending on the pitch. Determining which came first, the music or the language, would be impossible, and their co-development is more likely, with each importing aspects of the other as they evolve.

Now, the increasing numbers of musicians in societies begin to interact with each other musically. Coordinating musical patterns leads to increased cooperation and social development. Societies with a tighter social bond are more likely to survive in a harsh world.

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c	d	e	f#	g#	a	b
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g	a	b	c	d	e	f
f	g	a	b	c	d	e
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c	d	e	f	g	a	b
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ABSTRACT

Secondary Dominants Based on viio

Secondary Dominants Based on vii_o

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