Kite Guitar chord shapes (downmajor tuning)

Overview

There are many chords on the Kite Guitar to explore, but the obvious place to start is with those of intervallic odd-limit 9 or less. These chords are mostly subsets of the 4:5:6:7:9 pentad or the 9/(9:7:6:5:4) pentad. Thus most of these chords can be classified as either harmonic or subharmonic. The ^m7 and vm7 chords (and their homonyms v6 and ^6) are classified as stacked chords, because they are formed by stacking complimentary 3rds. Many chords fall outside these 3 categories.

These tables list all chords of odd-limit 9, plus a few with downmajor 7ths that are odd-limit 15. The chord shapes are written in tablature, using fret numbers. The root is placed arbitrarily on the 4th fret. In these tables, the interval between open strings is always a downmajor 3rd. This makes the Kite guitar isomorphic, thus a tab like 4 6 3 5 can start on the 6th, 5th or 4th string, and of course any fret of that string. A skipped string is indicated by a period. Alternate fingerings are possible, especially for 2-finger and 3-finger chords.

Chords are named using ups and down notation, see also the notation guide for edos 5-72. Briefly, an up or down in the chord name immediately after the root affects the 3rd, 6th, 7th and/or the 11th, but not the 5th, 9th or 13th. Thus Gv9 is G vB D vF A. Alterations are enclosed in parentheses, as in Cvm7(b5). Additions are set off with a comma (the punctuation mark, not the interval!). In general, the comma is spoken as "add", e.g. Cv,9 = "C down add-9" = C vE G D. Chord progressions are written as Cv7 - vEb^m6 - Fv7 or Iv7 - vbIII^m6 - IVv7.

In general, an odd-limit 15 chord has only one 15-limit interval, and most of the others are much lower odd-limit. For example, the downmajor seven chord has intervals of odd-limit 3, 3, 5, 5, 5 and 15. The many low-limit intervals serve as "glue" to hold together the chord, despite the one 15-limit interval. This is the rationale for focusing on odd-limit 15 chords here and not those of odd-limit 11 or 13, for those chords have multiple intervals of high odd-limit. But see below, at the very end of this section.

These tables are fairly exhaustive. Don't get overwhelmed! The most essential chords are in the first two tables (triads and seventh chords). Here's a printer-friendly chart to get you started, with and without fingerings:

Triads

The alternate names for the voicings are explained in the next section. Other voicings are possible; these are just the most convenient ones. The upmajor chord is a particularly dissonant triad. See "Innate-comma chords" below for augmented triads. Added ninths are shown in parentheses. Adding a major 9th (ratio 9/4, example note D) to the up or down triad increases the intervallic odd-limit only slightly if at all. The up chord is arguably improved by adding a 9th.

chord type ---->	sus4	up or upmajor (up add 9)	down or downmajor (down add 9)	upminor	downminor	sus2	updim	downdim
example, with homonym	C4 = F2	C^ (C^,9)	Cv (Cv,9)	C^m	Cvm	C2 = G4	C^dim or C^o	Cvdim or Cvo
example notes	C F G	C ^E G	C vE G	C ^Eb G	C vEb G	C D G	C ^Eb Gb	C vEb Gb
ratio of the 3rd	P4 = 4/3	^M3 = 9/7	vM3 = 5/4	^m3 = 6/5	vm3 = 7/6	M2 = 9/8	^m3 = 6/5	vm3 = 7/6
ratio of the 5th	P5 = 3/2	P5 = 3/2	P5 = 3/2	P5 = 3/2	P5 = 3/2	P5 = 3/2	d5 = 7/5	d5 = 7/5
close voicing R 3 5 8 (9)
frets	4 6 3 5	4 5 3 5 (2)	4 4 3 5 (2)	4 3 3 5	4 2 3 5	4 1 3 5	4 3 1 5	4 2 1 5
suggested fingerings	2 4 1 3	2 3 1 4	2 3 1 4 2 2 1 4	2 1 1 3	3 1 2 4	3 1 2 4	3 2 1 4	3 2 1 4
open or high-3 voicing R 5 8 3
frets	4 . 3 5 7	4 . 3 5 6	4 . 3 5 5	4 . 3 5 4	4 . 3 5 3	4 . 3 5 2	4 . 1 5 4	4 . 1 5 3
suggested fingerings	2 . 1 3 4	2 . 1 3 4	2 . 1 3 4 2 . 1 4 4	2 . 1 4 3	3 . 1 4 2	3 . 2 4 1	2 . 1 4 3	3 . 1 4 2
high-R voicing 3 5 8 (9) (1st inversion)
frets	5 2 4	4 2 4 (1)	3 2 4 (1)	2 2 4	1 2 4	0 2 4	2 0 4	1 0 4
suggested fingerings	4 1 3	2 1 3	2 1 3	1 1 3	1 2 4	1 2 4	2 1 4	2 1 4
low-5 voicing 5 R 3 5 (2nd inversion)
frets	2 4 6 3	2 4 5 3	2 4 4 3	2 4 3 3	2 4 2 3	2 4 1 3	(difficult)	(difficult)
suggested fingerings	1 3 4 2	1 3 4 2	1 3 4 2	1 4 2 3 1 3 2 2	1 3 1 2	2 4 1 3

Seventh chords

It's generally impossible to voice 7th chords in 1st, 2nd or 3rd inversion close voicings, because the 7th occurs on the same string as the 8ve. Instead voicings are named as close (root position, R 3 5 7), high-3 (3rd raised an 8ve) and low-5 (5th lowered an 8ve). A high-3 low-5 voicing (5 R 7 3) uses all 6 strings, thus is only sometimes possible. A high-3-7 voicing (R 5 3 7) requires 7 strings. Half-dim chords can alternatively be named as dim add-7 chords, e.g. the uphalfdim chord is C^dim^7 or C^o^7, spoken as updim-upseven.

See "Innate-comma chords" below for dim7 chords. The upmajor7 chord C^M7 = C ^E G ^B is a possibility, but it's quite dissonant, with ^M7 = 27/14.

9ths are shown in parentheses. Adding a major 9th (ratio 9/4, example note D) to any of the first 4 tetrads increases the intervallic odd-limit only slightly if at all. The up-7 chord is arguably improved by adding a 9th. The no3, no5 and no7 (i.e. add9) versions of the ^9 and v9 chords are all 9-odd-limit chords.

11th chords include vM11 (4 4 3 3 2 0), v11 (4 4 3 1 2 0), and ^m11 (4 3 3 2 2 1). All these chords contain a wolf 11th. Rather than 8/3, the vM11 and v11 chords have 21/8, and the ^m11 chord has 27/10. The mid-11th, ratio 11/4, is also available. However 4:5:6:7:9:11 is very difficult to play.

chord type ---->	downmajor7 (downmajor9)	up7 (up9)	down7 (down9)	upminor7 (upminor9)	downminor7	uphalfdim	downhalfdim
example, with homonym	CvM7 (CvM9)	C^7 (C^9)	Cv7 (Cv9)	C^m7 = ^Ebv6 (C^m9)	Cvm7 = vEb^6	C^m7(b5) = ^Ebvm6	Cvm7(b5) = vEb^m6
example notes	C vE G vB	C ^E G ^Bb	C vE G vBb	C ^Eb G ^Bb	C vEb G vBb	C ^Eb Gb ^Bb	C vEb Gb vBb
ratio of the 3rd	vM3 = 5/4	^M3 = 9/7	vM3 = 5/4	^m3 = 6/5	vm3 = 7/6	^m3 = 6/5	vm3 = 7/6
ratio of the 5th	P5 = 3/2	P5 = 3/2	P5 = 3/2	P5 = 3/2	P5 = 3/2	d5 = 7/5	d5 = 7/5
ratio of the 7th	vM7 =15/8	^m7 = 9/5	vm7 = 7/4	^m7 = 9/5	vm7 = 7/4	^m7 = 9/5	vm7 = 7/4
close voicing R 3 5 7 (9)
frets	4 4 3 3 (2)	4 5 3 2 (2)	4 4 3 1 (2)	4 3 3 2 (2)	4 2 3 1	4 3 1 2	4 2 1 1
suggested fingerings	3 4 2 2 (1) 3 3 2 2 (1) 1 1 1 1 (1)	3 4 2 1 (1)	3 4 2 1 4 4 3 1 (2)	4 2 3 1 (1) 4 3 2 1 (1) 3 2 2 1 (1)	4 2 3 1	4 3 1 2	4 2 1 1
high-3 voicing R 5 7 3
frets	4 . 3 3 5	4 . 3 2 6	4 . 3 1 5	4 . 3 2 4	4 . 3 1 3	4 . 1 2 4	4 . 1 1 3
suggested fingerings	2 . 1 1 3	3 . 2 1 4	3 . 2 1 4	3 . 2 1 4 2 . 1 1 4	4 . 2 1 3	3 . 1 2 4	4 . 1 1 3
low-5 voicing 5 R 3 7 (9)
frets	2 4 4 . 3 (2)	2 4 5 . 2 (2)	2 4 4 . 1 (2)	2 4 3 . 2 (2)	2 4 2 . 1	(difficult)	(difficult)
suggested fingerings	1 3 4 . 2 (1)	1 3 4 . 2 (2) 1 3 4 . 1 (1)	2 3 4 . 1 2 3 4 . 2 (1)	1 4 3 . 2 (2) 1 3 2 . 1 (1)	2 4 2 . 1

Flat-nine chords are possible. The plain minor 9th is 21/10, which is the sum of 7/5 and 3/2, thus a m9 works with either a perfect or diminished 5th. Examples:

• the upminor-7 flat-9 chord = C^m7,b9 = C ^Eb G ^Bb Db = 4 3 3 2 0
• the upminor-7 flat-5 flat-9 chord = C^m7(b5)b9 = C ^Eb Gb ^Bb Db = 4 3 1 2 0
• the downminor-7 flat-9 chord = Cvm7,b9 = C vEb G vBb Db = 4 2 3 1 0
• the downminor-7 flat-5 flat-9 chord = Cvm7(b5)b9 = C vEb Gb vBb Db = 4 2 1 1 0

The upminor 9th (15/7) is also possible, but hard to play, Example: the downmajor-7 upflat-9 chord = CvM7,^b9 = C vE G vB ^Db. Note that ^Db is enharmonically equivalent to C#, the augmented 8ve. Thus this chord's homonym is vE^m6/C.

Sixth chords

Sixth chords are hard to voice. A close voicing in root position is generally impossible, because the 6th occurs on the same string as the 5th. One solution is to play a riff that alternates between the 5th and the 6th (3/6 in the tab indicates alternating between the 3rd and 6th fret). Another is to omit the 5th, but then the chord can be mistaken for a triad in 1st inversion. It helps to double the root at the octave, i.e. play R 3 6 8 not R 3 6. Another voicing is the low-6 (6 R 3 5) i.e. the 3rd inversion. But this is the same as the close voicing of its 7th chord homonym, and again the chord can be mistaken. A non-ambiguous voicing is low-5 (5 R 3 6), but it can be a difficult stretch. Also the 9th from the 5th to the 6th is usually not a plain 9th, and can be dissonant. The best voicing is high-3-5 (R 6 3 5 or R 6 8 3 5), but with only 6 strings, it often isn't possible. Other possibilities are high-3-6 (R 5 3 6), high-5 (R 3 6 5 or R 3 6 8 5) and high-6 (R 3 5 8 6).

The up-6 chord is particularly dissonant, unless voiced as its homonym, the vm7 chord.

Adding a major 9th (ratio 9/4) to any of these chords will make a wolf 4th with the 6th. A 9th that is a P4 above the 6th (^M9 or vM9) will clash with the 5th. It can be safely added if the 5th is omitted, but then the chord becomes ambiguous. Cv6,v9no5 is the same as vD^9no3 (or vD^m9no3). C^6,^9no5 is ^Dv9no3. C^m6,^9no5 and Cvm6,v9no5 both have an awkward interval from the 3rd up to the 9th: a M7 = 40/21.

Adding an 11th (ratio 8/3) to either the ^m6 or the vm6 chord won't increase the intervallic odd-limit above 9. But a Cvm6,11 chord is the same as an Fv9 chord, and every easy fingering puts the F in the bass, so it's hardly a distinct chord. Adding an 11th to a Cv6 chord makes Cv6,11, which is an FvM9 chord. Again, every easy fingering has F in the bass, and Cv6,11 isn't a distinct chord.

chord type ---->	up-6 or upmajor-6	down-6 or downmajor-6	upminor-6 (upminor-6 add-11)	downminor-6
example, with homonym	C^6 = ^Avm7	Cv6 = vA^m7	C^m6 = ^Avm7(b5) (C^m6,11 = F^9)	Cvm6 = vA^m7(b5)
example notes	C ^E G ^A	C vE G vA	C ^Eb G ^A	C vEb G vA
ratio of the 3rd	^M3 = 9/7	vM3 = 5/4	^m3 = 6/5	vm3 = 7/6
ratio of the 5th	P5 = 3/2	P5 = 3/2	P5 = 3/2	P5 = 3/2
ratio of the 6th	^M6 = 12/7	vM6 = 5/3	^M6 = 12/7	vM6 = 5/3
close voicing for riffing R 3 5/6 (8)
frets	4 5 3/7	4 4 3/6	4 3 3/7 (5)	4 2 3/6
suggested fingerings	2 3 1/4	2 3 1/4	2 1 1/4 (3)	3 1 2/4
close no-5th voicing R 3 6 8
homonyms	C^6no5 = ^Avm	Cv6no5 = vA^m	C^m6no5 = ^Avdim	Cvm6no5 = vA^dim
frets	4 5 7 5	4 4 6 5	4 3 7 5	4 2 6 5
suggested fingerings	1 2 4 3	1 1 3 2	2 1 4 3	2 1 4 3
low-6 voicing 6 R 3 5 (11)
frets	6 4 5 3	5 4 4 3	6 4 3 3 . (7)	5 4 2 3
suggested fingerings	4 2 3 1	4 2 3 1	4 2 1 1 3 2 1 1 . (4)	4 3 1 2
low-5 voicing 5 R 3 6
frets	2 4 5 7	2 4 4 6	2 4 3 7	2 4 2 6
suggested fingerings	1 2 3 4	1 2 3 4	1 3 2 4	1 3 1 4
high-3-5 voicing R 6 (8) 3 5
frets	4 . 7 . 6 4	4 . 6 (5) 5 4	4 . 7 (5) 4 4	4 . 6 (5) 3 4
suggested fingerings	1 . 4 . 3 2 1 . 4 . 3 1	1 . 4 . 3 2 1 . 4 (2) 3 1	1 . 4 (2) 1 1	???

Innate-comma chords

We've covered every chord that maps to a JI chord of intervallic odd-limit 9. However there are many Kite guitar chords that don't, although their individual intervals do. These chords are called innate-comma chords aka essentially tempered chords. Such chords often have a mysterious sound. Almost every easily reachable interval on the fretboard is odd-limit 9. The only exceptions are ~4, ~5, vM7, ^M7, vm9 and ^m9. Thus the majority of Kite guitar chord shapes are intervallic odd-limit 9.

For example, the downadd7no5 chord has 5/4 and 16/9. The interval from 5/4 up to 16/9 is 64/45. But because 41edo tempers out the Ruyoyo comma of only 8¢, 64/45 is equivalent to 10/7. The high-3 voicing inverts this into an even smoother 7/5. This dom7 chord is often appropriate for translating 12-edos V7 -- I cadence: relaxed but not too relaxed. Note that adding the 5th would increase the odd-limit to 27.

The downmajor7sus4 chord (odd-limit 15) also has an innate Ruyoyo comma. The chord is quite striking in close voicing. The interval from 4/3 up to 15/8 is 45/32, equivalent to 7/5. The homonym of CvM7(4) is the sus2addb5 chord F2,b5 = F G Cb C. In 41-edo, Cb is enharmonically equivalent to vB. In chord names, "(b5)" means alter the 5th by flattening it, but ",b5" means add a flat 5th alongside the perfect 5th.

The down7flat5 chord (odd-limit 9) is also innate-ruyoyo. The interval from 5/4 up to 7/5 is 28/25, equivalent to 9/8. The homonym of Cv7(b5) is the Gb downadd7upflat5 chord Gbv,7(^b5) = Gb vBb ^Dbb Fb. Enharmonic equivalences: ^Dbb = C, Fb = vE, and upflat 5th = aug 4th = 10/7.

All three of these chords contain the chord shape 4 1 1. This 3-note "nugget" implies the Ruyoyo comma: 9/8 plus 5/4 equals 7/5. By itself, it's the v,7no5 chord in low-7 voicing. The v7(b5) chord in close voicing (4 4 1 1) also contains the octave inverse of this nugget, 4 4 1. By itself, this inverse nugget makes Cv(b5) = C vE Gb (odd-limit 9). Beware, "C-down flat-5" = Cv(b5) sounds much like "C downflat-5" = C(vb5) = C E vGb = C E ^^F. Fortunately, the latter chord is very unlikely.

The downaddflat5 chord Cv,b5 (odd-limit 15) has both a perfect and a diminished 5th. This chord is best voiced low-5. In other voicings, the two 5ths are on the same string, and one must play a riff that alternates between the two (indicated as 1/3 in the tab, 1st and 3rd fret).

When the added b5 is voiced an 8ve higher, it becomes a v#11, and suggests the downmajor7downsharp11 and downmajor9downsharp11 chords (both odd-limit 15). No need to omit the 3rd, it makes a pleasant M9 = 9/4 with the 11th.

chord type ---->	downadd7no5	downmaj7sus4	down7flat5	down-flat5	downaddflat5	downmaj9down#11
example	Cv,7no5 = Bb2(b5)	CvM7(4) = F2,b5	Cv7(b5) = Gbv,7(^b5)	Cv(b5)	Cv,b5	CvM9,v#11
	C vE Bb	C F G vB	C vE Gb vBb	C vE Gb	C vE Gb G	C vE G vB D vF#
ratio of the 3rd	vM3 = 5/4	P4 = 4/3	vM3 = 5/4	vM3 = 5/4	vM3 = 5/4	vM3 = 5/4
ratio of the 5th	------	P5 = 3/2	d5 = 7/5	d5 = 7/5	P5 = 3/2	P5 = 3/2
ratio of the 7th	m7 = 16/9	vM7 = 15/8	vm7 = 7/4	------	------	vM7 =15/8
other	------	------	------	------	d5 = 7/5	v#11 = b12 = 14/5
close voicing R 3 5 7 (8)
frets	4 4 8 (5)	4 6 3 3	4 4 1 1	4 4 1 (5)	4 4 1/3	4 4 3 3 2 2
suggested fingerings	1 1 4 (2)	2 4 1 1	3 4 1 1	3 4 1 2 3 1 (4)	3 4 1/2	3 4 2 2 1 1
high-3 voicing R 5 7 (8) 3
frets	4 . 8 (5) 5	4 . 3 3 7	4 . 1 1 5	4 . 1 (5) 5	4 . 1/3 (5) 5	4 . 3 3 5 2 (no 9th)
suggested fingerings	1 . 4 . 2 1 . 4 (1) 1	2 . 1 1 4	3 . 1 1 4	2 . 1 (3) 4	3 . 1/2 . 4 2 . 1/3 (3) 4	2 . 3 3 4 1
low-5 voicing 5 R 3 7
frets	(N/A)	(difficult)	0 4 4 . 1	(difficult)	2 4 4 1	2 4 4 . 3 2 2 (7 strings)
suggested fingerings			1 3 4 . 2		2 3 4 1	1 3 4 . 2 1 1
low-7 voicing 7 R 3 (7)
frets	7 4 4 (8)	(N/A)	(N/A)	(N/A)	(N/A)	(N/A)
suggested fingerings	3 1 1 (4)

The Ruyoyo comma implies augmented chords because 5/4, 5/4 and 9/7 add up to 2/1. There are three types of aug chord: upaug, downaug and down-halfaug. (Logically, the last one should be called down-doubledownsharp5 or down-double-up5, but those names are too long.) Each one is odd-limit 9, and each one is an inversion of the others.

The up-halfaug chord has ^M3 and ^^5. Its innate comma is the Zozoyo comma, which equates the octave with 9/7 plus 6/5 plus 9/7. In 1st inversion, it's the upminor-halfaug chord C^m(^^5). In 2nd inversion, it's the up-sesquiaug chord. All three chords are odd-limit 9.

Another possible aug chord is 7:9:11 = up-downsharp5 = C^(v#5) = C ^E vG#. Unfortunately it's very difficult to finger.

The updim7 and downdim7 chords are formed from stacked 6/5's and 7/6's, alternating to make 7/5's. The 7ths are rather dissonant. The updim7 chord has an innate Ruyoyo comma which equates its ^d7 = 42/25 to a M6 = 27/16. The downdim7 chord has an innate Thuzozogu comma which equates vd7 = 49/30 with ~6 = 13/8. Thus its odd-limit and prime-limit are both 13.

chord -->	upaug	downaug	downhalfaug	uphalfaug	upminor-halfaug	up-sesquiaug	updim7	downdim7
example	C^aug	Cvaug	Cv(vv#5)	C^(^^5)	C^m(^^5)	C^(^^#5)	C^o7	Cvo7
	C ^E G#	C vE G#	C vE vvG#	C ^E ^^G	C ^Eb ^^G	C ^E ^^G#	C ^Eb Gb ^Bbb	C vEb Gb vBbb
3rd	^M3 = 9/7	vM3 = 5/4	vM3 = 5/4	^M3 = 9/7	^m3 = 6/5	^M3 = 9/7	^m3 = 6/5	vm3 = 7/6
5th	A5 = ^m6 = 8/5	A5 = ^m6 = 8/5	vvA5 = vm6 = 14/9	^^5 = vm6 = 14/9	^^5 = vm6 = 14/9	^^A5 = vM6 = 5/3	d5 = 7/5	d5 = 7/5
7th	------	------	------	------	------	------	^d7 = M6 = 27/16	vd7 = ~6 = 13/8
close voicing R 3 5 7 (8)
frets	4 5 5 (5)	4 4 5 (5)	4 4 4 (5)	4 5 4 (5)	4 3 4 (5)	4 5 6 (5)	4 3 1 0	4 2 1 -1
fingerings	1 2 2 (2)	1 1 2 (2)	1 1 1 (2)	1 3 2 (4)	2 1 3 (4)	1 2 4 (3)	4 3 2 1	4 3 2 1
high-3 voicing R 5 7 (8) 3
frets	4 . 5 (5) 6	4 . 5 (5) 5	4 . 4 (5) 5	4 . 4 (5) 6	4 . 4 (5) 4	4 . 6 (5) 6	4 . 1 0 4	4 . 1 -1 3
fingerings	1 . 2 (2) 3	1 . 2 (2) 2	1 . 1 (2) 2	1 . 1 (2) 3	1 . 2 (4) 3	1 . 3 (2) 4	3 . 2 1 4	3 . 2 1 4
low-5 voicing 5 R 3 7
	see vaug	see v(vv#5)	see ^aug	see ^(^^#5)	see ^(^^5)	see ^m(^^5)	(difficult)	(difficult)

At the beginning of this article, chords of prime-limit 11 or 13 were dismissed because "those chords have multiple intervals of high odd-limit." But when innate-comma chords are allowed, this no longer holds true. For example, the mid-5th can be interpreted as either 16/11 or 13/9. Each of the following chords contain this interval, but all the other intervals in the chord are at most odd-limit 5, 7 or 9, depending on the chord. The one exception is the vM7(~5) chord, odd-limit 15.

• the downminor mid-5 chord = Cvm(~5) = C vEb vvG = 4 2 2
• the downminor-7 mid-5 chord = Cvm7(~5) = C vEb vvG vBb = 4 2 2 1 or 4 . 2 1 3
• the down up-six chord = Cv,^6 = C vE G ^A = 4 . 7 . 5 4, a homonym of ^Avm7(~5)
• the upminor mid-5 chord = C^m(~5) = C ^Eb ^^Gb = 4 3 2
• the upminor-7 mid-5 chord = C^m7(~5) = C ^Eb ^^Gb ^Bb = 4 3 2 2 or 4 . 2 2 4
• the upminor down-6 chord = C^m,v6 = C ^Eb G vA, a homonym of vA^m7(~5)
• the downmajor mid-5 chord = Cv(~5) = C vE vvG = 4 4 2
• the downmajor-7 mid-5 chord = CvM7(~5) = C vE vvG vB = 4 4 2 3 or 4 . 2 3 5

Note that the mid-5th is spelled as a double-up dim 5th from the chord root (^^Gb) if the 3rd is upped, but as a double-down 5th (vvG) if the 3rd is downed. This avoids the interval from the 3rd to the 5th being spelled with a triple up or down.

Rough draft of an upcoming article

I don't want to create the page until I know the title, but I won't know the title until I write the article!

Categories of scales on the Kite Guitar

A review of 12-edo scales

There are three broad categories of 12-edo scales: pentatonic, diatonic and chromatic:

scale type -->	pentatonic		diatonic			chromatic
scale steps	M2	m3	m2	M2	(A2)	A1 or m2	(M2)
semitones per scale step	2	3	1	2	(3)	1	(2)
example scale	C D E G A C		C D E F G A B C			C Db D Eb E F F# G Ab A Bb B C
scale steps in semitones	2 2 3 2 3		2 2 1 2 2 2 1			1 1 1 1 1 1 1 1 1 1 1 1

Strictly speaking, "diatonic" means a maximally even 5L2s scale, but here it's used more loosely. For example, the harmonic minor scale is considered to be diatonic. The maximally even requirement is relaxed. But in fact, common diatonic scales tend to avoid adjacent minor 2nds. Likewise, pentatonic scales tend to avoid adjacent minor 3rds. Hemitonic pentatonic scales such as C D Eb G Ab C and C E F G B C are considered to be diatonic scales with missing notes, as are most hexatonic scales.

Some music falls entirely within one category. Much world music is entirely pentatonic, much folk music is entirely diatonic, and serial music is entirely chromatic. But often a piece of music falls into two categories. The melody of "Ash Grove" is mostly diatonic, but the occasional augmented 4th makes it somewhat chromatic. Likewise, "Greensleeves" has an occasional major 7th. The melody of "Let It Be" is mostly pentatonic, but the occasional perfect 4th makes it somewhat diatonic. The blues scale 1 b3 4 b5 5 b7 8 is likewise somewhat pentatonic and somewhat chromatic, as are most blues melodies. This overlap in categories is why diatonic scales avoid adjacent minor 2nds, because otherwise it would become a diatonic/chromatic scale.

In addition, there are unconventional scales like the whole tone, diminished and Tcherepnin scales. But these are much less common, because they don't map to a compact shape in the JI lattice. (See "Convexity and the well-formedness of musical objects", Aline Honingh and Rens Bod, Journal of New Music Research, 2005.)

41-edo scales

41-edo has an enormous variety of scales. There are many thousands of unconventional scales, but here I will focus on the ones that map compactly to the JI lattice. These are scales that contain numerous perfect 5ths.

Imperfect degrees in 12-edo have two qualities, major and minor. In 41-edo, there are 7 qualities, and each one implies a color.

quality	downminor	minor	upminor	mid	downmajor	major	upmajor
color	zo	4thwd wa	gu	lo/lu/tho/thu	yo	5thwd wa	ru
prime	7-over	3-under	5-under	11-over/under, 13-over/under	5-over	3-over	7-under

Two notes a perfect fifth apart generally have the same quality. So compact scales use only a few qualities, and thus a small prime subgroup. In color notation, these subgroups are named wa = 2.3, ya = 2.3.5, za = 2.3.7, and ila = 2.3.11. 41-edo doesn't distinguish between the ila subgroup and the tha subgroup 2.3.13, so tha is lumped in with ila.

In practice, 41-edo scales tend to be "fuzzy", meaning that one or two scale notes may sometimes shift by an edostep. For example, a major scale may contain both a M2 and a vM2, and use whichever one is required by the harmony at the moment.

There are four basic categories of pentatonic scales, one for each of the prime subgroups:

scale type -->	wa pentatonic		ya pentatonic				za pentatonic				ila pentatonic
scale steps	M2	m3	vM2	M2	m3	^m3	M2	^M2	vm3	m3	~2	M2	m3	~3
edosteps per scale step	7	10	6	7	10	11	7	8	9	10	5	7	10	12
example scale	C D E G A C		C D vE G vA C				C vEb F G vBb C				C vvE F G vvB C
scale steps in edosteps	7 7 10 7 10		7 6 11 6 11				9 8 7 9 8				12 5 7 12 5

A scale needn't have every single step size on the list in order to be in the category, just most of them. In practice, a non-wa pentatonic scale will often lack a m3 step, as in the examples. But a "fuzzy" pentatonic scale often will have a m3, e.g. C D vE G vA/A C. Ya and za scales generally contain a wolf 5th (either an ^5 or a v5), and would likely become fuzzy to avoid the wolf.

These subgroups can be combined to make another four subgroups. Yala pentatonic scales tend to have wolf 5ths, and thus tend to be fuzzy. A yazala pentatonic scale must be fuzzy, in order to contain so many different step sizes.

scale type -->	yaza pentatonic						yala pentatonic						zala pentatonic						yazala pentatonic
scale steps	vM2	M2	^M2	vm3	m3	^m3	~2	vM2	M2	m3	^m3	~3	~2	M2	^M2	vm3	m3	~3	~2	vM2	M2	^M2	vm3	m3	^m3	~3
edosteps	6	7	8	9	10	11	5	6	7	10	11	12	5	7	8	9	10	12	5	6	7	8	9	10	11	12
example	C D vE G vBb C						C D vE G vvB C						C vEb F G vvB C						C ^Eb/vvE F G vBb C
edosteps	7 6 11 9 8						7 6 11 12 5						9 8 7 12 5						11/12 6/5 7 9 8

There are four basic categories of diatonic scales. In practice, a non-wa scale will often lack a m2 step, unless it's fuzzy. The ya and za diatonic scales have wolf 5ths, and thus tend to be fuzzy.

scale type -->	wa diatonic		ya diatonic				za diatonic				ila diatonic
scale steps	m2	M2	m2	^m2	vM2	M2	vm2	m2	M2	^M2	m2	~2	M2
edosteps	3	7	3	4	6	7	2	3	7	8	3	5	7
example	C D E F G A B C		C D vE F G vA vB C				C D vEb F G vAb vBb C				C vvD Eb F G vvA Bb C
edosteps	7 7 3 7 7 7 3		7 6 4 7 6 7 4				7 2 8 7 2 7 8				5 5 7 7 5 5 7

There are four "combo" categories of diatonic scales:

scale type -->	yaza diatonic						yala diatonic					zala diatonic					yazala diatonic
scale steps	vm2	m2	^m2	vM2	M2	^M2	m2	^m2	~2	vM2	M2	vm2	m2	~2	M2	^M2	vm2	m2	^m2	~2	vM2	M2	^M2
edosteps	2	3	4	6	7	8	3	4	5	6	7	2	3	5	7	8	2	3	4	5	6	7	8
example	C vD vEb F G vAb vBb C						C vvD ^Eb F G ^Ab Bb C					C vvD Eb F G vAb vBb C					C D vE ^^F G vvA vBb C
edosteps	6 3 8 7 2 7 8						5 6 6 7 4 6 7					5 5 7 7 2 7 8					7 6 6 5 5 4 8

Most 41-edo intervals suggest a specific ratio, but those only a few edosteps wide don't. Thus the remaining categories don't imply any prime subgroups. On the Kite guitar, playing a run of notes one fret apart would seem to inherit the term "chromatic" from 12-edo. But we must have a term for 12-edo's chromaticism, which translates to runs played on every other fret. Category names are tentative:

scale type -->	old chromatic (dodecatonic?)		new chromatic (chromatic?)		microtonal (commatic?)
scale steps	m2	A1	vm2	m2	^1	vm2
edosteps	3	4	2	3	1	2

What to call scales with 2, 3 and 4? Or 1, 2 and 3? Or 3, 4 and 5? Maybe have

• wa or ya dodecatonic = 3, 4
• za or yaza dodecatonic = 2, 3, 4
• ila dodecatonic = 3, 4, 5 (but hard to get 12 notes, 10 is easier)

Harmonic scales

In Western music, harmonies often require notes that the melody doesn't. For example, "Auld Lang Syne" has a pentatonic melody but diatonic harmonies. Often the melody is diatonic but the harmonies are at least somewhat chromatic. The score will have accidentals in the piano part but not the vocal part.

41-edo allows 7-limit harmonies, which are almost never diatonic, and tend to be dodecatonic.

otonal (yo and zo): P1 vm2 vM2/M2 vm3 vM3 P4 d5 P5 vm6 vM6 vm7/m7 vM7 P8

scale steps: vm2 A1/~2 vm2/m2 A1 ^m2 m2 A1 vm2 A1 m2/^m2 m2/A1 ^m2 = 2 4/5 2/3 4 4 3 4 2 4 3/4 3/4 4

In 12-edo, a song is generally in a major or minor key, and uses a major or minor scale. 41-edo allows the use of 7-limit chords such as 4:5:6:7. If this is one's tonic chord, both major and minor are used simultaneously. A simple Iv7 - IVv7 progression has both a downmajor and a downminor 3rd. Clearly the major/minor duality no longer applies. Instead, there is an up/down duality.

12-edo scales: choose a 7-note subset, and let the imperfect degrees be either major or minor, or some combination

41-edo scales: choose a 12-note subset, and let all but the tonic, 4th and 5th be either upped or downed.

12-edo: implies a ya subgroup

41-edo: implies both ya and za, and even ila

ya implies heptatonic, za implies pentatonic

ya: 5-over maps to major, so major is more otonal than minor, and for a scale using A B C D E F G, C is the obvious tonic.

za: 7-over maps to minor, so minor is otonal, and for a C D E G A scale, A is the obvious tonic.

ila: 11-over and 11-under both map to neutral, so no obvious tonic.

yaza: D is the obvious tonic for either scale