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Kite's ups and downs notation: Difference between revisions

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=__EDOs with an inaccurate 3/2__=


Not counting the trivial edos 2, 3, 4 and 6, there are only seven such edos. As seen in the above diagram, they are the ones to the left of the heptatonic kite's spine, plus the ones to the right of the pentatonic kite's spine. The ones on the left edge of the heptatonic kite are the fourthward ones like 16edo, and have been dealt with already. 23edo can be notated similarly to 16edo by using a fifth of 13\23 instead of 14\23. That leaves only four edos: 8, 11, 13, and 18.
=__**Scale Fragments**__=


There are two strategies for notating these "oddball" EDOs, besides heptatonic fifth-based notation with ups and downs. One is to switch from heptatonic notation to some other type. The orange kite contains edos for which pentatonic notation is a natural fit, in the sense that no ups or downs are needed. This includes 8edo, 13edo and 18edo.
Besides the trivial EDOs, 1, 2, 3, 4 and 6, which can be notated with standard notation as a subset of 12-EDO, there are five EDO categories, based on the size of the fifth:
"Fifth-less" EDOs (8, 11, 13 and 18)
"Fourthward" EDOs (9, 16 and 23)
"Perfect" EDOs (7, 14, 21, 28 and 35)
"Pentatonic" EDOs (5, 10, 15, 20, 25 and 30)
"Regular" EDOs (all others)
The first two categories never use ups and downs, the next two always do (except for 5edo and 7edo). The regular EDOs may or may not.


The other approach is to use some interval other than the fifth to generate the notation. Earlier I said notating 22edo using an even distribution of note names such as C _ _ D _ _ E _ _ F _ _ _ G _ _ A _ _ B _ _ C was a bad idea because the G-D and the A-E fifths looked perfect but were actually diminished. The reasoning is that 3/2 is an important ratio, and any decent approximation of 3/2 should look like a perfect fifth. But these EDOs don't approximate 3/2 well, so they can be thought of as having both a major fifth and a minor fifth. This negates any expectations of what a fifth should look like.
To summarize an EDO, a scale fragment from C to D is shown, including C# and Db. Examples:
C C#/Db D (12edo)
C Db C# D (17edo)
C C# Db D (19edo)
C C# _ Db D (26edo)
C _ C# Db _ D (31edo)


The scale fragment concisely conveys the "flavor" of the EDO's notation. The C-C# interval is the augmented unison, and if the 2nd key in the fragment isn't C#, ups and downs are required. The only exception is 7edo. For most EDOs, the C-Db interval is the minor 2nd and the C-D interval is the major 2nd. For perfect EDOs, C-Db = d2 and C-D = P2. For fourthward EDOs, C-Db = d2 and C-D = m2. D# is included for these EDOs because C-D# is a M2 just like E-F. For fifthless EDOs, the scale fragment isn't as helpful because you can't deduce the entire keyboard layout from it.


__**Theoretical alternatives for 8edo, 11edo, 13edo and 18edo**__
Every EDO contains a unique scale fragment, and every scale fragment implies a unique EDO. Furthermore, this uniqueness applies to EDOs with alternate fifths: "wide-fifth" 35edo (which uses 21\35 as a fifth) has a different scale fragment than "narrow-fifth" 35edo with 20\35. If an EDO has a fifth of keyspan F and an octave of keyspan O (i.e. it's O-EDO), the minor 2nd's keyspan is m2 = -5F + 3O, and the augmented unison's is A1 = 7F - 4O. These equations can be reversed: F = 4(m2) + 3(A1) and O = 7(m2) + 5(A1). (For perfect and fourthwards EDOs, substitute M2 for m2.)


8edo octatonic (every note is a generator)
In the chart below, 13edo and 18edo use the narrower fifth.
D E F G H A B C D
||= 5edo ||= pentatonic ||=  ||= C/Db ||= C#/D ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||
P1 - P2 - P3 - P4 - P5 - P6 - P7 - P8 - P9
||= 6edo ||= fifthless ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||
requires learning octatonic interval arithmetic and staff notation
||= 7edo ||= perfect ||=  ||= C/C# ||= Db/D ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||
 
||= 8edo ||= fifthless ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||
11edo heptatonic narrow-fifth-based, fourthwards, # is vv (3/2 maps to 6\11 perfect 5th):
||= 9edo ||= fourthward ||=  ||= C/Db ||= C#/D ||= D# ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||
D E * * F G A B * * C D = D E F# F~ F G A B B~ Bb C D
||= 10edo ||= pentatonic ||=  ||= C/Db ||= * ||= C#/D ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||
fourthwards chain of fifths: m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7
||= 11edo ||= fourthward ||=  ||= C ||= D ||= C# ||= D# ||=  ||=  ||=  ||=  ||=  ||=  ||
P1 - M2 - ~2/M3 - m2/~3 - m3 - P4 - P5 - M6 - ~6/M7 - m6/~7 - m7 - P8
||= 12edo ||= regular ||=  ||= C ||= C#/Db ||= D ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||
problematic because M3 = 2\11 is narrower than m2 = 3\11
||= 13edo ||= fourthward ||=  ||= C ||= D ||= * ||= C# ||= D# ||=  ||=  ||=  ||=  ||=  ||
 
||= 14edo ||= perfect ||=  ||= C/C# ||= * ||= Db/D ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||
11edo nonotonic narrow-fifth-based, fifthwards with no ups and downs (3/2 maps to 6\11 = perfect 6th):
||= 15edo ||= pentatonic ||=  ||= C/Db ||= * ||= * ||= C#/D ||=  ||=  ||=  ||=  ||=  ||=  ||
nonotonic fifthwards chain of sixths: M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9
||= 16edo ||= fourthward ||=  ||= C ||= C#/Db ||= D ||= D# ||=  ||=  ||=  ||=  ||=  ||=  ||
P1 m2 M2/m3 M3/m4 M4 P5 P6 m7 M7/m8 M8/m9 M9 P8
||= 17edo ||= regular ||=  ||= C ||= Db ||= C# ||= D ||=  ||=  ||=  ||=  ||=  ||=  ||
requires learning nonotonic interval arithmetic and staff notation
||= 18edo ||= fourthward ||=  ||= C/Db ||= * ||= C#/D ||= * ||= D# ||=  ||=  ||=  ||=  ||=  ||
 
||= 19edo ||= regular ||=  ||= C ||= C# ||= Db ||= D ||=  ||=  ||=  ||=  ||=  ||=  ||
11edo pentatonic wide-fifth-based, fifthwards, # is ^^ (3/2 maps to 7\11 6th):
||= 20edo ||= pentatonic ||=  ||= C/Db ||= * ||= * ||= * ||= C#/D ||=  ||=  ||=  ||=  ||=  ||
D * * E G * * A C * * D
||= 21edo ||= perfect ||=  ||= C/C# ||= * ||= * ||= Db/D ||=  ||=  ||=  ||=  ||=  ||=  ||
pentatonic fifthwards chain of fifthoids: m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7
||= 22edo ||= regular ||=  ||= C ||= Db ||= * ||= C# ||= D ||=  ||=  ||=  ||=  ||=   ||
P1 - ms3 - ^ms3/vMs3 - Ms3 - P4d - ^P4d/d5d - A4d/vP5d - P5d - ms7 - ^ms7/vMs7 - Ms7 - P8d
||= 23edo ||= fourthward ||=  ||= C ||= C# ||= Db ||= D ||= D# ||=  ||=  ||=  ||=  ||=  ||
pentatonic plus ups and downs is doubly confusing!
||= 24edo ||= regular ||=  ||= C ||= * ||= C#/Db ||= * ||= D ||=  ||=  ||=   ||=  ||=  ||
 
||= 25edo ||= pentatonic ||=  ||= C/Db ||= * ||= * ||= * ||= * ||= C#/D ||=  ||=  ||=  ||=  ||
11edo octatonic wide-fifth-based, fifthwards, no ^/v (3/2 maps to 7\11 = perfect 6th):
||= 26edo ||= regular ||=  ||= C ||= C# ||= * ||= Db ||= D ||=  ||=  ||=  ||=  ||=  ||
octatonic chain of 6ths: m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7
||= 27edo ||= regular ||=  ||= C ||= Db ||= * ||= * ||= C# ||= D ||=  ||=  ||=  ||=  ||
P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9
||= 28edo ||= perfect ||=  ||= C/C# ||= * ||= * ||= * ||= Db/D ||=  ||=  ||=  ||=  ||=  ||
requires learning octatonic interval arithmetic and notation
||= 29edo ||= regular ||=  ||= C ||= * ||= Db ||= C# ||= * ||= D ||=  ||=  ||=  ||=  ||
 
||= 30edo ||= pentatonic ||=  ||= C/Db ||= * ||= * ||= * ||= * ||= * ||= C#/D ||=  ||=  ||=  ||
13edo heptatonic narrow-fifth-based, fourthwards, sharp = ^^^ (3/2 maps to 7\13 perfect 5th):
||= 31edo ||= regular ||=  ||= C ||= * ||= C# ||= Db ||= * ||= D ||=  ||=  ||=  ||=  ||
D E * * * F G A B * * * C D
||= 32edo ||= regular ||=  ||= C ||= Db ||= * ||= * ||= * ||= C# ||= D ||=  ||=  ||=  ||
fourthwards chain of fifths: M2 - M6 - M3 - M7 - P4 - P1 - P5 - m2 - m6 - m3 - m7
||= 33edo ||= regular ||=  ||= C ||= C# ||= * ||= * ||= Db ||= D ||=  ||=  ||=  ||=  ||
P1 - m2 - m3 - vM2/^m3 - M2 - M3 - P4 - P5 - m6 - m7 - vM6/^m7 - M6 - M7 - P8
||= 34edo ||= regular ||=  ||= C ||= * ||= Db ||= * ||= C# ||= * ||= D ||=  ||=  ||=  ||
problematic because m3 = 2\13 is narrower than M2 = 4\13
||= 35edo ||= perfect ||=  ||= C/C# ||= * ||= * ||= * ||= * ||= Db/D ||=  ||=  ||=  ||=  ||
||= 36edo ||= regular ||=  ||= C ||= * ||= * ||= C#/Db ||= * ||= * ||= D ||=  ||=  ||=  ||
||= 37edo ||= " ||=  ||= C ||= Db ||= * ||= * ||= * ||= * ||= C# ||= D ||=  ||=  ||
||= 38edo ||= " ||=  ||= C ||= * ||= C# ||= * ||= Db ||= * ||= D ||=  ||=  ||=  ||
||= 39edo ||= " ||=  ||= C ||= * ||= Db ||= * ||= * ||= C# ||= * ||= D ||=  ||=  ||
||= 40edo ||= " ||=  ||= C ||= C# ||= * ||= * ||= * ||= Db ||= D ||=  ||=  ||=  ||
||= 41edo ||= " ||=  ||= C ||= * ||= * ||= Db ||= C# ||= * ||= * ||= D ||=  ||=  ||
||= 42edo ||= " ||=  ||= C ||= Db ||= * ||= * ||= * ||= * ||= * ||= C# ||= D ||=  ||
||= 43edo ||= " ||=  ||= C ||= * ||= * ||= C# ||= Db ||= * ||= * ||= D ||=  ||=   ||
||= 44ddo ||= " ||=  ||= C ||= * ||= Db ||= * ||= * ||= * ||= C# ||= * ||= D ||=  ||
||= 45edo ||= " ||=  ||= C ||= * ||= C# ||= * ||= * ||= Db ||= * ||= D ||=  ||=  ||
||= 46edo ||= " ||=  ||= C ||= * ||= * ||= Db ||= * ||= C# ||= * ||= * ||= D ||=  ||
||= 47edo ||= " ||=  ||= C ||= C# ||= * ||= * ||= * ||= * ||= Db ||= D ||=  ||=  ||
||= 48edo ||= " ||=  ||= C ||= * ||= * ||= * ||= C#/Db ||= * ||= * ||= * ||= D ||=  ||
||= 49edo ||= " ||=  ||= C ||= * ||= Db ||= * ||= * ||= * ||= * ||= C# ||= * ||= D ||
||= 50edo ||= " ||=  ||= C ||= * ||= * ||= C# ||= * ||= Db ||= * ||= * ||= D ||=  ||
||= 51edo ||= " ||=  ||= C ||= * ||= * ||= Db ||= * ||= * ||= C# ||= * ||= * ||= D ||
||= 52edo ||= " ||=  ||= C ||= * ||= C# ||= * ||= * ||= * ||= Db ||= * ||= D ||=  ||
||= 53edo ||= " ||=  ||= C ||= * ||= * ||= * ||= Db ||= C# ||= * ||= * ||= * ||= D ||
 
=__**Summary of EDO notation**__=  


13edo undecatonic narrow-fifth-based, fourthwards, 3/2 maps to 7\13 = perfect 7th
undecatonic sixthwards chain of sevenths:
M2 - M8 - M3 - M9 - M4 - M10 - M5 - M11 - P6 - P1 - P7 - m2 - m8 - m3 - m9 - m4 - m10 - m5 - m11
P1 - m2 - M2/m3 - M3/m4 - M4/m5 - M5 - P6 - P7 - m8 - M8/m9 - M9/m10 - M10/m11 - M11 - P12
requires learning undecatonic interval arithmetic and notation


13edo octatonic wide-fifth-based, fourthwards, 3/2 maps to 8\13 = perfect 6th
===__"Regular" EDOs (12, 17, 19, 22, 24, 26, 27, 29, 31-34, and all edos 36 or higher)__===
octotonic chain of sixths: M3 - M8 - M5 - M2 - M7 - P4 - P1 - P6 - m3 - m8 - m5 - m2 - m7
P1 - m2 - M2 - m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7 - m8 - M8 - P9
requires learning octatonic interval arithmetic and notation


18edo heptatonic narrow-fifth-based, fourthwards, sharp = ^^ (3/2 maps to 10\18 perfect 5th)
All regular EDOs use the usual chain of fifths: m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7 etc.
D * E * * * F * G * A * B * * * C * D
Fb - Cb - Gb - Db - Ab - Eb - Bb - F - C - G - D - A - E - B - F# - C# - G# - D# - A# - E# - B# etc.
P1 - vm2 - m2 - vM2 - M2/m3 - vM3 - M3 - ^M3 - P4 - ^P4/vP5 - P5 - vm6 - m6 - vM6 - M6/m7 - vM7 - M7 - ^M7 - P8
fourthwards plus ups and downs plus closed is triply confusing!


18edo nonatonic narrow-fifth-based (3/2 maps to 10\18 = perfect 6th)
**__12edo__:** sharp/flat = 1 key, no ups and downs: C C#/Db D
P1 - vP2 - P2 - vP3 - P3 - vP4- P4 - vP5 - P5 - vP6 - P6 - vP7 - P7 - vP8 - P8 - vP9 - P9 - vP10 - P10
D * E F * G * A * B C * D
requires learning nonotonic interval arithmetic and staff notation
D - D#/Eb - E - F - F#/Gb - G - G#/Ab - A - A#/Bb - B - C - C#/Db - D
P1 - m2 - M2 - m3 - M3 - P4 - A4/d5 - P5 - m6 - M6 - m7 - M7 - P8
perfect = white, major = red, yellow and fifthward white, minor = green, blue and fourthwards white


**__17edo__:** sharp = 2 keys: C Db C# D
D * * E F * * G * * A * * B C * * D
D - D^/Eb - D#/Ev - Eb - E - F - F^/Gb - F#/Gv - G - G^/Ab - G#/Av - A - A^/Bb - A#/Bv - B - C - C^/Db - C#/Dv - D
P1 - m2 - ~2 - M2 - m3 - ~3 - M3 - P4 - ^P4/d5 - A4/vP5 - P5 - m6 - ~6 - M6 - m7 - ~7 - M7 - P8


__**Alternate notation for other edos:**__
**__19edo__:** no ups and downs C C# Db D
23edo pentatonic wide-fifth-based, fifthwards, 3/2 maps to 14\23 = perfect fifthoid:
D * * E * F * * G * * A * * B * C * * D
D * * * * E * * * G * * * * A * * * C * * * * D
D - D# - Eb - E - E#/Fb - F - F# - Gb - G - G# - Ab - A - A# - Bb - B - B#/Cb - C - C# - Db - D
35edo heptatonic wide-fifth-based, sharp = seven ups, 3/2 maps to 21\35 = perfect fifth:
P1 - A1/d2 - m2 - M2 - A2/d3 - m3 - M3 - A3/d4 - P4 - A4 - d5 - P5 - A5/d6 - m6 - M6 - A6/d7 - m7 - M7 - A7/d8 - P8
D * * * * * * E/F * * * * * * G * * * * * * A * * * * * * B/C * * * * * * D
perfect = white, major = yellow and fifthward white, minor = green and fourthward white, aug/dim = red/blue.
42edo heptatonic narrow-fifth-based, sharp = six ups, 3/2 maps to 24\42 = perfect fifth:
 
D * * * * * E * * * * * F * * * * * G * * * * * A * * * * * B * * * * * C * * * * * D
**__22edo__:** sharp = 3 keys: C Db * C# D
D * * * E F * * * G * * * A * * * B C * * * D
D - D^/Eb - D#v/Eb^ - D#/Ev - E - F - F^/Gb - F#v/Gb^ - F#/Gv - G - G^/Ab - G#v/Ab^ - G#/Av - A etc.
P1 - m2 - ^m2 - vM2 - M2 - m3 - ^m3 - vM3 - M3 - P4 - ^P4/d5 - vA4/^d5 - A4/vP5 - P5 etc.


**__24edo__:** sharp = 2 keys: C * C#/Db * D
D * * * E * F * * * G * * * A * * * B * C * * * D
D - D^/Ebv - D#/Eb - D#^/Ev - E - E^/Fv - F - F^/Gbv - F#/Gb - F#^/Gv - G - G^/Abv - G#/Ab - G#^/Av - A etc.
P1 - ^P1/vm2 - m2 - ~2 - M2 - ^M2/vm3 - m3 - ~3 - M3 - ^M3/vP4 - P4 - ^P4/vd5 - A4/d5 - ^A4/vP5 - P5 etc.


=__**Scale Fragments**__=  
===__"Perfect" EDOs (7, 14, 21, 28 and 35)__===  


Besides the trivial EDOs, 1, 2, 3, 4 and 6, which can be notated with standard notation as a subset of 12-EDO, there are five EDO categories, based on the size of the fifth:
All perfect EDOs use the same circle of 7 fifths: P4 - P1 - P5 - P2 - P6 - P3 - P7 - P4 - P1 etc.
"Fifth-less" EDOs (8, 11, 13 and 18)
F - C - G - D - A - E - B - F - C - G - D - A - E - B etc.
"Fourthward" EDOs (9, 16 and 23)
Sharp and flat are zero keys because C and C# are the same note.
"Perfect" EDOs (7, 14, 21, 28 and 35)
 
"Pentatonic" EDOs (5, 10, 15, 20, 25 and 30)
**__7edo__:** C/C# Db/D
"Regular" EDOs (all others)
D E F G A B C D
The first two categories never use ups and downs, the next two always do (except for 5edo and 7edo). The regular EDOs may or may not.
P1 - P2 - P3 - P4 - P5 - P6 - P7 - P8
Because everything is perfect, the quality can be omitted: 1 - 2 - 3 - 4 - 5 - 6 - 7 - 8
 
**__14edo__:** C/C# * Db/D
D * E * F * G * A * B * C * D
D - D^/Ev - E - E/ Fv - F - F^/Gv - G - G^/Av - A - A^/Bv - B - B^/Cv - C - C^/Dv - D
1 - ^1/v2 - 2 - ^2/v3 - 3 - ^3/v4 - 4 - ^4/v5 - 5 - ^5/v6 - 6 - ^6/v7 - 7 - ^7/v8 - 8


To summarize an EDO, a scale fragment from C to D is shown, including C# and Db. Examples:
**__21edo__:** C/C# * * Db/D
C C#/Db D (12edo)
D * * E * * F * * G * * A * * B * * C * * D
C Db C# D (17edo)
D - D^ - Ev - E - E^ - Fv - F - F^ - Gv - G - G^ - Av - A - A^ - Bv - B - B^ - Cv - C - C^ - Dv - D
C C# Db D (19edo)
1 - ^1 - v2 - 2 - ^2 - v3 - 3 - ^3 - v4 - 4 - ^4 - v5 - 5 - ^5 - v6 - 6 - ^6 - v7 - 7 - ^7 - v8 - 8
C C# _ Db D (26edo)
C _ C# Db _ D (31edo)


The scale fragment concisely conveys the "flavor" of the EDO's notation. The C-C# interval is the augmented unison, and if the 2nd key in the fragment isn't C#, ups and downs are required. The only exception is 7edo. For most EDOs, the C-Db interval is the minor 2nd and the C-D interval is the major 2nd. For perfect EDOs, C-Db = d2 and C-D = P2. For fourthward EDOs, C-Db = d2 and C-D = m2. D# is included for these EDOs because C-D# is a M2 just like E-F. For fifthless EDOs, the scale fragment isn't as helpful because you can't deduce the entire keyboard layout from it.
**__28edo__:** C/C# * * * Db/D
D * * * E * * * F * * * G * * * A * * * B * * * C * * * D
D - D^ - D^^/Evv - Ev - E - E^ - E^^/Fvv - Fv - F - F^ - F^^/Gvv - Gv - G - G^ - G^^/Avv - Av - A etc.
1 - ^1 - ^^1/vv2 - v2 - 2 - ^2 - ^^2/vv3 - v3 - 3 - ^3 - ^^3/vv4 - v4 - 4 - ^4 - ^^4/vv5 - v5 - 5 etc.


Every EDO contains a unique scale fragment, and every scale fragment implies a unique EDO. Furthermore, this uniqueness applies to EDOs with alternate fifths: "wide-fifth" 35edo (which uses 21\35 as a fifth) has a different scale fragment than "narrow-fifth" 35edo with 20\35. If an EDO has a fifth of keyspan F and an octave of keyspan O (i.e. it's O-EDO), the minor 2nd's keyspan is m2 = -5F + 3O, and the augmented unison's is A1 = 7F - 4O. These equations can be reversed: F = 4(m2) + 3(A1) and O = 7(m2) + 5(A1). (For perfect and fourthwards EDOs, substitute M2 for m2.)
**__35edo__:** C/C# * * * * Db/D
D * * * * E * * * * F * * * * G * * * * A * * * * B * * * * C * * * * D
D - D^ - D^^ - Evv - Ev - E - E^ - E^^ - Fvv - Fv - F - F^ - F^^ - Gvv - Gv - G - G^ - G^^ - Avv - Av - A etc.
1 - ^1 - ^^1 - vv2 - v2 - 2 - ^2 - ^^2 - vv3 - v3 - 3 - ^3 - ^^3 - vv4 - v4 - 4 - ^4 - ^^4 - vv5 - v5 - 5 etc.


In the chart below, 13edo and 18edo use the narrower fifth.
===__Fourthward EDOs (9, 11, 13b, 16, 18b and 23)__===  
||= 5edo ||= pentatonic ||=   ||= C/Db ||= C#/D ||=   ||=   ||=  ||=  ||=  ||=  ||=  ||=  ||
 
||= 6edo ||= fifthless ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||
All fourthwards EDOs use the usual chain of fifths: m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7 etc.
||= 7edo ||= perfect ||=  ||= C/C# ||= Db/D ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||
Fb - Cb - Gb - Db - Ab - Eb - Bb - F - C - G - D - A - E - B - F# - C# - G# - D# - A# - E# - B# etc.
||= 8edo ||= fifthless ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||
 
||= 9edo ||= fourthward ||=  ||= C/Db ||= C#/D ||= D# ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||
**__9edo__:** C/D# Cb/D, # = v
||= 10edo ||= pentatonic ||=  ||= C/Db ||= * ||= C#/D ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||
D E * F G A B * C D
||= 11edo ||= fourthward ||=  ||= C ||= D ||= C# ||= D# ||=  ||=  ||=  ||=  ||=  ||=  ||
D - E - Eb/F# - F - G - A - B - Bb/C# - C - D
||= 12edo ||= regular ||=  ||= C ||= C#/Db ||= D ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||
P1 - M2 - m2/M3 - m3 - P4 - P5 - M6 - m6/M7 - m7 - P8
||= 13edo ||= fourthward ||=  ||= C ||= D ||= * ||= C# ||= D# ||=  ||=  ||=  ||=  ||=  ||
 
||= 14edo ||= perfect ||=  ||= C/C# ||= * ||= Db/D ||=  ||=  ||=  ||=  ||=  ||=  ||=  ||
**__11edo__:** C D Cb Db, # = vv
||= 15edo ||= pentatonic ||=  ||= C/Db ||= * ||= * ||= C#/D ||=  ||=  ||=  ||=  ||=  ||=  ||
D E * * F G A B * * C D
||= 16edo ||= fourthward ||=  ||= C ||= C#/Db ||= D ||= D# ||=  ||=  ||=  ||=  ||=  ||=   ||
D - E - E^/F# - Eb/Fv - F - G - A - B - B^/C# - Bb/Cv - C - D
||= 17edo ||= regular ||=  ||= C ||= Db ||= C# ||= D ||=  ||=  ||=  ||=  ||=  ||=  ||
P1 - M2 - ~2/M3 - m2/~3 - m3 - P4 - P5 - M6 - ~6/M7 - m6/~7 - m7 - P8
||= 18edo ||= fourthward ||=  ||= C/Db ||= * ||= C#/D ||= * ||= D# ||=  ||=  ||=  ||=  ||=  ||
 
||= 19edo ||= regular ||=  ||= C ||= C# ||= Db ||= D ||=  ||=  ||=  ||=  ||=  ||=  ||
**__13b-edo__:** C D * Cb Db, # = vvv
||= 20edo ||= pentatonic ||=  ||= C/Db ||= * ||= * ||= * ||= C#/D ||=  ||=  ||=  ||=  ||=   ||
D E * * * F G A B * * * C D
||= 21edo ||= perfect ||=  ||= C/C# ||= * ||= * ||= Db/D ||=  ||=  ||=  ||=  ||=  ||=  ||
D - E - E^/F# - Ebv/F#^ - Eb/Fv - F - G - A - B - B^/C# - Bbv/C#^ - Bb/Cv - C - D
||= 22edo ||= regular ||=  ||= C ||= Db ||= * ||= C# ||= D ||=  ||=  ||=  ||=  ||=  ||
P1 - M2 - ^M2/M3 - vm2/^M3 - m2/vm3 - m3 - P4 - P5 - M6 - ^M6/M7 - vm6/^M7 - m6/vm7 - m7 - P8
||= 23edo ||= fourthward ||=  ||= C ||= C# ||= Db ||= D ||= D# ||=  ||=  ||=  ||=  ||=  ||
 
||= 24edo ||= regular ||=  ||= C ||= * ||= C#/Db ||= * ||= D ||=  ||=  ||=  ||=  ||=  ||
**__16edo__:** C Cb/D# D, # = v
||= 25edo ||= pentatonic ||=  ||= C/Db ||= * ||= * ||= * ||= * ||= C#/D ||=  ||=  ||=  ||=  ||
D * E * * F * G * A * B * * C * D
||= 26edo ||= regular ||=  ||= C ||= C# ||= * ||= Db ||= D ||=  ||=  ||=  ||=  ||=  ||
D - Db/E# - E - Eb - F# - F - Fb/G# - G - Gb/A# - A - Ab/B# - B - Bb - C# - C - Cb/D# - D
||= 27edo ||= regular ||=  ||= C ||= Db ||= * ||= * ||= C# ||= D ||=  ||=  ||=  ||=  ||
P1 - d1/A2 - M2 - m2 - M3 - m3 - d3/A4 - P4 - d4/A5 - P5 - d5/A6 - M6 - m6 - M7 - m7 - d7/A8 - P8
||= 28edo ||= perfect ||=  ||= C/C# ||= * ||= * ||= * ||= Db/D ||=  ||=  ||=  ||=  ||=  ||
 
||= 29edo ||= regular ||=  ||= C ||= * ||= Db ||= C# ||= * ||= D ||=  ||=  ||=  ||=  ||
**__18b-edo__:** C/D# * Cb/D, # = vv
||= 30edo ||= pentatonic ||=  ||= C/Db ||= * ||= * ||= * ||= * ||= * ||= C#/D ||=  ||=  ||=  ||
D * E * * * F * G * A * B * * * C * D
||= 31edo ||= regular ||=  ||= C ||= * ||= C# ||= Db ||= * ||= D ||=  ||=  ||=  ||=   ||
D - D^/Ev - E - E^/F#v - Eb/F# - Eb^/Fv - F - F^/Gv - G - G^/Av - A - A^/Bv - B - B^/C# - Bbv/C#^ - Bb/Cv - C - D
||= 32edo ||= regular ||=  ||= C ||= Db ||= * ||= * ||= * ||= C# ||= D ||=  ||=  ||=  ||
P1 - M2 - ^M2/M3 - vm2/^M3 - m2/vm3 - m3 - P4 - P5 - M6 - ^M6/M7 - vm6/^M7 - m6/vm7 - m7 - P8
||= 33edo ||= regular ||=  ||= C ||= C# ||= * ||= * ||= Db ||= D ||=  ||=  ||=  ||=  ||
 
||= 34edo ||= regular ||=  ||= C ||= * ||= Db ||= * ||= C# ||= * ||= D ||=  ||=  ||=  ||
**__23edo__:** C Cb * D# D, # = v
||= 35edo ||= perfect ||=  ||= C/C# ||= * ||= * ||= * ||= * ||= Db/D ||=  ||=  ||=  ||=  ||
D * * E * * * F * * G * * A * * B * * * C * * D
||= 36edo ||= regular ||=  ||= C ||= * ||= * ||= C#/Db ||= * ||= * ||= D ||=  ||=  ||=  ||
D - Db - E# - E - Eb - Ebb/Fx - F# - F - Fb - G# - G - Gb - A# - A - Ab - B# - B - Bb - Bbb/Cx - C# - C - Cb - D# - D
||= 37edo ||= " ||=  ||= C ||= Db ||= * ||= * ||= * ||= * ||= C# ||= D ||=  ||=  ||
P1 - d1 - A2 - M2 - m2 - d2/A3 - M3 - m3 - d3 - A4 - P4 - d4 - A5 - P5 - d5 - A6 - M6 - m6 - d6/A7 - M7 - m7 - d7 - A8 - P8
||= 38edo ||= " ||=  ||= C ||= * ||= C# ||= * ||= Db ||= * ||= D ||=   ||=  ||=  ||
 
||= 39edo ||= " ||=  ||= C ||= * ||= Db ||= * ||= * ||= C# ||= * ||= D ||=  ||=  ||
 
||= 40edo ||= " ||=  ||= C ||= C# ||= * ||= * ||= * ||= Db ||= D ||=  ||=  ||=  ||
===__Pentatonic EDOs (5, 10, 15, 20, 25 and 30)__===  
||= 41edo ||= " ||=  ||= C ||= * ||= * ||= Db ||= C# ||= * ||= * ||= D ||=  ||=  ||
 
||= 42edo ||= " ||=  ||= C ||= Db ||= * ||= * ||= * ||= * ||= * ||= C# ||= D ||=  ||
All pentatonic EDOs use the usual chain of fifths: m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7 etc.
||= 43edo ||= " ||=  ||= C ||= * ||= * ||= C# ||= Db ||= * ||= * ||= D ||=  ||=  ||
Fb - Cb - Gb - Db - Ab - Eb - Bb - F - C - G - D - A - E - B - F# - C# - G# - D# - A# - E# - B# etc.
||= 44ddo ||= " ||=  ||= C ||= * ||= Db ||= * ||= * ||= * ||= C# ||= * ||= D ||=   ||
In all pentatonic EDOs, the minor 2nd = the unison and the major 3rd = the perfect fourth.
||= 45edo ||= " ||=  ||= C ||= * ||= C# ||= * ||= * ||= Db ||= * ||= D ||=  ||=  ||
||= 46edo ||= " ||=  ||= C ||= * ||= * ||= Db ||= * ||= C# ||= * ||= * ||= D ||=  ||
||= 47edo ||= " ||=  ||= C ||= C# ||= * ||= * ||= * ||= * ||= Db ||= D ||=  ||=  ||
||= 48edo ||= " ||=  ||= C ||= * ||= * ||= * ||= C#/Db ||= * ||= * ||= * ||= D ||=  ||
||= 49edo ||= " ||=  ||= C ||= * ||= Db ||= * ||= * ||= * ||= * ||= C# ||= * ||= D ||
||= 50edo ||= " ||=  ||= C ||= * ||= * ||= C# ||= * ||= Db ||= * ||= * ||= D ||=  ||
||= 51edo ||= " ||=   ||= C ||= * ||= * ||= Db ||= * ||= * ||= C# ||= * ||= * ||= D ||
||= 52edo ||= " ||=  ||= C ||= * ||= C# ||= * ||= * ||= * ||= Db ||= * ||= D ||=  ||
||= 53edo ||= " ||=  ||= C ||= * ||= * ||= * ||= Db ||= C# ||= * ||= * ||= * ||= D ||


=__**Summary of EDO notation**__=
**__5edo__:** C/Db C#/D
D E/F G A B/C D
P1 - M2/m3 - P4 - P5 - M6/m7 - P8


===__**"Fifth-less" EDOs (8, 11, 13 and 18)**__===
**__10edo__:** 2 keys per sharp/flat: C/Db * C#/D
D * E/F * G * A * B/C * D
D - D^/Ev - E/F - F^/Gv - G - G^/Av - A - A^/Bv - B/C - C^/Dv - D
P1/m2 - ^m2/vM2 - M2/m3 - ^m3/vM3 - M3/P4 - ^P4/vP5 - P5/m6 - ^m6/vM6 - M6/m7 - ^m7/vM7 - P8


**__8edo__:** (generator = 1\8 = perfect 2nd = 150¢)
**__15edo__:** 3 keys per sharp/flat: C/Db [*] [*] C#/D
D E F G * A B C D
D * * E/F * * G * * A * * B/C * * D
D - E - F - G - G#/Ab - A -B - C - D
D - D^ - Ev - E/F - F^ - Gv - G - G^ - Av - A - A^ - Bv - B/C - C^ - Dv - D
P1 - P2 - m3 - M3/m4 - M4/m5 - M5/m6 - M6 - P7 - P8
P1/m2 - ^m2 - vM2 - M2/m3 - ^m3 - vM3 - M3/P4 - ^P4 - vP5 - P5/m6 - ^m6 - vM6 - M6/m7 - ^m7 - vM7 - P8
seventhwards chain of seconds: M3 - M4 - M5 - M6 - P7 - P1 - P2 - m3 - m4 - m5 - m6 - d7 etc.
A# - B# - C# - D# - E# - F# - G# - A - B - C - D - E - F - G - Ab - Bb - Cb - Db - Eb - Fb - Gb etc.


__**11edo**__: (generator = 3\11 = perfect 3rd)
**__20edo__:** 4 keys per sharp/flat: C/Db * * * C#/D
D * E F * G A * B C * D
D * * * E/F * * * G * * * A * * * B/C * * * D
D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D
D - D^ - D^^/Evv - Ev - E/F - F^ - F^^/Gvv - Gv - G - G^ - G^^/Avv - Av - A - A^ - A^^/Bvv - Bv - B/C - C^ - C^^/Dvv - Dv - D
P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8
P1/m2 - ^m2 - ^^m2/vvM2 - vM2 - M2/m3 - ^m3 - ^^m3/vvM3 - vM3 - M3/P4 - ^P4 - ^^P4/vvP5 - vP5 - P5/m6 - ^m6 - ^^m6/vvM6 - vM6 - M6/m7 - ^m7 - ^^m7/vvM7 - vM7 - P8
sixthwards chain of thirds: M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6 etc.
E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb


__**13edo**__**:** (generator = 2\13 = perfect 2nd)
**__25edo__:** 5 keys per sharp/flat: C/Db * * * * C#/D
D * E * F * G A * B * C * D
D * * * * E/F * * * * G * * * * A * * * * B/C * * * * D
D - D#/Eb - E - E#/Fb - F - F#/Gb - G - A - A#/Bb - B - B#/Cb - C - C#/Db - D
D - D^ - D^^ - Evv - Ev - E/F - F^ - F^^ - Gvv - Gv - G - G^ - G^^ - Avv - Av - A - A^ - A^^ - Bvv - Bv - B/C - C^ - C^^ - Dvv - Dv - D
P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - m6 - M6 - P7 - A7/d8 - P8
P1/m2 - ^m2 - ^^m2 - vvM2 - vM2 - M2/m3 - ^m3 - ^^m3 - vvM3 - vM3 - M3/P4 - ^P4 - ^^P4 - vvP5 - vP5 - P5/m6 - ^m6 - ^^m6 - vvM6 - vM6 - M6/m7 - ^m7 - ^^m7 - vvM7 - vM7 - P8
secondwards chain of seconds: m3 - m4 - m5 - m6 - P7 - P1 - P2 - M3 - M4 - M5 - M6 - A7 etc.
Ab - Bb - Cb - Db - Eb - Fb - Gb - A - B - C - D - E - F - G - A# - B# - C# - D# - E# - F# - G#


**__18edo__:** (generator = 5\18 = perfect 3rd)
**__30edo__:** 6 keys per sharp/flat: C/Db * * * * * C#/D
D * * E * F * * G * A * * B * C * * D
D * * * * * E/F * * * * * G * * * * * A * * * * * B/C * * * * * D
D - D# - Eb - E - E#/Fb - F - F# - Gb - G - G#/Ab - A - A# - Bb - B - B#/Cb - C - C# - Db - D
D - D^ - D^^ - Evv - Ev - E/F - F^ - F^^ - Gvv - Gv - G - G^ - G^^ - Avv - Av - A - A^ - A^^ - Bvv - Bv - B/C - C^ - C^^ - Dvv - Dv - D
P1 - A1/d2 - m2 - M2 - A2/d3 - P3 - A3/d4 - m4 - M4 - A4/d5 - m5 - M5 - A5/d6 - P6 - A6/d7 - m7 - M7 - A7/d8 - P8
P1/m2 - ^m2 - ^^m2 - vvM2 - vM2 - M2/m3 - ^m3 - ^^m3 - vvM3 - vM3 - M3/P4 - ^P4 - ^^P4 - vvP5 - vP5 - P5/m6 - ^m6 - ^^m6 - vvM6 - vM6 - M6/m7 - ^m7 - ^^m7 - vvM7 - vM7 - P8
sixthwards chain of thirds: M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6 etc.
E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb


===__**Alternate pentatonic notation for EDOs 8, 13 and 18**__===  
===__Alternative pentatonic notation for pentatonic EDOs:__===  


All three EDOs use the same pentatonic fifthwards chain of fifths: ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d etc.
Pentatonic fourthwards chain of fifthoids: Ms3 - Ms7 - P4d - P1 - P5d - ms3 - ms7 - d4d etc.
Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E# etc.
C# - G# - D# - A# - E# - C - G - D - A - E - Cb - Gb - Db - Ab - Eb etc.
All intervals are perfect, so quality can be omitted.


__**8edo**__**:** (generator = 5\8 = perfect 5thoid) C C#/Db D
__**5edo**__**:** zero keys per sharp/flat: C/C# Db/D
D * E G * A C * D
D E G A C D
D - D#/Eb - E - G - G#/Ab - A - C - C#/Db - D
1 - s3 - 4d - 5d - s7 - 8d
P1 - ms3 - Ms3 - P4d - A4d/d5d - P5d - ms7 - Ms7 - P8d


__**13edo**__**:** (generator = 8\13 = perfect 5thoid) C C# Db D
__**10edo**__**:** zero keys per sharp/flat: C/C# * Db/D
D * * E * G * * A * C * * D
D * E * G * A * C * D
D - D# - Eb - E - E#/Gb - G - G# - Ab - A - A#/Cb - C - C# - Db - D
D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D
P1 - A1/ds3 - ms3 - Ms3 - As3/d4d - P4d - A4d - d5d - P5d - A5d/ds7 - ms7 - Ms7 - As7/d8d - P8d
1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d


__**18edo**__**:** (generator = 11\18 = perfect 5thoid) C C# * Db D
__**15edo**__**:** zero keys per sharp/flat: C/C# * * Db/D
D * * * E * * G * * * A * * C * * * D
D * * E * * G * * A * * C * * D
D - D# - Dx/Ebb - Eb - E - E# - Gb - G - G# - Gx/Abb - Ab - A - A# - Cb - C - C# - Cx/Dbb - Db - D
D - D^ - Ev - E - E^ - Gv - G - G^ - Av - A - A^ - Cv - C - C^ - Dv - D
P1 - A1 - ds3 - ms3 - Ms3 - As3 - d4d - P4d - A4d - AA4d/dd5d - d5d - P5d - A5d - ds7 - ms7 - Ms7 - As7 - d8d - P8d
1 - ^1 - vs3 - s3 - ^s3 - v4d - 4d - ^4d - v5d - 5d - ^5d - vs7 - s7 - ^s7 - v8d - 8d
etc.


===__**"Fifth-less" EDOs (8, 11b, 13 and 18)**__===


===__Fourthward EDOs (9, 16 and 23)__===
**__8edo__:** (generator = 1\8 = perfect 2nd = 150¢)
D E F G * A B C D
D - E - F - G - G#/Ab - A -B - C - D
P1 - P2 - m3 - M3/m4 - M4/m5 - M5/m6 - M6 - P7 - P8
seventhwards chain of seconds: M3 - M4 - M5 - M6 - P7 - P1 - P2 - m3 - m4 - m5 - m6 - d7 etc.
A# - B# - C# - D# - E# - F# - G# - A - B - C - D - E - F - G - Ab - Bb - Cb - Db - Eb - Fb - Gb etc.


All fourthwards EDOs use the same chain of fifths: M2 - M6 - M3 - M7 - P4 - P1 - P5 - m2 - m6 - m3 - m7 - A4 etc.
__**11edo**__: (generator = 3\11 = perfect 3rd)
F# - C# - G# - D# - A# - E# - B# - F - C - G - D - A - E - B - Fb - Cb - Gb - Db - Ab - Eb - Bb - Fbb etc.
D * E F * G A * B C * D
D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D
P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8
sixthwards chain of thirds: M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6 etc.
E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb


**__9edo__:** C/Db C#/D
__**13edo**__**:** (generator = 2\13 = perfect 2nd)
D E * F G A B * C D
D * E * F * G A * B * C * D
D - E - E#/Fb - F - G - A - B - B#/Cb - C - D
D - D#/Eb - E - E#/Fb - F - F#/Gb - G - A - A#/Bb - B - B#/Cb - C - C#/Db - D
P1 - m2 - M2/m3 - M3 - P4 - P5 - m6 - M6/m7 - M7 - P8
P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - m6 - M6 - P7 - A7/d8 - P8
secondwards chain of seconds: m3 - m4 - m5 - m6 - P7 - P1 - P2 - M3 - M4 - M5 - M6 - A7 etc.
Ab - Bb - Cb - Db - Eb - Fb - Gb - A - B - C - D - E - F - G - A# - B# - C# - D# - E# - F# - G#


**__16edo__:** C C#/Db D
**__18edo__:** (generator = 5\18 = perfect 3rd)
D * E * * F * G * A * B * * C * D
D * * E * F * * G * A * * B * C * * D
D - D#/Eb - E - E# - Fb - F F#/Gb - G - G#/Ab - A - A#/Bb - B - B# - Cb - C - C#/Db - D
D - D# - Eb - E - E#/Fb - F - F# - Gb - G - G#/Ab - A - A# - Bb - B - B#/Cb - C - C# - Db - D
P1 - A1/d2 - m2 - M2 - m3 - M3 - A3/d4 - P4 - A4/d5 - P5 - A5/d6 - m6 - M6 - m7 - M7 - A7/d8 - P8
P1 - A1/d2 - m2 - M2 - A2/d3 - P3 - A3/d4 - m4 - M4 - A4/d5 - m5 - M5 - A5/d6 - P6 - A6/d7 - m7 - M7 - A7/d8 - P8
sixthwards chain of thirds: M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6 etc.
E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb


**__23edo__:** C C# * Db D
===__**Alternate pentatonic notation for EDOs 8, 13 and 18**__===
D * * E * * * F * * G * * A * * B * * * C * * D
D - D# - Eb - E - E# - Ex/Fbb - Fb - F - F# - Gb - G - G# - Ab - A - A# - Bb - B - B# - Bx/Cbb - Cb - C - C# - Db - D
P1 - A1 - d2 - m2 - M2 - A2/d3 - m3 - M3 - A3 - d4 - P4 - A4 - d5 - P5 - A5 - d6 - m6 - M6 - A6/d7 - m7 - M7 - A7 - d8 - P8


All three EDOs use the same pentatonic fifthwards chain of fifths: ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d etc.
Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E# etc.


===__"Perfect" EDOs (7, 14, 21, 28 and 35)__===  
__**8edo**__**:** (generator = 5\8 = perfect 5thoid) C C#/Db D
D * E G * A C * D
D - D#/Eb - E - G - G#/Ab - A - C - C#/Db - D
P1 - ms3 - Ms3 - P4d - A4d/d5d - P5d - ms7 - Ms7 - P8d


All perfect EDOs use the same chain of fifths: P2 - P6 - P3 - P7 - P4 - P1 - P5 - P2 - P6 - P3 - P7 etc.
__**13edo**__**:** (generator = 8\13 = perfect 5thoid) C C# Db D
F - C - G - D - A - E - B - F - C - G - D - A - E - B etc.
D * * E * G * * A * C * * D
D - D# - Eb - E - E#/Gb - G - G# - Ab - A - A#/Cb - C - C# - Db - D
P1 - A1/ds3 - ms3 - Ms3 - As3/d4d - P4d - A4d - d5d - P5d - A5d/ds7 - ms7 - Ms7 - As7/d8d - P8d


**__7edo__:** C/Db C#/D
__**18edo**__**:** (generator = 11\18 = perfect 5thoid) C C# * Db D
D E F G A B C D
D * * * E * * G * * * A * * C * * * D
P1 - P2 - P3 - P4 - P5 - P6 - P7 - P8
D - D# - Dx/Ebb - Eb - E - E# - Gb - G - G# - Gx/Abb - Ab - A - A# - Cb - C - C# - Cx/Dbb - Db - D
Because everything is perfect, the quality can be omitted: 1 - 2 - 3 - 4 - 5 - 6 - 7 - 8
P1 - A1 - ds3 - ms3 - Ms3 - As3 - d4d - P4d - A4d - AA4d/dd5d - d5d - P5d - A5d - ds7 - ms7 - Ms7 - As7 - d8d - P8d


**__14edo__:** 2 keys per sharp/flat: C/C# * Db/D
D * E * F * G * A * B * C * D
D - D^/Ev - E - E/ Fv - F - F^/Gv - G - G^/Av - A - A^/Bv - B - B^/Cv - C - C^/Dv - D
1 - ^1/v2 - 2 - ^2/v3 - 3 - ^3/v4 - 4 - ^4/v5 - 5 - ^5/v6 - 6 - ^6/v7 - 7 - ^7/v8 - 8


**__21edo__:** 3 keys per sharp/flat: C/C# * * Db/D
=__EDOs with an inaccurate 3/2__=
D * * E * * F * * G * * A * * B * * C * * D
D - D^ - Ev - E - E^ - Fv - F - F^ - Gv - G - G^ - Av - A - A^ - Bv - B - B^ - Cv - C - C^ - Dv - D
1 - ^1 - v2 - 2 - ^2 - v3 - 3 - ^3 - v4 - 4 - ^4 - v5 - 5 - ^5 - v6 - 6 - ^6 - v7 - 7 - ^7 - v8 - 8


**__28edo__:** 4 keys per sharp/flat: C/C# * * * Db/D
Not counting the trivial edos 2, 3, 4 and 6, there are only seven such edos. As seen in the above diagram, they are the ones to the left of the heptatonic kite's spine, plus the ones to the right of the pentatonic kite's spine. The ones on the left edge of the heptatonic kite are the fourthward ones like 16edo, and have been dealt with already. 23edo can be notated similarly to 16edo by using a fifth of 13\23 instead of 14\23. That leaves only four edos: 8, 11, 13, and 18.
D * * * E * * * F * * * G * * * A * * * B * * * C * * * D
D - D^ - D^^/Evv - Ev - E - E^ - E^^/Fvv - Fv - F - F^ - F^^/Gvv - Gv - G - G^ - G^^/Avv - Av - A etc.
1 - ^1 - ^^1/vv2 - v2 - 2 - ^2 - ^^2/vv3 - v3 - 3 - ^3 - ^^3/vv4 - v4 - 4 - ^4 - ^^4/vv5 - v5 - 5 etc.


**__35edo__:** 5 keys per sharp/flat: C/C# * * * * Db/D
There are two strategies for notating these "oddball" EDOs, besides heptatonic fifth-based notation with ups and downs. One is to switch from heptatonic notation to some other type. The orange kite contains edos for which pentatonic notation is a natural fit, in the sense that no ups or downs are needed. This includes 8edo, 13edo and 18edo.
D * * * * E * * * * F * * * * G * * * * A * * * * B * * * * C * * * * D
D - D^ - D^^ - Evv - Ev - E - E^ - E^^ - Fvv - Fv - F - F^ - F^^ - Gvv - Gv - G - G^ - G^^ - Avv - Av - A etc.
1 - ^1 - ^^1 - vv2 - v2 - 2 - ^2 - ^^2 - vv3 - v3 - 3 - ^3 - ^^3 - vv4 - v4 - 4 - ^4 - ^^4 - vv5 - v5 - 5 etc.


The other approach is to use some interval other than the fifth to generate the notation. Earlier I said notating 22edo using an even distribution of note names such as C _ _ D _ _ E _ _ F _ _ _ G _ _ A _ _ B _ _ C was a bad idea because the G-D and the A-E fifths looked perfect but were actually diminished. The reasoning is that 3/2 is an important ratio, and any decent approximation of 3/2 should look like a perfect fifth. But these EDOs don't approximate 3/2 well, so they can be thought of as having both a major fifth and a minor fifth. This negates any expectations of what a fifth should look like.


===__Pentatonic EDOs (5, 10, 15, 20, 25 and 30)__===


All pentatonic EDOs use the usual chain of fifths: m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7 etc.
__**Theoretical alternatives for 8edo, 11edo, 13edo and 18edo**__
Fb - Cb - Gb - Db - Ab - Eb - Bb - F - C - G - D - A - E - B - F# - C# - G# - D# - A# - E# - B# etc.
In all pentatonic EDOs, the minor 2nd = the unison and the major 3rd = the perfect fourth.


**__5edo__:** C/Db C#/D
8edo octatonic (every note is a generator)
D E/F G A B/C D
D E F G H A B C D
P1 - M2/m3 - P4 - P5 - M6/m7 - P8
P1 - P2 - P3 - P4 - P5 - P6 - P7 - P8 - P9
requires learning octatonic interval arithmetic and staff notation


**__10edo__:** 2 keys per sharp/flat: C/Db * C#/D
11edo heptatonic narrow-fifth-based, fourthwards, # is vv (3/2 maps to 6\11 perfect 5th):
D * E/F * G * A * B/C * D
D E * * F G A B * * C D = D E F# F~ F G A B B~ Bb C D
D - D^/Ev - E/F - F^/Gv - G - G^/Av - A - A^/Bv - B/C - C^/Dv - D
fourthwards chain of fifths: m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7
P1/m2 - ^m2/vM2 - M2/m3 - ^m3/vM3 - M3/P4 - ^P4/vP5 - P5/m6 - ^m6/vM6 - M6/m7 - ^m7/vM7 - P8
P1 - M2 - ~2/M3 - m2/~3 - m3 - P4 - P5 - M6 - ~6/M7 - m6/~7 - m7 - P8
problematic because M3 = 2\11 is narrower than m2 = 3\11


**__15edo__:** 3 keys per sharp/flat: C/Db [*] [*] C#/D
11edo nonotonic narrow-fifth-based, fifthwards with no ups and downs (3/2 maps to 6\11 = perfect 6th):
D * * E/F * * G * * A * * B/C * * D
nonotonic fifthwards chain of sixths: M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9
D - D^ - Ev - E/F - F^ - Gv - G - G^ - Av - A - A^ - Bv - B/C - C^ - Dv - D
P1 m2 M2/m3 M3/m4 M4 P5 P6 m7 M7/m8 M8/m9 M9 P8
P1/m2 - ^m2 - vM2 - M2/m3 - ^m3 - vM3 - M3/P4 - ^P4 - vP5 - P5/m6 - ^m6 - vM6 - M6/m7 - ^m7 - vM7 - P8
requires learning nonotonic interval arithmetic and staff notation


**__20edo__:** 4 keys per sharp/flat: C/Db * * * C#/D
11edo pentatonic wide-fifth-based, fifthwards, # is ^^ (3/2 maps to 7\11 6th):
D * * * E/F * * * G * * * A * * * B/C * * * D
D * * E G * * A C * * D
D - D^ - D^^/Evv - Ev - E/F - F^ - F^^/Gvv - Gv - G - G^ - G^^/Avv - Av - A - A^ - A^^/Bvv - Bv - B/C - C^ - C^^/Dvv - Dv - D
pentatonic fifthwards chain of fifthoids: m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7
P1/m2 - ^m2 - ^^m2/vvM2 - vM2 - M2/m3 - ^m3 - ^^m3/vvM3 - vM3 - M3/P4 - ^P4 - ^^P4/vvP5 - vP5 - P5/m6 - ^m6 - ^^m6/vvM6 - vM6 - M6/m7 - ^m7 - ^^m7/vvM7 - vM7 - P8
P1 - ms3 - ^ms3/vMs3 - Ms3 - P4d - ^P4d/d5d - A4d/vP5d - P5d - ms7 - ^ms7/vMs7 - Ms7 - P8d
pentatonic plus ups and downs is doubly confusing!


**__25edo__:** 5 keys per sharp/flat: C/Db * * * * C#/D
11edo octatonic wide-fifth-based, fifthwards, no ^/v (3/2 maps to 7\11 = perfect 6th):
D * * * * E/F * * * * G * * * * A * * * * B/C * * * * D
octatonic chain of 6ths: m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7
D - D^ - D^^ - Evv - Ev - E/F - F^ - F^^ - Gvv - Gv - G - G^ - G^^ - Avv - Av - A - A^ - A^^ - Bvv - Bv - B/C - C^ - C^^ - Dvv - Dv - D
P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9
P1/m2 - ^m2 - ^^m2 - vvM2 - vM2 - M2/m3 - ^m3 - ^^m3 - vvM3 - vM3 - M3/P4 - ^P4 - ^^P4 - vvP5 - vP5 - P5/m6 - ^m6 - ^^m6 - vvM6 - vM6 - M6/m7 - ^m7 - ^^m7 - vvM7 - vM7 - P8
requires learning octatonic interval arithmetic and notation


**__30edo__:** 6 keys per sharp/flat: C/Db * * * * * C#/D
13edo heptatonic narrow-fifth-based, fourthwards, sharp = ^^^ (3/2 maps to 7\13 perfect 5th):
D * * * * * E/F * * * * * G * * * * * A * * * * * B/C * * * * * D
D E * * * F G A B * * * C D
D - D^ - D^^ - Evv - Ev - E/F - F^ - F^^ - Gvv - Gv - G - G^ - G^^ - Avv - Av - A - A^ - A^^ - Bvv - Bv - B/C - C^ - C^^ - Dvv - Dv - D
fourthwards chain of fifths: M2 - M6 - M3 - M7 - P4 - P1 - P5 - m2 - m6 - m3 - m7
P1/m2 - ^m2 - ^^m2 - vvM2 - vM2 - M2/m3 - ^m3 - ^^m3 - vvM3 - vM3 - M3/P4 - ^P4 - ^^P4 - vvP5 - vP5 - P5/m6 - ^m6 - ^^m6 - vvM6 - vM6 - M6/m7 - ^m7 - ^^m7 - vvM7 - vM7 - P8
P1 - m2 - m3 - vM2/^m3 - M2 - M3 - P4 - P5 - m6 - m7 - vM6/^m7 - M6 - M7 - P8
problematic because m3 = 2\13 is narrower than M2 = 4\13


===__Alternative pentatonic notation for pentatonic EDOs:__===
13edo undecatonic narrow-fifth-based, fourthwards, 3/2 maps to 7\13 = perfect 7th
undecatonic sixthwards chain of sevenths:
M2 - M8 - M3 - M9 - M4 - M10 - M5 - M11 - P6 - P1 - P7 - m2 - m8 - m3 - m9 - m4 - m10 - m5 - m11
P1 - m2 - M2/m3 - M3/m4 - M4/m5 - M5 - P6 - P7 - m8 - M8/m9 - M9/m10 - M10/m11 - M11 - P12
requires learning undecatonic interval arithmetic and notation


Pentatonic fourthwards chain of fifthoids: Ms3 - Ms7 - P4d - P1 - P5d - ms3 - ms7 - d4d etc.
13edo octatonic wide-fifth-based, fourthwards, 3/2 maps to 8\13 = perfect 6th
C# - G# - D# - A# - E# - C - G - D - A - E - Cb - Gb - Db - Ab - Eb etc.
octotonic chain of sixths: M3 - M8 - M5 - M2 - M7 - P4 - P1 - P6 - m3 - m8 - m5 - m2 - m7
All intervals are perfect, so quality can be omitted.
P1 - m2 - M2 - m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7 - m8 - M8 - P9
requires learning octatonic interval arithmetic and notation


__**5edo**__**:** zero keys per sharp/flat: C/C# Db/D
18edo heptatonic narrow-fifth-based, fourthwards, sharp = ^^ (3/2 maps to 10\18 perfect 5th)
D E G A C D
D * E * * * F * G * A * B * * * C * D
1 - s3 - 4d - 5d - s7 - 8d
P1 - vm2 - m2 - vM2 - M2/m3 - vM3 - M3 - ^M3 - P4 - ^P4/vP5 - P5 - vm6 - m6 - vM6 - M6/m7 - vM7 - M7 - ^M7 - P8
fourthwards plus ups and downs plus closed is triply confusing!


__**10edo**__**:** zero keys per sharp/flat: C/C# * Db/D
18edo nonatonic narrow-fifth-based (3/2 maps to 10\18 = perfect 6th)
D * E * G * A * C * D
P1 - vP2 - P2 - vP3 - P3 - vP4- P4 - vP5 - P5 - vP6 - P6 - vP7 - P7 - vP8 - P8 - vP9 - P9 - vP10 - P10
D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D
requires learning nonotonic interval arithmetic and staff notation
1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d


__**15edo**__**:** zero keys per sharp/flat: C/C# * * Db/D
D * * E * * G * * A * * C * * D
D - D^ - Ev - E - E^ - Gv - G - G^ - Av - A - A^ - Cv - C - C^ - Dv - D
1 - ^1 - vs3 - s3 - ^s3 - v4d - 4d - ^4d - v5d - 5d - ^5d - vs7 - s7 - ^s7 - v8d - 8d


etc.
__**Alternate notation for other edos:**__
23edo pentatonic wide-fifth-based, fifthwards, 3/2 maps to 14\23 = perfect fifthoid:
D * * * * E * * * G * * * * A * * * C * * * * D
35edo heptatonic wide-fifth-based, sharp = seven ups, 3/2 maps to 21\35 = perfect fifth:
D * * * * * * E/F * * * * * * G * * * * * * A * * * * * * B/C * * * * * * D
42edo heptatonic narrow-fifth-based, sharp = six ups, 3/2 maps to 24\42 = perfect fifth:
D * * * * * E * * * * * F * * * * * G * * * * * A * * * * * B * * * * * C * * * * * D


=__Ups and downs solfege__=


===__"Regular" EDOs (12, 17, 19, 22, 24, 26, 27, 29, 31-34, and all edos 36 or higher)__===
Solfege (do-re-mi) can be adapted to indicate sharp/flat and up/down:
The initial consonant remains as before: D, R, M, F, S, L and T
The first vowel indicates sharp or flat: a = natural, e = #, i = ##, o = b, u = bb
The vowels are pronounced as in Spanish or Italian
The pitch from ## to bb follows the natural vowel spectrum i-e-a-o-u
The optional 2nd vowel indicates up/down: a = ^^^, e = ^, i = ^^, o = v, u = vv
The 2nd vowel is separated from the first by either a glottal stop, an "h", a "w", or a "y"
Thus C#v is Deo, pronounced as De'o or Deho or Dewo or Deyo.
This suffices for many but not all edos, as some require triple sharps or quadruple ups.


All regular EDOs use the usual chain of fifths: m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7 etc.
Fixed-do solfege:
Fb - Cb - Gb - Db - Ab - Eb - Bb - F - C - G - D - A - E - B - F# - C# - G# - D# - A# - E# - B# etc.
Da = C, De = C#, Di = C##, Do = Cb, Du =Cbb
Da = C, Da'e = C^, Da'i = C^^, Da'o = Cv, Da'u = Cvv, Da'a = C^^^
De = C#, De'e = C#^, De'i = C#^^, De'o = C#v, De'u = C#vv, De'a = C#^^^
etc.


**__12edo__:** sharp/flat = 1 key, no ups and downs: C C#/Db D
Moveable-do solfege:
D * E F * G * A * B C * D
The 2nd vowel is as before. The 1st vowel's meaning depends on the interval.
D - D#/Eb - E - F - F#/Gb - G - G#/Ab - A - A#/Bb - B - C - C#/Db - D
Perfect intervals (tonic, 4th, 5th and octave): a = perfect, e= aug, i = double-aug, o = dim, u = double-dim
P1 - m2 - M2 - m3 - M3 - P4 - A4/d5 - P5 - m6 - M6 - m7 - M7 - P8
Da = P1, De = A1, Di = AA1, Do = d1, Du = dd1
perfect = white, major = red, yellow and fifthward white, minor = green, blue and fourthwards white
Da'e = ^P1, Da'i = ^^P1, Da'o = vP1, Da'u = vvP1, Da'a = ^^^P1
etc.


**__17edo__:** sharp = 2 keys: C Db C# D
Imperfect intervals (2nd, 3rd, 6th and 7th): a = major, e = aug, i = double-aug, o = minor, u = dim
D * * E F * * G * * A * * B C * * D
Ra = M2, Re = A2, Ri = AA2, Ro = m2, Ru = d2
D - D^/Eb - D#/Ev - Eb - E - F - F^/Gb - F#/Gv - G - G^/Ab - G#/Av - A - A^/Bb - A#/Bv - B - C - C^/Db - C#/Dv - D
Ra'e = ^M2, Ra'i = ^^M2, Ra'o = vM2, Ra'u = vvM2, Ra'a = ^^^M2
P1 - m2 - ^m2/vM2 - M2 - m3 - ^m3/vM3 - M3 - P4 - ^P4/d5 - A4/vP5 - P5 - m6 - ^m6/vM6 - M6 - m7 ^m7/vM7 - M7 - P8
etc.


**__19edo__:** no ups and downs C C# Db D
D * * E * F * * G * * A * * B * C * * D
D - D# - Eb - E - E#/Fb - F - F# - Gb - G - G# - Ab - A - A# - Bb - B - B#/Cb - C - C# - Db - D
P1 - A1/d2 - m2 - M2 - A2/d3 - m3 - M3 - A3/d4 - P4 - A4 - d5 - P5 - A5/d6 - m6 - M6 - A6/d7 - m7 - M7 - A7/d8 - P8
perfect = white, major = yellow and fifthward white, minor = green and fourthward white, aug/dim = red/blue.


**__22edo__:** sharp = 3 keys: C Db * C# D
=__Rank-2 Notation__=
D * * * E F * * * G * * * A * * * B C * * * D
D - D^/Eb - D#v/Eb^ - D#/Ev - E - F - F^/Gb - F#v/Gb^ - F#/Gv - G - G^/Ab - G#v/Ab^ - G#/Av - A etc.
P1 - m2 - ^m2 - vM2 - M2 - m3 - ^m3 - vM3 - M3 - P4 - ^P4/d5 - vA4/^d5 - A4/vP5 - P5 etc.


**__24edo__:** sharp = 2 keys: C * C#/Db * D
Ups and downs can be used to notate rank-2 scales. First we must distinguish between edos and sizing frameworks. For example, keyboards with 7 white keys and 5 black keys, and fretted instruments with 12 frets per octave, predate the use of 12edo by many centuries. Such instruments use a 12-tone framework. Traditional Western notation uses a 7-note naming framework and a 12-tone sizing framework. (See the first chapter of part V of Kite's book for more on frameworks.)
D * * * E * F * * * G * * * A * * * B * C * * * D
D - D^/Ebv - D#/Eb - D#^/Ev - E - E^/Fv - F - F^/Gbv - F#/Gb - F#^/Gv - G - G^/Abv - G#/Ab - G#^/Av - A etc.
P1 - ^P1/vm2 - m2 - ^m2/vM2 - M2 - ^M2/vm3 - m3 - ^m3/vM3 - M3 - ^M3/vP4 - P4 - ^P4/vd5 - A4/d5 - ^A4/vP5 - P5 etc.


 
Let's start with fifth-generated tunings. For large frameworks, we'll need a long genchain:
=__Ups and downs solfege__=
 
Solfege (do-re-mi) can be adapted to indicate sharp/flat and up/down:
The initial consonant remains as before: D, R, M, F, S, L and T
The first vowel indicates sharp or flat: a = natural, e = #, i = ##, o = b, u = bb
The vowels are pronounced as in Spanish or Italian
The pitch from ## to bb follows the natural vowel spectrum i-e-a-o-u
The optional 2nd vowel indicates up/down: a = ^^^, e = ^, i = ^^, o = v, u = vv
The 2nd vowel is separated from the first by either a glottal stop, an "h", a "w", or a "y"
Thus C#v is Deo, pronounced as De'o or Deho or Dewo or Deyo.
This suffices for many but not all edos, as some require triple sharps or quadruple ups.
 
Fixed-do solfege:
Da = C, De = C#, Di = C##, Do = Cb, Du =Cbb
Da = C, Da'e = C^, Da'i = C^^, Da'o = Cv, Da'u = Cvv, Da'a = C^^^
De = C#, De'e = C#^, De'i = C#^^, De'o = C#v, De'u = C#vv, De'a = C#^^^
etc.
 
Moveable-do solfege:
The 2nd vowel is as before. The 1st vowel's meaning depends on the interval.
Perfect intervals (tonic, 4th, 5th and octave): a = perfect, e= aug, i = double-aug, o = dim, u = double-dim
Da = P1, De = A1, Di = AA1, Do = d1, Du = dd1
Da'e = ^P1, Da'i = ^^P1, Da'o = vP1, Da'u = vvP1, Da'a = ^^^P1
etc.
 
Imperfect intervals (2nd, 3rd, 6th and 7th): a = major, e = aug, i = double-aug, o = minor, u = dim
Ra = M2, Re = A2, Ri = AA2, Ro = m2, Ru = d2
Ra'e = ^M2, Ra'i = ^^M2, Ra'o = vM2, Ra'u = vvM2, Ra'a = ^^^M2
etc.
 
 
=__Rank-2 Notation__=
 
Ups and downs can be used to notate rank-2 scales. First we must distinguish between edos and sizing frameworks. For example, keyboards with 7 white keys and 5 black keys, and fretted instruments with 12 frets per octave, predate the use of 12edo by many centuries. Such instruments use a 12-tone framework. Traditional Western notation uses a 7-note naming framework and a 12-tone sizing framework. (See the first chapter of part V of Kite's book for more on frameworks.)
 
Let's start with fifth-generated tunings. For large frameworks, we'll need a long genchain:
...Fbb Cbb Gbb Dbb Abb Ebb Bbb Fb Cb Gb Db Ab Eb Bb F C G D A E B F# C# G# D# A# E# B# Fx Cx Gx Dx Ax Ex Bx...
...Fbb Cbb Gbb Dbb Abb Ebb Bbb Fb Cb Gb Db Ab Eb Bb F C G D A E B F# C# G# D# A# E# B# Fx Cx Gx Dx Ax Ex Bx...


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&lt;!-- ws:end:WikiTextTocRule:57 --&gt;&lt;!-- ws:start:WikiTextTocRule:58: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Cross-EDO considerations"&gt;Cross-EDO considerations&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:58 --&gt;&lt;!-- ws:start:WikiTextTocRule:59: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#EDOs with an inaccurate 3/2"&gt;EDOs with an inaccurate 3/2&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:58 --&gt;&lt;!-- ws:start:WikiTextTocRule:59: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Scale Fragments"&gt;Scale Fragments&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:59 --&gt;&lt;!-- ws:start:WikiTextTocRule:60: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Scale Fragments"&gt;Scale Fragments&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:59 --&gt;&lt;!-- ws:start:WikiTextTocRule:60: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Summary of EDO notation"&gt;Summary of EDO notation&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:60 --&gt;&lt;!-- ws:start:WikiTextTocRule:61: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Summary of EDO notation"&gt;Summary of EDO notation&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:60 --&gt;&lt;!-- ws:start:WikiTextTocRule:61: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Summary of EDO notation--&amp;quot;Regular&amp;quot; EDOs (12, 17, 19, 22, 24, 26, 27, 29, 31-34, and all edos 36 or higher)"&gt;&amp;quot;Regular&amp;quot; EDOs (12, 17, 19, 22, 24, 26, 27, 29, 31-34, and all edos 36 or higher)&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:61 --&gt;&lt;!-- ws:start:WikiTextTocRule:62: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Summary of EDO notation--&amp;quot;Fifth-less&amp;quot; EDOs (8, 11, 13 and 18)"&gt;&amp;quot;Fifth-less&amp;quot; EDOs (8, 11, 13 and 18)&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:61 --&gt;&lt;!-- ws:start:WikiTextTocRule:62: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Summary of EDO notation--&amp;quot;Perfect&amp;quot; EDOs (7, 14, 21, 28 and 35)"&gt;&amp;quot;Perfect&amp;quot; EDOs (7, 14, 21, 28 and 35)&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:62 --&gt;&lt;!-- ws:start:WikiTextTocRule:63: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Summary of EDO notation--Alternate pentatonic notation for EDOs 8, 13 and 18"&gt;Alternate pentatonic notation for EDOs 8, 13 and 18&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:62 --&gt;&lt;!-- ws:start:WikiTextTocRule:63: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Summary of EDO notation--Fourthward EDOs (9, 11, 13b, 16, 18b and 23)"&gt;Fourthward EDOs (9, 11, 13b, 16, 18b and 23)&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:63 --&gt;&lt;!-- ws:start:WikiTextTocRule:64: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Summary of EDO notation--Fourthward EDOs (9, 16 and 23)"&gt;Fourthward EDOs (9, 16 and 23)&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:63 --&gt;&lt;!-- ws:start:WikiTextTocRule:64: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Summary of EDO notation--Pentatonic EDOs (5, 10, 15, 20, 25 and 30)"&gt;Pentatonic EDOs (5, 10, 15, 20, 25 and 30)&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:64 --&gt;&lt;!-- ws:start:WikiTextTocRule:65: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Summary of EDO notation--&amp;quot;Perfect&amp;quot; EDOs (7, 14, 21, 28 and 35)"&gt;&amp;quot;Perfect&amp;quot; EDOs (7, 14, 21, 28 and 35)&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:64 --&gt;&lt;!-- ws:start:WikiTextTocRule:65: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Summary of EDO notation--Alternative pentatonic notation for pentatonic EDOs:"&gt;Alternative pentatonic notation for pentatonic EDOs:&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:65 --&gt;&lt;!-- ws:start:WikiTextTocRule:66: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Summary of EDO notation--Pentatonic EDOs (5, 10, 15, 20, 25 and 30)"&gt;Pentatonic EDOs (5, 10, 15, 20, 25 and 30)&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:65 --&gt;&lt;!-- ws:start:WikiTextTocRule:66: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Summary of EDO notation--&amp;quot;Fifth-less&amp;quot; EDOs (8, 11b, 13 and 18)"&gt;&amp;quot;Fifth-less&amp;quot; EDOs (8, 11b, 13 and 18)&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:66 --&gt;&lt;!-- ws:start:WikiTextTocRule:67: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Summary of EDO notation--Alternative pentatonic notation for pentatonic EDOs:"&gt;Alternative pentatonic notation for pentatonic EDOs:&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:66 --&gt;&lt;!-- ws:start:WikiTextTocRule:67: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Summary of EDO notation--Alternate pentatonic notation for EDOs 8, 13 and 18"&gt;Alternate pentatonic notation for EDOs 8, 13 and 18&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:67 --&gt;&lt;!-- ws:start:WikiTextTocRule:68: --&gt;&lt;div style="margin-left: 3em;"&gt;&lt;a href="#Summary of EDO notation--&amp;quot;Regular&amp;quot; EDOs (12, 17, 19, 22, 24, 26, 27, 29, 31-34, and all edos 36 or higher)"&gt;&amp;quot;Regular&amp;quot; EDOs (12, 17, 19, 22, 24, 26, 27, 29, 31-34, and all edos 36 or higher)&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:67 --&gt;&lt;!-- ws:start:WikiTextTocRule:68: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#EDOs with an inaccurate 3/2"&gt;EDOs with an inaccurate 3/2&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:68 --&gt;&lt;!-- ws:start:WikiTextTocRule:69: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Ups and downs solfege"&gt;Ups and downs solfege&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:68 --&gt;&lt;!-- ws:start:WikiTextTocRule:69: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Ups and downs solfege"&gt;Ups and downs solfege&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:69 --&gt;&lt;!-- ws:start:WikiTextTocRule:70: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Rank-2 Notation"&gt;Rank-2 Notation&lt;/a&gt;&lt;/div&gt;
&lt;!-- ws:end:WikiTextTocRule:69 --&gt;&lt;!-- ws:start:WikiTextTocRule:70: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Rank-2 Notation"&gt;Rank-2 Notation&lt;/a&gt;&lt;/div&gt;
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&lt;!-- ws:start:WikiTextHeadingRule:20:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc10"&gt;&lt;a name="Scale Fragments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt;&lt;u&gt;&lt;strong&gt;Scale Fragments&lt;/strong&gt;&lt;/u&gt;&lt;/h1&gt;
  &lt;br /&gt;
  &lt;br /&gt;
Not counting the trivial edos 2, 3, 4 and 6, there are only seven such edos. As seen in the above diagram, they are the ones to the left of the heptatonic kite's spine, plus the ones to the right of the pentatonic kite's spine. The ones on the left edge of the heptatonic kite are the fourthward ones like 16edo, and have been dealt with already. 23edo can be notated similarly to 16edo by using a fifth of 13\23 instead of 14\23. That leaves only four edos: 8, 11, 13, and 18.&lt;br /&gt;
Besides the trivial EDOs, 1, 2, 3, 4 and 6, which can be notated with standard notation as a subset of 12-EDO, there are five EDO categories, based on the size of the fifth:&lt;br /&gt;
&amp;quot;Fifth-less&amp;quot; EDOs (8, 11, 13 and 18)&lt;br /&gt;
&amp;quot;Fourthward&amp;quot; EDOs (9, 16 and 23)&lt;br /&gt;
&amp;quot;Perfect&amp;quot; EDOs (7, 14, 21, 28 and 35)&lt;br /&gt;
&amp;quot;Pentatonic&amp;quot; EDOs (5, 10, 15, 20, 25 and 30)&lt;br /&gt;
&amp;quot;Regular&amp;quot; EDOs (all others)&lt;br /&gt;
The first two categories never use ups and downs, the next two always do (except for 5edo and 7edo). The regular EDOs may or may not.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
There are two strategies for notating these &amp;quot;oddball&amp;quot; EDOs, besides heptatonic fifth-based notation with ups and downs. One is to switch from heptatonic notation to some other type. The orange kite contains edos for which pentatonic notation is a natural fit, in the sense that no ups or downs are needed. This includes 8edo, 13edo and 18edo.&lt;br /&gt;
To summarize an EDO, a scale fragment from C to D is shown, including C# and Db. Examples:&lt;br /&gt;
C C#/Db D (12edo)&lt;br /&gt;
C Db C# D (17edo)&lt;br /&gt;
C C# Db D (19edo)&lt;br /&gt;
C C# _ Db D (26edo)&lt;br /&gt;
C _ C# Db _ D (31edo)&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
The other approach is to use some interval other than the fifth to generate the notation. Earlier I said notating 22edo using an even distribution of note names such as C _ _ D _ _ E _ _ F _ _ _ G _ _ A _ _ B _ _ C was a bad idea because the G-D and the A-E fifths looked perfect but were actually diminished. The reasoning is that 3/2 is an important ratio, and any decent approximation of 3/2 should look like a perfect fifth. But these EDOs don't approximate 3/2 well, so they can be thought of as having both a major fifth and a minor fifth. This negates any expectations of what a fifth should look like.&lt;br /&gt;
The scale fragment concisely conveys the &amp;quot;flavor&amp;quot; of the EDO's notation. The C-C# interval is the augmented unison, and if the 2nd key in the fragment isn't C#, ups and downs are required. The only exception is 7edo. For most EDOs, the C-Db interval is the minor 2nd and the C-D interval is the major 2nd. For perfect EDOs, C-Db = d2 and C-D = P2. For fourthward EDOs, C-Db = d2 and C-D = m2. D# is included for these EDOs because C-D# is a M2 just like E-F. For fifthless EDOs, the scale fragment isn't as helpful because you can't deduce the entire keyboard layout from it.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
Every EDO contains a unique scale fragment, and every scale fragment implies a unique EDO. Furthermore, this uniqueness applies to EDOs with alternate fifths: &amp;quot;wide-fifth&amp;quot; 35edo (which uses 21\35 as a fifth) has a different scale fragment than &amp;quot;narrow-fifth&amp;quot; 35edo with 20\35. If an EDO has a fifth of keyspan F and an octave of keyspan O (i.e. it's O-EDO), the minor 2nd's keyspan is m2 = -5F + 3O, and the augmented unison's is A1 = 7F - 4O. These equations can be reversed: F = 4(m2) + 3(A1) and O = 7(m2) + 5(A1). (For perfect and fourthwards EDOs, substitute M2 for m2.)&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;u&gt;&lt;strong&gt;Theoretical alternatives for 8edo, 11edo, 13edo and 18edo&lt;/strong&gt;&lt;/u&gt;&lt;br /&gt;
In the chart below, 13edo and 18edo use the narrower fifth.&lt;br /&gt;
&lt;br /&gt;
 
8edo octatonic (every note is a generator)&lt;br /&gt;
 
D E F G H A B C D&lt;br /&gt;
&lt;table class="wiki_table"&gt;
P1 - P2 - P3 - P4 - P5 - P6 - P7 - P8 - P9&lt;br /&gt;
    &lt;tr&gt;
requires learning octatonic interval arithmetic and staff notation&lt;br /&gt;
        &lt;td style="text-align: center;"&gt;5edo&lt;br /&gt;
&lt;br /&gt;
&lt;/td&gt;
11edo heptatonic narrow-fifth-based, fourthwards, # is vv (3/2 maps to 6\11 perfect 5th):&lt;br /&gt;
        &lt;td style="text-align: center;"&gt;pentatonic&lt;br /&gt;
D E * * F G A B * * C D = D E F# F~ F G A B B~ Bb C D&lt;br /&gt;
&lt;/td&gt;
fourthwards chain of fifths: m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7&lt;br /&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
P1 - M2 - ~2/M3 - m2/~3 - m3 - P4 - P5 - M6 - ~6/M7 - m6/~7 - m7 - P8&lt;br /&gt;
&lt;/td&gt;
problematic because M3 = 2\11 is narrower than m2 = 3\11&lt;br /&gt;
        &lt;td style="text-align: center;"&gt;C/Db&lt;br /&gt;
&lt;br /&gt;
&lt;/td&gt;
11edo nonotonic narrow-fifth-based, fifthwards with no ups and downs (3/2 maps to 6\11 = perfect 6th):&lt;br /&gt;
        &lt;td style="text-align: center;"&gt;C#/D&lt;br /&gt;
nonotonic fifthwards chain of sixths: M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9&lt;br /&gt;
&lt;/td&gt;
P1 m2 M2/m3 M3/m4 M4 P5 P6 m7 M7/m8 M8/m9 M9 P8&lt;br /&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
requires learning nonotonic interval arithmetic and staff notation&lt;br /&gt;
&lt;/td&gt;
&lt;br /&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
11edo pentatonic wide-fifth-based, fifthwards, # is ^^ (3/2 maps to 7\11 6th):&lt;br /&gt;
&lt;/td&gt;
D * * E G * * A C * * D&lt;br /&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
pentatonic fifthwards chain of fifthoids: m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7&lt;br /&gt;
&lt;/td&gt;
P1 - ms3 - ^ms3/vMs3 - Ms3 - P4d - ^P4d/d5d - A4d/vP5d - P5d - ms7 - ^ms7/vMs7 - Ms7 - P8d&lt;br /&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
pentatonic plus ups and downs is doubly confusing!&lt;br /&gt;
&lt;/td&gt;
&lt;br /&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
11edo octatonic wide-fifth-based, fifthwards, no ^/v (3/2 maps to 7\11 = perfect 6th):&lt;br /&gt;
&lt;/td&gt;
octatonic chain of 6ths: m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7&lt;br /&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9&lt;br /&gt;
&lt;/td&gt;
requires learning octatonic interval arithmetic and notation&lt;br /&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;/td&gt;
13edo heptatonic narrow-fifth-based, fourthwards, sharp = ^^^ (3/2 maps to 7\13 perfect 5th):&lt;br /&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
D E * * * F G A B * * * C D&lt;br /&gt;
&lt;/td&gt;
fourthwards chain of fifths: M2 - M6 - M3 - M7 - P4 - P1 - P5 - m2 - m6 - m3 - m7&lt;br /&gt;
    &lt;/tr&gt;
P1 - m2 - m3 - vM2/^m3 - M2 - M3 - P4 - P5 - m6 - m7 - vM6/^m7 - M6 - M7 - P8&lt;br /&gt;
    &lt;tr&gt;
problematic because m3 = 2\13 is narrower than M2 = 4\13&lt;br /&gt;
        &lt;td style="text-align: center;"&gt;6edo&lt;br /&gt;
&lt;br /&gt;
&lt;/td&gt;
13edo undecatonic narrow-fifth-based, fourthwards, 3/2 maps to 7\13 = perfect 7th&lt;br /&gt;
        &lt;td style="text-align: center;"&gt;fifthless&lt;br /&gt;
undecatonic sixthwards chain of sevenths:&lt;br /&gt;
&lt;/td&gt;
M2 - M8 - M3 - M9 - M4 - M10 - M5 - M11 - P6 - P1 - P7 - m2 - m8 - m3 - m9 - m4 - m10 - m5 - m11&lt;br /&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
P1 - m2 - M2/m3 - M3/m4 - M4/m5 - M5 - P6 - P7 - m8 - M8/m9 - M9/m10 - M10/m11 - M11 - P12&lt;br /&gt;
&lt;/td&gt;
requires learning undecatonic interval arithmetic and notation&lt;br /&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;/td&gt;
13edo octatonic wide-fifth-based, fourthwards, 3/2 maps to 8\13 = perfect 6th&lt;br /&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
octotonic chain of sixths: M3 - M8 - M5 - M2 - M7 - P4 - P1 - P6 - m3 - m8 - m5 - m2 - m7&lt;br /&gt;
&lt;/td&gt;
P1 - m2 - M2 - m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7 - m8 - M8 - P9&lt;br /&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
requires learning octatonic interval arithmetic and notation&lt;br /&gt;
&lt;/td&gt;
&lt;br /&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
18edo heptatonic narrow-fifth-based, fourthwards, sharp = ^^ (3/2 maps to 10\18 perfect 5th)&lt;br /&gt;
&lt;/td&gt;
D * E * * * F * G * A * B * * * C * D&lt;br /&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
P1 - vm2 - m2 - vM2 - M2/m3 - vM3 - M3 - ^M3 - P4 - ^P4/vP5 - P5 - vm6 - m6 - vM6 - M6/m7 - vM7 - M7 - ^M7 - P8&lt;br /&gt;
&lt;/td&gt;
fourthwards plus ups and downs plus closed is triply confusing!&lt;br /&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;/td&gt;
18edo nonatonic narrow-fifth-based (3/2 maps to 10\18 = perfect 6th)&lt;br /&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
P1 - vP2 - P2 - vP3 - P3 - vP4- P4 - vP5 - P5 - vP6 - P6 - vP7 - P7 - vP8 - P8 - vP9 - P9 - vP10 - P10&lt;br /&gt;
&lt;/td&gt;
requires learning nonotonic interval arithmetic and staff notation&lt;br /&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;/td&gt;
&lt;br /&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;u&gt;&lt;strong&gt;Alternate notation for other edos:&lt;/strong&gt;&lt;/u&gt;&lt;br /&gt;
&lt;/td&gt;
23edo pentatonic wide-fifth-based, fifthwards, 3/2 maps to 14\23 = perfect fifthoid:&lt;br /&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
D * * * * E * * * G * * * * A * * * C * * * * D&lt;br /&gt;
&lt;/td&gt;
35edo heptatonic wide-fifth-based, sharp = seven ups, 3/2 maps to 21\35 = perfect fifth:&lt;br /&gt;
    &lt;/tr&gt;
D * * * * * * E/F * * * * * * G * * * * * * A * * * * * * B/C * * * * * * D&lt;br /&gt;
    &lt;tr&gt;
42edo heptatonic narrow-fifth-based, sharp = six ups, 3/2 maps to 24\42 = perfect fifth:&lt;br /&gt;
        &lt;td style="text-align: center;"&gt;7edo&lt;br /&gt;
D * * * * * E * * * * * F * * * * * G * * * * * A * * * * * B * * * * * C * * * * * D&lt;br /&gt;
&lt;/td&gt;
&lt;br /&gt;
        &lt;td style="text-align: center;"&gt;perfect&lt;br /&gt;
&lt;br /&gt;
&lt;/td&gt;
&lt;!-- ws:start:WikiTextHeadingRule:22:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc11"&gt;&lt;a name="Scale Fragments"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:22 --&gt;&lt;u&gt;&lt;strong&gt;Scale Fragments&lt;/strong&gt;&lt;/u&gt;&lt;/h1&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;/td&gt;
Besides the trivial EDOs, 1, 2, 3, 4 and 6, which can be notated with standard notation as a subset of 12-EDO, there are five EDO categories, based on the size of the fifth:&lt;br /&gt;
        &lt;td style="text-align: center;"&gt;C/C#&lt;br /&gt;
&amp;quot;Fifth-less&amp;quot; EDOs (8, 11, 13 and 18)&lt;br /&gt;
&lt;/td&gt;
&amp;quot;Fourthward&amp;quot; EDOs (9, 16 and 23)&lt;br /&gt;
        &lt;td style="text-align: center;"&gt;Db/D&lt;br /&gt;
&amp;quot;Perfect&amp;quot; EDOs (7, 14, 21, 28 and 35)&lt;br /&gt;
&lt;/td&gt;
&amp;quot;Pentatonic&amp;quot; EDOs (5, 10, 15, 20, 25 and 30)&lt;br /&gt;
&amp;quot;Regular&amp;quot; EDOs (all others)&lt;br /&gt;
The first two categories never use ups and downs, the next two always do (except for 5edo and 7edo). The regular EDOs may or may not.&lt;br /&gt;
&lt;br /&gt;
To summarize an EDO, a scale fragment from C to D is shown, including C# and Db. Examples:&lt;br /&gt;
C C#/Db D (12edo)&lt;br /&gt;
C Db C# D (17edo)&lt;br /&gt;
C C# Db D (19edo)&lt;br /&gt;
C C# _ Db D (26edo)&lt;br /&gt;
C _ C# Db _ D (31edo)&lt;br /&gt;
&lt;br /&gt;
The scale fragment concisely conveys the &amp;quot;flavor&amp;quot; of the EDO's notation. The C-C# interval is the augmented unison, and if the 2nd key in the fragment isn't C#, ups and downs are required. The only exception is 7edo. For most EDOs, the C-Db interval is the minor 2nd and the C-D interval is the major 2nd. For perfect EDOs, C-Db = d2 and C-D = P2. For fourthward EDOs, C-Db = d2 and C-D = m2. D# is included for these EDOs because C-D# is a M2 just like E-F. For fifthless EDOs, the scale fragment isn't as helpful because you can't deduce the entire keyboard layout from it.&lt;br /&gt;
&lt;br /&gt;
Every EDO contains a unique scale fragment, and every scale fragment implies a unique EDO. Furthermore, this uniqueness applies to EDOs with alternate fifths: &amp;quot;wide-fifth&amp;quot; 35edo (which uses 21\35 as a fifth) has a different scale fragment than &amp;quot;narrow-fifth&amp;quot; 35edo with 20\35. If an EDO has a fifth of keyspan F and an octave of keyspan O (i.e. it's O-EDO), the minor 2nd's keyspan is m2 = -5F + 3O, and the augmented unison's is A1 = 7F - 4O. These equations can be reversed: F = 4(m2) + 3(A1) and O = 7(m2) + 5(A1). (For perfect and fourthwards EDOs, substitute M2 for m2.)&lt;br /&gt;
&lt;br /&gt;
In the chart below, 13edo and 18edo use the narrower fifth.&lt;br /&gt;
 
 
&lt;table class="wiki_table"&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;5edo&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;pentatonic&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;C/Db&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;C#/D&lt;br /&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
Line 2,107: Line 2,103:
     &lt;/tr&gt;
     &lt;/tr&gt;
     &lt;tr&gt;
     &lt;tr&gt;
         &lt;td style="text-align: center;"&gt;6edo&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;8edo&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;fifthless&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;fifthless&lt;br /&gt;
Line 2,135: Line 2,131:
     &lt;/tr&gt;
     &lt;/tr&gt;
     &lt;tr&gt;
     &lt;tr&gt;
         &lt;td style="text-align: center;"&gt;7edo&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;9edo&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;perfect&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;fourthward&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;C/C#&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;C/Db&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;Db/D&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;C#/D&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;D#&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
Line 2,163: Line 2,159:
     &lt;/tr&gt;
     &lt;/tr&gt;
     &lt;tr&gt;
     &lt;tr&gt;
         &lt;td style="text-align: center;"&gt;8edo&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;10edo&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;fifthless&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;pentatonic&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;C/Db&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;C#/D&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
Line 2,191: Line 2,187:
     &lt;/tr&gt;
     &lt;/tr&gt;
     &lt;tr&gt;
     &lt;tr&gt;
         &lt;td style="text-align: center;"&gt;9edo&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;11edo&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;fourthward&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;fourthward&lt;br /&gt;
Line 2,197: Line 2,193:
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;C/Db&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;C&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;D&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;C#/D&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;C#&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;D#&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;D#&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
Line 2,219: Line 2,215:
     &lt;/tr&gt;
     &lt;/tr&gt;
     &lt;tr&gt;
     &lt;tr&gt;
         &lt;td style="text-align: center;"&gt;10edo&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;12edo&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;pentatonic&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;regular&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;C/Db&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;C&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;C#/Db&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;C#/D&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;D&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
Line 2,247: Line 2,243:
     &lt;/tr&gt;
     &lt;/tr&gt;
     &lt;tr&gt;
     &lt;tr&gt;
         &lt;td style="text-align: center;"&gt;11edo&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;13edo&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;fourthward&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;fourthward&lt;br /&gt;
Line 2,256: Line 2,252:
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;D&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;D&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;C#&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;C#&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;D#&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;D#&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
Line 2,275: Line 2,271:
     &lt;/tr&gt;
     &lt;/tr&gt;
     &lt;tr&gt;
     &lt;tr&gt;
         &lt;td style="text-align: center;"&gt;12edo&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;14edo&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;regular&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;perfect&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;C&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;C/C#&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;C#/Db&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;D&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;Db/D&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
Line 2,303: Line 2,299:
     &lt;/tr&gt;
     &lt;/tr&gt;
     &lt;tr&gt;
     &lt;tr&gt;
         &lt;td style="text-align: center;"&gt;13edo&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;15edo&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;fourthward&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;pentatonic&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;C&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;C/Db&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;D&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;C#&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;C#/D&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;D#&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
Line 2,331: Line 2,327:
     &lt;/tr&gt;
     &lt;/tr&gt;
     &lt;tr&gt;
     &lt;tr&gt;
         &lt;td style="text-align: center;"&gt;14edo&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;16edo&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;perfect&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;fourthward&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;C/C#&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;C&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;C#/Db&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;Db/D&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;D&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;D#&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
Line 2,359: Line 2,355:
     &lt;/tr&gt;
     &lt;/tr&gt;
     &lt;tr&gt;
     &lt;tr&gt;
         &lt;td style="text-align: center;"&gt;15edo&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;17edo&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;pentatonic&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;regular&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;C/Db&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;C&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;Db&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;C#&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;C#/D&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;D&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
Line 2,387: Line 2,383:
     &lt;/tr&gt;
     &lt;/tr&gt;
     &lt;tr&gt;
     &lt;tr&gt;
         &lt;td style="text-align: center;"&gt;16edo&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;18edo&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;fourthward&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;fourthward&lt;br /&gt;
Line 2,393: Line 2,389:
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;C&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;C/Db&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;C#/Db&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;C#/D&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;D&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;D#&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;D#&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
Line 2,415: Line 2,411:
     &lt;/tr&gt;
     &lt;/tr&gt;
     &lt;tr&gt;
     &lt;tr&gt;
         &lt;td style="text-align: center;"&gt;17edo&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;19edo&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;regular&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;regular&lt;br /&gt;
Line 2,422: Line 2,418:
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;C&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;C&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;C#&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;Db&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;Db&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;C#&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;D&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;D&lt;br /&gt;
Line 2,443: Line 2,439:
     &lt;/tr&gt;
     &lt;/tr&gt;
     &lt;tr&gt;
     &lt;tr&gt;
         &lt;td style="text-align: center;"&gt;18edo&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;20edo&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;fourthward&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;pentatonic&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
Line 2,453: Line 2,449:
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;C#/D&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;D#&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;C#/D&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
Line 2,471: Line 2,467:
     &lt;/tr&gt;
     &lt;/tr&gt;
     &lt;tr&gt;
     &lt;tr&gt;
         &lt;td style="text-align: center;"&gt;19edo&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;21edo&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;regular&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;perfect&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;C&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;C/C#&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;C#&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;Db&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;D&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;Db/D&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
Line 2,499: Line 2,495:
     &lt;/tr&gt;
     &lt;/tr&gt;
     &lt;tr&gt;
     &lt;tr&gt;
         &lt;td style="text-align: center;"&gt;20edo&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;22edo&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;pentatonic&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;regular&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;C/Db&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;C&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;Db&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;C#&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;C#/D&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;D&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
Line 2,527: Line 2,523:
     &lt;/tr&gt;
     &lt;/tr&gt;
     &lt;tr&gt;
     &lt;tr&gt;
         &lt;td style="text-align: center;"&gt;21edo&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;23edo&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;perfect&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;fourthward&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;C/C#&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;C&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;C#&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;Db&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;Db/D&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;D&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;D#&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
Line 2,555: Line 2,551:
     &lt;/tr&gt;
     &lt;/tr&gt;
     &lt;tr&gt;
     &lt;tr&gt;
         &lt;td style="text-align: center;"&gt;22edo&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;24edo&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;regular&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;regular&lt;br /&gt;
Line 2,562: Line 2,558:
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;C&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;C&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Db&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;C#&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;C#/Db&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;D&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;D&lt;br /&gt;
Line 2,583: Line 2,579:
     &lt;/tr&gt;
     &lt;/tr&gt;
     &lt;tr&gt;
     &lt;tr&gt;
         &lt;td style="text-align: center;"&gt;23edo&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;25edo&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;fourthward&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;pentatonic&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;C&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;C/Db&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;C#&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;Db&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;D&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;D#&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;C#/D&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
Line 2,611: Line 2,607:
     &lt;/tr&gt;
     &lt;/tr&gt;
     &lt;tr&gt;
     &lt;tr&gt;
         &lt;td style="text-align: center;"&gt;24edo&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;26edo&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;regular&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;regular&lt;br /&gt;
Line 2,618: Line 2,614:
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;C&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;C&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;C#&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;C#/Db&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;Db&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;D&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;D&lt;br /&gt;
Line 2,639: Line 2,635:
     &lt;/tr&gt;
     &lt;/tr&gt;
     &lt;tr&gt;
     &lt;tr&gt;
         &lt;td style="text-align: center;"&gt;25edo&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;27edo&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;pentatonic&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;regular&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;C/Db&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;C&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;Db&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
Line 2,653: Line 2,649:
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;C#&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;C#/D&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;D&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
Line 2,667: Line 2,663:
     &lt;/tr&gt;
     &lt;/tr&gt;
     &lt;tr&gt;
     &lt;tr&gt;
         &lt;td style="text-align: center;"&gt;26edo&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;28edo&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;regular&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;perfect&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;C&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;C/C#&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;C#&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;Db&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;D&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;Db/D&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
Line 2,695: Line 2,691:
     &lt;/tr&gt;
     &lt;/tr&gt;
     &lt;tr&gt;
     &lt;tr&gt;
         &lt;td style="text-align: center;"&gt;27edo&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;29edo&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;regular&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;regular&lt;br /&gt;
Line 2,702: Line 2,698:
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;C&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;C&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;Db&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;Db&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;C#&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;C#&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;D&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;D&lt;br /&gt;
Line 2,723: Line 2,719:
     &lt;/tr&gt;
     &lt;/tr&gt;
     &lt;tr&gt;
     &lt;tr&gt;
         &lt;td style="text-align: center;"&gt;28edo&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;30edo&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;perfect&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;pentatonic&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;C/C#&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;C/Db&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
Line 2,737: Line 2,733:
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;Db/D&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;C#/D&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
Line 2,751: Line 2,747:
     &lt;/tr&gt;
     &lt;/tr&gt;
     &lt;tr&gt;
     &lt;tr&gt;
         &lt;td style="text-align: center;"&gt;29edo&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;31edo&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;regular&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;regular&lt;br /&gt;
Line 2,760: Line 2,756:
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;C#&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;Db&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;Db&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;C#&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
Line 2,779: Line 2,775:
     &lt;/tr&gt;
     &lt;/tr&gt;
     &lt;tr&gt;
     &lt;tr&gt;
         &lt;td style="text-align: center;"&gt;30edo&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;32edo&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;pentatonic&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;regular&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;C/Db&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;C&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;Db&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
Line 2,795: Line 2,791:
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;C#&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;C#/D&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;D&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
Line 2,807: Line 2,803:
     &lt;/tr&gt;
     &lt;/tr&gt;
     &lt;tr&gt;
     &lt;tr&gt;
         &lt;td style="text-align: center;"&gt;31edo&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;33edo&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;regular&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;regular&lt;br /&gt;
Line 2,814: Line 2,810:
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;C&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;C&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;C#&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;C#&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;Db&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;Db&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;D&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;D&lt;br /&gt;
Line 2,835: Line 2,831:
     &lt;/tr&gt;
     &lt;/tr&gt;
     &lt;tr&gt;
     &lt;tr&gt;
         &lt;td style="text-align: center;"&gt;32edo&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;34edo&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;regular&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;regular&lt;br /&gt;
Line 2,842: Line 2,838:
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;C&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;C&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;Db&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;Db&lt;br /&gt;
Line 2,847: Line 2,845:
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;C#&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;C#&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;D&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;D&lt;br /&gt;
Line 2,863: Line 2,859:
     &lt;/tr&gt;
     &lt;/tr&gt;
     &lt;tr&gt;
     &lt;tr&gt;
         &lt;td style="text-align: center;"&gt;33edo&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;35edo&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;regular&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;perfect&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;C&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;C/C#&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;C#&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
Line 2,877: Line 2,873:
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;Db&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;D&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;Db/D&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
Line 2,891: Line 2,887:
     &lt;/tr&gt;
     &lt;/tr&gt;
     &lt;tr&gt;
     &lt;tr&gt;
         &lt;td style="text-align: center;"&gt;34edo&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;36edo&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;regular&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;regular&lt;br /&gt;
Line 2,901: Line 2,897:
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;Db&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;C#/Db&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;C#&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
Line 2,919: Line 2,915:
     &lt;/tr&gt;
     &lt;/tr&gt;
     &lt;tr&gt;
     &lt;tr&gt;
         &lt;td style="text-align: center;"&gt;35edo&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;37edo&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;perfect&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;C/C#&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;C&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Db&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
Line 2,935: Line 2,933:
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;Db/D&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;C#&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;D&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
Line 2,947: Line 2,943:
     &lt;/tr&gt;
     &lt;/tr&gt;
     &lt;tr&gt;
     &lt;tr&gt;
         &lt;td style="text-align: center;"&gt;36edo&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;38edo&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;regular&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
Line 2,956: Line 2,952:
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;C#&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;C#/Db&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;Db&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
Line 2,975: Line 2,971:
     &lt;/tr&gt;
     &lt;/tr&gt;
     &lt;tr&gt;
     &lt;tr&gt;
         &lt;td style="text-align: center;"&gt;37edo&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;39edo&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;
Line 2,982: Line 2,978:
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;C&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;C&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;Db&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;Db&lt;br /&gt;
Line 2,989: Line 2,987:
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;C#&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;C#&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;D&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;D&lt;br /&gt;
Line 3,003: Line 2,999:
     &lt;/tr&gt;
     &lt;/tr&gt;
     &lt;tr&gt;
     &lt;tr&gt;
         &lt;td style="text-align: center;"&gt;38edo&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;40edo&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;
Line 3,010: Line 3,006:
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;C&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;C&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;C#&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;C#&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;Db&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;Db&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;D&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;D&lt;br /&gt;
Line 3,031: Line 3,027:
     &lt;/tr&gt;
     &lt;/tr&gt;
     &lt;tr&gt;
     &lt;tr&gt;
         &lt;td style="text-align: center;"&gt;39edo&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;41edo&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;
Line 3,038: Line 3,034:
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;C&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;C&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
Line 3,043: Line 3,041:
         &lt;td style="text-align: center;"&gt;Db&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;Db&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;C#&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;C#&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
Line 3,059: Line 3,055:
     &lt;/tr&gt;
     &lt;/tr&gt;
     &lt;tr&gt;
     &lt;tr&gt;
         &lt;td style="text-align: center;"&gt;40edo&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;42edo&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;
Line 3,067: Line 3,063:
         &lt;td style="text-align: center;"&gt;C&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;C&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;C#&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;Db&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
Line 3,075: Line 3,071:
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;Db&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;D&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;C#&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;D&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
Line 3,087: Line 3,083:
     &lt;/tr&gt;
     &lt;/tr&gt;
     &lt;tr&gt;
     &lt;tr&gt;
         &lt;td style="text-align: center;"&gt;41edo&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;43edo&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;
Line 3,098: Line 3,094:
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;C#&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;Db&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;Db&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;C#&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
Line 3,115: Line 3,111:
     &lt;/tr&gt;
     &lt;/tr&gt;
     &lt;tr&gt;
     &lt;tr&gt;
         &lt;td style="text-align: center;"&gt;42edo&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;44ddo&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;
Line 3,122: Line 3,118:
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;C&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;C&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;Db&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;Db&lt;br /&gt;
Line 3,131: Line 3,129:
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;C#&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;C#&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;D&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;D&lt;br /&gt;
Line 3,143: Line 3,139:
     &lt;/tr&gt;
     &lt;/tr&gt;
     &lt;tr&gt;
     &lt;tr&gt;
         &lt;td style="text-align: center;"&gt;43edo&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;45edo&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;
Line 3,152: Line 3,148:
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;C#&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;C#&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;Db&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;Db&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
Line 3,171: Line 3,167:
     &lt;/tr&gt;
     &lt;/tr&gt;
     &lt;tr&gt;
     &lt;tr&gt;
         &lt;td style="text-align: center;"&gt;44ddo&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;46edo&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;
Line 3,178: Line 3,174:
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;C&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;C&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
Line 3,185: Line 3,183:
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;C#&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;C#&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
Line 3,199: Line 3,195:
     &lt;/tr&gt;
     &lt;/tr&gt;
     &lt;tr&gt;
     &lt;tr&gt;
         &lt;td style="text-align: center;"&gt;45edo&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;47edo&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;
Line 3,206: Line 3,202:
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;C&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;C&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;C#&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;C#&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
Line 3,216: Line 3,214:
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;Db&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;Db&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;D&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;D&lt;br /&gt;
Line 3,227: Line 3,223:
     &lt;/tr&gt;
     &lt;/tr&gt;
     &lt;tr&gt;
     &lt;tr&gt;
         &lt;td style="text-align: center;"&gt;46edo&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;48edo&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;
Line 3,239: Line 3,235:
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;Db&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;C#/Db&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;C#&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
Line 3,255: Line 3,251:
     &lt;/tr&gt;
     &lt;/tr&gt;
     &lt;tr&gt;
     &lt;tr&gt;
         &lt;td style="text-align: center;"&gt;47edo&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;49edo&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;
Line 3,263: Line 3,259:
         &lt;td style="text-align: center;"&gt;C&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;C&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;C#&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Db&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
Line 3,273: Line 3,271:
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;Db&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;C#&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;D&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;D&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
     &lt;/tr&gt;
     &lt;/tr&gt;
     &lt;tr&gt;
     &lt;tr&gt;
         &lt;td style="text-align: center;"&gt;48edo&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;50edo&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;
Line 3,294: Line 3,290:
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;C#&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;C#/Db&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;Db&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
Line 3,311: Line 3,307:
     &lt;/tr&gt;
     &lt;/tr&gt;
     &lt;tr&gt;
     &lt;tr&gt;
         &lt;td style="text-align: center;"&gt;49edo&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;51edo&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;
Line 3,318: Line 3,314:
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;C&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;C&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
Line 3,327: Line 3,325:
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;C#&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;C#&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
Line 3,339: Line 3,335:
     &lt;/tr&gt;
     &lt;/tr&gt;
     &lt;tr&gt;
     &lt;tr&gt;
         &lt;td style="text-align: center;"&gt;50edo&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;52edo&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;
Line 3,348: Line 3,344:
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;C#&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;C#&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;Db&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;Db&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
Line 3,367: Line 3,363:
     &lt;/tr&gt;
     &lt;/tr&gt;
     &lt;tr&gt;
     &lt;tr&gt;
         &lt;td style="text-align: center;"&gt;51edo&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;53edo&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;
Line 3,374: Line 3,370:
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;C&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;C&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
Line 3,380: Line 3,378:
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;Db&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;Db&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;C#&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;C#&lt;br /&gt;
Line 3,390: Line 3,384:
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;D&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td style="text-align: center;"&gt;52edo&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;C&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;C#&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Db&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
         &lt;td style="text-align: center;"&gt;D&lt;br /&gt;
         &lt;td style="text-align: center;"&gt;D&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
&lt;/td&gt;
     &lt;/tr&gt;
     &lt;/tr&gt;
    &lt;tr&gt;
&lt;/table&gt;
        &lt;td style="text-align: center;"&gt;53edo&lt;br /&gt;
 
&lt;/td&gt;
&lt;br /&gt;
        &lt;td style="text-align: center;"&gt;&amp;quot;&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:22:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc11"&gt;&lt;a name="Summary of EDO notation"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:22 --&gt;&lt;u&gt;&lt;strong&gt;Summary of EDO notation&lt;/strong&gt;&lt;/u&gt;&lt;/h1&gt;
&lt;/td&gt;
&lt;br /&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;br /&gt;
&lt;/td&gt;
&lt;!-- ws:start:WikiTextHeadingRule:24:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc12"&gt;&lt;a name="Summary of EDO notation--&amp;quot;Regular&amp;quot; EDOs (12, 17, 19, 22, 24, 26, 27, 29, 31-34, and all edos 36 or higher)"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:24 --&gt;&lt;u&gt;&amp;quot;Regular&amp;quot; EDOs (12, 17, 19, 22, 24, 26, 27, 29, 31-34, and all edos 36 or higher)&lt;/u&gt;&lt;/h3&gt;
        &lt;td style="text-align: center;"&gt;C&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Db&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;C#&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;*&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;D&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;
 
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:24:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc12"&gt;&lt;a name="Summary of EDO notation"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:24 --&gt;&lt;u&gt;&lt;strong&gt;Summary of EDO notation&lt;/strong&gt;&lt;/u&gt;&lt;/h1&gt;
  &lt;br /&gt;
  &lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:26:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc13"&gt;&lt;a name="Summary of EDO notation--&amp;quot;Fifth-less&amp;quot; EDOs (8, 11, 13 and 18)"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:26 --&gt;&lt;u&gt;&lt;strong&gt;&amp;quot;Fifth-less&amp;quot; EDOs (8, 11, 13 and 18)&lt;/strong&gt;&lt;/u&gt;&lt;/h3&gt;
All regular EDOs use the usual chain of fifths: m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7 etc.&lt;br /&gt;
&lt;br /&gt;
Fb - Cb - Gb - Db - Ab - Eb - Bb - F - C - G - D - A - E - B - F# - C# - G# - D# - A# - E# - B# etc.&lt;br /&gt;
&lt;strong&gt;&lt;u&gt;8edo&lt;/u&gt;:&lt;/strong&gt; (generator = 1\8 = perfect 2nd = 150¢)&lt;br /&gt;
D E F G * A B C D&lt;br /&gt;
D - E - F - G - G#/Ab - A -B - C - D&lt;br /&gt;
P1 - P2 - m3 - M3/m4 - M4/m5 - M5/m6 - M6 - P7 - P8&lt;br /&gt;
seventhwards chain of seconds: M3 - M4 - M5 - M6 - P7 - P1 - P2 - m3 - m4 - m5 - m6 - d7 etc.&lt;br /&gt;
A# - B# - C# - D# - E# - F# - G# - A - B - C - D - E - F - G - Ab - Bb - Cb - Db - Eb - Fb - Gb etc.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;u&gt;&lt;strong&gt;11edo&lt;/strong&gt;&lt;/u&gt;: (generator = 3\11 = perfect 3rd)&lt;br /&gt;
&lt;strong&gt;&lt;u&gt;12edo&lt;/u&gt;:&lt;/strong&gt; sharp/flat = 1 key, no ups and downs: C C#/Db D&lt;br /&gt;
D * E F * G A * B C * D&lt;br /&gt;
D * E F * G * A * B C * D&lt;br /&gt;
D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D&lt;br /&gt;
D - D#/Eb - E - F - F#/Gb - G - G#/Ab - A - A#/Bb - B - C - C#/Db - D&lt;br /&gt;
P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8&lt;br /&gt;
P1 - m2 - M2 - m3 - M3 - P4 - A4/d5 - P5 - m6 - M6 - m7 - M7 - P8&lt;br /&gt;
sixthwards chain of thirds: M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6 etc.&lt;br /&gt;
perfect = white, major = red, yellow and fifthward white, minor = green, blue and fourthwards white&lt;br /&gt;
E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb&lt;br /&gt;
&lt;br /&gt;
&lt;strong&gt;&lt;u&gt;17edo&lt;/u&gt;:&lt;/strong&gt; sharp = 2 keys: C Db C# D&lt;br /&gt;
D * * E F * * G * * A * * B C * * D&lt;br /&gt;
D - D^/Eb - D#/Ev - Eb - E - F - F^/Gb - F#/Gv - G - G^/Ab - G#/Av - A - A^/Bb - A#/Bv - B - C - C^/Db - C#/Dv - D&lt;br /&gt;
P1 - m2 - ~2 - M2 - m3 - ~3 - M3 - P4 - ^P4/d5 - A4/vP5 - P5 - m6 - ~6 - M6 - m7 - ~7 - M7 - P8&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;u&gt;&lt;strong&gt;13edo&lt;/strong&gt;&lt;/u&gt;&lt;strong&gt;:&lt;/strong&gt; (generator = 2\13 = perfect 2nd)&lt;br /&gt;
&lt;strong&gt;&lt;u&gt;19edo&lt;/u&gt;:&lt;/strong&gt; no ups and downs C C# Db D&lt;br /&gt;
D * E * F * G A * B * C * D&lt;br /&gt;
D * * E * F * * G * * A * * B * C * * D&lt;br /&gt;
D - D#/Eb - E - E#/Fb - F - F#/Gb - G - A - A#/Bb - B - B#/Cb - C - C#/Db - D&lt;br /&gt;
D - D# - Eb - E - E#/Fb - F - F# - Gb - G - G# - Ab - A - A# - Bb - B - B#/Cb - C - C# - Db - D&lt;br /&gt;
P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - m6 - M6 - P7 - A7/d8 - P8&lt;br /&gt;
P1 - A1/d2 - m2 - M2 - A2/d3 - m3 - M3 - A3/d4 - P4 - A4 - d5 - P5 - A5/d6 - m6 - M6 - A6/d7 - m7 - M7 - A7/d8 - P8&lt;br /&gt;
secondwards chain of seconds: m3 - m4 - m5 - m6 - P7 - P1 - P2 - M3 - M4 - M5 - M6 - A7 etc.&lt;br /&gt;
perfect = white, major = yellow and fifthward white, minor = green and fourthward white, aug/dim = red/blue.&lt;br /&gt;
Ab - Bb - Cb - Db - Eb - Fb - Gb - A - B - C - D - E - F - G - A# - B# - C# - D# - E# - F# - G#&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;strong&gt;&lt;u&gt;18edo&lt;/u&gt;:&lt;/strong&gt; (generator = 5\18 = perfect 3rd)&lt;br /&gt;
&lt;strong&gt;&lt;u&gt;22edo&lt;/u&gt;:&lt;/strong&gt; sharp = 3 keys: C Db * C# D&lt;br /&gt;
D * * E * F * * G * A * * B * C * * D&lt;br /&gt;
D * * * E F * * * G * * * A * * * B C * * * D&lt;br /&gt;
D - D# - Eb - E - E#/Fb - F - F# - Gb - G - G#/Ab - A - A# - Bb - B - B#/Cb - C - C# - Db - D&lt;br /&gt;
D - D^/Eb - D#v/Eb^ - D#/Ev - E - F - F^/Gb - F#v/Gb^ - F#/Gv - G - G^/Ab - G#v/Ab^ - G#/Av - A etc.&lt;br /&gt;
P1 - A1/d2 - m2 - M2 - A2/d3 - P3 - A3/d4 - m4 - M4 - A4/d5 - m5 - M5 - A5/d6 - P6 - A6/d7 - m7 - M7 - A7/d8 - P8&lt;br /&gt;
P1 - m2 - ^m2 - vM2 - M2 - m3 - ^m3 - vM3 - M3 - P4 - ^P4/d5 - vA4/^d5 - A4/vP5 - P5 etc.&lt;br /&gt;
sixthwards chain of thirds: M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6 etc.&lt;br /&gt;
E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:28:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc14"&gt;&lt;a name="Summary of EDO notation--Alternate pentatonic notation for EDOs 8, 13 and 18"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:28 --&gt;&lt;u&gt;&lt;strong&gt;Alternate pentatonic notation for EDOs 8, 13 and 18&lt;/strong&gt;&lt;/u&gt;&lt;/h3&gt;
&lt;strong&gt;&lt;u&gt;24edo&lt;/u&gt;:&lt;/strong&gt; sharp = 2 keys: C * C#/Db * D&lt;br /&gt;
&lt;br /&gt;
D * * * E * F * * * G * * * A * * * B * C * * * D&lt;br /&gt;
All three EDOs use the same pentatonic fifthwards chain of fifths: ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d etc.&lt;br /&gt;
D - D^/Ebv - D#/Eb - D#^/Ev - E - E^/Fv - F - F^/Gbv - F#/Gb - F#^/Gv - G - G^/Abv - G#/Ab - G#^/Av - A etc.&lt;br /&gt;
Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E# etc.&lt;br /&gt;
P1 - ^P1/vm2 - m2 - ~2 - M2 - ^M2/vm3 - m3 - ~3 - M3 - ^M3/vP4 - P4 - ^P4/vd5 - A4/d5 - ^A4/vP5 - P5 etc.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;u&gt;&lt;strong&gt;8edo&lt;/strong&gt;&lt;/u&gt;&lt;strong&gt;:&lt;/strong&gt; (generator = 5\8 = perfect 5thoid) C C#/Db D&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:26:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc13"&gt;&lt;a name="Summary of EDO notation--&amp;quot;Perfect&amp;quot; EDOs (7, 14, 21, 28 and 35)"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:26 --&gt;&lt;u&gt;&amp;quot;Perfect&amp;quot; EDOs (7, 14, 21, 28 and 35)&lt;/u&gt;&lt;/h3&gt;
D * E G * A C * D&lt;br /&gt;
D - D#/Eb - E - G - G#/Ab - A - C - C#/Db - D&lt;br /&gt;
P1 - ms3 - Ms3 - P4d - A4d/d5d - P5d - ms7 - Ms7 - P8d&lt;br /&gt;
&lt;br /&gt;
&lt;u&gt;&lt;strong&gt;13edo&lt;/strong&gt;&lt;/u&gt;&lt;strong&gt;:&lt;/strong&gt; (generator = 8\13 = perfect 5thoid) C C# Db D&lt;br /&gt;
D * * E * G * * A * C * * D&lt;br /&gt;
D - D# - Eb - E - E#/Gb - G - G# - Ab - A - A#/Cb - C - C# - Db - D&lt;br /&gt;
P1 - A1/ds3 - ms3 - Ms3 - As3/d4d - P4d - A4d - d5d - P5d - A5d/ds7 - ms7 - Ms7 - As7/d8d - P8d&lt;br /&gt;
&lt;br /&gt;
&lt;u&gt;&lt;strong&gt;18edo&lt;/strong&gt;&lt;/u&gt;&lt;strong&gt;:&lt;/strong&gt; (generator = 11\18 = perfect 5thoid) C C# * Db D&lt;br /&gt;
D * * * E * * G * * * A * * C * * * D&lt;br /&gt;
D - D# - Dx/Ebb - Eb - E - E# - Gb - G - G# - Gx/Abb - Ab - A - A# - Cb - C - C# - Cx/Dbb - Db - D&lt;br /&gt;
P1 - A1 - ds3 - ms3 - Ms3 - As3 - d4d - P4d - A4d - AA4d/dd5d - d5d - P5d - A5d - ds7 - ms7 - Ms7 - As7 - d8d - P8d&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:30:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc15"&gt;&lt;a name="Summary of EDO notation--Fourthward EDOs (9, 16 and 23)"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:30 --&gt;&lt;u&gt;Fourthward EDOs (9, 16 and 23)&lt;/u&gt;&lt;/h3&gt;
  &lt;br /&gt;
  &lt;br /&gt;
All fourthwards EDOs use the same chain of fifths: M2 - M6 - M3 - M7 - P4 - P1 - P5 - m2 - m6 - m3 - m7 - A4 etc.&lt;br /&gt;
All perfect EDOs use the same circle of 7 fifths: P4 - P1 - P5 - P2 - P6 - P3 - P7 - P4 - P1 etc.&lt;br /&gt;
F# - C# - G# - D# - A# - E# - B# - F - C - G - D - A - E - B - Fb - Cb - Gb - Db - Ab - Eb - Bb - Fbb etc.&lt;br /&gt;
&lt;br /&gt;
&lt;strong&gt;&lt;u&gt;9edo&lt;/u&gt;:&lt;/strong&gt; C/Db C#/D&lt;br /&gt;
D E * F G A B * C D&lt;br /&gt;
D - E - E#/Fb - F - G - A - B - B#/Cb - C - D&lt;br /&gt;
P1 - m2 - M2/m3 - M3 - P4 - P5 - m6 - M6/m7 - M7 - P8&lt;br /&gt;
&lt;br /&gt;
&lt;strong&gt;&lt;u&gt;16edo&lt;/u&gt;:&lt;/strong&gt; C C#/Db D&lt;br /&gt;
D * E * * F * G * A * B * * C * D&lt;br /&gt;
D - D#/Eb - E - E# - Fb - F F#/Gb - G - G#/Ab - A - A#/Bb - B - B# - Cb - C - C#/Db - D&lt;br /&gt;
P1 - A1/d2 - m2 - M2 - m3 - M3 - A3/d4 - P4 - A4/d5 - P5 - A5/d6 - m6 - M6 - m7 - M7 - A7/d8 - P8&lt;br /&gt;
&lt;br /&gt;
&lt;strong&gt;&lt;u&gt;23edo&lt;/u&gt;:&lt;/strong&gt; C C# * Db D&lt;br /&gt;
D * * E * * * F * * G * * A * * B * * * C * * D&lt;br /&gt;
D - D# - Eb - E - E# - Ex/Fbb - Fb - F - F# - Gb - G - G# - Ab - A - A# - Bb - B - B# - Bx/Cbb - Cb - C - C# - Db - D&lt;br /&gt;
P1 - A1 - d2 - m2 - M2 - A2/d3 - m3 - M3 - A3 - d4 - P4 - A4 - d5 - P5 - A5 - d6 - m6 - M6 - A6/d7 - m7 - M7 - A7 - d8 - P8&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:32:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc16"&gt;&lt;a name="Summary of EDO notation--&amp;quot;Perfect&amp;quot; EDOs (7, 14, 21, 28 and 35)"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:32 --&gt;&lt;u&gt;&amp;quot;Perfect&amp;quot; EDOs (7, 14, 21, 28 and 35)&lt;/u&gt;&lt;/h3&gt;
&lt;br /&gt;
All perfect EDOs use the same chain of fifths: P2 - P6 - P3 - P7 - P4 - P1 - P5 - P2 - P6 - P3 - P7 etc.&lt;br /&gt;
F - C - G - D - A - E - B - F - C - G - D - A - E - B etc.&lt;br /&gt;
F - C - G - D - A - E - B - F - C - G - D - A - E - B etc.&lt;br /&gt;
Sharp and flat are zero keys because C and C# are the same note.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;strong&gt;&lt;u&gt;7edo&lt;/u&gt;:&lt;/strong&gt; C/Db C#/D&lt;br /&gt;
&lt;strong&gt;&lt;u&gt;7edo&lt;/u&gt;:&lt;/strong&gt; C/C# Db/D&lt;br /&gt;
D E F G A B C D&lt;br /&gt;
D E F G A B C D&lt;br /&gt;
P1 - P2 - P3 - P4 - P5 - P6 - P7 - P8&lt;br /&gt;
P1 - P2 - P3 - P4 - P5 - P6 - P7 - P8&lt;br /&gt;
Because everything is perfect, the quality can be omitted: 1 - 2 - 3 - 4 - 5 - 6 - 7 - 8&lt;br /&gt;
Because everything is perfect, the quality can be omitted: 1 - 2 - 3 - 4 - 5 - 6 - 7 - 8&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;strong&gt;&lt;u&gt;14edo&lt;/u&gt;:&lt;/strong&gt; 2 keys per sharp/flat: C/C# * Db/D&lt;br /&gt;
&lt;strong&gt;&lt;u&gt;14edo&lt;/u&gt;:&lt;/strong&gt; C/C# * Db/D&lt;br /&gt;
D * E * F * G * A * B * C * D&lt;br /&gt;
D * E * F * G * A * B * C * D&lt;br /&gt;
D - D^/Ev - E - E/ Fv - F - F^/Gv - G - G^/Av - A - A^/Bv - B - B^/Cv - C - C^/Dv - D&lt;br /&gt;
D - D^/Ev - E - E/ Fv - F - F^/Gv - G - G^/Av - A - A^/Bv - B - B^/Cv - C - C^/Dv - D&lt;br /&gt;
1 - ^1/v2 - 2 - ^2/v3 - 3 - ^3/v4 - 4 - ^4/v5 - 5 - ^5/v6 - 6 - ^6/v7 - 7 - ^7/v8 - 8&lt;br /&gt;
1 - ^1/v2 - 2 - ^2/v3 - 3 - ^3/v4 - 4 - ^4/v5 - 5 - ^5/v6 - 6 - ^6/v7 - 7 - ^7/v8 - 8&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;strong&gt;&lt;u&gt;21edo&lt;/u&gt;:&lt;/strong&gt; 3 keys per sharp/flat: C/C# * * Db/D&lt;br /&gt;
&lt;strong&gt;&lt;u&gt;21edo&lt;/u&gt;:&lt;/strong&gt; C/C# * * Db/D&lt;br /&gt;
D * * E * * F * * G * * A * * B * * C * * D&lt;br /&gt;
D * * E * * F * * G * * A * * B * * C * * D&lt;br /&gt;
D - D^ - Ev - E - E^ - Fv - F - F^ - Gv - G - G^ - Av - A - A^ - Bv - B - B^ - Cv - C - C^ - Dv - D&lt;br /&gt;
D - D^ - Ev - E - E^ - Fv - F - F^ - Gv - G - G^ - Av - A - A^ - Bv - B - B^ - Cv - C - C^ - Dv - D&lt;br /&gt;
1 - ^1 - v2 - 2 - ^2 - v3 - 3 - ^3 - v4 - 4 - ^4 - v5 - 5 - ^5 - v6 - 6 - ^6 - v7 - 7 - ^7 - v8 - 8&lt;br /&gt;
1 - ^1 - v2 - 2 - ^2 - v3 - 3 - ^3 - v4 - 4 - ^4 - v5 - 5 - ^5 - v6 - 6 - ^6 - v7 - 7 - ^7 - v8 - 8&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;strong&gt;&lt;u&gt;28edo&lt;/u&gt;:&lt;/strong&gt; 4 keys per sharp/flat: C/C# * * * Db/D&lt;br /&gt;
&lt;strong&gt;&lt;u&gt;28edo&lt;/u&gt;:&lt;/strong&gt; C/C# * * * Db/D&lt;br /&gt;
D * * * E * * * F * * * G * * * A * * * B * * * C * * * D&lt;br /&gt;
D * * * E * * * F * * * G * * * A * * * B * * * C * * * D&lt;br /&gt;
D - D^ - D^^/Evv - Ev - E - E^ - E^^/Fvv - Fv - F - F^ - F^^/Gvv - Gv - G - G^ - G^^/Avv - Av - A etc.&lt;br /&gt;
D - D^ - D^^/Evv - Ev - E - E^ - E^^/Fvv - Fv - F - F^ - F^^/Gvv - Gv - G - G^ - G^^/Avv - Av - A etc.&lt;br /&gt;
1 - ^1 - ^^1/vv2 - v2 - 2 - ^2 - ^^2/vv3 - v3 - 3 - ^3 - ^^3/vv4 - v4 - 4 - ^4 - ^^4/vv5 - v5 - 5 etc.&lt;br /&gt;
1 - ^1 - ^^1/vv2 - v2 - 2 - ^2 - ^^2/vv3 - v3 - 3 - ^3 - ^^3/vv4 - v4 - 4 - ^4 - ^^4/vv5 - v5 - 5 etc.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;strong&gt;&lt;u&gt;35edo&lt;/u&gt;:&lt;/strong&gt; 5 keys per sharp/flat: C/C# * * * * Db/D&lt;br /&gt;
&lt;strong&gt;&lt;u&gt;35edo&lt;/u&gt;:&lt;/strong&gt; C/C# * * * * Db/D&lt;br /&gt;
D * * * * E * * * * F * * * * G * * * * A * * * * B * * * * C * * * * D&lt;br /&gt;
D * * * * E * * * * F * * * * G * * * * A * * * * B * * * * C * * * * D&lt;br /&gt;
D - D^ - D^^ - Evv - Ev - E - E^ - E^^ - Fvv - Fv - F - F^ - F^^ - Gvv - Gv - G - G^ - G^^ - Avv - Av - A etc.&lt;br /&gt;
D - D^ - D^^ - Evv - Ev - E - E^ - E^^ - Fvv - Fv - F - F^ - F^^ - Gvv - Gv - G - G^ - G^^ - Avv - Av - A etc.&lt;br /&gt;
1 - ^1 - ^^1 - vv2 - v2 - 2 - ^2 - ^^2 - vv3 - v3 - 3 - ^3 - ^^3 - vv4 - v4 - 4 - ^4 - ^^4 - vv5 - v5 - 5 etc.&lt;br /&gt;
1 - ^1 - ^^1 - vv2 - v2 - 2 - ^2 - ^^2 - vv3 - v3 - 3 - ^3 - ^^3 - vv4 - v4 - 4 - ^4 - ^^4 - vv5 - v5 - 5 etc.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:28:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc14"&gt;&lt;a name="Summary of EDO notation--Fourthward EDOs (9, 11, 13b, 16, 18b and 23)"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:28 --&gt;&lt;u&gt;Fourthward EDOs (9, 11, 13b, 16, 18b and 23)&lt;/u&gt;&lt;/h3&gt;
&lt;!-- ws:start:WikiTextHeadingRule:34:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc17"&gt;&lt;a name="Summary of EDO notation--Pentatonic EDOs (5, 10, 15, 20, 25 and 30)"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:34 --&gt;&lt;u&gt;Pentatonic EDOs (5, 10, 15, 20, 25 and 30)&lt;/u&gt;&lt;/h3&gt;
  &lt;br /&gt;
  &lt;br /&gt;
All pentatonic EDOs use the usual chain of fifths: m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7 etc.&lt;br /&gt;
All fourthwards EDOs use the usual chain of fifths: m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7 etc.&lt;br /&gt;
Fb - Cb - Gb - Db - Ab - Eb - Bb - F - C - G - D - A - E - B - F# - C# - G# - D# - A# - E# - B# etc.&lt;br /&gt;
Fb - Cb - Gb - Db - Ab - Eb - Bb - F - C - G - D - A - E - B - F# - C# - G# - D# - A# - E# - B# etc.&lt;br /&gt;
In all pentatonic EDOs, the minor 2nd = the unison and the major 3rd = the perfect fourth.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;strong&gt;&lt;u&gt;5edo&lt;/u&gt;:&lt;/strong&gt; C/Db C#/D&lt;br /&gt;
&lt;strong&gt;&lt;u&gt;9edo&lt;/u&gt;:&lt;/strong&gt; C/D# Cb/D, # = v&lt;br /&gt;
D E/F G A B/C D&lt;br /&gt;
D E * F G A B * C D&lt;br /&gt;
P1 - M2/m3 - P4 - P5 - M6/m7 - P8&lt;br /&gt;
D - E - Eb/F# - F - G - A - B - Bb/C# - C - D&lt;br /&gt;
P1 - M2 - m2/M3 - m3 - P4 - P5 - M6 - m6/M7 - m7 - P8&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;strong&gt;&lt;u&gt;10edo&lt;/u&gt;:&lt;/strong&gt; 2 keys per sharp/flat: C/Db * C#/D&lt;br /&gt;
&lt;strong&gt;&lt;u&gt;11edo&lt;/u&gt;:&lt;/strong&gt; C D Cb Db, # = vv&lt;br /&gt;
D * E/F * G * A * B/C * D&lt;br /&gt;
D E * * F G A B * * C D&lt;br /&gt;
D - D^/Ev - E/F - F^/Gv - G - G^/Av - A - A^/Bv - B/C - C^/Dv - D&lt;br /&gt;
D - E - E^/F# - Eb/Fv - F - G - A - B - B^/C# - Bb/Cv - C - D&lt;br /&gt;
P1/m2 - ^m2/vM2 - M2/m3 - ^m3/vM3 - M3/P4 - ^P4/vP5 - P5/m6 - ^m6/vM6 - M6/m7 - ^m7/vM7 - P8&lt;br /&gt;
P1 - M2 - ~2/M3 - m2/~3 - m3 - P4 - P5 - M6 - ~6/M7 - m6/~7 - m7 - P8&lt;br /&gt;
&lt;br /&gt;
&lt;strong&gt;&lt;u&gt;13b-edo&lt;/u&gt;:&lt;/strong&gt; C D * Cb Db, # = vvv&lt;br /&gt;
D E * * * F G A B * * * C D&lt;br /&gt;
D - E - E^/F# - Ebv/F#^ - Eb/Fv - F - G - A - B - B^/C# - Bbv/C#^ - Bb/Cv - C - D&lt;br /&gt;
P1 - M2 - ^M2/M3 - vm2/^M3 - m2/vm3 - m3 - P4 - P5 - M6 - ^M6/M7 - vm6/^M7 - m6/vm7 - m7 - P8&lt;br /&gt;
&lt;br /&gt;
&lt;strong&gt;&lt;u&gt;16edo&lt;/u&gt;:&lt;/strong&gt; C Cb/D# D, # = v&lt;br /&gt;
D * E * * F * G * A * B * * C * D&lt;br /&gt;
D - Db/E# - E - Eb - F# - F - Fb/G# - G - Gb/A# - A - Ab/B# - B - Bb - C# - C - Cb/D# - D&lt;br /&gt;
P1 - d1/A2 - M2 - m2 - M3 - m3 - d3/A4 - P4 - d4/A5 - P5 - d5/A6 - M6 - m6 - M7 - m7 - d7/A8 - P8&lt;br /&gt;
&lt;br /&gt;
&lt;strong&gt;&lt;u&gt;18b-edo&lt;/u&gt;:&lt;/strong&gt; C/D# * Cb/D, # = vv&lt;br /&gt;
D * E * * * F * G * A * B * * * C * D&lt;br /&gt;
D - D^/Ev - E - E^/F#v - Eb/F# - Eb^/Fv - F - F^/Gv - G - G^/Av - A - A^/Bv - B - B^/C# - Bbv/C#^ - Bb/Cv - C - D&lt;br /&gt;
P1 - M2 - ^M2/M3 - vm2/^M3 - m2/vm3 - m3 - P4 - P5 - M6 - ^M6/M7 - vm6/^M7 - m6/vm7 - m7 - P8&lt;br /&gt;
&lt;br /&gt;
&lt;strong&gt;&lt;u&gt;23edo&lt;/u&gt;:&lt;/strong&gt; C Cb * D# D, # = v&lt;br /&gt;
D * * E * * * F * * G * * A * * B * * * C * * D&lt;br /&gt;
D - Db - E# - E - Eb - Ebb/Fx - F# - F - Fb - G# - G - Gb - A# - A - Ab - B# - B - Bb - Bbb/Cx - C# - C - Cb - D# - D&lt;br /&gt;
P1 - d1 - A2 - M2 - m2 - d2/A3 - M3 - m3 - d3 - A4 - P4 - d4 - A5 - P5 - d5 - A6 - M6 - m6 - d6/A7 - M7 - m7 - d7 - A8 - P8&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:30:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc15"&gt;&lt;a name="Summary of EDO notation--Pentatonic EDOs (5, 10, 15, 20, 25 and 30)"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:30 --&gt;&lt;u&gt;Pentatonic EDOs (5, 10, 15, 20, 25 and 30)&lt;/u&gt;&lt;/h3&gt;
&lt;br /&gt;
All pentatonic EDOs use the usual chain of fifths: m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7 etc.&lt;br /&gt;
Fb - Cb - Gb - Db - Ab - Eb - Bb - F - C - G - D - A - E - B - F# - C# - G# - D# - A# - E# - B# etc.&lt;br /&gt;
In all pentatonic EDOs, the minor 2nd = the unison and the major 3rd = the perfect fourth.&lt;br /&gt;
&lt;br /&gt;
&lt;strong&gt;&lt;u&gt;5edo&lt;/u&gt;:&lt;/strong&gt; C/Db C#/D&lt;br /&gt;
D E/F G A B/C D&lt;br /&gt;
P1 - M2/m3 - P4 - P5 - M6/m7 - P8&lt;br /&gt;
&lt;br /&gt;
&lt;strong&gt;&lt;u&gt;10edo&lt;/u&gt;:&lt;/strong&gt; 2 keys per sharp/flat: C/Db * C#/D&lt;br /&gt;
D * E/F * G * A * B/C * D&lt;br /&gt;
D - D^/Ev - E/F - F^/Gv - G - G^/Av - A - A^/Bv - B/C - C^/Dv - D&lt;br /&gt;
P1/m2 - ^m2/vM2 - M2/m3 - ^m3/vM3 - M3/P4 - ^P4/vP5 - P5/m6 - ^m6/vM6 - M6/m7 - ^m7/vM7 - P8&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;strong&gt;&lt;u&gt;15edo&lt;/u&gt;:&lt;/strong&gt; 3 keys per sharp/flat: C/Db [*] [*] C#/D&lt;br /&gt;
&lt;strong&gt;&lt;u&gt;15edo&lt;/u&gt;:&lt;/strong&gt; 3 keys per sharp/flat: C/Db [*] [*] C#/D&lt;br /&gt;
Line 3,593: Line 3,530:
P1/m2 - ^m2 - ^^m2 - vvM2 - vM2 - M2/m3 - ^m3 - ^^m3 - vvM3 - vM3 - M3/P4 - ^P4 - ^^P4 - vvP5 - vP5 - P5/m6 - ^m6 - ^^m6 - vvM6 - vM6 - M6/m7 - ^m7 - ^^m7 - vvM7 - vM7 - P8&lt;br /&gt;
P1/m2 - ^m2 - ^^m2 - vvM2 - vM2 - M2/m3 - ^m3 - ^^m3 - vvM3 - vM3 - M3/P4 - ^P4 - ^^P4 - vvP5 - vP5 - P5/m6 - ^m6 - ^^m6 - vvM6 - vM6 - M6/m7 - ^m7 - ^^m7 - vvM7 - vM7 - P8&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:36:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc18"&gt;&lt;a name="Summary of EDO notation--Alternative pentatonic notation for pentatonic EDOs:"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:36 --&gt;&lt;u&gt;Alternative pentatonic notation for pentatonic EDOs:&lt;/u&gt;&lt;/h3&gt;
&lt;!-- ws:start:WikiTextHeadingRule:32:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc16"&gt;&lt;a name="Summary of EDO notation--Alternative pentatonic notation for pentatonic EDOs:"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:32 --&gt;&lt;u&gt;Alternative pentatonic notation for pentatonic EDOs:&lt;/u&gt;&lt;/h3&gt;
  &lt;br /&gt;
  &lt;br /&gt;
Pentatonic fourthwards chain of fifthoids: Ms3 - Ms7 - P4d - P1 - P5d - ms3 - ms7 - d4d etc.&lt;br /&gt;
Pentatonic fourthwards chain of fifthoids: Ms3 - Ms7 - P4d - P1 - P5d - ms3 - ms7 - d4d etc.&lt;br /&gt;
Line 3,612: Line 3,549:
D - D^ - Ev - E - E^ - Gv - G - G^ - Av - A - A^ - Cv - C - C^ - Dv - D&lt;br /&gt;
D - D^ - Ev - E - E^ - Gv - G - G^ - Av - A - A^ - Cv - C - C^ - Dv - D&lt;br /&gt;
1 - ^1 - vs3 - s3 - ^s3 - v4d - 4d - ^4d - v5d - 5d - ^5d - vs7 - s7 - ^s7 - v8d - 8d&lt;br /&gt;
1 - ^1 - vs3 - s3 - ^s3 - v4d - 4d - ^4d - v5d - 5d - ^5d - vs7 - s7 - ^s7 - v8d - 8d&lt;br /&gt;
&lt;br /&gt;
etc.&lt;br /&gt;
etc.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:34:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc17"&gt;&lt;a name="Summary of EDO notation--&amp;quot;Fifth-less&amp;quot; EDOs (8, 11b, 13 and 18)"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:34 --&gt;&lt;u&gt;&lt;strong&gt;&amp;quot;Fifth-less&amp;quot; EDOs (8, 11b, 13 and 18)&lt;/strong&gt;&lt;/u&gt;&lt;/h3&gt;
&lt;!-- ws:start:WikiTextHeadingRule:38:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc19"&gt;&lt;a name="Summary of EDO notation--&amp;quot;Regular&amp;quot; EDOs (12, 17, 19, 22, 24, 26, 27, 29, 31-34, and all edos 36 or higher)"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:38 --&gt;&lt;u&gt;&amp;quot;Regular&amp;quot; EDOs (12, 17, 19, 22, 24, 26, 27, 29, 31-34, and all edos 36 or higher)&lt;/u&gt;&lt;/h3&gt;
  &lt;br /&gt;
  &lt;br /&gt;
All regular EDOs use the usual chain of fifths: m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7 etc.&lt;br /&gt;
&lt;strong&gt;&lt;u&gt;8edo&lt;/u&gt;:&lt;/strong&gt; (generator = 1\8 = perfect 2nd = 150¢)&lt;br /&gt;
Fb - Cb - Gb - Db - Ab - Eb - Bb - F - C - G - D - A - E - B - F# - C# - G# - D# - A# - E# - B# etc.&lt;br /&gt;
D E F G * A B C D&lt;br /&gt;
D - E - F - G - G#/Ab - A -B - C - D&lt;br /&gt;
P1 - P2 - m3 - M3/m4 - M4/m5 - M5/m6 - M6 - P7 - P8&lt;br /&gt;
seventhwards chain of seconds: M3 - M4 - M5 - M6 - P7 - P1 - P2 - m3 - m4 - m5 - m6 - d7 etc.&lt;br /&gt;
A# - B# - C# - D# - E# - F# - G# - A - B - C - D - E - F - G - Ab - Bb - Cb - Db - Eb - Fb - Gb etc.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;strong&gt;&lt;u&gt;12edo&lt;/u&gt;:&lt;/strong&gt; sharp/flat = 1 key, no ups and downs: C C#/Db D&lt;br /&gt;
&lt;u&gt;&lt;strong&gt;11edo&lt;/strong&gt;&lt;/u&gt;: (generator = 3\11 = perfect 3rd)&lt;br /&gt;
D * E F * G * A * B C * D&lt;br /&gt;
D * E F * G A * B C * D&lt;br /&gt;
D - D#/Eb - E - F - F#/Gb - G - G#/Ab - A - A#/Bb - B - C - C#/Db - D&lt;br /&gt;
D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D&lt;br /&gt;
P1 - m2 - M2 - m3 - M3 - P4 - A4/d5 - P5 - m6 - M6 - m7 - M7 - P8&lt;br /&gt;
P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8&lt;br /&gt;
perfect = white, major = red, yellow and fifthward white, minor = green, blue and fourthwards white&lt;br /&gt;
sixthwards chain of thirds: M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6 etc.&lt;br /&gt;
E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;strong&gt;&lt;u&gt;17edo&lt;/u&gt;:&lt;/strong&gt; sharp = 2 keys: C Db C# D&lt;br /&gt;
&lt;u&gt;&lt;strong&gt;13edo&lt;/strong&gt;&lt;/u&gt;&lt;strong&gt;:&lt;/strong&gt; (generator = 2\13 = perfect 2nd)&lt;br /&gt;
D * * E F * * G * * A * * B C * * D&lt;br /&gt;
D * E * F * G A * B * C * D&lt;br /&gt;
D - D^/Eb - D#/Ev - Eb - E - F - F^/Gb - F#/Gv - G - G^/Ab - G#/Av - A - A^/Bb - A#/Bv - B - C - C^/Db - C#/Dv - D&lt;br /&gt;
D - D#/Eb - E - E#/Fb - F - F#/Gb - G - A - A#/Bb - B - B#/Cb - C - C#/Db - D&lt;br /&gt;
P1 - m2 - ^m2/vM2 - M2 - m3 - ^m3/vM3 - M3 - P4 - ^P4/d5 - A4/vP5 - P5 - m6 - ^m6/vM6 - M6 - m7 ^m7/vM7 - M7 - P8&lt;br /&gt;
P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - m6 - M6 - P7 - A7/d8 - P8&lt;br /&gt;
secondwards chain of seconds: m3 - m4 - m5 - m6 - P7 - P1 - P2 - M3 - M4 - M5 - M6 - A7 etc.&lt;br /&gt;
Ab - Bb - Cb - Db - Eb - Fb - Gb - A - B - C - D - E - F - G - A# - B# - C# - D# - E# - F# - G#&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;strong&gt;&lt;u&gt;19edo&lt;/u&gt;:&lt;/strong&gt; no ups and downs C C# Db D&lt;br /&gt;
&lt;strong&gt;&lt;u&gt;18edo&lt;/u&gt;:&lt;/strong&gt; (generator = 5\18 = perfect 3rd)&lt;br /&gt;
D * * E * F * * G * * A * * B * C * * D&lt;br /&gt;
D * * E * F * * G * A * * B * C * * D&lt;br /&gt;
D - D# - Eb - E - E#/Fb - F - F# - Gb - G - G# - Ab - A - A# - Bb - B - B#/Cb - C - C# - Db - D&lt;br /&gt;
D - D# - Eb - E - E#/Fb - F - F# - Gb - G - G#/Ab - A - A# - Bb - B - B#/Cb - C - C# - Db - D&lt;br /&gt;
P1 - A1/d2 - m2 - M2 - A2/d3 - m3 - M3 - A3/d4 - P4 - A4 - d5 - P5 - A5/d6 - m6 - M6 - A6/d7 - m7 - M7 - A7/d8 - P8&lt;br /&gt;
P1 - A1/d2 - m2 - M2 - A2/d3 - P3 - A3/d4 - m4 - M4 - A4/d5 - m5 - M5 - A5/d6 - P6 - A6/d7 - m7 - M7 - A7/d8 - P8&lt;br /&gt;
perfect = white, major = yellow and fifthward white, minor = green and fourthward white, aug/dim = red/blue.&lt;br /&gt;
sixthwards chain of thirds: M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6 etc.&lt;br /&gt;
E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;strong&gt;&lt;u&gt;22edo&lt;/u&gt;:&lt;/strong&gt; sharp = 3 keys: C Db * C# D&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:36:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc18"&gt;&lt;a name="Summary of EDO notation--Alternate pentatonic notation for EDOs 8, 13 and 18"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:36 --&gt;&lt;u&gt;&lt;strong&gt;Alternate pentatonic notation for EDOs 8, 13 and 18&lt;/strong&gt;&lt;/u&gt;&lt;/h3&gt;
D * * * E F * * * G * * * A * * * B C * * * D&lt;br /&gt;
&lt;br /&gt;
D - D^/Eb - D#v/Eb^ - D#/Ev - E - F - F^/Gb - F#v/Gb^ - F#/Gv - G - G^/Ab - G#v/Ab^ - G#/Av - A etc.&lt;br /&gt;
All three EDOs use the same pentatonic fifthwards chain of fifths: ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d etc.&lt;br /&gt;
P1 - m2 - ^m2 - vM2 - M2 - m3 - ^m3 - vM3 - M3 - P4 - ^P4/d5 - vA4/^d5 - A4/vP5 - P5 etc.&lt;br /&gt;
Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E# etc.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;strong&gt;&lt;u&gt;24edo&lt;/u&gt;:&lt;/strong&gt; sharp = 2 keys: C * C#/Db * D&lt;br /&gt;
&lt;u&gt;&lt;strong&gt;8edo&lt;/strong&gt;&lt;/u&gt;&lt;strong&gt;:&lt;/strong&gt; (generator = 5\8 = perfect 5thoid) C C#/Db D&lt;br /&gt;
D * * * E * F * * * G * * * A * * * B * C * * * D&lt;br /&gt;
D * E G * A C * D&lt;br /&gt;
D - D^/Ebv - D#/Eb - D#^/Ev - E - E^/Fv - F - F^/Gbv - F#/Gb - F#^/Gv - G - G^/Abv - G#/Ab - G#^/Av - A etc.&lt;br /&gt;
D - D#/Eb - E - G - G#/Ab - A - C - C#/Db - D&lt;br /&gt;
P1 - ^P1/vm2 - m2 - ^m2/vM2 - M2 - ^M2/vm3 - m3 - ^m3/vM3 - M3 - ^M3/vP4 - P4 - ^P4/vd5 - A4/d5 - ^A4/vP5 - P5 etc.&lt;br /&gt;
P1 - ms3 - Ms3 - P4d - A4d/d5d - P5d - ms7 - Ms7 - P8d&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;u&gt;&lt;strong&gt;13edo&lt;/strong&gt;&lt;/u&gt;&lt;strong&gt;:&lt;/strong&gt; (generator = 8\13 = perfect 5thoid) C C# Db D&lt;br /&gt;
D * * E * G * * A * C * * D&lt;br /&gt;
D - D# - Eb - E - E#/Gb - G - G# - Ab - A - A#/Cb - C - C# - Db - D&lt;br /&gt;
P1 - A1/ds3 - ms3 - Ms3 - As3/d4d - P4d - A4d - d5d - P5d - A5d/ds7 - ms7 - Ms7 - As7/d8d - P8d&lt;br /&gt;
&lt;br /&gt;
&lt;u&gt;&lt;strong&gt;18edo&lt;/strong&gt;&lt;/u&gt;&lt;strong&gt;:&lt;/strong&gt; (generator = 11\18 = perfect 5thoid) C C# * Db D&lt;br /&gt;
D * * * E * * G * * * A * * C * * * D&lt;br /&gt;
D - D# - Dx/Ebb - Eb - E - E# - Gb - G - G# - Gx/Abb - Ab - A - A# - Cb - C - C# - Cx/Dbb - Db - D&lt;br /&gt;
P1 - A1 - ds3 - ms3 - Ms3 - As3 - d4d - P4d - A4d - AA4d/dd5d - d5d - P5d - A5d - ds7 - ms7 - Ms7 - As7 - d8d - P8d&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
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Not counting the trivial edos 2, 3, 4 and 6, there are only seven such edos. As seen in the above diagram, they are the ones to the left of the heptatonic kite's spine, plus the ones to the right of the pentatonic kite's spine. The ones on the left edge of the heptatonic kite are the fourthward ones like 16edo, and have been dealt with already. 23edo can be notated similarly to 16edo by using a fifth of 13\23 instead of 14\23. That leaves only four edos: 8, 11, 13, and 18.&lt;br /&gt;
&lt;br /&gt;
There are two strategies for notating these &amp;quot;oddball&amp;quot; EDOs, besides heptatonic fifth-based notation with ups and downs. One is to switch from heptatonic notation to some other type. The orange kite contains edos for which pentatonic notation is a natural fit, in the sense that no ups or downs are needed. This includes 8edo, 13edo and 18edo.&lt;br /&gt;
&lt;br /&gt;
The other approach is to use some interval other than the fifth to generate the notation. Earlier I said notating 22edo using an even distribution of note names such as C _ _ D _ _ E _ _ F _ _ _ G _ _ A _ _ B _ _ C was a bad idea because the G-D and the A-E fifths looked perfect but were actually diminished. The reasoning is that 3/2 is an important ratio, and any decent approximation of 3/2 should look like a perfect fifth. But these EDOs don't approximate 3/2 well, so they can be thought of as having both a major fifth and a minor fifth. This negates any expectations of what a fifth should look like.&lt;br /&gt;
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&lt;u&gt;&lt;strong&gt;Theoretical alternatives for 8edo, 11edo, 13edo and 18edo&lt;/strong&gt;&lt;/u&gt;&lt;br /&gt;
&lt;br /&gt;
8edo octatonic (every note is a generator)&lt;br /&gt;
D E F G H A B C D&lt;br /&gt;
P1 - P2 - P3 - P4 - P5 - P6 - P7 - P8 - P9&lt;br /&gt;
requires learning octatonic interval arithmetic and staff notation&lt;br /&gt;
&lt;br /&gt;
11edo heptatonic narrow-fifth-based, fourthwards, # is vv (3/2 maps to 6\11 perfect 5th):&lt;br /&gt;
D E * * F G A B * * C D = D E F# F~ F G A B B~ Bb C D&lt;br /&gt;
fourthwards chain of fifths: m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7&lt;br /&gt;
P1 - M2 - ~2/M3 - m2/~3 - m3 - P4 - P5 - M6 - ~6/M7 - m6/~7 - m7 - P8&lt;br /&gt;
problematic because M3 = 2\11 is narrower than m2 = 3\11&lt;br /&gt;
&lt;br /&gt;
11edo nonotonic narrow-fifth-based, fifthwards with no ups and downs (3/2 maps to 6\11 = perfect 6th):&lt;br /&gt;
nonotonic fifthwards chain of sixths: M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9&lt;br /&gt;
P1 m2 M2/m3 M3/m4 M4 P5 P6 m7 M7/m8 M8/m9 M9 P8&lt;br /&gt;
requires learning nonotonic interval arithmetic and staff notation&lt;br /&gt;
&lt;br /&gt;
11edo pentatonic wide-fifth-based, fifthwards, # is ^^ (3/2 maps to 7\11 6th):&lt;br /&gt;
D * * E G * * A C * * D&lt;br /&gt;
pentatonic fifthwards chain of fifthoids: m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7&lt;br /&gt;
P1 - ms3 - ^ms3/vMs3 - Ms3 - P4d - ^P4d/d5d - A4d/vP5d - P5d - ms7 - ^ms7/vMs7 - Ms7 - P8d&lt;br /&gt;
pentatonic plus ups and downs is doubly confusing!&lt;br /&gt;
&lt;br /&gt;
11edo octatonic wide-fifth-based, fifthwards, no ^/v (3/2 maps to 7\11 = perfect 6th):&lt;br /&gt;
octatonic chain of 6ths: m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7&lt;br /&gt;
P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9&lt;br /&gt;
requires learning octatonic interval arithmetic and notation&lt;br /&gt;
&lt;br /&gt;
13edo heptatonic narrow-fifth-based, fourthwards, sharp = ^^^ (3/2 maps to 7\13 perfect 5th):&lt;br /&gt;
D E * * * F G A B * * * C D&lt;br /&gt;
fourthwards chain of fifths: M2 - M6 - M3 - M7 - P4 - P1 - P5 - m2 - m6 - m3 - m7&lt;br /&gt;
P1 - m2 - m3 - vM2/^m3 - M2 - M3 - P4 - P5 - m6 - m7 - vM6/^m7 - M6 - M7 - P8&lt;br /&gt;
problematic because m3 = 2\13 is narrower than M2 = 4\13&lt;br /&gt;
&lt;br /&gt;
13edo undecatonic narrow-fifth-based, fourthwards, 3/2 maps to 7\13 = perfect 7th&lt;br /&gt;
undecatonic sixthwards chain of sevenths:&lt;br /&gt;
M2 - M8 - M3 - M9 - M4 - M10 - M5 - M11 - P6 - P1 - P7 - m2 - m8 - m3 - m9 - m4 - m10 - m5 - m11&lt;br /&gt;
P1 - m2 - M2/m3 - M3/m4 - M4/m5 - M5 - P6 - P7 - m8 - M8/m9 - M9/m10 - M10/m11 - M11 - P12&lt;br /&gt;
requires learning undecatonic interval arithmetic and notation&lt;br /&gt;
&lt;br /&gt;
13edo octatonic wide-fifth-based, fourthwards, 3/2 maps to 8\13 = perfect 6th&lt;br /&gt;
octotonic chain of sixths: M3 - M8 - M5 - M2 - M7 - P4 - P1 - P6 - m3 - m8 - m5 - m2 - m7&lt;br /&gt;
P1 - m2 - M2 - m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7 - m8 - M8 - P9&lt;br /&gt;
requires learning octatonic interval arithmetic and notation&lt;br /&gt;
&lt;br /&gt;
18edo heptatonic narrow-fifth-based, fourthwards, sharp = ^^ (3/2 maps to 10\18 perfect 5th)&lt;br /&gt;
D * E * * * F * G * A * B * * * C * D&lt;br /&gt;
P1 - vm2 - m2 - vM2 - M2/m3 - vM3 - M3 - ^M3 - P4 - ^P4/vP5 - P5 - vm6 - m6 - vM6 - M6/m7 - vM7 - M7 - ^M7 - P8&lt;br /&gt;
fourthwards plus ups and downs plus closed is triply confusing!&lt;br /&gt;
&lt;br /&gt;
18edo nonatonic narrow-fifth-based (3/2 maps to 10\18 = perfect 6th)&lt;br /&gt;
P1 - vP2 - P2 - vP3 - P3 - vP4- P4 - vP5 - P5 - vP6 - P6 - vP7 - P7 - vP8 - P8 - vP9 - P9 - vP10 - P10&lt;br /&gt;
requires learning nonotonic interval arithmetic and staff notation&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
&lt;u&gt;&lt;strong&gt;Alternate notation for other edos:&lt;/strong&gt;&lt;/u&gt;&lt;br /&gt;
23edo pentatonic wide-fifth-based, fifthwards, 3/2 maps to 14\23 = perfect fifthoid:&lt;br /&gt;
D * * * * E * * * G * * * * A * * * C * * * * D&lt;br /&gt;
35edo heptatonic wide-fifth-based, sharp = seven ups, 3/2 maps to 21\35 = perfect fifth:&lt;br /&gt;
D * * * * * * E/F * * * * * * G * * * * * * A * * * * * * B/C * * * * * * D&lt;br /&gt;
42edo heptatonic narrow-fifth-based, sharp = six ups, 3/2 maps to 24\42 = perfect fifth:&lt;br /&gt;
D * * * * * E * * * * * F * * * * * G * * * * * A * * * * * B * * * * * C * * * * * D&lt;br /&gt;
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