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Kite's ups and downs notation: Difference between revisions

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autopatrolled, Sysops
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autopatrolled, Sysops
523 edits
m single style="text-align:center;" per table
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Interval arithmetic is done using a simple trick: first reverse everything, then perform normal arithmetic, then reverse everything again. Reversing means exchanging major for minor, aug for dim, and sharp for flat. Perfect and natural are unaffected. Examples:
Interval arithmetic is done using a simple trick: first reverse everything, then perform normal arithmetic, then reverse everything again. Reversing means exchanging major for minor, aug for dim, and sharp for flat. Perfect and natural are unaffected. Examples:


{| class="wikitable"
{| class="wikitable" style="text-align:center;"
|-
|-
| style="text-align:center;" | initial question
| | initial question
| style="text-align:center;" | reverse everything
| | reverse everything
| style="text-align:center;" | do the math
| | do the math
| style="text-align:center;" | reverse again
| | reverse again
|-
|-
| style="text-align:center;" | M2 + M2
| | M2 + M2
| style="text-align:center;" | m2 + m2
| | m2 + m2
| style="text-align:center;" | dim3
| | dim3
| style="text-align:center;" | aug3
| | aug3
|-
|-
| style="text-align:center;" | D to F#
| | D to F#
| style="text-align:center;" | D to Fb
| | D to Fb
| style="text-align:center;" | dim3
| | dim3
| style="text-align:center;" | aug3
| | aug3
|-
|-
| style="text-align:center;" | D to F
| | D to F
| style="text-align:center;" | D to F
| | D to F
| style="text-align:center;" | m3
| | m3
| style="text-align:center;" | M3
| | M3
|-
|-
| style="text-align:center;" | Eb + m3
| | Eb + m3
| style="text-align:center;" | E# + M3
| | E# + M3
| style="text-align:center;" | G##
| | G##
| style="text-align:center;" | Gbb
| | Gbb
|-
|-
| style="text-align:center;" | Eb + P5
| | Eb + P5
| style="text-align:center;" | E# + P5
| | E# + P5
| style="text-align:center;" | B#
| | B#
| style="text-align:center;" | Bb
| | Bb
|-
|-
| style="text-align:center;" | A minor chord
| | A minor chord
| style="text-align:center;" | A major
| | A major
| style="text-align:center;" | A C# E
| | A C# E
| style="text-align:center;" | A Cb E
| | A Cb E
|-
|-
| style="text-align:center;" | Eb major chord
| | Eb major chord
| style="text-align:center;" | E# minor
| | E# minor
| style="text-align:center;" | E# G# B#
| | E# G# B#
| style="text-align:center;" | Eb Gb Db
| | Eb Gb Db
|-
|-
| style="text-align:center;" | Gm7 = G + m3 + P5 + m7
| | Gm7 = G + m3 + P5 + m7
| style="text-align:center;" | G + M3 + P5 + M7
| | G + M3 + P5 + M7
| style="text-align:center;" | G B D F#
| | G B D F#
| style="text-align:center;" | G B D Fb
| | G B D Fb
|-
|-
| style="text-align:center;" | Ab7aug = Ab + M3 + A5 + m7
| | Ab7aug = Ab + M3 + A5 + m7
| style="text-align:center;" | A# + m3 + d5 + M7
| | A# + m3 + d5 + M7
| style="text-align:center;" | A# C# E G##
| | A# C# E G##
| style="text-align:center;" | Ab Cb E Gbb
| | Ab Cb E Gbb
|-
|-
| style="text-align:center;" | what chord is D F A#?
| | what chord is D F A#?
| style="text-align:center;" | D F Ab
| | D F Ab
| style="text-align:center;" | D + m3 + d5
| | D + m3 + d5
| style="text-align:center;" | D + M3 + A5 = Daug
| | D + M3 + A5 = Daug
|-
|-
| style="text-align:center;" | what chord is C E Gb Bb?
| | what chord is C E Gb Bb?
| style="text-align:center;" | C E G# B#
| | C E G# B#
| style="text-align:center;" | C + M3 + A5 + A7
| | C + M3 + A5 + A7
| style="text-align:center;" | C + m3 + d5 + d7 = Cdim7
| | C + m3 + d5 + d7 = Cdim7
|-
|-
| style="text-align:center;" | C major scale = C + M2 + M3
| | C major scale = C + M2 + M3


+ P4 + P5 + M6 + M7 + P8
+ P4 + P5 + M6 + M7 + P8
| style="text-align:center;" | C + m2 + m3 + P4
| | C + m2 + m3 + P4


+ P5 + m6 + m7 + P8
+ P5 + m6 + m7 + P8
| style="text-align:center;" | C Db Eb F
| | C Db Eb F


G Ab Bb C
G Ab Bb C
| style="text-align:center;" | C D# E# F
| | C D# E# F


G A# B# C
G A# B# C
|-
|-
| style="text-align:center;" | C minor scale = C + M2 + m3
| | C minor scale = C + M2 + m3


+ P4 + P5 + m6 + m7 + P8
+ P4 + P5 + m6 + m7 + P8
| style="text-align:center;" | C + m2 + M3 + P4
| | C + m2 + M3 + P4


+ P5 + M6 + M7 + P8
+ P5 + M6 + M7 + P8
| style="text-align:center;" | C Db E F
| | C Db E F


G A B C
G A B C
| style="text-align:center;" | C D# E F
| | C D# E F


G A B C
G A B C
|-
|-
| style="text-align:center;" | what scale is A B# Cb D
| | what scale is A B# Cb D


E F Gb A?
E F Gb A?
| style="text-align:center;" | A Bb C# D
| | A Bb C# D


E F G# A
E F G# A
| style="text-align:center;" | A + m2 + M3 + P4
| | A + m2 + M3 + P4


+ P5 + m6 + M7
+ P5 + m6 + M7
| style="text-align:center;" | A + M2 + m3 + P4
| | A + M2 + m3 + P4


+ P5 + M6 + m7 = A dorian
+ P5 + M6 + m7 = A dorian
Line 1,098: Line 1,098:
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 superflat EDOs, substitute M2 for m2.)
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 superflat EDOs, substitute M2 for m2.)


{| class="wikitable"
{| class="wikitable" style="text-align:center;"
|-
|-
| style="text-align:center;" | 5edo
| | 5edo
| style="text-align:center;" | pentatonic
| | pentatonic
| style="text-align:center;" |  
| |  
| style="text-align:center;" | C/Db
| | C/Db
| style="text-align:center;" | C#/D
| | C#/D
| style="text-align:center;" |  
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" |  
| |  
|-
|-
| style="text-align:center;" | 6edo
| | 6edo
| style="text-align:center;" | supersharp
| | supersharp
| style="text-align:center;" |  
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" |  
| |  
|-
|-
| style="text-align:center;" | 7edo
| | 7edo
| style="text-align:center;" | perfect
| | perfect
| style="text-align:center;" |  
| |  
| style="text-align:center;" | C/C#
| | C/C#
| style="text-align:center;" | Db/D
| | Db/D
| style="text-align:center;" |  
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" |
| |  
|-
| style="text-align:center;" | 8edo
| style="text-align:center;" | supersharp
| style="text-align:center;" |
| style="text-align:center;" |
| style="text-align:center;" |
| style="text-align:center;" |
| style="text-align:center;" |
| style="text-align:center;" |
| style="text-align:center;" |
| style="text-align:center;" |
| style="text-align:center;" |
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 9edo
| style="text-align:center;" | superflat
| style="text-align:center;" |
| style="text-align:center;" | C/Db
| style="text-align:center;" | C#/D
| style="text-align:center;" | D#
| style="text-align:center;" |
| style="text-align:center;" |
| style="text-align:center;" |
| style="text-align:center;" |
| style="text-align:center;" |
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 10edo
| style="text-align:center;" | pentatonic
| style="text-align:center;" |
| style="text-align:center;" | C/Db
| style="text-align:center;" | *
| style="text-align:center;" | C#/D
| style="text-align:center;" |
| style="text-align:center;" |
| style="text-align:center;" |
| style="text-align:center;" |
| style="text-align:center;" |
| style="text-align:center;" |
| style="text-align:center;" |  
|-
|-
| style="text-align:center;" | 11edo
| | 8edo
| style="text-align:center;" | superflat
| | supersharp
| style="text-align:center;" |  
| |  
| style="text-align:center;" | C
| |  
| style="text-align:center;" | D
| |  
| style="text-align:center;" | C#
| |  
| style="text-align:center;" | D#
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" |  
| |  
|-
|-
| style="text-align:center;" | 12edo
| | 9edo
| style="text-align:center;" |diatonic
| | superflat
| style="text-align:center;" |  
| |  
| style="text-align:center;" | C
| | C/Db
| style="text-align:center;" | C#/Db
| | C#/D
| style="text-align:center;" | D
| | D#
| style="text-align:center;" |  
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" |  
| |  
|-
|-
| style="text-align:center;" | 13b-edo
| | 10edo
| style="text-align:center;" | superflat
| | pentatonic
| style="text-align:center;" |  
| |  
| style="text-align:center;" | C
| | C/Db
| style="text-align:center;" | D
| | *
| style="text-align:center;" | *
| | C#/D
| style="text-align:center;" | C#
| |  
| style="text-align:center;" | D#
| |  
| style="text-align:center;" |
| |  
| style="text-align:center;" |
| |  
| style="text-align:center;" |
| |  
| style="text-align:center;" |
| |  
| style="text-align:center;" |
| |  
|-
| style="text-align:center;" | 14edo
| style="text-align:center;" | perfect
| style="text-align:center;" |
| style="text-align:center;" | C/C#
| style="text-align:center;" | *
| style="text-align:center;" | Db/D
| style="text-align:center;" |
| style="text-align:center;" |
| style="text-align:center;" |
| style="text-align:center;" |
| style="text-align:center;" |
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 15edo
| style="text-align:center;" | pentatonic
| style="text-align:center;" |
| style="text-align:center;" | C/Db
| style="text-align:center;" | *
| style="text-align:center;" | *
| style="text-align:center;" | C#/D
| style="text-align:center;" |  
| style="text-align:center;" |  
| style="text-align:center;" |  
| style="text-align:center;" |  
| style="text-align:center;" |  
| style="text-align:center;" |  
|-
|-
| style="text-align:center;" | 16edo
| | 11edo
| style="text-align:center;" | superflat
| | superflat
| style="text-align:center;" |  
| |  
| style="text-align:center;" | C
| | C
| style="text-align:center;" | C#/Db
| | D
| style="text-align:center;" | D
| | C#
| style="text-align:center;" | D#
| | D#
| style="text-align:center;" |  
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" |  
| |  
|-
|-
| style="text-align:center;" | 17edo
| | 12edo
| style="text-align:center;" |diatonic
| |diatonic
| style="text-align:center;" |  
| |  
| style="text-align:center;" | C
| | C
| style="text-align:center;" | Db
| | C#/Db
| style="text-align:center;" | C#
| | D
| style="text-align:center;" | D
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" |  
| |  
|-
|-
| style="text-align:center;" | 18b-edo
| | 13b-edo
| style="text-align:center;" | superflat
| | superflat
| style="text-align:center;" |  
| |  
| style="text-align:center;" | C/Db
| | C
| style="text-align:center;" | *
| | D
| style="text-align:center;" | C#/D
| | *
| style="text-align:center;" | *
| | C#
| style="text-align:center;" | D#
| | D#
| style="text-align:center;" |
| |  
| style="text-align:center;" |
| |  
| style="text-align:center;" |
| |  
| style="text-align:center;" |
| |  
| style="text-align:center;" |
| |  
|-
| style="text-align:center;" | 19edo
| style="text-align:center;" | diatonic
| style="text-align:center;" |
| style="text-align:center;" | C
| style="text-align:center;" | C#
| style="text-align:center;" | Db
| style="text-align:center;" | D
| style="text-align:center;" |  
| style="text-align:center;" |  
| style="text-align:center;" |  
| style="text-align:center;" |  
| style="text-align:center;" |
| style="text-align:center;" |  
|-
|-
| style="text-align:center;" | 20edo
| | 14edo
| style="text-align:center;" | pentatonic
| | perfect
| style="text-align:center;" |  
| |  
| style="text-align:center;" | C/Db
| | C/C#
| style="text-align:center;" | *
| | *
| style="text-align:center;" | *
| | Db/D
| style="text-align:center;" | *
| |  
| style="text-align:center;" | C#/D
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" |  
| |  
|-
|-
| style="text-align:center;" | 21edo
| | 15edo
| style="text-align:center;" | perfect
| | pentatonic
| style="text-align:center;" |  
| |  
| style="text-align:center;" | C/C#
| | C/Db
| style="text-align:center;" | *
| | *
| style="text-align:center;" | *
| | *
| style="text-align:center;" | Db/D
| | C#/D
| style="text-align:center;" |  
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" |  
| |  
|-
|-
| style="text-align:center;" | 22edo
| | 16edo
| style="text-align:center;" | diatonic
| | superflat
| style="text-align:center;" |
| |  
| style="text-align:center;" | C
| | C
| style="text-align:center;" | Db
| | C#/Db
| style="text-align:center;" | *
| | D
| style="text-align:center;" | C#
| | D#
| style="text-align:center;" | D
| |  
| style="text-align:center;" |
| |  
| style="text-align:center;" |
| |  
| style="text-align:center;" |
| |  
| style="text-align:center;" |
| |  
| style="text-align:center;" |
| |  
|-
| style="text-align:center;" | 23edo
| style="text-align:center;" | superflat
| style="text-align:center;" |  
| style="text-align:center;" | C
| style="text-align:center;" | C#
| style="text-align:center;" | Db
| style="text-align:center;" | D
| style="text-align:center;" | D#
| style="text-align:center;" |
| style="text-align:center;" |
| style="text-align:center;" |
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 24edo
| style="text-align:center;" | diatonic
| style="text-align:center;" |
| style="text-align:center;" | C
| style="text-align:center;" | *
| style="text-align:center;" | C#/Db
| style="text-align:center;" | *
| style="text-align:center;" | D
| style="text-align:center;" |
| style="text-align:center;" |  
| style="text-align:center;" |  
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 25edo
| style="text-align:center;" | pentatonic
| style="text-align:center;" |
| style="text-align:center;" | C/Db
| style="text-align:center;" | *
| style="text-align:center;" | *
| style="text-align:center;" | *
| style="text-align:center;" | *
| style="text-align:center;" | C#/D
| style="text-align:center;" |  
| style="text-align:center;" |  
| style="text-align:center;" |  
| style="text-align:center;" |  
|-
|-
| style="text-align:center;" | 26edo
| | 17edo
| style="text-align:center;" | diatonic
| |diatonic
| style="text-align:center;" |  
| |  
| style="text-align:center;" | C
| | C
| style="text-align:center;" | C#
| | Db
| style="text-align:center;" | *
| | C#
| style="text-align:center;" | Db
| | D
| style="text-align:center;" | D
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" |  
| |  
|-
|-
| style="text-align:center;" | 27edo
| | 18b-edo
| style="text-align:center;" | diatonic
| | superflat
| style="text-align:center;" |  
| |  
| style="text-align:center;" | C
| | C/Db
| style="text-align:center;" | Db
| | *
| style="text-align:center;" | *
| | C#/D
| style="text-align:center;" | *
| | *
| style="text-align:center;" | C#
| | D#
| style="text-align:center;" | D
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" |  
| |  
|-
|-
| style="text-align:center;" | 28edo
| | 19edo
| style="text-align:center;" | perfect
| | diatonic
| style="text-align:center;" |
| |  
| style="text-align:center;" | C/C#
| | C
| style="text-align:center;" | *
| | C#
| style="text-align:center;" | *
| | Db
| style="text-align:center;" | *
| | D
| style="text-align:center;" | Db/D
| |  
| style="text-align:center;" |
| |  
| style="text-align:center;" |
| |  
| style="text-align:center;" |
| |  
| style="text-align:center;" |
| |  
| style="text-align:center;" |
| |  
|-
| style="text-align:center;" | 29edo
| style="text-align:center;" | diatonic
| style="text-align:center;" |  
| style="text-align:center;" | C
| style="text-align:center;" | *
| style="text-align:center;" | Db
| style="text-align:center;" | C#
| style="text-align:center;" | *
| style="text-align:center;" | D
| style="text-align:center;" |
| style="text-align:center;" |
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 30edo
| style="text-align:center;" | pentatonic
| style="text-align:center;" |
| style="text-align:center;" | C/Db
| style="text-align:center;" | *
| style="text-align:center;" | *
| style="text-align:center;" | *
| style="text-align:center;" | *
| style="text-align:center;" | *
| style="text-align:center;" | C#/D
| style="text-align:center;" |
| style="text-align:center;" |  
| style="text-align:center;" |  
|-
| style="text-align:center;" | 31edo
| style="text-align:center;" | diatonic
| style="text-align:center;" |
| style="text-align:center;" | C
| style="text-align:center;" | *
| style="text-align:center;" | C#
| style="text-align:center;" | Db
| style="text-align:center;" | *
| style="text-align:center;" | D
| style="text-align:center;" |  
| style="text-align:center;" |  
| style="text-align:center;" |  
| style="text-align:center;" |  
|-
|-
| style="text-align:center;" | 32edo
| | 20edo
| style="text-align:center;" | "
| | pentatonic
| style="text-align:center;" |  
| |  
| style="text-align:center;" | C
| | C/Db
| style="text-align:center;" | Db
| | *
| style="text-align:center;" | *
| | *
| style="text-align:center;" | *
| | *
| style="text-align:center;" | *
| | C#/D
| style="text-align:center;" | C#
| |
| style="text-align:center;" | D
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" |  
| |  
|-
|-
| style="text-align:center;" | 33edo
| | 21edo
| style="text-align:center;" | "
| | perfect
| style="text-align:center;" |  
| |  
| style="text-align:center;" | C
| | C/C#
| style="text-align:center;" | C#
| | *
| style="text-align:center;" | *
| | *
| style="text-align:center;" | *
| | Db/D
| style="text-align:center;" | Db
| |
| style="text-align:center;" | D
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" |  
| |  
|-
|-
| style="text-align:center;" | 34edo
| | 22edo
| style="text-align:center;" | "
| | diatonic
| style="text-align:center;" |  
| |  
| style="text-align:center;" | C
| | C
| style="text-align:center;" | *
| | Db
| style="text-align:center;" | Db
| | *
| style="text-align:center;" | *
| | C#
| style="text-align:center;" | C#
| | D
| style="text-align:center;" | *
| |
| style="text-align:center;" | D
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" |  
| |  
|-
|-
| style="text-align:center;" | 35edo
| | 23edo
| style="text-align:center;" | perfect
| | superflat
| style="text-align:center;" |  
| |  
| style="text-align:center;" | C/C#
| | C
| style="text-align:center;" | *
| | C#
| style="text-align:center;" | *
| | Db
| style="text-align:center;" | *
| | D
| style="text-align:center;" | *
| | D#
| style="text-align:center;" | Db/D
| |  
| style="text-align:center;" |
| |  
| style="text-align:center;" |
| |  
| style="text-align:center;" |
| |  
| style="text-align:center;" |
| |  
|-
| style="text-align:center;" | 36edo
| style="text-align:center;" | diatonic
| style="text-align:center;" |
| style="text-align:center;" | C
| style="text-align:center;" | *
| style="text-align:center;" | *
| style="text-align:center;" | C#/Db
| style="text-align:center;" | *
| style="text-align:center;" | *
| style="text-align:center;" | D
| style="text-align:center;" |  
| style="text-align:center;" |  
| style="text-align:center;" |  
|-
|-
| style="text-align:center;" | 37edo
| | 24edo
| style="text-align:center;" | "
| | diatonic
| style="text-align:center;" |  
| |  
| style="text-align:center;" | C
| | C
| style="text-align:center;" | Db
| | *
| style="text-align:center;" | *
| | C#/Db
| style="text-align:center;" | *
| | *
| style="text-align:center;" | *
| | D
| style="text-align:center;" | *
| |  
| style="text-align:center;" | C#
| |  
| style="text-align:center;" | D
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" |  
| |  
|-
|-
| style="text-align:center;" | 38edo
| | 25edo
| style="text-align:center;" | "
| | pentatonic
| style="text-align:center;" |  
| |  
| style="text-align:center;" | C
| | C/Db
| style="text-align:center;" | *
| | *
| style="text-align:center;" | C#
| | *
| style="text-align:center;" | *
| | *
| style="text-align:center;" | Db
| | *
| style="text-align:center;" | *
| | C#/D
| style="text-align:center;" | D
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" |  
| |  
|-
|-
| style="text-align:center;" | 39edo
| | 26edo
| style="text-align:center;" | "
| | diatonic
| style="text-align:center;" |  
| |  
| style="text-align:center;" | C
| | C
| style="text-align:center;" | *
| | C#
| style="text-align:center;" | Db
| | *
| style="text-align:center;" | *
| | Db
| style="text-align:center;" | *
| | D
| style="text-align:center;" | C#
| |  
| style="text-align:center;" | *
| |  
| style="text-align:center;" | D
| |  
| style="text-align:center;" |
| |  
| style="text-align:center;" |
| |  
|-
| style="text-align:center;" | 40edo
| style="text-align:center;" | "
| style="text-align:center;" |
| style="text-align:center;" | C
| style="text-align:center;" | C#
| style="text-align:center;" | *
| style="text-align:center;" | *
| style="text-align:center;" | *
| style="text-align:center;" | Db
| style="text-align:center;" | D
| style="text-align:center;" |
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 41edo
| style="text-align:center;" | "
| style="text-align:center;" |
| style="text-align:center;" | C
| style="text-align:center;" | *
| style="text-align:center;" | *
| style="text-align:center;" | Db
| style="text-align:center;" | C#
| style="text-align:center;" | *
| style="text-align:center;" | *
| style="text-align:center;" | D
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 42edo
| style="text-align:center;" | "
| style="text-align:center;" |
| style="text-align:center;" | C
| style="text-align:center;" | Db
| style="text-align:center;" | *
| style="text-align:center;" | *
| style="text-align:center;" | *
| style="text-align:center;" | *
| style="text-align:center;" | *
| style="text-align:center;" | C#
| style="text-align:center;" | D
| style="text-align:center;" |
|-
| style="text-align:center;" | 43edo
| style="text-align:center;" | "
| style="text-align:center;" |
| style="text-align:center;" | C
| style="text-align:center;" | *
| style="text-align:center;" | *
| style="text-align:center;" | C#
| style="text-align:center;" | Db
| style="text-align:center;" | *
| style="text-align:center;" | *
| style="text-align:center;" | D
| style="text-align:center;" |  
| style="text-align:center;" |  
|-
|-
| style="text-align:center;" | 44ddo
| | 27edo
| style="text-align:center;" | "
| | diatonic
| style="text-align:center;" |  
| |  
| style="text-align:center;" | C
| | C
| style="text-align:center;" | *
| | Db
| style="text-align:center;" | Db
| | *
| style="text-align:center;" | *
| | *
| style="text-align:center;" | *
| | C#
| style="text-align:center;" | *
| | D
| style="text-align:center;" | C#
| |  
| style="text-align:center;" | *
| |  
| style="text-align:center;" | D
| |  
| style="text-align:center;" |  
| |  
|-
|-
| style="text-align:center;" | 45edo
| | 28edo
| style="text-align:center;" | "
| | perfect
| style="text-align:center;" |  
| |  
| style="text-align:center;" | C
| | C/C#
| style="text-align:center;" | *
| | *
| style="text-align:center;" | C#
| | *
| style="text-align:center;" | *
| | *
| style="text-align:center;" | *
| | Db/D
| style="text-align:center;" | Db
| |  
| style="text-align:center;" | *
| |  
| style="text-align:center;" | D
| |  
| style="text-align:center;" |  
| |  
| style="text-align:center;" |  
| |  
|-
|-
| style="text-align:center;" | 46edo
| | 29edo
| style="text-align:center;" | "
| | diatonic
| style="text-align:center;" |  
| |  
| style="text-align:center;" | C
| | C
| style="text-align:center;" | *
| | *
| style="text-align:center;" | *
| | Db
| style="text-align:center;" | Db
| | C#
| style="text-align:center;" | *
| | *
| style="text-align:center;" | C#
| | D
| style="text-align:center;" | *
| |  
| style="text-align:center;" | *
| |  
| style="text-align:center;" | D
| |  
| style="text-align:center;" |  
| |  
|-
|-
| style="text-align:center;" | 47edo
| | 30edo
| style="text-align:center;" | "
| | pentatonic
| style="text-align:center;" |  
| |  
| style="text-align:center;" | C
| | C/Db
| style="text-align:center;" | C#
| | *
| style="text-align:center;" | *
| | *
| style="text-align:center;" | *
| | *
| style="text-align:center;" | *
| | *
| style="text-align:center;" | *
| | *
| style="text-align:center;" | Db
| | C#/D
| style="text-align:center;" | D
| |  
| style="text-align:center;" |
| |  
| style="text-align:center;" |
| |  
|-
| style="text-align:center;" | 48edo
| style="text-align:center;" | "
| style="text-align:center;" |
| style="text-align:center;" | C
| style="text-align:center;" | *
| style="text-align:center;" | *
| style="text-align:center;" | *
| style="text-align:center;" | C#/Db
| style="text-align:center;" | *
| style="text-align:center;" | *
| style="text-align:center;" | *
| style="text-align:center;" | D
| style="text-align:center;" |  
|-
|-
| style="text-align:center;" | 49edo
| | 31edo
| style="text-align:center;" | "
| | diatonic
| style="text-align:center;" |  
| |  
| style="text-align:center;" | C
| | C
| style="text-align:center;" | *
| | *
| style="text-align:center;" | Db
| | C#
| style="text-align:center;" | *
| | Db
| style="text-align:center;" | *
| | *
| style="text-align:center;" | *
| | D
| style="text-align:center;" | *
| |  
| style="text-align:center;" | C#
| |  
| style="text-align:center;" | *
| |  
| style="text-align:center;" | D
| |  
|-
|-
| style="text-align:center;" | 50edo
| | 32edo
| style="text-align:center;" | "
| | "
| style="text-align:center;" |  
| |  
| style="text-align:center;" | C
| | C
| style="text-align:center;" | *
| | Db
| style="text-align:center;" | *
| | *
| style="text-align:center;" | C#
| | *
| style="text-align:center;" | *
| | *
| style="text-align:center;" | Db
| | C#
| style="text-align:center;" | *
| | D
| style="text-align:center;" | *
| |  
| style="text-align:center;" | D
| |  
| style="text-align:center;" |  
| |  
|-
|-
| style="text-align:center;" | 51edo
| | 33edo
| style="text-align:center;" | "
| | "
| style="text-align:center;" |  
| |  
| style="text-align:center;" | C
| | C
| style="text-align:center;" | *
| | C#
| style="text-align:center;" | *
| | *
| style="text-align:center;" | Db
| | *
| style="text-align:center;" | *
| | Db
| style="text-align:center;" | *
| | D
| style="text-align:center;" | C#
| |  
| style="text-align:center;" | *
| |  
| style="text-align:center;" | *
| |  
| style="text-align:center;" | D
| |  
|-
|-
| style="text-align:center;" | 52edo
| | 34edo
| style="text-align:center;" | "
| | "
| style="text-align:center;" |  
| |  
| style="text-align:center;" | C
| | C
| style="text-align:center;" | *
| | *
| style="text-align:center;" | C#
| | Db
| style="text-align:center;" | *
| | *
| style="text-align:center;" | *
| | C#
| style="text-align:center;" | *
| | *
| style="text-align:center;" | Db
| | D
| style="text-align:center;" | *
| |
| style="text-align:center;" | D
| |
| style="text-align:center;" |  
| |
|-
|-
| style="text-align:center;" | 53edo
| | 35edo
| style="text-align:center;" | "
| | perfect
| style="text-align:center;" |  
| |
| style="text-align:center;" | C
| | C/C#
| style="text-align:center;" | *
| | *
| style="text-align:center;" | *
| | *
| style="text-align:center;" | *
| | *
| style="text-align:center;" | Db
| | *
| style="text-align:center;" | C#
| | Db/D
| style="text-align:center;" | *
| |
| style="text-align:center;" | *
| |
| style="text-align:center;" | *
| |
| style="text-align:center;" | D
| |  
|}
|-
 
| | 36edo
=<u>'''Summary of EDO notation'''</u>=
| | diatonic
 
| |  
==<u>Diatonic EDOs</u>==
| | C
(12, 17, 19, 22, 24, 26, 27, 29, 31-34, and all edos 36 or higher)
| | *
 
| | *
All diatonic EDOs use the usual chain of fifths: m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7 etc.
| | C#/Db
 
| | *
Fb - Cb - Gb - Db - Ab - Eb - Bb - F - C - G - D - A - E - B - F# - C# - G# - D# - A# - E# - B# etc.
| | *
 
| | D
'''<u>12edo</u>:''' sharp/flat = 1 key, no ups and downs: C C#/Db D
| |
 
| |
D * E F * G * A * B C * D
| |
 
|-
D - D#/Eb - E - F - F#/Gb - G - G#/Ab - A - A#/Bb - B - C - C#/Db - D
| | 37edo
 
| | "
P1 - m2 - M2 - m3 - M3 - P4 - A4/d5 - P5 - m6 - M6 - m7 - M7 - P8
| |  
 
| | C
perfect = wa, major = ru, yo and fifthward wa, minor = gu, zo and fourthwards wa
| | Db
 
| | *
'''<u>17edo</u>:''' sharp = 2 keys: C Db C# D
| | *
 
| | *
D * * E F * * G * * A * * B C * * D
| | *
 
| | C#
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
| | D
 
| |
P1 - m2 - ~2 - M2 - m3 - ~3 - M3 - P4 - ^P4/d5 - A4/vP5 - P5 - m6 - ~6 - M6 - m7 - ~7 - M7 - P8
| |
 
|-
'''<u>19edo</u>:''' no ups and downs C C# Db D
| | 38edo
 
| | "
D * * E * F * * G * * A * * B * C * * D
| |
 
| | C
D - D# - Eb - E - E#/Fb - F - F# - Gb - G - G# - Ab - A - A# - Bb - B - B#/Cb - C - C# - Db - D
| | *
 
| | C#
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
| | *
 
| | Db
perfect = wa, major = yo and fifthward wa, minor = gu and fourthward wa, aug/dim = ru/zo.
| | *
 
| | D
'''<u>22edo</u>:''' 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.
| | 39edo
 
| | "
P1 - m2 - ^m2 - vM2 - M2 - m3 - ^m3 - vM3 - M3 - P4 - ^P4/d5 - vA4/^d5 - A4/vP5 - P5 etc.
| |
 
| | C
'''<u>24edo</u>:''' sharp = 2 keys: C * C#/Db * D
| | *
 
| | Db
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.
| | C#
 
| | *
P1 - ^P1/vm2 - m2 - ~2 - M2 - ^M2/vm3 - m3 - ~3 - M3 - ^M3/vP4 - P4 - ^P4/vd5 - A4/d5 - ^A4/vP5 - P5 etc.
| | D
 
| |
etc.
| |
 
|-
==<u>Perfect EDOs</u>==
| | 40edo
(7, 14, 21, 28 and 35)
| | "
 
| |
All perfect EDOs use the same circle of 7 fifths: P4 - P1 - P5 - P2 - P6 - P3 - P7 - P4 - P1 etc.
| | C
 
| | C#
F - C - G - D - A - E - B - F - C - G - D - A - E - B etc.
| | *
 
| | *
Sharp and flat are zero keys because C and C# are the same note.
| | *
 
| | Db
'''<u>7edo</u>:''' C/C# Db/D
| | D
 
| |
D E F G A B C D
| |
 
| |
P1 - P2 - P3 - P4 - P5 - P6 - P7 - P8
|-
 
| | 41edo
Because everything is perfect, the quality can be omitted: 1 - 2 - 3 - 4 - 5 - 6 - 7 - 8
| | "
 
| |
'''<u>14edo</u>:''' C/C# * Db/D
| | C
 
| | *
D * E * F * G * A * B * C * D
| | *
 
| | Db
D - D^/Ev - E - E/ Fv - F - F^/Gv - G - G^/Av - A - A^/Bv - B - B^/Cv - C - C^/Dv - D
| | C#
 
| | *
1 - ^1/v2 - 2 - ^2/v3 - 3 - ^3/v4 - 4 - ^4/v5 - 5 - ^5/v6 - 6 - ^6/v7 - 7 - ^7/v8 - 8
| | *
 
| | D
'''<u>21edo</u>:''' C/C# * * Db/D
| |
 
| |
D * * E * * F * * G * * A * * B * * C * * D
|-
 
| | 42edo
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
| | C
 
| | Db
'''<u>28edo</u>:''' 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.
| | *
 
| | C#
1 - ^1 - ^^1/vv2 - v2 - 2 - ^2 - ^^2/vv3 - v3 - 3 - ^3 - ^^3/vv4 - v4 - 4 - ^4 - ^^4/vv5 - v5 - 5 etc.
| | D
 
| |
'''<u>35edo</u>:''' C/C# * * * * Db/D
|-
 
| | 43edo
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.
| | C
 
| | *
1 - ^1 - ^^1 - vv2 - v2 - 2 - ^2 - ^^2 - vv3 - v3 - 3 - ^3 - ^^3 - vv4 - v4 - 4 - ^4 - ^^4 - vv5 - v5 - 5 etc.
| | *
 
| | C#
==<u>Superflat EDOs</u>==
| | Db
(9, 11, 13b, 16, 18b and 23)
| | *
| | *
| | 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
|}
 
=<u>'''Summary of EDO notation'''</u>=
 
==<u>Diatonic EDOs</u>==
(12, 17, 19, 22, 24, 26, 27, 29, 31-34, and all edos 36 or higher)


If sharp is lower than flat, the chain of fifths is m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7 etc.
All diatonic EDOs use the usual chain of fifths: m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7 etc.


Fb - Cb - Gb - Db - Ab - Eb - Bb - F - C - G - D - A - E - B - F# - C# - G# - D# - A# - E# - B# etc.
Fb - Cb - Gb - Db - Ab - Eb - Bb - F - C - G - D - A - E - B - F# - C# - G# - D# - A# - E# - B# etc.


If sharp is higher than flat, the chain of fifths is M2 - M6 - M3 - M7 - P4 - P1 - P5 - m2 - m6 - m3 - m7 etc.
'''<u>12edo</u>:''' sharp/flat = 1 key, no ups and downs: C C#/Db D


F# - C# - G# - D# - A# - E# - B# - F - C - G - D - A - E - B - Fb - Cb - Gb - Db - Ab - Eb - Bb etc.
D * E F * G * A * B C * D


Edos 11 and 13b and problematic. See "Supersharp EDOs" below for alternate notations for them.
D - D#/Eb - E - F - F#/Gb - G - G#/Ab - A - A#/Bb - B - C - C#/Db - D


'''<u>9edo</u>:''' C/D# Cb/D (# = v) with sharp lowering the pitch, and major/aug narrower than minor/dim
P1 - m2 - M2 - m3 - M3 - P4 - A4/d5 - P5 - m6 - M6 - m7 - M7 - P8


D E * F G A B * C D
perfect = wa, major = ru, yo and fifthward wa, minor = gu, zo and fourthwards wa


D - E - Eb/F# - F - G - A - B - Bb/C# - C - D
'''<u>17edo</u>:''' sharp = 2 keys: C Db C# D


P1 - M2 - m2/M3 - m3 - P4 - P5 - M6 - m6/M7 - m7 - P8
D * * E F * * G * * A * * B C * * D


C/Db C#/D (# = ^) with sharp raising the pitch, and major/aug wider than minor/dim
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


D - E - E#/Fb - F - G - A - B - B#/Cb - C - D
P1 - m2 - ~2 - M2 - m3 - ~3 - M3 - P4 - ^P4/d5 - A4/vP5 - P5 - m6 - ~6 - M6 - m7 - ~7 - M7 - P8


P1 - m2 - M2/m3 - M3 - P4 - P5 - m6 - M6/m7 - M7 - P8
'''<u>19edo</u>:''' no ups and downs C C# Db D


'''<u>11edo</u>:''' C D Cb Db (# = vv) with sharp lowering the pitch, and major/aug narrower than minor/dim
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 - E - E^/F# - Eb/Fv - F - G - A - B - B^/C# - Bb/Cv - C - 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


P1 - M2 - ~2/M3 - m2/~3 - m3 - P4 - P5 - M6 - ~6/M7 - m6/~7 - m7 - P8
perfect = wa, major = yo and fifthward wa, minor = gu and fourthward wa, aug/dim = ru/zo.


problematic because M3 is narrower than m2
'''<u>22edo</u>:''' sharp = 3 keys: C Db * C# D


C D C# D# (# = ^^) with sharp raising the pitch, and major/aug wider than minor/dim
D * * * E F * * * G * * * A * * * B C * * * D


D - E - E^/Fb - E#/Fv - F - G - A - B - B^/Cb - B#/Cv - 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 - ~2/m3 - M2/~3 - M3 - P4 - P5 - m6 - ~6/m7 - M6/~7 - M7 - P8
P1 - m2 - ^m2 - vM2 - M2 - m3 - ^m3 - vM3 - M3 - P4 - ^P4/d5 - vA4/^d5 - A4/vP5 - P5 etc.


problematic because m3 is narrower than M2
'''<u>24edo</u>:''' sharp = 2 keys: C * C#/Db * D


'''<u>13b-edo</u>:''' C D * Cb Db (# = vvv) with sharp lowering the pitch, and major/aug narrower than minor/dim
D * * * E * F * * * G * * * A * * * B * C * * * 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.


D - E - E^/F# - Ebv/F#^ - Eb/Fv - F - G - A - B - B^/C# - Bbv/C#^ - Bb/Cv - C - D
P1 - ^P1/vm2 - m2 - ~2 - M2 - ^M2/vm3 - m3 - ~3 - M3 - ^M3/vP4 - P4 - ^P4/vd5 - A4/d5 - ^A4/vP5 - P5 etc.


P1 - M2 - ^M2/M3 - vm2/^M3 - m2/vm3 - m3 - P4 - P5 - M6 - ^M6/M7 - vm6/^M7 - m6/vm7 - m7 - P8
etc.


problematic because M3 is narrower than m2
==<u>Perfect EDOs</u>==
(7, 14, 21, 28 and 35)


C D * C# D# (# = ^^^) with sharp raising the pitch, and major/aug wider than minor/dim
All perfect EDOs use the same circle of 7 fifths: P4 - P1 - P5 - P2 - P6 - P3 - P7 - P4 - P1 etc.


D - E - E^/Fb - E#v/Fb^ - E#/Fv - F - G - A - B - B^/Cb - B#v/Cb^ - B#/Cv - C - D
F - C - G - D - A - E - B - F - C - G - D - A - E - B etc.


P1 - m2 - ^m2/m3 - vM2/^m3 - M2/vM3 - M3 - P4 - P5 - m6 - ^m6/m7 - vM6/^m7 - M6/vM7 - M7 - P8
Sharp and flat are zero keys because C and C# are the same note.


problematic because m3 is narrower than M2
'''<u>7edo</u>:''' C/C# Db/D


'''<u>16edo</u>:''' C Cb/D# D (# = v) with sharp lowering the pitch, and major/aug narrower than minor/dim
D E F G A B C D


D * E * * F * G * A * B * * C * D
P1 - P2 - P3 - P4 - P5 - P6 - P7 - P8


D - Db/E# - E - Eb - F# - F - Fb/G# - G - Gb/A# - A - Ab/B# - B - Bb - C# - C - Cb/D# - D
Because everything is perfect, the quality can be omitted: 1 - 2 - 3 - 4 - 5 - 6 - 7 - 8


P1 - d1/A2 - M2 - m2 - M3 - m3 - d3/A4 - P4 - d4/A5 - P5 - d5/A6 - M6 - m6 - M7 - m7 - d7/A8 - P8
'''<u>14edo</u>:''' C/C# * Db/D


C C#/Db D (# = ^) with sharp raising the pitch, and major/aug wider than minor/dim
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^/Ev - E - E/ Fv - F - F^/Gv - G - G^/Av - A - A^/Bv - B - B^/Cv - C - C^/Dv - D


P1 - A1/d2 - m2 - M2 - m3 - M3 - A3/d4 - P4 - A4/d5 - P5 - A5/d6 - m6 - M6 - m7 - M7 - A7/d8 - P8
1 - ^1/v2 - 2 - ^2/v3 - 3 - ^3/v4 - 4 - ^4/v5 - 5 - ^5/v6 - 6 - ^6/v7 - 7 - ^7/v8 - 8


'''<u>18b-edo</u>:''' C/D# * Cb/D (# = vv) with sharp lowering the pitch, and major/aug narrower than minor/dim
'''<u>21edo</u>:''' C/C# * * Db/D


D * E * * * F * G * A * B * * * C * D
D * * E * * F * * G * * A * * B * * C * * D


D - D^/Ev - E - E^ - Eb/F# - Fv - F - F^/Gv - G - G^/Av - A - A^/Bv - B - B^ - Bb/C# - Cv - C - C^/Dv - D
D - D^ - Ev - E - E^ - Fv - F - F^ - Gv - G - G^ - Av - A - A^ - Bv - B - B^ - Cv - C - C^ - Dv - D


P1 - ^P1/vM2 - M2 - ~2 - m2/M3 - ~3 - m3 - ^m3/vP4 - P4 - ^P4/vP5 - P5 - ^P5/vM6 - M6 - ~6 - m6/M7 - ~7 - m7 - ^m2/d8 - P8
1 - ^1 - v2 - 2 - ^2 - v3 - 3 - ^3 - v4 - 4 - ^4 - v5 - 5 - ^5 - v6 - 6 - ^6 - v7 - 7 - ^7 - v8 - 8


Mid "~" is midway between major and minor, and replaces both upmajor and downminor.
'''<u>28edo</u>:''' C/C# * * * Db/D


C/Db * C#/D (# = ^^) with sharp raising the pitch, and major/aug wider than minor/dim
D * * * E * * * F * * * G * * * A * * * B * * * C * * * D


D - D^/Ev - E - E^ - E#/Fb - Fv - F - F^/Gv - G - G^/Av - A - A^/Bv - B - B^ - B#/Cb - Cv - C - C^/Dv - D
D - D^ - D^^/Evv - Ev - E - E^ - E^^/Fvv - Fv - F - F^ - F^^/Gvv - Gv - G - G^ - G^^/Avv - Av - A etc.


P1 - ^P1/vm2 - m2 - ~2 - mM2/m3 - ~3 - M3 - ^M3/vP4 - P4 - ^P4/vP5 - P5 - ^P5/vm6 - m6 - ~6 - M6/m7 - ~7 - M7 - ^M7/d8 - P8
1 - ^1 - ^^1/vv2 - v2 - 2 - ^2 - ^^2/vv3 - v3 - 3 - ^3 - ^^3/vv4 - v4 - 4 - ^4 - ^^4/vv5 - v5 - 5 etc.


'''<u>23edo</u>:''' C Cb * D# D (# = v) with sharp lowering the pitch, and major/aug narrower than minor/dim
'''<u>35edo</u>:''' C/C# * * * * Db/D


D * * E * * * F * * G * * A * * B * * * C * * D
D * * * * E * * * * F * * * * G * * * * A * * * * B * * * * C * * * * 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
D - D^ - D^^ - Evv - Ev - E - E^ - E^^ - Fvv - Fv - F - F^ - F^^ - Gvv - Gv - G - G^ - G^^ - Avv - Av - A etc.


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
1 - ^1 - ^^1 - vv2 - v2 - 2 - ^2 - ^^2 - vv3 - v3 - 3 - ^3 - ^^3 - vv4 - v4 - 4 - ^4 - ^^4 - vv5 - v5 - 5 etc.


C C# * Db D (# = ^) with sharp raising the pitch, and major/aug wider than minor/dim
==<u>Superflat EDOs</u>==
(9, 11, 13b, 16, 18b and 23)


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
If sharp is lower than flat, the chain of fifths is m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7 etc.


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
Fb - Cb - Gb - Db - Ab - Eb - Bb - F - C - G - D - A - E - B - F# - C# - G# - D# - A# - E# - B# etc.


==<u>Pentatonic EDOs</u>==
If sharp is higher than flat, the chain of fifths is M2 - M6 - M3 - M7 - P4 - P1 - P5 - m2 - m6 - m3 - m7 etc.
(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.
F# - C# - G# - D# - A# - E# - B# - F - C - G - D - A - E - B - Fb - Cb - Gb - Db - Ab - Eb - Bb etc.


Fb - Cb - Gb - Db - Ab - Eb - Bb - F - C - G - D - A - E - B - F# - C# - G# - D# - A# - E# - B# etc.
Edos 11 and 13b and problematic. See "Supersharp EDOs" below for alternate notations for them.


In all pentatonic EDOs, the minor 2nd = the unison and the major 3rd = the perfect fourth.
'''<u>9edo</u>:''' C/D# Cb/D (# = v) with sharp lowering the pitch, and major/aug narrower than minor/dim


'''<u>5edo</u>:''' C/Db C#/D
D E * F G A B * C D


D E/F G A B/C D
D - E - Eb/F# - F - G - A - B - Bb/C# - C - D


P1 - M2/m3 - P4 - P5 - M6/m7 - P8
P1 - M2 - m2/M3 - m3 - P4 - P5 - M6 - m6/M7 - m7 - P8


'''<u>10edo</u>:''' 2 keys per sharp/flat: C/Db * C#/D
C/Db C#/D (# = ^) with sharp raising the pitch, and major/aug wider than minor/dim


D * E/F * G * A * B/C * D
D - E - E#/Fb - F - G - A - B - B#/Cb - C - D


D - D^/Ev - E/F - F^/Gv - G - G^/Av - A - A^/Bv - B/C - C^/Dv - D
P1 - m2 - M2/m3 - M3 - P4 - P5 - m6 - M6/m7 - M7 - P8


P1/m2 - ^m2/vM2 - M2/m3 - ^m3/vM3 - M3/P4 - ^P4/vP5 - P5/m6 - ^m6/vM6 - M6/m7 - ^m7/vM7 - P8
'''<u>11edo</u>:''' C D Cb Db (# = vv) with sharp lowering the pitch, and major/aug narrower than minor/dim


'''<u>15edo</u>:''' 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 - E^/F# - Eb/Fv - F - G - A - B - B^/C# - Bb/Cv - C - D


D - D^ - Ev - E/F - F^ - Gv - G - G^ - Av - A - A^ - Bv - B/C - C^ - Dv - D
P1 - M2 - ~2/M3 - m2/~3 - m3 - P4 - P5 - M6 - ~6/M7 - m6/~7 - m7 - P8


P1/m2 - ^m2 - vM2 - M2/m3 - ^m3 - vM3 - M3/P4 - ^P4 - vP5 - P5/m6 - ^m6 - vM6 - M6/m7 - ^m7 - vM7 - P8
problematic because M3 is narrower than m2


'''<u>20edo</u>:''' 4 keys per sharp/flat: C/Db * * * C#/D
C D C# D# (# = ^^) with sharp raising the pitch, and major/aug wider than minor/dim


D * * * E/F * * * G * * * A * * * B/C * * * D
D - E - E^/Fb - E#/Fv - F - G - A - B - B^/Cb - B#/Cv - 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
P1 - m2 - ~2/m3 - M2/~3 - M3 - P4 - P5 - m6 - ~6/m7 - M6/~7 - M7 - P8


P1/m2 - ^m2 - ~2 - vM2 - M2/m3 - ^m3 - ~3 - vM3 - M3/P4 - ^P4 - ^^P4/vvP5 - vP5 - P5/m6 - ^m6 - ~6 - vM6 - M6/m7 - ^m7 - ~7 - vM7 - P8
problematic because m3 is narrower than M2


'''<u>25edo</u>:''' 5 keys per sharp/flat: C/Db * * * * C#/D
'''<u>13b-edo</u>:''' C D * Cb Db (# = vvv) with sharp lowering the pitch, and major/aug narrower than minor/dim


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
D - E - E^/F# - Ebv/F#^ - Eb/Fv - F - G - A - B - B^/C# - Bbv/C#^ - Bb/Cv - C - D


P1/m2 - ^m2 - v~2 - ^~2 - vM2 - M2/m3 - ^m3 - v~3 - ^~3 - vM3 - M3/P4 - ^P4 - ^^P4 - vvP5 - vP5 - P5/m6 - ^m6 - v~6 - ^~6 - vM6 - M6/m7 - ^m7 - v~7 - ^~7 - vM7 - P8
P1 - M2 - ^M2/M3 - vm2/^M3 - m2/vm3 - m3 - P4 - P5 - M6 - ^M6/M7 - vm6/^M7 - m6/vm7 - m7 - P8


'''<u>30edo</u>:''' 6 keys per sharp/flat: C/Db * * * * * C#/D
problematic because M3 is narrower than m2


D * * * * * E/F * * * * * G * * * * * A * * * * * B/C * * * * * D
C D * C# D# (# = ^^^) with sharp raising the pitch, and major/aug wider than minor/dim


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
D - E - E^/Fb - E#v/Fb^ - E#/Fv - F - G - A - B - B^/Cb - B#v/Cb^ - B#/Cv - C - D


P1/m2 - ^m2 - v~2 - ^~2 - vM2 - M2/m3 - ^m3 - v~3 - ^~3 - vM3 - M3/P4 - ^P4 - ^^P4 - vvP5 - vP5 - P5/m6 - ^m6 - v~6 - ^~6 - vM6 - M6/m7 - ^m7 - v~7 - ^~7 - vM7 - P8
P1 - m2 - ^m2/m3 - vM2/^m3 - M2/vM3 - M3 - P4 - P5 - m6 - ^m6/m7 - vM6/^m7 - M6/vM7 - M7 - P8


Alternatively, pentatonic notation can be used:
problematic because m3 is narrower than M2


Pentatonic chain of fifthoids: Ms3 - Ms7 - P4d - P1 - P5d - ms3 - ms7 - d4d etc.
'''<u>16edo</u>:''' C Cb/D# D (# = v) with sharp lowering the pitch, and major/aug narrower than minor/dim


C# - G# - D# - A# - E# - C - G - D - A - E - Cb - Gb - Db - Ab - Eb etc.
D * E * * F * G * A * B * * C * D


All intervals are perfect, so quality can be omitted.
D - Db/E# - E - Eb - F# - F - Fb/G# - G - Gb/A# - A - Ab/B# - B - Bb - C# - C - Cb/D# - D


s3 = subthird, 4d = fourthoid, 5d = fifthoid, s7 = subseventh, 8d = octoid.
P1 - d1/A2 - M2 - m2 - M3 - m3 - d3/A4 - P4 - d4/A5 - P5 - d5/A6 - M6 - m6 - M7 - m7 - d7/A8 - P8


<u>'''5edo'''</u>''':''' zero keys per sharp/flat: C/C# Db/D
C C#/Db D (# = ^) with sharp raising the pitch, and major/aug wider than minor/dim


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


1 - s3 - 4d - 5d - s7 - 8d
P1 - A1/d2 - m2 - M2 - m3 - M3 - A3/d4 - P4 - A4/d5 - P5 - A5/d6 - m6 - M6 - m7 - M7 - A7/d8 - P8


<u>'''10edo'''</u>''':''' zero keys per sharp/flat: C/C# * Db/D
'''<u>18b-edo</u>:''' C/D# * Cb/D (# = vv) with sharp lowering the pitch, and major/aug narrower than minor/dim


D * E * G * A * C * D
D * E * * * F * G * A * B * * * C * D


D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D
D - D^/Ev - E - E^ - Eb/F# - Fv - F - F^/Gv - G - G^/Av - A - A^/Bv - B - B^ - Bb/C# - Cv - C - C^/Dv - D


1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d
P1 - ^P1/vM2 - M2 - ~2 - m2/M3 - ~3 - m3 - ^m3/vP4 - P4 - ^P4/vP5 - P5 - ^P5/vM6 - M6 - ~6 - m6/M7 - ~7 - m7 - ^m2/d8 - P8


<u>'''15edo'''</u>''':''' zero keys per sharp/flat: C/C# * * Db/D
Mid "~" is midway between major and minor, and replaces both upmajor and downminor.


D * * E * * G * * A * * C * * D
C/Db * C#/D (# = ^^) with sharp raising the pitch, and major/aug wider than minor/dim


D - D^ - Ev - E - E^ - Gv - G - G^ - Av - A - A^ - Cv - C - C^ - Dv - D
D - D^/Ev - E - E^ - E#/Fb - Fv - F - F^/Gv - G - G^/Av - A - A^/Bv - B - B^ - B#/Cb - Cv - C - C^/Dv - D


1 - ^1 - vs3 - s3 - ^s3 - v4d - 4d - ^4d - v5d - 5d - ^5d - vs7 - s7 - ^s7 - v8d - 8d
P1 - ^P1/vm2 - m2 - ~2 - mM2/m3 - ~3 - M3 - ^M3/vP4 - P4 - ^P4/vP5 - P5 - ^P5/vm6 - m6 - ~6 - M6/m7 - ~7 - M7 - ^M7/d8 - P8


etc.
'''<u>23edo</u>:''' C Cb * D# D (# = v) with sharp lowering the pitch, and major/aug narrower than minor/dim


==<u>'''Supersharp EDOs'''</u>==
D * * E * * * F * * G * * A * * B * * * C * * D
(8, 11b, 13 and 18)


There are three strategies for notating these EDOs. The best one is to convert them to superflat EDOs by using an alternate fifth, as discussed above. This doesn't work for 8edo.
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


Another is to switch from heptatonic notation to some other type. Pentatonic notation is a natural fit, in the sense that no ups or downs are needed, for 8edo, 13edo and 18edo, but not 11edo.
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


The third approach is to use the natural generator, see the next section.
C C# * Db D (# = ^) with sharp raising the pitch, and major/aug wider than minor/dim


<u>'''Pentatonic notation for 8edo, 11b-edo, 13edo and 18edo'''</u>
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


All four EDOs use the same pentatonic fifthwards chain of fifths: ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d etc.
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


Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E# etc.
==<u>Pentatonic EDOs</u>==
(5, 10, 15, 20, 25 and 30)


<u>'''8edo'''</u>''':''' (generator = 5\8 = perfect 5thoid) C C#/Db D
All pentatonic EDOs use the usual chain of fifths: m2 - m6 - m3 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7 etc.


D * E G * A C * D
Fb - Cb - Gb - Db - Ab - Eb - Bb - F - C - G - D - A - E - B - F# - C# - G# - D# - A# - E# - B# etc.


D - D#/Eb - E - G - G#/Ab - A - C - C#/Db - D
In all pentatonic EDOs, the minor 2nd = the unison and the major 3rd = the perfect fourth.


P1 - ms3 - Ms3 - P4d - A4d/d5d - P5d - ms7 - Ms7 - P8d
'''<u>5edo</u>:''' C/Db C#/D


<u>'''11b-edo'''</u>''':''' (generator = 7\11 = perfect 5thoid) C Db C# D, # is ^^
D E/F G A B/C D


D * * E G * * A C * * D
P1 - M2/m3 - P4 - P5 - M6/m7 - P8


P1 - ms3 - ~s3 - Ms3 - P4d - ^P4d/d5d - A4d/vP5d - P5d - ms7 - ~s7 - Ms7 - P8d
'''<u>10edo</u>:''' 2 keys per sharp/flat: C/Db * C#/D


<u>'''13edo'''</u>''':''' (generator = 8\13 = perfect 5thoid) C C# Db D
D * E/F * G * A * B/C * D


D * * E * G * * A * C * * D
D - D^/Ev - E/F - F^/Gv - G - G^/Av - A - A^/Bv - B/C - C^/Dv - D


D - D# - Eb - E - E#/Gb - G - G# - Ab - A - A#/Cb - C - C# - Db - D
P1/m2 - ^m2/vM2 - M2/m3 - ^m3/vM3 - M3/P4 - ^P4/vP5 - P5/m6 - ^m6/vM6 - M6/m7 - ^m7/vM7 - P8


P1 - A1/ds3 - ms3 - Ms3 - As3/d4d - P4d - A4d - d5d - P5d - A5d/ds7 - ms7 - Ms7 - As7/d8d - P8d
'''<u>15edo</u>:''' 3 keys per sharp/flat: C/Db * * C#/D


<u>'''18edo'''</u>''':''' (generator = 11\18 = perfect 5thoid) C C# * Db D
D * * E/F * * G * * A * * B/C * * D


D * * * E * * G * * * A * * C * * * D
D - D^ - Ev - E/F - F^ - Gv - G - G^ - Av - A - A^ - Bv - B/C - C^ - Dv - D


D - D# - Dx/Ebb - Eb - E - E# - Gb - G - G# - Gx/Abb - Ab - A - A# - Cb - C - C# - Cx/Dbb - Db - D
P1/m2 - ^m2 - vM2 - M2/m3 - ^m3 - vM3 - M3/P4 - ^P4 - vP5 - P5/m6 - ^m6 - vM6 - M6/m7 - ^m7 - vM7 - P8


P1 - A1 - ds3 - ms3 - Ms3 - As3 - d4d - P4d - A4d - AA4d/dd5d - d5d - P5d - A5d - ds7 - ms7 - Ms7 - As7 - d8d - P8d
'''<u>20edo</u>:''' 4 keys per sharp/flat: C/Db * * * C#/D


<u>'''Other non-heptatonic notations for 8edo, 11edo, 13edo and 18edo'''</u>
D * * * E/F * * * G * * * A * * * B/C * * * D


<u>'''8edo'''</u> octatonic (every note is a generator)
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


D E F G H A B C D
P1/m2 - ^m2 - ~2 - vM2 - M2/m3 - ^m3 - ~3 - vM3 - M3/P4 - ^P4 - ^^P4/vvP5 - vP5 - P5/m6 - ^m6 - ~6 - vM6 - M6/m7 - ^m7 - ~7 - vM7 - P8


P1 - P2 - P3 - P4 - P5 - P6 - P7 - P8 - P9
'''<u>25edo</u>:''' 5 keys per sharp/flat: C/Db * * * * C#/D


requires learning octatonic interval arithmetic and staff notation
D * * * * E/F * * * * G * * * * A * * * * B/C * * * * D


<u>'''11edo'''</u> nonatonic (narrow 3/2 maps to 6\11 = perfect 6th)
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


nonotonic genchain of sixths: M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9
P1/m2 - ^m2 - v~2 - ^~2 - vM2 - M2/m3 - ^m3 - v~3 - ^~3 - vM3 - M3/P4 - ^P4 - ^^P4 - vvP5 - vP5 - P5/m6 - ^m6 - v~6 - ^~6 - vM6 - M6/m7 - ^m7 - v~7 - ^~7 - vM7 - P8


A B C * D E F G * H J A
'''<u>30edo</u>:''' 6 keys per sharp/flat: C/Db * * * * * C#/D


P1 m2 M2/m3 M3/m4 M4 P5 P6 m7 M7/m8 M8/m9 M9 P8
D * * * * * E/F * * * * * G * * * * * A * * * * * B/C * * * * * D


requires learning nonotonic interval arithmetic and staff notation
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


<u>'''11b-edo'''</u> octatonic (wide 3/2 maps to 7\11 = perfect 6th)
P1/m2 - ^m2 - v~2 - ^~2 - vM2 - M2/m3 - ^m3 - v~3 - ^~3 - vM3 - M3/P4 - ^P4 - ^^P4 - vvP5 - vP5 - P5/m6 - ^m6 - v~6 - ^~6 - vM6 - M6/m7 - ^m7 - v~7 - ^~7 - vM7 - P8


A B * C D * E F G * H A
Alternatively, pentatonic notation can be used:


octatonic genchain of sixths: m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7
Pentatonic chain of fifthoids: Ms3 - Ms7 - P4d - P1 - P5d - ms3 - ms7 - d4d etc.


P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9
C# - G# - D# - A# - E# - C - G - D - A - E - Cb - Gb - Db - Ab - Eb etc.


requires learning octatonic interval arithmetic and notation
All intervals are perfect, so quality can be omitted.


<u>'''13edo'''</u> octatonic (wide 3/2 maps to 8\13 = perfect 6th)
s3 = subthird, 4d = fourthoid, 5d = fifthoid, s7 = subseventh, 8d = octoid.


octotonic genchain of sixths: M3 - M8 - M5 - M2 - M7 - P4 - P1 - P6 - m3 - m8 - m5 - m2 - m7
<u>'''5edo'''</u>''':''' zero keys per sharp/flat: C/C# Db/D


P1 - m2 - M2 - m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7 - m8 - M8 - P9
D E G A C D


requires learning octatonic interval arithmetic and notation
1 - s3 - 4d - 5d - s7 - 8d


<u>'''18b-edo'''</u> nonatonic (narrow 3/2 maps to 10\18 = perfect 6th)
<u>'''10edo'''</u>''':''' zero keys per sharp/flat: 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 * G * A * C * D


requires learning nonotonic interval arithmetic and staff notation
D - D^/Ev - E - E^/Gv - G - G^/Av - A - A^/Cv - C - C^/Dv - D


=<u>Natural Generators</u>=
1 - ^1/vs3 - s3 - ^s3/v4d - 4d - ^4d/v5d - 5d - ^5d/vs7 - s7 - ^s7/v8d - 8d


Ups and downs can be avoided entirely by using 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 EDOs like 8, 11, 13 and 18 don't approximate 3/2 well, so they can be thought of as having both a major fifth and a minor fifth.
<u>'''15edo'''</u>''':''' zero keys per sharp/flat: C/C# * * Db/D


Every non-perfect EDO has a "natural" heptatonic generator. For 13-edo, it's a 2\13 2nd (and its octave inverse of course), because seven 2\13's falls only one EDOstep away from the octave. Thus the sharp means "sharpened by one EDO-step", major is one EDO-step wider than minor, and ups and downs aren't needed.
D * * E * * G * * A * * C * * D


The usual genchain of fifths runs ...d5 - m2 - m6 - m4 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7 - A4... This can be generalized to any generator: The generator is always perfect, so the middle three intervals of the genchain are always perfect. One side of the genchain is always major or augmented, and the other side is always minor or diminished. For heptatonic notation, there are four major and four minor intervals. For pentatonic, there's two of each. In general, N-3 of each. The major side is usually chosen so that major is wider than minor. The only exception is for fifth-generated notation of superflat EDOs, when major may be on the left even when it should be on the right, in order to preserve familiar interval arithmetic.
D - D^ - Ev - E - E^ - Gv - G - G^ - Av - A - A^ - Cv - C - C^ - Dv - D


For 13edo, the genchain runs in 2nds: ...5 - 6 - 7 - 1 - 2 - 3 - 4 - 5... The righthand 5th is the sum of four perfect 2nds, and equals 4 * (2\13) = 8\13. The lefthand 5th is the octave minus three perfect 2nds, and equals 13\13 - 3 * (2\13) = 7\13. The righthand one is larger and therefore major. Thus the 13edo genchain is ...d2 - m3 - m4 - m5 - m6 - P7 - P1 - P2 - M3 - M4 - M5 - M6 - A7...
1 - ^1 - vs3 - s3 - ^s3 - v4d - 4d - ^4d - v5d - 5d - ^5d - vs7 - s7 - ^s7 - v8d - 8d


<u>'''Natural generators for 8edo, 11edo, 13edo and 18edo'''</u>
etc.


'''<u>8edo</u>:''' (generator = 1\8 = perfect 2nd = 150¢)
==<u>'''Supersharp EDOs'''</u>==
(8, 11b, 13 and 18)


D E F G * A B C D
There are three strategies for notating these EDOs. The best one is to convert them to superflat EDOs by using an alternate fifth, as discussed above. This doesn't work for 8edo.


D - E - F - G - G#/Ab - A -B - C - D
Another is to switch from heptatonic notation to some other type. Pentatonic notation is a natural fit, in the sense that no ups or downs are needed, for 8edo, 13edo and 18edo, but not 11edo.


P1 - P2 - m3 - M3/m4 - M4/m5 - M5/m6 - M6 - P7 - P8
The third approach is to use the natural generator, see the next section.


genchain of seconds: M3 - M4 - M5 - M6 - P7 - P1 - P2 - m3 - m4 - m5 - m6 - d7 etc.
<u>'''Pentatonic notation for 8edo, 11b-edo, 13edo and 18edo'''</u>


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


<u>'''11edo'''</u>: (generator = 3\11 = perfect 3rd)
Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E# etc.


D * E F * G A * B C * D
<u>'''8edo'''</u>''':''' (generator = 5\8 = perfect 5thoid) C C#/Db D


D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D
D * E G * A C * D


P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8
D - D#/Eb - E - G - G#/Ab - A - C - C#/Db - D


genchain of thirds: M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6 etc.
P1 - ms3 - Ms3 - P4d - A4d/d5d - P5d - ms7 - Ms7 - P8d


E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb
<u>'''11b-edo'''</u>''':''' (generator = 7\11 = perfect 5thoid) C Db C# D, # is ^^


<u>'''13edo'''</u>''':''' (generator = 2\13 = perfect 2nd)
D * * E G * * A C * * D


D * E * F * G A * B * C * D
P1 - ms3 - ~s3 - Ms3 - P4d - ^P4d/d5d - A4d/vP5d - P5d - ms7 - ~s7 - Ms7 - P8d


D - D#/Eb - E - E#/Fb - F - F#/Gb - G - A - A#/Bb - B - B#/Cb - C - C#/Db - D
<u>'''13edo'''</u>''':''' (generator = 8\13 = perfect 5thoid) C C# Db D


P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - m6 - M6 - P7 - A7/d8 - P8
D * * E * G * * A * C * * D


genchain of seconds: d2 - m3 - m4 - m5 - m6 - P7 - P1 - P2 - M3 - M4 - M5 - M6 - A7 etc.
D - D# - Eb - E - E#/Gb - G - G# - Ab - A - A#/Cb - C - C# - Db - D


Ab - Bb - Cb - Db - Eb - Fb - Gb - A - B - C - D - E - F - G - A# - B# - C# - D# - E# - F# - G#
P1 - A1/ds3 - ms3 - Ms3 - As3/d4d - P4d - A4d - d5d - P5d - A5d/ds7 - ms7 - Ms7 - As7/d8d - P8d


'''<u>18edo</u>:''' (generator = 5\18 = perfect 3rd)
<u>'''18edo'''</u>''':''' (generator = 11\18 = perfect 5thoid) C C# * Db D


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
D - D# - Dx/Ebb - Eb - E - E# - Gb - G - G# - Gx/Abb - Ab - A - A# - Cb - C - C# - Cx/Dbb - Db - 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 - A1 - ds3 - ms3 - Ms3 - As3 - d4d - P4d - A4d - AA4d/dd5d - d5d - P5d - A5d - ds7 - ms7 - Ms7 - As7 - d8d - P8d


genchain of thirds: M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6 etc.
<u>'''Other non-heptatonic notations for 8edo, 11edo, 13edo and 18edo'''</u>


E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb
<u>'''8edo'''</u> octatonic (every note is a generator)


Natural generators can be used for other EDOs as well. For pentatonic EDOs, they avoid E and F naming the same note. For other EDOs, they make notating certain MOS scales easier, such as 22edo's Porcupine [7] scale. However, using any generator besides the fifth completely changes interval arithmetic. Naming chords and scales becomes very complicated. So this notation is <u>'''not recommended for edos'''</u> except as an alternate, composer-oriented notation.
D E F G H A B C D


For all EDOs with sharpness 1, -1 or 0, the natural generator is the fifth, the same as standard notation. For all sharp-2 and sharp-5 edos, the natural generator is a 3rd. For sharp-3 and sharp-4, it's a 2nd. For sharp-6, -7 or -8, it's the fifth closest to 7-edo's fifth, not the one closest to 3/2. This is the down-fifth in standard notation. For 42-edo, vP5 = 24\42. For 72-edo, vP5 = 41\72.
P1 - P2 - P3 - P4 - P5 - P6 - P7 - P8 - P9


<u>'''Sharp-2 edos:'''</u>
requires learning octatonic interval arithmetic and staff notation


genchain of thirds: m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1 etc.
<u>'''11edo'''</u> nonatonic (narrow 3/2 maps to 6\11 = perfect 6th)


Eb - Gb - Bb - Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D# - F# - A# - C#
nonotonic genchain of sixths: M2 - M7 - M3 - M8 - M4 - M9 - P5 - P1 - P6 - m2 - m7 - m3 - m8 - m4 - m9


"Every Good Boy Deserves Fudge And Candy"
A B C * D E F G * H J A


<u>'''10-edo'''</u>: (generator = 3\10 = perfect 3rd)
P1 m2 M2/m3 M3/m4 M4 P5 P6 m7 M7/m8 M8/m9 M9 P8


D E * F G * A B * C D
requires learning nonotonic interval arithmetic and staff notation


P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8
<u>'''11b-edo'''</u> octatonic (wide 3/2 maps to 7\11 = perfect 6th)


<u>'''17-edo'''</u>: (generator = 5\17 = perfect 3rd)
A B * C D * E F G * H A


D * E * * F * G * * A * B * * C * D
octatonic genchain of sixths: m3 - m8 - m5 - m2 - m7 - P4 - P1 - P6 - M3 - M8 - M5 - M2 - M7


P1 - A1/d2 - m2 - M2 - A2/d3 - P3 - A3/d4 - m4 - M4 - m5 - M5 - A5/d6 - P6 - A6/d7 - m7 - M7 - A7/d8 - P8
P1 - m2 - M2/m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7/m8 - M8 - P9


<u>'''24-edo'''</u>: (generator = 7\24 = perfect 3rd)
requires learning octatonic interval arithmetic and notation


D * * E * * * F * * G * * * A * * B * * * C * * D
<u>'''13edo'''</u> octatonic (wide 3/2 maps to 8\13 = perfect 6th)


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
octotonic genchain of sixths: M3 - M8 - M5 - M2 - M7 - P4 - P1 - P6 - m3 - m8 - m5 - m2 - m7


<u>'''31-edo'''</u>: (generator = 9\31 = perfect 3rd)
P1 - m2 - M2 - m3 - M3 - P4 - m5 - M5 - P6 - m7 - M7 - m8 - M8 - P9


D * * * E * * * * F * * * G * * * * A * * * B * * * * C * * * D
requires learning octatonic interval arithmetic and notation


P1 - A1 - AA1/dd2 - d2 - m2 - M2 - A2 - AA2/dd3 - d3 - P3 - A3 - AA3/dd4 - d4 - m4 - M4 - A4 - d5 - m5 - M5 - A5...
<u>'''18b-edo'''</u> nonatonic (narrow 3/2 maps to 10\18 = perfect 6th)


etc.
P1 - vP2 - P2 - vP3 - P3 - vP4- P4 - vP5 - P5 - vP6 - P6 - vP7 - P7 - vP8 - P8 - vP9 - P9 - vP10 - P10


<u>'''Sharp-3 edos'''</u>''':'''
requires learning nonotonic interval arithmetic and staff notation


genchain of seconds: A1 - A2 - M3 - M4 - M5 - M6 - P7 - P1 - P2 - m3 - m4 - m5 - m6 - d7 - d8 etc.
=<u>Natural Generators</u>=


D# - E# - F# - G# - A - B - C - D - E - F - G - Ab - Bb - Cb - Db etc.
Ups and downs can be avoided entirely by using 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 EDOs like 8, 11, 13 and 18 don't approximate 3/2 well, so they can be thought of as having both a major fifth and a minor fifth.


'''<u>15-edo</u>:''' (generator = 2\15 = perfect 2nd)
Every non-perfect EDO has a "natural" heptatonic generator. For 13-edo, it's a 2\13 2nd (and its octave inverse of course), because seven 2\13's falls only one EDOstep away from the octave. Thus the sharp means "sharpened by one EDO-step", major is one EDO-step wider than minor, and ups and downs aren't needed.


D * E * F * G * * A * B * C * D
The usual genchain of fifths runs ...d5 - m2 - m6 - m4 - m7 - P4 - P1 - P5 - M2 - M6 - M3 - M7 - A4... This can be generalized to any generator: The generator is always perfect, so the middle three intervals of the genchain are always perfect. One side of the genchain is always major or augmented, and the other side is always minor or diminished. For heptatonic notation, there are four major and four minor intervals. For pentatonic, there's two of each. In general, N-3 of each. The major side is usually chosen so that major is wider than minor. The only exception is for fifth-generated notation of superflat EDOs, when major may be on the left even when it should be on the right, in order to preserve familiar interval arithmetic.


P1 - A1/d2 - P2 - A2/d3 - m3 - M3 - m4 - M4 - m5 - M5 - m6 - M6 - A6/d7 - P7 - A7/d8 - P8
For 13edo, the genchain runs in 2nds: ...5 - 6 - 7 - 1 - 2 - 3 - 4 - 5... The righthand 5th is the sum of four perfect 2nds, and equals 4 * (2\13) = 8\13. The lefthand 5th is the octave minus three perfect 2nds, and equals 13\13 - 3 * (2\13) = 7\13. The righthand one is larger and therefore major. Thus the 13edo genchain is ...d2 - m3 - m4 - m5 - m6 - P7 - P1 - P2 - M3 - M4 - M5 - M6 - A7...


'''<u>22-edo</u>:''' (generator = 3\22 = perfect 2nd)
<u>'''Natural generators for 8edo, 11edo, 13edo and 18edo'''</u>


D * * E * * F * * G * * * A * * B * * C * * D
'''<u>8edo</u>:''' (generator = 1\8 = perfect 2nd = 150¢)


P1 - A1 - d2 - P2 - A2 - d3 - m3 - M3 - A3/d4 - m4 - M4 - A4/d5 - m5 - M5 - A5/d6 - m6 - M6 - A6 - d7 - P7 - A7 - d8 - P8
D E F G * A B C D


etc.
D - E - F - G - G#/Ab - A -B - C - D


<u>'''Sharp-4 edos'''</u>''':'''
P1 - P2 - m3 - M3/m4 - M4/m5 - M5/m6 - M6 - P7 - P8


genchain of seconds: d8 - d2 - m3 - m4 - m5 - m6 - P7 - P1 - P2 - M3 - M4 - M5 - M6 - A7 - A1 etc.
genchain of seconds: M3 - M4 - M5 - M6 - P7 - P1 - P2 - m3 - m4 - m5 - m6 - d7 etc.


Db - Eb - Fb - Gb - A - B - C - D - E - F - G - A# - B# - C# - D# etc.
A# - B# - C# - D# - E# - F# - G# - A - B - C - D - E - F - G - Ab - Bb - Cb - Db - Eb - Fb - Gb etc.


'''<u>20-edo</u>:''' (generator = 3\20 = perfect 2nd)
<u>'''11edo'''</u>: (generator = 3\11 = perfect 3rd)


D * * E * * F * * G * A * * B * * C * * D
D * E F * G A * B C * D


P1 - A1 - d2 - P2 - A2/d3 - m3 - M3 - A3/d4 - m4 - M4 - A4/d5 - m5 - M5 - A5/d6 - m6 - M6 - A6/d7 - P7 - A7 - d8 - P8
D - D#/Eb - E - F - F#/Gb - G - A - A#/Bb - B - C - C#/Db - D


'''<u>27-edo</u>:''' (generator = 4\27 = perfect 2nd)
P1 - m2 - M2 - P3 - m4 - M4 - m5 - M5 - P6 - m7 - M7 - P8


D * * * E * * * F * * * G * * A * * * B * * * C * * * D
genchain of thirds: M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6 etc.


P1 - A1 - AA1/dd2 - d2 - P2 - A2 - d3 - m3 - M3 - A3 - d4 - m4 - M4 - A4 - d5 - m5 - M5 - A5 etc.
E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb


etc.
<u>'''13edo'''</u>''':''' (generator = 2\13 = perfect 2nd)


<u>'''Sharp-5 edos:'''</u>
D * E * F * G A * B * C * D


genchain of thirds: A1 - A3 - M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6 - d8 etc.
D - D#/Eb - E - E#/Fb - F - F#/Gb - G - A - A#/Bb - B - B#/Cb - C - C#/Db - D


E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb
P1 - A1/d2 - P2 - m3 - M3 - m4 - M4 - m5 - M5 - m6 - M6 - P7 - A7/d8 - P8


"Every Good Boy Deserves Fudge And Candy"
genchain of seconds: d2 - m3 - m4 - m5 - m6 - P7 - P1 - P2 - M3 - M4 - M5 - M6 - A7 etc.


<u>'''25-edo'''</u>: (generator = 7\25 = perfect 3rd)
Ab - Bb - Cb - Db - Eb - Fb - Gb - A - B - C - D - E - F - G - A# - B# - C# - D# - E# - F# - G#


D * * * E * * F * * * G * * A * * * B * * C * * * D
'''<u>18edo</u>:''' (generator = 5\18 = perfect 3rd)


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
D * * E * F * * G * A * * B * C * * D


etc.
D - D# - Eb - E - E#/Fb - F - F# - Gb - G - G#/Ab - A - A# - Bb - B - B#/Cb - C - C# - Db - D


=<u>Ups and downs solfege</u>=
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


Solfege (do-re-mi) can be adapted to indicate sharp/flat and up/down:
genchain of thirds: M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6 etc.


The initial consonant remains as before: D, R, M, F, S, L and T
E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb


The first vowel indicates sharp or flat: a = natural, e = #, i = ##, o = b, u = bb
Natural generators can be used for other EDOs as well. For pentatonic EDOs, they avoid E and F naming the same note. For other EDOs, they make notating certain MOS scales easier, such as 22edo's Porcupine [7] scale. However, using any generator besides the fifth completely changes interval arithmetic. Naming chords and scales becomes very complicated. So this notation is <u>'''not recommended for edos'''</u> except as an alternate, composer-oriented notation.


The vowels are pronounced as in Spanish or Italian
For all EDOs with sharpness 1, -1 or 0, the natural generator is the fifth, the same as standard notation. For all sharp-2 and sharp-5 edos, the natural generator is a 3rd. For sharp-3 and sharp-4, it's a 2nd. For sharp-6, -7 or -8, it's the fifth closest to 7-edo's fifth, not the one closest to 3/2. This is the down-fifth in standard notation. For 42-edo, vP5 = 24\42. For 72-edo, vP5 = 41\72.


The pitch from ## to bb follows the natural vowel spectrum i-e-a-o-u
<u>'''Sharp-2 edos:'''</u>


The optional 2nd vowel indicates up/down: a = ^^^, e = ^, i = ^^, o = v, u = vv
genchain of thirds: m5 - m7 - m2 - m4 - P6 - P1 - P3 - M5 - M7 - M2 - M4 - A6 - A1 etc.


The 2nd vowel is separated from the first by an "h", a "w", or a "y"
Eb - Gb - Bb - Db - Fb - Ab - Cb - E - G - B - D - F - A - C - E# - G# - B# - D# - F# - A# - C#
 
"Every Good Boy Deserves Fudge And Candy"


Thus C#v is Deo, pronounced as Deho or Dewo or Deyo.
<u>'''10-edo'''</u>: (generator = 3\10 = perfect 3rd)


This suffices for many but not all edos, as some require triple sharps or quadruple ups.
D E * F G * A B * C D


Fixed-do solfege:
P1 - m2 - M2 - P3 - m4 - M4/m5 - M5 - P6 - m7 - M7 - P8


Da = C, De = C#, Di = C##, Do = Cb, Du =Cbb
<u>'''17-edo'''</u>: (generator = 5\17 = perfect 3rd)


Da = C, Dae = C^, Dai = C^^, Dao = Cv, Dau = Cvv, Daa = C^^^
D * E * * F * G * * A * B * * C * D


De = C#, Dee = C#^, Dei = C#^^, Deo = C#v, Deu = C#vv, Dea = C#^^^
P1 - A1/d2 - m2 - M2 - A2/d3 - P3 - A3/d4 - m4 - M4 - m5 - M5 - A5/d6 - P6 - A6/d7 - m7 - M7 - A7/d8 - P8


etc.
<u>'''24-edo'''</u>: (generator = 7\24 = perfect 3rd)


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.
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


Perfect intervals (tonic, 4th, 5th and octave): a = perfect, e= aug, i = double-aug, o = dim, u = double-dim
<u>'''31-edo'''</u>: (generator = 9\31 = perfect 3rd)


Da = P1, De = A1, Di = AA1, Do = d1, Du = dd1
D * * * E * * * * F * * * G * * * * A * * * B * * * * C * * * D


Dae = ^1, Dai = ^^1, Dao = v1, Dau = vv1, Daa = ^^^1
P1 - A1 - AA1/dd2 - d2 - m2 - M2 - A2 - AA2/dd3 - d3 - P3 - A3 - AA3/dd4 - d4 - m4 - M4 - A4 - d5 - m5 - M5 - A5...


etc.
etc.


Imperfect intervals (2nd, 3rd, 6th and 7th): a = mid, e = major, i = aug, o = minor, u = dim
<u>'''Sharp-3 edos'''</u>''':'''


Ra = ~2, Re = M2, Ri = A2, Ro = m2, Ru = d2
genchain of seconds: A1 - A2 - M3 - M4 - M5 - M6 - P7 - P1 - P2 - m3 - m4 - m5 - m6 - d7 - d8 etc.


Ree = ^M2, Rei = ^^M2, Reo = vM2, Reu = vvM2, Rea = ^^^M2
D# - E# - F# - G# - A - B - C - D - E - F - G - Ab - Bb - Cb - Db etc.


etc.
'''<u>15-edo</u>:''' (generator = 2\15 = perfect 2nd)


=<u>Rank-2 Scales: 8ve Periods (OBSOLETE)</u>=
D * E * F * G * * A * B * C * D


<span style="font-size: 150%;">'''<big>This section is obsolete, see the [[pergen|pergens]] page instead.</big>'''</span>
P1 - A1/d2 - P2 - A2/d3 - m3 - M3 - m4 - M4 - m5 - M5 - m6 - M6 - A6/d7 - P7 - A7/d8 - P8


Ups and downs can be used to notate rank-2 scales as well. Instead of edos like 12-edo, we'll be talking about '''frameworks''' like 12-tone. The generator chain is called a '''genchain'''. Fifth-generated rank-2 tunings can be notated without ups and downs in any framework on either side of the 4\7 kite (sharp-1 or flat-1):
'''<u>22-edo</u>:''' (generator = 3\22 = perfect 2nd)


12-tone genchain Eb Bb F C G D A E B F# C# G# makes this scale: C C# D Eb E F F# G G# A Bb B C
D * * E * * F * * G * * * A * * B * * C * * D


12-tone genchain F C G D A E B F# C# G# D# A# makes this scale: C C# D D# E F F# G G# A A# B C
P1 - A1 - d2 - P2 - A2 - d3 - m3 - M3 - A3/d4 - m4 - M4 - A4/d5 - m5 - M5 - A5/d6 - m6 - M6 - A6 - d7 - P7 - A7 - d8 - P8


When the notes selected from the genchain don't make a continuous chain, you get a MODMOS, easily notated:
etc.


7-tone: Eb * F C G D A * B = C D Eb F G A B C
<u>'''Sharp-4 edos'''</u>''':'''


5-tone: Bb * C G D * E = C D E G Bb C
genchain of seconds: d8 - d2 - m3 - m4 - m5 - m6 - P7 - P1 - P2 - M3 - M4 - M5 - M6 - A7 - A1 etc.


12-tone: Gb Db * * Bb F C G D A E B * * G# D# = C Db D D# E F Gb G G# A Bb B C
Db - Eb - Fb - Gb - A - B - C - D - E - F - G - A# - B# - C# - D# etc.


For a rank-2 temperament to work with a given framework, the keyspans of the generator and the period must be coprime. Otherwise the genchain won't reach all the notes. The framework must be single-ring, i.e. not on the spine of a kite. For example, fifth-generated tunings like meantone and pythagorean are compatible with 12-tone, but not with 15-tone or 24-tone. Likewise a third-generated tuning like dicot or mohajira is incompatible with 12-tone, but compatible with 24-tone. In the region of the scale tree near the 2\7 kite, 12-tone is multi-ring and 24 isn't.
'''<u>20-edo</u>:''' (generator = 3\20 = perfect 2nd)


All supersharp frameworks are incompatible with fifth-generated heptatonic notation, since the minor 2nd becomes a descending interval. All perfect and pentatonic frameworks, except for 5-tone and 7-tone, are incompatible with fifth-generated rank-2 tunings. We need only consider single-ring diatonic frameworks with sharpness &gt; 1 or &lt; -1. If these are notated without ups and downs, the notes run out of order:
D * * E * * F * * G * A * * B * * C * * D


17-tone: Gb Db Ab Eb Bb F C G D A E B F# C# G# D# A# = C Db C# D Eb D# E F Gb F# G Ab G# A Bb A# B C
P1 - A1 - d2 - P2 - A2/d3 - m3 - M3 - A3/d4 - m4 - M4 - A4/d5 - m5 - M5 - A5/d6 - m6 - M6 - A6/d7 - P7 - A7 - d8 - P8


To extend ups and downs to rank-2 tunings, the up symbol is assigned not only a '''keyspan''' (always +1) but also a '''genspan''', which indicates how many steps forward or backwards along the genchain one must travel to find the interval. The sharp is always genspan +7, and the flat is always genspan -7. By adding the genspans of the sharps/flats to the genspans of the ups/downs attached to a note, we can determine the exact location of the note on the genchain, and thus its exact tuning.
'''<u>27-edo</u>:''' (generator = 4\27 = perfect 2nd)
 
D * * * E * * * F * * * G * * A * * * B * * * C * * * D
 
P1 - A1 - AA1/dd2 - d2 - P2 - A2 - d3 - m3 - M3 - A3 - d4 - m4 - M4 - A4 - d5 - m5 - M5 - A5 etc.
 
etc.
 
<u>'''Sharp-5 edos:'''</u>
 
genchain of thirds: A1 - A3 - M5 - M7 - M2 - M4 - P6 - P1 - P3 - m5 - m7 - m2 - m4 - d6 - d8 etc.


Every single-ring node on the scale tree heads up a kite and is on the side of two other kites. These two other kites can be used to find the rank-2 interval with keyspan of 1. For example, the 10\17 node is on the side of the 7\12 kite and the 3\5 kite (its two stern-brocot ancestors). Because it's on the <u>right</u> (fifthward) side of the 7\12 kite, we know that 12 <u>fifths</u> add up to 1\17. Because it's on the <u>left</u> (fourthward) side of the 3\5 kite, 5 <u>fourths</u> add up to 1\17. Between the two, choose the interval with the smaller genspan for simplicity, which is always the kite closest to the top of the diagram. Thus in the 17-tone framework, up has a genspan of -5, corresponding to five stacked fourths, octave-reduced, which equals a tempered pythagorean minor 2nd of 256/243. Because a minor 2nd equals an up, a downminor 2nd (vm2) equals no change, and can be freely added to or subtracted from any note to change its name. To avoid out-of-order notes, either rewrite C# as C# + vm2 = Dv, or rewrite Db as Db - vm2 = C^ (subtracting a down equals adding an up).
E# - G# - B# - D# - F# - A# - C# - E - G - B - D - F - A - C - Eb - Gb - Bb - Db - Fb - Ab - Cb


17-tone Gb - A# genchain = C C^ C# D D^ D# E F F^ F# G G^ G# A A^ A# B C = C Db Dv D Eb Ev E F Gb Gv G Ab Av A Bb Bv B C
"Every Good Boy Deserves Fudge And Candy"


Substituting E# for Gb in the genchain gives us E# + vm2 = F#v in place of F^ or Gb. Unlike 17edo, F#v is not equivalent to F^, even though they occupy the same key on the keyboard, just as C# equals Db in 12-edo but not 12-tone.
<u>'''25-edo'''</u>: (generator = 7\25 = perfect 3rd)


22-tone also has a pentatonic ancestor, and vm2 still equals a unison. The 22-tone genchain:
D * * * E * * F * * * G * * A * * * B * * C * * * D


{| class="wikitable"
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
|-
 
| style="text-align:center;" | genspan from C
etc.
| style="text-align:center;" | keyspan from C
 
| |
=<u>Ups and downs solfege</u>=
| |
 
| |
Solfege (do-re-mi) can be adapted to indicate sharp/flat and up/down:
| style="text-align:center;" |
 
| style="text-align:center;" |
The initial consonant remains as before: D, R, M, F, S, L and T
|-
 
| style="text-align:center;" | -13
The first vowel indicates sharp or flat: a = natural, e = #, i = ##, o = b, u = bb
| style="text-align:center;" | 7
 
| |
The vowels are pronounced as in Spanish or Italian
| |
 
| | Gbb
The pitch from ## to bb follows the natural vowel spectrum i-e-a-o-u
| style="text-align:center;" | Fb^
 
| style="text-align:center;" | Eb^^
The optional 2nd vowel indicates up/down: a = ^^^, e = ^, i = ^^, o = v, u = vv
|-
 
| style="text-align:center;" | -12
The 2nd vowel is separated from the first by an "h", a "w", or a "y"
| style="text-align:center;" | 20
 
| |
Thus C#v is Deo, pronounced as Deho or Dewo or Deyo.
| |
 
| | Dbb
This suffices for many but not all edos, as some require triple sharps or quadruple ups.
| style="text-align:center;" | Cb^
 
| style="text-align:center;" | Bb^^
Fixed-do solfege:
|-
 
| style="text-align:center;" | -11
Da = C, De = C#, Di = C##, Do = Cb, Du =Cbb
| style="text-align:center;" | 11
 
| |
Da = C, Dae = C^, Dai = C^^, Dao = Cv, Dau = Cvv, Daa = C^^^
| |
 
| | Abb
De = C#, Dee = C#^, Dei = C#^^, Deo = C#v, Deu = C#vv, Dea = C#^^^
| style="text-align:center;" | Gb^
 
| style="text-align:center;" | F^^
etc.
|-
 
| style="text-align:center;" | -10
Moveable-do solfege:
| style="text-align:center;" | 2
 
| |
The 2nd vowel is as before. The 1st vowel's meaning depends on the interval.
| |
| | Ebb
| style="text-align:center;" | Db^
| style="text-align:center;" | C^^
|-
| style="text-align:center;" | -9
| style="text-align:center;" | 15
| |
| |
| | Bbb
| style="text-align:center;" | Ab^
| style="text-align:center;" | G^^
|-
| style="text-align:center;" | -8
| style="text-align:center;" | 6
| style="text-align:center;" |
| style="text-align:center;" |
| style="text-align:center;" | Fb
| style="text-align:center;" | Eb^
| style="text-align:center;" | D^^
|-
| style="text-align:center;" | -7
| style="text-align:center;" | 19
| style="text-align:center;" |
| style="text-align:center;" |
| style="text-align:center;" | Cb
| style="text-align:center;" | Bb^
| style="text-align:center;" | A^^
|-
| style="text-align:center;" | -6
| style="text-align:center;" | 10
| style="text-align:center;" |
| style="text-align:center;" |
| style="text-align:center;" | Gb
| style="text-align:center;" | F^
| style="text-align:center;" | E^^
|-
| style="text-align:center;" | -5
| style="text-align:center;" | 1
| style="text-align:center;" |
| style="text-align:center;" |
| style="text-align:center;" | Db
| style="text-align:center;" | C^
| style="text-align:center;" | B^^
|-
| style="text-align:center;" | -4
| style="text-align:center;" | 14
| style="text-align:center;" |
| style="text-align:center;" |
| style="text-align:center;" | Ab
| style="text-align:center;" | G^
| style="text-align:center;" | F#^^
|-
| style="text-align:center;" | -3
| style="text-align:center;" | 5
| style="text-align:center;" |
| style="text-align:center;" | Fbv
| style="text-align:center;" | Eb
| style="text-align:center;" | D^
| style="text-align:center;" | C#^^
|-
| style="text-align:center;" | -2
| style="text-align:center;" | 18
| style="text-align:center;" |
| style="text-align:center;" | Cbv
| style="text-align:center;" | Bb
| style="text-align:center;" | A^
| style="text-align:center;" | G#^^
|-
| style="text-align:center;" | -1
| style="text-align:center;" | 9
| style="text-align:center;" |
| style="text-align:center;" | Gbv
| style="text-align:center;" | F
| style="text-align:center;" | E^
| style="text-align:center;" | D#^^
|-
| style="text-align:center;" | 0
| style="text-align:center;" | 0
| style="text-align:center;" |
| style="text-align:center;" | Dbv
| style="text-align:center;" | C
| style="text-align:center;" | B^
| style="text-align:center;" | A#^^
|-
| style="text-align:center;" | 1
| style="text-align:center;" | 13
| style="text-align:center;" |
| style="text-align:center;" | Abv
| style="text-align:center;" | G
| style="text-align:center;" | F#^
| style="text-align:center;" | E#^^
|-
| style="text-align:center;" | 2
| style="text-align:center;" | 4
| style="text-align:center;" | Fbvv
| style="text-align:center;" | Ebv
| style="text-align:center;" | D
| style="text-align:center;" | C#^
| style="text-align:center;" | B#^^
|-
| style="text-align:center;" | 3
| style="text-align:center;" | 17
| style="text-align:center;" | Cbvv
| style="text-align:center;" | Bbv
| style="text-align:center;" | A
| style="text-align:center;" | G#^
| style="text-align:center;" |
|-
| style="text-align:center;" | 4
| style="text-align:center;" | 8
| style="text-align:center;" | Gbvv
| style="text-align:center;" | Fv
| style="text-align:center;" | E
| style="text-align:center;" | D#^
| style="text-align:center;" |
|-
| style="text-align:center;" | 5
| style="text-align:center;" | 21
| style="text-align:center;" | Dbvv
| style="text-align:center;" | Cv
| style="text-align:center;" | B
| style="text-align:center;" | A#^
| style="text-align:center;" |
|-
| style="text-align:center;" | 6
| style="text-align:center;" | 12
| style="text-align:center;" | Abvv
| style="text-align:center;" | Gv
| style="text-align:center;" | F#
| style="text-align:center;" | E#^
| style="text-align:center;" |
|-
| style="text-align:center;" | 7
| style="text-align:center;" | 3
| style="text-align:center;" | Ebvv
| style="text-align:center;" | Dv
| style="text-align:center;" | C#
| style="text-align:center;" | B#^
| style="text-align:center;" |
|-
| style="text-align:center;" | 8
| style="text-align:center;" | 16
| style="text-align:center;" | Bbvv
| style="text-align:center;" | Av
| style="text-align:center;" | G#
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 9
| style="text-align:center;" | 7
| style="text-align:center;" | Fvv
| style="text-align:center;" | Ev
| style="text-align:center;" | D#
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 10
| style="text-align:center;" | 20
| style="text-align:center;" | Cvv
| style="text-align:center;" | Bv
| style="text-align:center;" | A#
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 11
| style="text-align:center;" | 11
| style="text-align:center;" | Gvv
| style="text-align:center;" | F#v
| style="text-align:center;" | E#
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 12
| style="text-align:center;" | 2
| style="text-align:center;" | Dvv
| style="text-align:center;" | C#v
| style="text-align:center;" | B#
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 13
| style="text-align:center;" | 15
| style="text-align:center;" | Avv
| style="text-align:center;" | G#v
| style="text-align:center;" | Fx
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 14
| style="text-align:center;" | 6
| style="text-align:center;" | Evv
| style="text-align:center;" | D#v
| style="text-align:center;" | Cx
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 15
| style="text-align:center;" | 19
| style="text-align:center;" | Bvv
| style="text-align:center;" | A#v
| style="text-align:center;" | Gx
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 16
| style="text-align:center;" | 10
| style="text-align:center;" | F#vv
| style="text-align:center;" | E#v
| style="text-align:center;" | Dx
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 17
| style="text-align:center;" | 1
| style="text-align:center;" | C#vv
| style="text-align:center;" | B#v
| style="text-align:center;" | Ax
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" |
| style="text-align:center;" |
| |
| |
| | etc.
| style="text-align:center;" |
| style="text-align:center;" |
|}


The 22-tone keyboard, with alternate tunings for the black keys:
Perfect intervals (tonic, 4th, 5th and octave): a = perfect, e= aug, i = double-aug, o = dim, u = double-dim


{| class="wikitable"
Da = P1, De = A1, Di = AA1, Do = d1, Du = dd1
|-
| style="text-align:center;" | keyspan from C
| style="text-align:center;" | genspan from C
| style="text-align:center;" | note
| style="text-align:center;" | genspan from C
| style="text-align:center;" | note
|-
| style="text-align:center;" | 0
| style="text-align:center;" | 0
| style="text-align:center;" | C
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 1
| style="text-align:center;" | -5
| style="text-align:center;" | Db = C^
| style="text-align:center;" | +17
| style="text-align:center;" | C#vv = Dv<span style="vertical-align: super;">3</span>
|-
| style="text-align:center;" | 2
| style="text-align:center;" | -10
| style="text-align:center;" | Db^ = C^^
| style="text-align:center;" | +12
| style="text-align:center;" | C#v = Dvv
|-
| style="text-align:center;" | 3
| style="text-align:center;" | -15
| style="text-align:center;" | Db^^ = C^<span style="vertical-align: super;">3</span>
| style="text-align:center;" | +7
| style="text-align:center;" | C# = Dv
|-
| style="text-align:center;" | 4
| style="text-align:center;" | +2
| style="text-align:center;" | D
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 5
| style="text-align:center;" | -3
| style="text-align:center;" | Eb = D^
| style="text-align:center;" | +19
| style="text-align:center;" | D#vv = Ev<span style="vertical-align: super;">3</span>
|-
| style="text-align:center;" | 6
| style="text-align:center;" | -8
| style="text-align:center;" | Eb^ = D^^
| style="text-align:center;" | +14
| style="text-align:center;" | D#v = Evv
|-
| style="text-align:center;" | 7
| style="text-align:center;" | -13
| style="text-align:center;" | Eb^^ = D^<span style="vertical-align: super;">3</span>
| style="text-align:center;" | +9
| style="text-align:center;" | D# = Ev
|-
| style="text-align:center;" | 8
| style="text-align:center;" | +4
| style="text-align:center;" | E
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 9
| style="text-align:center;" | -1
| style="text-align:center;" | F
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 10
| style="text-align:center;" | -6
| style="text-align:center;" | Gb = F^
| style="text-align:center;" | +16
| style="text-align:center;" | F#vv = Gv<span style="vertical-align: super;">3</span>
|-
| style="text-align:center;" | 11
| style="text-align:center;" | -11
| style="text-align:center;" | Gb^ = F^^
| style="text-align:center;" | +11
| style="text-align:center;" | F#v = Gvv
|-
| style="text-align:center;" | 12
| style="text-align:center;" | -16
| style="text-align:center;" | Gb^^ = F^<span style="vertical-align: super;">3</span>
| style="text-align:center;" | +6
| style="text-align:center;" | F# = Gv
|-
| style="text-align:center;" | 13
| style="text-align:center;" | +1
| style="text-align:center;" | G
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 14
| style="text-align:center;" | -4
| style="text-align:center;" | Ab = G^
| style="text-align:center;" | +18
| style="text-align:center;" | G#vv = Av<span style="vertical-align: super;">3</span>
|-
| style="text-align:center;" | 15
| style="text-align:center;" | -9
| style="text-align:center;" | Ab^ = G^^
| style="text-align:center;" | +13
| style="text-align:center;" | G#v = Avv
|-
| style="text-align:center;" | 16
| style="text-align:center;" | -14
| style="text-align:center;" | Ab^^ = G^<span style="vertical-align: super;">3</span>
| style="text-align:center;" | +8
| style="text-align:center;" | G# = Av
|-
| style="text-align:center;" | 17
| style="text-align:center;" | +3
| style="text-align:center;" | A
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 18
| style="text-align:center;" | -2
| style="text-align:center;" | Bb = A^
| style="text-align:center;" | +20
| style="text-align:center;" | A#vv = Bv<span style="vertical-align: super;">3</span>
|-
| style="text-align:center;" | 19
| style="text-align:center;" | -7
| style="text-align:center;" | Bb^ = A^^
| style="text-align:center;" | +15
| style="text-align:center;" | A#v = Bvv
|-
| style="text-align:center;" | 20
| style="text-align:center;" | -12
| style="text-align:center;" | Bb^^ = A^<span style="vertical-align: super;">3</span>
| style="text-align:center;" | +10
| style="text-align:center;" | A# = Bv
|-
| style="text-align:center;" | 21
| style="text-align:center;" | +5
| style="text-align:center;" | B
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 22
| style="text-align:center;" | 0
| style="text-align:center;" | C
| style="text-align:center;" |
| style="text-align:center;" |
|}


In 22-tone, positive genspans, which lie on the fifthward half of the genchain, create sharps and downs. Negative genspans, from the fourthward half of the genchain, create flats and ups.
Dae = ^1, Dai = ^^1, Dao = v1, Dau = vv1, Daa = ^^^1


The three black keys between C and D each have two names, one some version of C and the other some version of D. You can choose which one you want to keep the notes in order. Here are four possible tunings of these 3 keys, each written out in four ways:
etc.


C C#vv C#v C# D = C C#vv C#v Dv D = C C#vv Dvv Dv D = C Dv<span style="vertical-align: super;">3</span> Dvv Dv D
Imperfect intervals (2nd, 3rd, 6th and 7th): a = mid, e = major, i = aug, o = minor, u = dim


C C^ C#v C# D = C C^ C#v Dv D = C C^ Dvv Dv D = C Db Dvv Dv D
Ra = ~2, Re = M2, Ri = A2, Ro = m2, Ru = d2


C C^ C^^ C# D = C C^ C^^ Dv D = C C^ Db^ Dv D = C Db Db^ Dv D
Ree = ^M2, Rei = ^^M2, Reo = vM2, Reu = vvM2, Rea = ^^^M2
 
etc.
 
=<u>Rank-2 Scales: 8ve Periods (OBSOLETE)</u>=


C C^ C^^ C^<span style="vertical-align: super;">3</span> D = C C^ C^^ Db^^ D = C C^ Db^ Db^^ D = C Db Db^ Db^^ D
<span style="font-size: 150%;">'''<big>This section is obsolete, see the [[pergen|pergens]] page instead.</big>'''</span>


All four tunings could be part of a MOS. Here's one that requires a MODMOS:
Ups and downs can be used to notate rank-2 scales as well. Instead of edos like 12-edo, we'll be talking about '''frameworks''' like 12-tone. The generator chain is called a '''genchain'''. Fifth-generated rank-2 tunings can be notated without ups and downs in any framework on either side of the 4\7 kite (sharp-1 or flat-1):


C C^ C#v C^<span style="vertical-align: super;">3</span> D = C C^ C#v Db^^ D = C C^ Dvv Db^^ D = C Db Dvv Db^^ D
12-tone genchain Eb Bb F C G D A E B F# C# G# makes this scale: C C# D Eb E F F# G G# A Bb B C


There are hundreds of possibilities, and ups and downs can notate all of them.
12-tone genchain F C G D A E B F# C# G# D# A# makes this scale: C C# D D# E F F# G G# A A# B C


<u>'''Finding the up's genspan'''</u>
When the notes selected from the genchain don't make a continuous chain, you get a MODMOS, easily notated:


The genspan for the up symbol in 22-tone can be found from the scale tree. Or it can be derived more rigorously if calculated from the keyspans:
7-tone: Eb * F C G D A * B = C D Eb F G A B C


K(^) = +1, K(v) = -1 (by definition, the keyspan of an up is 1)
5-tone: Bb * C G D * E = C D E G Bb C


K(#) = c, K(b) = -c (c = sharpness = keyspan of a sharp = how many keys wide aug1 is. For 22-tone, c = 3)
12-tone: Gb Db * * Bb F C G D A E B * * G# D# = C Db D D# E F Gb G G# A Bb B C


K(#v<span style="vertical-align: super;">c</span>) = K(#) + c * K(v) = 0 (going up c keys using a sharp, then going down c keys using c downs, must cancel out)
For a rank-2 temperament to work with a given framework, the keyspans of the generator and the period must be coprime. Otherwise the genchain won't reach all the notes. The framework must be single-ring, i.e. not on the spine of a kite. For example, fifth-generated tunings like meantone and pythagorean are compatible with 12-tone, but not with 15-tone or 24-tone. Likewise a third-generated tuning like dicot or mohajira is incompatible with 12-tone, but compatible with 24-tone. In the region of the scale tree near the 2\7 kite, 12-tone is multi-ring and 24 isn't.


#v<span style="vertical-align: super;">c</span> means one sharp plus c downs. Zero keyspans in the genchain only occur on every Nth step for a N-tone framework. E.g., 12-tone keyspans:
All supersharp frameworks are incompatible with fifth-generated heptatonic notation, since the minor 2nd becomes a descending interval. All perfect and pentatonic frameworks, except for 5-tone and 7-tone, are incompatible with fifth-generated rank-2 tunings. We need only consider single-ring diatonic frameworks with sharpness &gt; 1 or &lt; -1. If these are notated without ups and downs, the notes run out of order:


{| class="wikitable"
17-tone: Gb Db Ab Eb Bb F C G D A E B F# C# G# D# A# = C Db C# D Eb D# E F Gb F# G Ab G# A Bb A# B C
|-
 
| | genchain of fifths
To extend ups and downs to rank-2 tunings, the up symbol is assigned not only a '''keyspan''' (always +1) but also a '''genspan''', which indicates how many steps forward or backwards along the genchain one must travel to find the interval. The sharp is always genspan +7, and the flat is always genspan -7. By adding the genspans of the sharps/flats to the genspans of the ups/downs attached to a note, we can determine the exact location of the note on the genchain, and thus its exact tuning.
| | C
 
| | G
Every single-ring node on the scale tree heads up a kite and is on the side of two other kites. These two other kites can be used to find the rank-2 interval with keyspan of 1. For example, the 10\17 node is on the side of the 7\12 kite and the 3\5 kite (its two stern-brocot ancestors). Because it's on the <u>right</u> (fifthward) side of the 7\12 kite, we know that 12 <u>fifths</u> add up to 1\17. Because it's on the <u>left</u> (fourthward) side of the 3\5 kite, 5 <u>fourths</u> add up to 1\17. Between the two, choose the interval with the smaller genspan for simplicity, which is always the kite closest to the top of the diagram. Thus in the 17-tone framework, up has a genspan of -5, corresponding to five stacked fourths, octave-reduced, which equals a tempered pythagorean minor 2nd of 256/243. Because a minor 2nd equals an up, a downminor 2nd (vm2) equals no change, and can be freely added to or subtracted from any note to change its name. To avoid out-of-order notes, either rewrite C# as C# + vm2 = Dv, or rewrite Db as Db - vm2 = C^ (subtracting a down equals adding an up).
| | D
 
| | A
17-tone Gb - A# genchain = C C^ C# D D^ D# E F F^ F# G G^ G# A A^ A# B C = C Db Dv D Eb Ev E F Gb Gv G Ab Av A Bb Bv B C
| | E
 
| | B
Substituting E# for Gb in the genchain gives us E# + vm2 = F#v in place of F^ or Gb. Unlike 17edo, F#v is not equivalent to F^, even though they occupy the same key on the keyboard, just as C# equals Db in 12-edo but not 12-tone.
| | F#
 
| | C#
22-tone also has a pentatonic ancestor, and vm2 still equals a unison. The 22-tone genchain:
| | G#
 
| | D#
{| class="wikitable" style="text-align:center;"
| | A#
| | E#
| | B#
|-
|-
| | genspan from C
| | genspan from C
| | 0
| | keyspan from C
| | 1
| |  
| | 2
| |  
| | 3
| |  
| | 4
| |  
| | 5
| |  
| | 6
|-
| | -13
| | 7
| | 7
| | 8
| |  
| | 9
| |  
| | 10
| | Gbb
| | Fb^
| | Eb^^
|-
| | -12
| | 20
| |
| |
| | Dbb
| | Cb^
| | Bb^^
|-
| | -11
| | 11
| | 11
| | 12
| |  
| |
| | Abb
| | Gb^
| | F^^
|-
|-
| | 12-tone keyspan from C
| | -10
| | 0
| | 7
| | 2
| | 2
| | 9
| |  
| | 4
| |
| | 11
| | Ebb
| | Db^
| | C^^
|-
| | -9
| | 15
| |
| |
| | Bbb
| | Ab^
| | G^^
|-
| | -8
| | 6
| | 6
| |
| |
| | Fb
| | Eb^
| | D^^
|-
| | -7
| | 19
| |
| |
| | Cb
| | Bb^
| | A^^
|-
| | -6
| | 10
| |
| |
| | Gb
| | F^
| | E^^
|-
| | -5
| | 1
| | 1
| | 8
| |  
| | 3
| |  
| | 10
| | Db
| | C^
| | B^^
|-
| | -4
| | 14
| |
| |
| | Ab
| | G^
| | F#^^
|-
| | -3
| | 5
| | 5
| | 0
| |  
|}
| | Fbv
B#, genspan 12, has a zero keyspan, as does Dbb, genspan -12, and A###, genspan 24. Thus the final equation means that the genspan resulting from going up a sharp and down c downs must be zero, N, -N, 2N, -2N, etc. Thus this genspan mod N must be zero.
| | Eb
 
| | D^
G(#) = 7 (by definition, the sharp's genspan = 7, since we're assuming heptatonic notation)
| | C#^^
 
|-
G(#v<span style="vertical-align: super;">c</span>) = G(#) + c * G(v) = G(#) - c * G(^) = 7 - c * G(^)
| | -2
 
| | 18
G(#v<span style="vertical-align: super;">c</span>) mod N = 0, thus G(#v<span style="vertical-align: super;">c</span>) = i * N for some integer i
| |
 
| | Cbv
7 - c * G(^) = i * N
| | Bb
 
| | A^
G(^) = - (i * N - 7) / c
| | G#^^
 
For 22-tone, N = 22 and c = 3. We choose i to be the smallest (least absolute value) number that avoids fractions, and produces an interval with a keyspan of 1. Thus i = 1, G(^) = -5, and ^ = min 2nd. In order to provide alternate names for each note, the ^ should always be a 2nd, or a descending 2nd which is a 7th. This means the up's genspan modulo 7 should be 2 or 5. However as we'll see, this isn't always possible.
 
All relevant frameworks of size 53 or less:
 
{| class="wikitable"
|-
|-
| style="text-align:center;" |
| | -1
| style="text-align:center;" | Keyspan of #
| | 9
| | value of i
| |  
| style="text-align:center;" | genspan of ^
| | Gbv
| style="text-align:center;" | example
| | F
| style="text-align:center;" | stepspan &amp;
| | E^
 
| | D#^^
quality of ^
| style="text-align:center;" | stepspan &amp;
 
quality of unison
|-
|-
| style="text-align:center;" | 11-tone
| | 0
| style="text-align:center;" | -2
| | 0
| style="text-align:center;" | 1
| |  
| style="text-align:center;" | 2
| | Dbv
| style="text-align:center;" | C^ = D
| | C
| style="text-align:center;" | maj 2nd
| | B^
| style="text-align:center;" | vM2
| | A#^^
|-
|-
| style="text-align:center;" | 13b-tone
| | 1
| style="text-align:center;" | -3
| | 13
| style="text-align:center;" | 1
| |  
| style="text-align:center;" | 2
| | Abv
| style="text-align:center;" | C^ = D
| | G
| style="text-align:center;" | maj 2nd
| | F#^
| style="text-align:center;" | vM2
| | E#^^
|-
| style="text-align:center;" | 17-tone
| style="text-align:center;" | 2
| style="text-align:center;" | 1
| style="text-align:center;" | -5
| style="text-align:center;" | C^ = Db
| style="text-align:center;" | min 2nd
| style="text-align:center;" | vm2
|-
|-
| style="text-align:center;" | 22-tone
| | 2
| style="text-align:center;" | 3
| | 4
| style="text-align:center;" | 1
| | Fbvv
| style="text-align:center;" | -5
| | Ebv
| style="text-align:center;" | C^ = Db
| | D
| style="text-align:center;" | min 2nd
| | C#^
| style="text-align:center;" | vm2
| | B#^^
|-
|-
| style="text-align:center;" | 27-tone
| | 3
| style="text-align:center;" | 4
| | 17
| style="text-align:center;" | 1
| | Cbvv
| style="text-align:center;" | -5
| | Bbv
| style="text-align:center;" | C^ = Db
| | A
| style="text-align:center;" | min 2nd
| | G#^
| style="text-align:center;" | vm2
| |  
|-
|-
| style="text-align:center;" | 29-tone
| | 4
| style="text-align:center;" | 3
| | 8
| style="text-align:center;" | -1
| | Gbvv
| style="text-align:center;" | +12
| | Fv
| style="text-align:center;" | C^ = B#
| | E
| style="text-align:center;" | '''desc''' dim 2nd
| | D#^
| style="text-align:center;" | ^d2
| |  
|-
|-
| style="text-align:center;" | 31-tone
| | 5
| style="text-align:center;" | 2
| | 21
| style="text-align:center;" | 1
| | Dbvv
| style="text-align:center;" | -12
| | Cv
| style="text-align:center;" | C^ = Dbb
| | B
| style="text-align:center;" | dim 2nd
| | A#^
| style="text-align:center;" | vd2
| |  
|-
|-
| style="text-align:center;" | 32-tone
| | 6
| style="text-align:center;" | 5
| | 12
| style="text-align:center;" | 1
| | Abvv
| style="text-align:center;" | -5
| | Gv
| style="text-align:center;" | C^ = Db
| | F#
| style="text-align:center;" | min 2nd
| | E#^
| style="text-align:center;" | vm2
| |  
|-
|-
| style="text-align:center;" | 37-tone
| | 7
| style="text-align:center;" | 6
| | 3
| style="text-align:center;" | 1
| | Ebvv
| style="text-align:center;" | -5
| | Dv
| style="text-align:center;" | C^ = Db
| | C#
| style="text-align:center;" | min 2nd
| | B#^
| style="text-align:center;" | vm2
| |  
|-
|-
| style="text-align:center;" | '''39-tone'''
| | 8
| style="text-align:center;" | 5
| | 16
| style="text-align:center;" | -2
| | Bbvv
| style="text-align:center;" | +17
| | Av
| style="text-align:center;" | C^ = Ax
| | G#
| style="text-align:center;" | '''desc''' double-dim '''3rd'''
| |  
| style="text-align:center;" | '''^dd3'''
| |  
|-
|-
| style="text-align:center;" | 41-tone
| | 9
| style="text-align:center;" | 4
| | 7
| style="text-align:center;" | -1
| | Fvv
| style="text-align:center;" | +12
| | Ev
| style="text-align:center;" | C^ = B#
| | D#
| style="text-align:center;" | desc dim 2nd
| |  
| style="text-align:center;" | ^d2
| |  
|-
| style="text-align:center;" | 42-tone
| style="text-align:center;" | 7
| style="text-align:center;" | 1
| style="text-align:center;" | -5
| style="text-align:center;" | C^ = Db
| style="text-align:center;" | min 2nd
| style="text-align:center;" | vm2
|-
|-
| style="text-align:center;" | 43-tone
| | 10
| style="text-align:center;" | 3
| | 20
| style="text-align:center;" | 1
| | Cvv
| style="text-align:center;" | -12
| | Bv
| style="text-align:center;" | C^ = Dbb
| | A#
| style="text-align:center;" | dim 2nd
| |  
| style="text-align:center;" | vd2
| |  
|-
|-
| style="text-align:center;" | 45-tone
| | 11
| style="text-align:center;" | 2
| | 11
| style="text-align:center;" | 1
| | Gvv
| style="text-align:center;" | -19
| | F#v
| | <span style="display: block; text-align: center;">C^ = Dbbb
| | E#
 
| |
</span>
| |
| style="text-align:center;" | double-dim 2nd
|-
| style="text-align:center;" | vdd2
| | 12
| | 2
| | Dvv
| | C#v
| | B#
| |  
| |  
|-
|-
| style="text-align:center;" | '''49-tone'''
| | 13
| style="text-align:center;" | 7
| | 15
| style="text-align:center;" | -3
| | Avv
| style="text-align:center;" | +22
| | G#v
| style="text-align:center;" | C^ = G###
| | Fx
| style="text-align:center;" | '''desc''' triple-dim '''4th'''
| |  
| style="text-align:center;" | '''^ddd4'''
| |  
|-
|-
| style="text-align:center;" | 50-tone
| | 14
| style="text-align:center;" | 3
| | 6
| style="text-align:center;" | -1
| | Evv
| style="text-align:center;" | +19
| | D#v
| style="text-align:center;" | C^ = Bx
| | Cx
| style="text-align:center;" | '''desc''' double-dim 2nd
| |  
| style="text-align:center;" | ^dd2
| |  
|-
|-
| style="text-align:center;" | 53-tone
| | 15
| style="text-align:center;" | 5
| | 19
| style="text-align:center;" | -1
| | Bvv
| style="text-align:center;" | +12
| | A#v
| style="text-align:center;" | C^ = B#
| | Gx
| style="text-align:center;" | desc dim 2nd
| |  
| style="text-align:center;" | vd2
| |  
|}
A look at the scale fragments reveals why 29-tone's up is a descending interval:
 
22-tone: C Db * C# D
 
27-tone: C Db * * C# D
 
29-tone: C * Db C# * D
 
The 22-tone and 27-tone frameworks all have Db adjacent to C, so that C^ equals Db. For 29-tone, Db = C^^. To find a D-something that is adjacent to C, we must use Dbb, which is one key <u>below</u> C. Thus Cv = Dbb, and C^ = B#, ^ is a descending dim 2nd, and the unison is an up-dim 2nd, ^d2. Descending ups are not a problem.
 
The 29-tone keyboard, with alternate tunings for the black keys:
 
{| class="wikitable"
|-
|-
| style="text-align:center;" | keyspan from C
| | 16
| style="text-align:center;" | genspan from C
| | 10
| style="text-align:center;" | note
| | F#vv
| style="text-align:center;" | genspan from C
| | E#v
| style="text-align:center;" | note
| | Dx
| |  
| |  
|-
|-
| style="text-align:center;" | 0
| | 17
| style="text-align:center;" | 0
| | 1
| style="text-align:center;" | C
| | C#vv
| style="text-align:center;" |  
| | B#v
| style="text-align:center;" |  
| | Ax
| |  
| |  
|-
|-
| style="text-align:center;" | 1
| |  
| style="text-align:center;" | -17
| |  
| style="text-align:center;" | Dbv = C#vv
| |
| style="text-align:center;" | +12
| |
| style="text-align:center;" | C^
| | etc.
| |
| |
|}
 
The 22-tone keyboard, with alternate tunings for the black keys:
 
{| class="wikitable" style="text-align:center;"
|-
|-
| style="text-align:center;" | 2
| | keyspan from C
| style="text-align:center;" | -5
| | genspan from C
| style="text-align:center;" | Db = C#v
| | note
| style="text-align:center;" | +24
| | genspan from C
| style="text-align:center;" | C^^ = Dbb^<span style="vertical-align: super;">3</span>
| | note
|-
| | 0
| | 0
| | C
| |
| |  
|-
| | 1
| | -5
| | Db = C^
| | +17
| | C#vv = Dv<span style="vertical-align: super;">3</span>
|-
| | 2
| | -10
| | Db^ = C^^
| | +12
| | C#v = Dvv
|-
| | 3
| | -15
| | Db^^ = C^<span style="vertical-align: super;">3</span>
| | +7
| | C# = Dv
|-
| | 4
| | +2
| | D
| |
| |
|-
|-
| style="text-align:center;" | 3
| | 5
| style="text-align:center;" | -22
| | -3
| style="text-align:center;" | Dvv = Cxv<span style="vertical-align: super;">3</span>
| | Eb = D^
| style="text-align:center;" | +7
| | +19
| style="text-align:center;" | C# = Db^
| | D#vv = Ev<span style="vertical-align: super;">3</span>
|-
|-
| style="text-align:center;" | 4
| | 6
| style="text-align:center;" | -10
| | -8
| style="text-align:center;" | Dv
| | Eb^ = D^^
| style="text-align:center;" | +19
| | +14
| style="text-align:center;" | C#^ = Db^^
| | D#v = Evv
|-
|-
| style="text-align:center;" | 5
| | 7
| style="text-align:center;" | +2
| | -13
| style="text-align:center;" | D
| | Eb^^ = D^<span style="vertical-align: super;">3</span>
| style="text-align:center;" |  
| | +9
| style="text-align:center;" |
| | D# = Ev
|-
|-
| style="text-align:center;" | etc.
| | 8
| | +4
| | E
| |
| |
|-
| | 9
| | -1
| | F
| |
| |
|-
| | 10
| | -6
| | Gb = F^
| | +16
| | F#vv = Gv<span style="vertical-align: super;">3</span>
|-
| | 11
| | -11
| | Gb^ = F^^
| | +11
| | F#v = Gvv
|-
| | 12
| | -16
| | Gb^^ = F^<span style="vertical-align: super;">3</span>
| | +6
| | F# = Gv
|-
| | 13
| | +1
| | G
| |
| |
|-
| | 14
| | -4
| | Ab = G^
| | +18
| | G#vv = Av<span style="vertical-align: super;">3</span>
|-
| | 15
| | -9
| | Ab^ = G^^
| | +13
| | G#v = Avv
|-
| | 16
| | -14
| | Ab^^ = G^<span style="vertical-align: super;">3</span>
| | +8
| | G# = Av
|-
| | 17
| | +3
| | A
| |
| |
|-
| | 18
| | -2
| | Bb = A^
| | +20
| | A#vv = Bv<span style="vertical-align: super;">3</span>
|-
| | 19
| | -7
| | Bb^ = A^^
| | +15
| | A#v = Bvv
|-
| | 20
| | -12
| | Bb^^ = A^<span style="vertical-align: super;">3</span>
| | +10
| | A# = Bv
|-
| | 21
| | +5
| | B
| |
| |
|-
| | 22
| | 0
| | C
| |
| |
|}
 
In 22-tone, positive genspans, which lie on the fifthward half of the genchain, create sharps and downs. Negative genspans, from the fourthward half of the genchain, create flats and ups.
 
The three black keys between C and D each have two names, one some version of C and the other some version of D. You can choose which one you want to keep the notes in order. Here are four possible tunings of these 3 keys, each written out in four ways:
 
C C#vv C#v C# D = C C#vv C#v Dv D = C C#vv Dvv Dv D = C Dv<span style="vertical-align: super;">3</span> Dvv Dv D
 
C C^ C#v C# D = C C^ C#v Dv D = C C^ Dvv Dv D = C Db Dvv Dv D
 
C C^ C^^ C# D = C C^ C^^ Dv D = C C^ Db^ Dv D = C Db Db^ Dv D
 
C C^ C^^ C^<span style="vertical-align: super;">3</span> D = C C^ C^^ Db^^ D = C C^ Db^ Db^^ D = C Db Db^ Db^^ D
 
All four tunings could be part of a MOS. Here's one that requires a MODMOS:
 
C C^ C#v C^<span style="vertical-align: super;">3</span> D = C C^ C#v Db^^ D = C C^ Dvv Db^^ D = C Db Dvv Db^^ D
 
There are hundreds of possibilities, and ups and downs can notate all of them.
 
<u>'''Finding the up's genspan'''</u>
 
The genspan for the up symbol in 22-tone can be found from the scale tree. Or it can be derived more rigorously if calculated from the keyspans:
 
K(^) = +1, K(v) = -1 (by definition, the keyspan of an up is 1)
 
K(#) = c, K(b) = -c (c = sharpness = keyspan of a sharp = how many keys wide aug1 is. For 22-tone, c = 3)
 
K(#v<span style="vertical-align: super;">c</span>) = K(#) + c * K(v) = 0 (going up c keys using a sharp, then going down c keys using c downs, must cancel out)
 
#v<span style="vertical-align: super;">c</span> means one sharp plus c downs. Zero keyspans in the genchain only occur on every Nth step for a N-tone framework. E.g., 12-tone keyspans:
 
{| class="wikitable"
|-
| | genchain of fifths
| | C
| | G
| | D
| | A
| | E
| | B
| | F#
| | C#
| | G#
| | D#
| | A#
| | E#
| | B#
|-
| | genspan from C
| | 0
| | 1
| | 2
| | 3
| | 4
| | 5
| | 6
| | 7
| | 8
| | 9
| | 10
| | 11
| | 12
|-
| | 12-tone keyspan from C
| | 0
| | 7
| | 2
| | 9
| | 4
| | 11
| | 6
| | 1
| | 8
| | 3
| | 10
| | 5
| | 0
|}
B#, genspan 12, has a zero keyspan, as does Dbb, genspan -12, and A###, genspan 24. Thus the final equation means that the genspan resulting from going up a sharp and down c downs must be zero, N, -N, 2N, -2N, etc. Thus this genspan mod N must be zero.
 
G(#) = 7 (by definition, the sharp's genspan = 7, since we're assuming heptatonic notation)
 
G(#v<span style="vertical-align: super;">c</span>) = G(#) + c * G(v) = G(#) - c * G(^) = 7 - c * G(^)
 
G(#v<span style="vertical-align: super;">c</span>) mod N = 0, thus G(#v<span style="vertical-align: super;">c</span>) = i * N for some integer i
 
7 - c * G(^) = i * N
 
G(^) = - (i * N - 7) / c
 
For 22-tone, N = 22 and c = 3. We choose i to be the smallest (least absolute value) number that avoids fractions, and produces an interval with a keyspan of 1. Thus i = 1, G(^) = -5, and ^ = min 2nd. In order to provide alternate names for each note, the ^ should always be a 2nd, or a descending 2nd which is a 7th. This means the up's genspan modulo 7 should be 2 or 5. However as we'll see, this isn't always possible.
 
All relevant frameworks of size 53 or less:
 
{| class="wikitable" style="text-align:center;"
|-
| |
| | Keyspan of #
| | value of i
| | genspan of ^
| | example
| | stepspan &amp;
 
quality of ^
| | stepspan &amp;
 
quality of unison
|-
| | 11-tone
| | -2
| | 1
| | 2
| | C^ = D
| | maj 2nd
| | vM2
|-
| | 13b-tone
| | -3
| | 1
| | 2
| | C^ = D
| | maj 2nd
| | vM2
|-
| | 17-tone
| | 2
| | 1
| | -5
| | C^ = Db
| | min 2nd
| | vm2
|-
| | 22-tone
| | 3
| | 1
| | -5
| | C^ = Db
| | min 2nd
| | vm2
|-
| | 27-tone
| | 4
| | 1
| | -5
| | C^ = Db
| | min 2nd
| | vm2
|-
| | 29-tone
| | 3
| | -1
| | +12
| | C^ = B#
| | '''desc''' dim 2nd
| | ^d2
|-
| | 31-tone
| | 2
| | 1
| | -12
| | C^ = Dbb
| | dim 2nd
| | vd2
|-
| | 32-tone
| | 5
| | 1
| | -5
| | C^ = Db
| | min 2nd
| | vm2
|-
| | 37-tone
| | 6
| | 1
| | -5
| | C^ = Db
| | min 2nd
| | vm2
|-
| | '''39-tone'''
| | 5
| | -2
| | +17
| | C^ = Ax
| | '''desc''' double-dim '''3rd'''
| | '''^dd3'''
|-
| | 41-tone
| | 4
| | -1
| | +12
| | C^ = B#
| | desc dim 2nd
| | ^d2
|-
| | 42-tone
| | 7
| | 1
| | -5
| | C^ = Db
| | min 2nd
| | vm2
|-
| | 43-tone
| | 3
| | 1
| | -12
| | C^ = Dbb
| | dim 2nd
| | vd2
|-
| | 45-tone
| | 2
| | 1
| | -19
| | <span style="display: block; text-align: center;">C^ = Dbbb
 
</span>
| | double-dim 2nd
| | vdd2
|-
| | '''49-tone'''
| | 7
| | -3
| | +22
| | C^ = G###
| | '''desc''' triple-dim '''4th'''
| | '''^ddd4'''
|-
| | 50-tone
| | 3
| | -1
| | +19
| | C^ = Bx
| | '''desc''' double-dim 2nd
| | ^dd2
|-
| | 53-tone
| | 5
| | -1
| | +12
| | C^ = B#
| | desc dim 2nd
| | vd2
|}
 
A look at the scale fragments reveals why 29-tone's up is a descending interval:
 
22-tone: C Db * C# D
 
27-tone: C Db * * C# D
 
29-tone: C * Db C# * D
 
The 22-tone and 27-tone frameworks all have Db adjacent to C, so that C^ equals Db. For 29-tone, Db = C^^. To find a D-something that is adjacent to C, we must use Dbb, which is one key <u>below</u> C. Thus Cv = Dbb, and C^ = B#, ^ is a descending dim 2nd, and the unison is an up-dim 2nd, ^d2. Descending ups are not a problem.
 
The 29-tone keyboard, with alternate tunings for the black keys:
 
{| class="wikitable" style="text-align:center;"
|-
| | keyspan from C
| | genspan from C
| | note
| | genspan from C
| | note
|-
| | 0
| | 0
| | C
| |
| |
|-
| | 1
| | -17
| | Dbv = C#vv
| | +12
| | C^
|-
| | 2
| | -5
| | Db = C#v
| | +24
| | C^^ = Dbb^<span style="vertical-align: super;">3</span>
|-
| | 3
| | -22
| | Dvv = Cxv<span style="vertical-align: super;">3</span>
| | +7
| | C# = Db^
|-
| | 4
| | -10
| | Dv
| | +19
| | C#^ = Db^^
|-
| | 5
| | +2
| | D
| |
| |
|-
| | etc.
| |  
| |  
| |  
| |  
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