User:Contribution/Exploring Selected Modes in 12-EDO: Difference between revisions

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== Commas ==
== Commas ==


=== Distincly tempered out commas ===
=== Distinctly tempered out commas ===


12edo is distincly consistent in the 5-odd-limit. It distinctly tempers out precisely 14 commas—ratios that vanish through a series of intervals in the distinct consistency odd-limit where each note is distinct, with the sole exception of the last note, which matches the first.
12edo is distinctly consistent in the 5-odd-limit. It distinctly tempers out precisely 14 commas—ratios that vanish through a series of intervals in the distinct consistency odd-limit where each note is distinct, with the sole exception of the last note, which matches the first.


{| class="wikitable center-all left-2 right-3 right-5 left-6"
{| class="wikitable center-all left-2 right-3 right-5 left-6"
|+style=white-space:nowrap|All commas tempered out in 12-tet throughout series of 5-odd-limit intervals with all notes distinct
|+style=white-space:nowrap|All commas vanishing in 12-tet throughout series of 5-odd-limit intervals with all notes distinct
! Ratio
! Ratio
! Factorization
! Factorization
Line 131: Line 131:
12edo remains consistent within the 9-odd-limit. Therefore, it's worthwhile to explore ratios tempered out in the 7-limit, particularly those with simple factorizations that facilitate quick harmonic operations.
12edo remains consistent within the 9-odd-limit. Therefore, it's worthwhile to explore ratios tempered out in the 7-limit, particularly those with simple factorizations that facilitate quick harmonic operations.
{| class="wikitable center-all left-2 right-3 right-5 left-6"
{| class="wikitable center-all left-2 right-3 right-5 left-6"
|+style=white-space:nowrap|7-limit commas tempered out in 12-tet within three 9-odd-limit intervals
|+style=white-space:nowrap|7-limit commas vanishing in 12-tet within three 9-odd-limit intervals
! Ratio
! Ratio
! Factorization
! Factorization
Line 222: Line 222:


== MOS series ==
== MOS series ==
Due to the octave equivalence principle inherent in odd-limits, the 5-odd-limit contains only two primes: 3 and 5. As a result, every ratio distinctly tempered in 12-tet possess at least one rank-2 MOS series of 5-odd-limit intervals that tempers them out.
Due to the octave equivalence principle inherent in odd-limits, the 5-odd-limit contains only two primes: 3 and 5. As a result, every ratio distinctly tempered in 12-tet possess at least one rank-2 MOS series of 5-odd-limit intervals that tempers them out.
{| class="wikitable center-3"
{| class="wikitable center-3"
Line 344: Line 345:
|3<sup>8</sup> • 5<sup>-11</sup>
|3<sup>8</sup> • 5<sup>-11</sup>
|}
|}
== Modes ==
== Modes ==
=== Modes of limited transposition ===
=== Modes of limited transposition ===


{| class="wikitable"
{| class="wikitable"
|+
|+
Modes of limited transposition with at least 6 notes
!Period
!Period
!Mode
!Mode
!Distincly tempered commas
!Distinctly tempered commas
|-
|-
|1\12
|1\12
|1 1 1 1 1 1 1 1 1 1 1 1
|1 1 1 1 1 1 1 1 1 1 1 1
|All commas ([[User:Contribution/Exploring Selected Modes in 12-EDO#Distincly_tempered_out_commas|see above]])
|All commas ([[User:Contribution/Exploring Selected Modes in 12-EDO#Distinctly tempered out commas|see above]])
|-
|-
|2\12
|2\12
Line 372: Line 376:
|-
|-
| rowspan="6" |6\12
| rowspan="6" |6\12
|1 4 1 1 4 1
|2048/2025
|-
|1 2 3 1 2 3
|1 2 3 1 2 3
| rowspan="2" |648/625
| rowspan="2" |648/625
Line 380: Line 381:
|1 3 2 1 3 2
|1 3 2 1 3 2
|-
|-
|1 1 3 1 1 1 3 1
|1 4 1 1 4 1
|81/80, 128/125, 2048/2025
|2048/2025
|-
|-
|1 2 2 1 1 2 2 1
|1 1 2 2 1 1 2 2
|81/80, 648/625, 2048/2025
|81/80, 648/625, 2048/2025
|-
|1 3 1 1 1 3 1 1
|81/80, 128/125, 2048/2025
|-
|-
|1 1 2 1 1 1 1 2 1 1
|1 1 2 1 1 1 1 2 1 1
Line 394: Line 398:
{| class="wikitable"
{| class="wikitable"
|+
|+
Circle of fourths and fifths with altered notes
!Alteration
!Alteration
!Modes
!Modes
!Distincly tempered commas
!Distinctly tempered commas
|-
|-
|Penta MOS
|Penta MOS
Line 402: Line 407:
|81/80
|81/80
|-
|-
|Penta b7
|Penta dom
|2 2 3 3 2
|3 3 2 2 2
|None
|None
|-
|-
|Penta #4 b7
|Penta app
|2 2 2 4 2
|2 2 2 4 2
|None
|None
|-
|-
|Ion
|Ion MOS
|2 2 1 2 2 2 1
|2 2 1 2 2 2 1
|81/80
|81/80
Line 441: Line 446:
|1 2 2 2 1 3 1
|1 2 2 2 1 3 1
|128/125
|128/125
|-
|Diaschisma
|2 2 1 2 1 1 2 1
|81/80, 128/125, 648/625, 2048/2025
|-
|-
|Schisma
|Schisma
Line 447: Line 456:
|}
|}


=== Blues scales ===
=== Pajara and Blues scales ===


{| class="wikitable"
{| class="wikitable"
|+
|+
Pajara
!Truncation
!Modes
! Distinctly tempered commas
|-
|Pajara[10]
|1 1 1 1 1 2 1 1 1 2
|81/80, 128/125, 648/625, 2048/2025, 6561/6250, 32805/32768, 82944/78125, 262144/253125
|-
|Pajara[8]
|1 1 1 1 3 1 1 3
|81/80, 648/625, 2048/2025
|-
|Pajara[8] mod
|1 1 1 2 2 1 2 2
|81/80
|-
|Pajara[6]
|1 1 1 4 1 4
|None
|}
{| class="wikitable"
|+
Penta MOS with added notes
!Added notes
!Added notes
!Modes
!Modes
! Distinctly tempered commas
! Distinctly tempered commas
|-
|None
|2 2 3 2 3
|81/80
|-
|-
|b3
|b3
|2 1 1 3 2 3
|2 1 1 3 2 3
|81/80
| rowspan="2" |81/80
|-
|-
|#1
|#1
|1 1 2 3 2 3
|1 1 2 3 2 3
|81/80
|-
|-
|#5/b6
|#5/b6
Line 473: Line 510:
|b3 #5/b6
|b3 #5/b6
|2 1 1 3 1 1 3
|2 1 1 3 1 1 3
|81/80, 648/625, 2048/2025
| rowspan="2" |81/80, 648/625, 2048/2025
|-
|-
|#1 #5/b6
|#1 #5/b6
|1 1 2 3 1 1 3
|1 1 2 3 1 1 3
|81/80, 648/625, 2048/2025
|-
|-
|#1 b3 #5/b6
|#1 b3 #5/b6
Line 483: Line 519:
|81/80, 648/625, 2048/2025
|81/80, 648/625, 2048/2025
|}
|}
== Scales notes ==
{| class="wikitable center-all"
|+style=white-space:nowrap|Diatonic modes & alterations
|+
!Ion MOS
!Ion b3
!Ion b6
!Ion b3 b6
!Ion b2
!Ion b2 b3
!Ion b2 b6
!Ion b2 b3 b6
|-
|Db Eb F Gb Ab Bb C
|Db Eb Fb Gb Ab Bb C
|C# D# E F# G# A# B#
|C# D# E F# G# A B#
|C# D E# F# G# A# B#
|C# D E F# G# A# B#
|C# D E# F# G# A B#
|C# D E F# G# A B#
|-
|F# G# A# B C# D# E#
|F# G# A B C# D# E#
|F# G# A# B C# D E#
|F# G# A B C# D E#
|F# G A# B C# D# E#
|F# G A B C# D# E#
|F# G A# B C# D E#
|F# G A B C# D E#
|-
|B C# D# E F# G# A#
|B C# D E F# G# A#
|B C# D# E F# G A#
|B C# D E F# G A#
|B C D# E F# G# A#
|B C D E F# G# A#
|B C D# E F# G A#
|B C D E F# G A#
|-
|E F# G# A B C# D#
|E F# G A B C# D#
|E F# G# A B C D#
|E F# G A B C D#
|E F G# A B C# D#
|E F G A B C# D#
|E F G# A B C D#
|E F G A B C D#
|-
|A B C# D E F# G#
|A B C D E F# G#
|A B C# D E F G#
|A B C D E F G#
|A Bb C# D E F# G#
|A Bb C D E F# G#
|A Bb C# D E F G#
|A Bb C D E F G#
|-
|D E F# G A B C#
|D E F G A B C#
|D E F# G A Bb C#
|D E F G A Bb C#
|D Eb F# G A B C#
|D Eb F G A B C#
|D Eb F# G A Bb C#
|D Eb F G A Bb C#
|-
|G A B C D E F#
|G A Bb C D E F#
|G A B C D Eb F#
|G A Bb C D Eb F#
|G Ab B C D E F#
|G Ab Bb C D E F#
|G Ab B C D Eb F#
|G Ab Bb C D Eb F#
|-
|C D E F G A B
|C D Eb F G A B
|C D E F G Ab B
|C D Eb F G Ab B
|C Db E F G A B
|C Db Eb F G A B
|C Db E F G Ab B
|C Db Eb F G Ab B
|-
|F G A Bb C D E
|F G Ab Bb C D E
|F G A Bb C Db E
|F G Ab Bb C Db E
|F Gb A Bb C D E
|F Gb Ab Bb C D E
|F Gb A Bb C Db E
|F Gb Ab Bb C Db E
|-
|Bb C D Eb F G A
|Bb C Db Eb F G A
|Bb C D Eb F Gb A
|Bb C Db Eb F Gb A
|Bb Cb D Eb F G A
|Bb Cb Db Eb F G A
|Bb Cb D Eb F Gb A
|Bb Cb Db Eb F Gb A
|-
|Eb F G Ab Bb C D
|Eb F Gb Ab Bb C D
|Eb F G Ab Bb Cb D
|Eb F Gb Ab Bb Cb D
|Eb Fb G Ab Bb C D
|Eb Fb Gb Ab Bb C D
|Eb Fb G Ab Bb Cb D
|Eb Fb Gb Ab Bb Cb D
|-
|Ab Bb C Db Eb F G
|Ab Bb Cb Db Eb F G
|Ab Bb C Db Eb Fb G
|G# A# B C# D# E Fx
|Ab Bbb C Db Eb F G
|Ab Bbb Cb Db Eb F G
|Ab Bbb C Db Eb Fb G
|G# A B C# D# E Fx
|}
{| class="wikitable center-all"
|+style=white-space:nowrap|
Pentatonic modes & diatonic extended
|+
!Penta MOS
!Penta dom
!Penta app
!2 2 1 1 1 2 1 1 1
!2 2 1 2 1 1 2 1
|-
|F# G# A# C# D#
|E# G# B C# D#
|F# G# A# B# E
|F# G# A# B B# C# D# E E#
|F# G# A# B C# D D# E#
|-
|B C# D# F# G#
|A# C# E F# G#
|B C# D# E# A
|B C# D# E E# F# G# A A#
|B C# D# E F# G G# A#
|-
|E F# G# B C#
|D# F# A B C#
|E F# G# A# D
|E F# G# A A# B C# D D#
|E F# G# A B C C# D#
|-
|A B C# E F#
|G# B D E F#
|A B C# D# G
|A B C# D D# E F# G G#
|A B C# D E F F# G#
|-
|D E F# A B
|C# E G A B
|D E F# G# C
|D E F# G G# A B C C#
|D E F# G A Bb B C#
|-
|G A B D E
|F# A C D E
|G A B C# F
|G A B C C# D E F F#
|G A B C D Eb E F#
|-
|C D E G A
|B D F G A
|C D E F# Bb
|C D E F F# G A Bb B
|C D E F G Ab A B
|-
|F G A C D
|E G Bb C D
|F G A B Eb
|F G A Bb B C D Eb E
|F G A Bb C Db D E
|-
|Bb C D F G
|A C Eb F G
|Bb C D E Ab
|Bb C D Eb E F G Ab A
|Bb C D Eb F Gb G A
|-
|Eb F G Bb C
|D F Ab Bb C
|Eb F G A Db
|Eb F G Ab A Bb C Db D
|Eb F G Ab Bb Cb C D
|-
|Ab Bb C Eb F
|G Bb Db Eb F
|Ab Bb C D Gb
|Ab Bb C Db D Eb F Gb G
|Ab Bb C Db Eb Fb F G
|-
|Db Eb F Ab Bb
|C Eb Gb Ab Bb
|Db Eb F G Cb
|Db Eb F Gb G Ab Bb Cb C
|C# D# E# F# G# A A# B#
|}
{| class="wikitable center-all"
|+style=white-space:nowrap| Modes of limited transposition
|+
!2 2 2 2 2 2
!1 2 1 2 1 2 1 2
!1 2 3 1 2 3
!1 3 2 1 3 2
!2 1 1 2 1 1 2 1 1
!3 1 3 1 3 1
!1 1 2 1 1 1 1 2 1 1
!1 3 1 1 1 3 1 1
!1 1 2 2 1 1 2 2
!1 4 1 1 4 1
!1 1 1 1 1 1 1 1 1 1 1 1
|-
|C D E F# G# Bb
|C Db Eb E F# G A Bb
|C Db Eb F# G A
|C Db E F# G Bb
|C D Eb E F# G Ab Bb B
|C Eb E G Ab B
|C Db D E F F# G Ab Bb B
|C Db E F F# G Bb B
|C Db D E F# G Ab Bb
|C Db F F# G B
|C Db D Eb E F F# G Ab A Bb B
|-
|C# Eb F G A B
|C# D E F G Ab Bb B
|C# D E G Ab Bb
|C# D F G Ab B
|C# D# E F G G# A B C
|C# E F G# A C
|C# D Eb F F# G Ab A B C
|C# D F F# G Ab B C
|C# D Eb F G Ab A B
|C# D F# G Ab C
|
|-
|
|D Eb F F# G# A B C
|D Eb F G# A B
|D Eb F# G# A C
|D E F F# G# A Bb C C#
|D F F# A Bb C#
|D Eb E F# G G# A Bb C C#
|D Eb F# G G# A C C#
|D Eb E F# G# A Bb C
|D Eb G G# A C#
|
|-
|
|
|D# E F# A Bb C
|D# E G A Bb C#
|Eb F F# G A Bb B C# D
|Eb F# G Bb B D
|D# E F G G# A Bb B C# D
|D# E G G# A Bb C# D
|D# E F G A Bb B C#
|D# E G# A Bb D
|
|-
|
|
|E F G A# B C#
|E F G# A# B D
|
|
|E F F# G# A A# B C D D#
|E F G# A A# B D D#
|E F F# G# A# B C D
|E F A A# B D#
|
|-
|
|
|F Gb Ab B C D
|F Gb A B C Eb
|
|
|F Gb G A Bb B C Db Eb E
|F Gb A Bb B C Eb E
|F Gb G A B C Db Eb
|F Gb Bb B C E
|
|}
{| class="wikitable center-all"
|+style=white-space:nowrap|Pajara modes
|+
!1 1 1 1 1 2 1 1 1 2
!1 1 1 1 3 1 1 3
!1 1 1 2 2 1 2 2
!1 1 1 4 1 4
|-
|F# G G# A A# B C# D D# E
|F# G G# A A# C# D D#
|F# G G# A B C# D E
|F# G G# A C# D
|-
|B C C# D D# E F# G G# A
|B C C# D D# F# G G#
|B C C# D E F# G A
|B C C# D F# G
|-
|E F F# G G# A B C C# D
|E F F# G G# B C C#
|E F F# G A B C D
|E F F# G B C
|-
|A Bb B C C# D E F F# G
|A Bb B C C# E F F#
|A Bb B C D E F G
|A Bb B C E F
|-
|D Eb E F F# G A Bb B C
|D Eb E F F# A Bb B
|D Eb E F G A Bb C
|D Eb E F A Bb
|-
|G Ab A Bb B C D Eb E F
|G Ab A Bb B D Eb E
|G Ab A Bb C D Eb F
|G Ab A Bb D Eb
|-
|C Db D Eb E F G Ab A Bb
|C Db D Eb E G Ab A
|C Db D Eb F G Ab Bb
|C Db D Eb G Ab
|-
|F Gb G Ab A Bb C Db D Eb
|F Gb G Ab A C Db D
|F Gb G Ab Bb C Db Eb
|F Gb G Ab C Db
|-
|Bb Cb C Db D Eb F Gb G Ab
|Bb Cb C Db D F Gb G
|Bb Cb C Db Eb F Gb Ab
|Bb Cb C Db F Gb
|-
|Eb Fb F Gb G Ab Bb Cb C Db
|Eb Fb F Gb G Bb Cb C
|Eb Fb F Gb Ab Bb Cb Db
|Eb Fb F Gb Bb Cb
|-
|G# A A# B B# C# D# E E# F#
|G# A A# B B# D# E E#
|G# A A# B C# D# E F#
|G# A A# B D# E
|-
|C# D D# E E# F# G# A A# B
|C# D D# E E# G# A A#
|C# D D# E F# G# A B
|C# D D# E G# A
|}
[[File:12edo modes.pdf|12 edo modes]]


== Modes series ==
== Modes series ==


=== Modes of limited transposition ===
=== Modes of limited transposition ===
{| class="wikitable"
{| class="wikitable"
|+
|+
Line 530: Line 932:
|-
|-
| rowspan="18" |4\12
| rowspan="18" |4\12
| rowspan="4" |1 3
| rowspan="4" |3 1
| rowspan="18" |5<sup>-3</sup>
| rowspan="18" |5<sup>-3</sup>
|3 5 3 5 3 5
|3 5 3 5 3 5
Line 546: Line 948:
|7 9 7 4 5 4
|7 9 7 4 5 4
|-
|-
| rowspan="14" |1 1 2
| rowspan="14" |2 1 1
|3 8 8 9 4 9 8 8 3
|3 8 8 9 4 9 8 8 3
|9 4 4 3 8 3 4 4 9
|9 4 4 3 8 3 4 4 9
Line 590: Line 992:
|-
|-
| rowspan="15" |6\12
| rowspan="15" |6\12
|1 1 4
|3<sup>-4</sup> • 5<sup>-2</sup>
|5 8 5 5 8 5
|[[2048/2025]]
|7 4 7 7 4 7
|3<sup>4</sup> • 5<sup>2</sup>
|-
|1 2 3
|1 2 3
| rowspan="2" |3<sup>-4</sup> • 5<sup>4</sup>
| rowspan="2" |3<sup>-4</sup> • 5<sup>4</sup>
Line 608: Line 1,003:
|7 3 8 7 3 8
|7 3 8 7 3 8
|-
|-
| rowspan="5" |1 1 1 3
|1 4 1
|3<sup>-4</sup> • 5<sup>-2</sup>
|5 8 5 5 8 5
|[[2048/2025]]
|7 4 7 7 4 7
|3<sup>4</sup> • 5<sup>2</sup>
|-
| rowspan="5" |1 3 1 1
|3<sup>-4</sup> • 5<sup>1</sup>
|3<sup>-4</sup> • 5<sup>1</sup>
|8 5 8 5 4 9 4 5
|8 5 8 5 4 9 4 5
Line 661: Line 1,063:
|7 4 4 3 7 4 4 3
|7 4 4 3 7 4 4 3
|-
|-
| colspan="2" |1 1 1 1 2
| colspan="2" |1 1 2 1 1
| colspan="4" |Too many (130 perfect circles, 130 plagal circles)
| colspan="4" |Too many (130 perfect circles, 130 plagal circles)
|}
|}
Line 683: Line 1,085:
|7 3 4 3 7
|7 3 4 3 7
|-
|-
|2 2 3 3 2
| colspan="2" |2 2 3 3 2 ;  2 2 2 4 2
| colspan="5" |None
| colspan="4" |None
|-
|2 2 2 4 2
| colspan="5" |None
|-
|-
| rowspan="9" |2 2 1 2 2 2 1
| rowspan="9" |2 2 1 2 2 2 1
Line 881: Line 1,280:
|8 9 5 8 7 3 8
|8 9 5 8 7 3 8
|4 9 5 4 7 3 4
|4 9 5 4 7 3 4
|-
| colspan="2" |2 2 1 2 1 1 2 1
| colspan="4" |Too many (41 perfect circles, 41 plagal circles)
|-
|-
| colspan="2" |2 2 1 1 1 2 1 1 1
| colspan="2" |2 2 1 1 1 2 1 1 1
Line 886: Line 1,288:
|}
|}


=== Blues scales ===
=== Pajara and Blues scales ===
{| class="wikitable"
{| class="wikitable"
|+
|+
Line 893: Line 1,295:
!Ratio
!Ratio
! colspan="2" |Plagal circle
! colspan="2" |Plagal circle
|-
| rowspan="2" |2 2 3 2 3
| rowspan="2" |3<sup>-4</sup> • 5<sup>1</sup>
|5 5 4 5 5
| rowspan="2" |[[81/80]]
|7 7 8 7 7
| rowspan="2" |3<sup>4</sup> • 5<sup>-1</sup>
|-
|5 9 8 9 5
|7 3 4 3 7
|-
|-
|2 1 1 3 2 3
|2 1 1 3 2 3
Line 927: Line 1,339:
| rowspan="5" |2 1 1 3 1 1 3
| rowspan="5" |2 1 1 3 1 1 3
|3<sup>-4</sup> • 5<sup>1</sup>
|3<sup>-4</sup> • 5<sup>1</sup>
|
|5 5 5 8 5 4 4
|[[81/80]]
|[[81/80]]
|
|8 8 7 4 7 7 7
|3<sup>4</sup> • 5<sup>-1</sup>
|3<sup>4</sup> • 5<sup>-1</sup>
|-
|-
|3<sup>-4</sup> • 5<sup>4</sup>
|3<sup>-4</sup> • 5<sup>4</sup>
|
|5 9 4 7 9 9 5
|[[648/625]]
|[[648/625]]
|
|7 3 3 5 8 3 7
|3<sup>4</sup> • 5<sup>-4</sup>
|3<sup>4</sup> • 5<sup>-4</sup>
|-
|-
| rowspan="3" |3<sup>-4</sup> • 5<sup>-2</sup>
| rowspan="3" |3<sup>-4</sup> • 5<sup>-2</sup>
|
|5 5 3 5 8 5 5
| rowspan="3" |[[2048/2025]]
| rowspan="3" |[[2048/2025]]
|
|7 7 4 7 9 7 7
| rowspan="3" |3<sup>4</sup> • 5<sup>2</sup>
| rowspan="3" |3<sup>4</sup> • 5<sup>2</sup>
|-
|-
|
|5 5 8 9 8 8 5
|
|7 4 4 3 4 7 7
|-
|-
|
|5 8 8 9 5 5 8
|
|4 7 7 3 4 4 7
|-
|-
| rowspan="5" |1 1 2 3 1 1 3
| rowspan="5" |1 1 2 3 1 1 3
|3<sup>-4</sup> • 5<sup>1</sup>
|3<sup>-4</sup> • 5<sup>1</sup>
|
|4 4 5 8 5 5 5
|[[81/80]]
|[[81/80]]
|
|7 7 7 4 7 8 8
|3<sup>4</sup> • 5<sup>-1</sup>
|3<sup>4</sup> • 5<sup>-1</sup>
|-
|-
|3<sup>-4</sup> • 5<sup>4</sup>
|3<sup>-4</sup> • 5<sup>4</sup>
|
|5 9 9 7 4 9 5
|[[648/625]]
|[[648/625]]
|
|7 3 8 5 3 3 7
|3<sup>4</sup> • 5<sup>-4</sup>
|3<sup>4</sup> • 5<sup>-4</sup>
|-
|-
| rowspan="3" |3<sup>-4</sup> • 5<sup>-2</sup>
| rowspan="3" |3<sup>-4</sup> • 5<sup>-2</sup>
|
|5 5 8 5 3 5 5
| rowspan="3" |[[2048/2025]]
| rowspan="3" |[[2048/2025]]
|
|7 7 9 7 4 7 7
| rowspan="3" |3<sup>4</sup> • 5<sup>2</sup>
| rowspan="3" |3<sup>4</sup> • 5<sup>2</sup>
|-
|-
|
|5 8 8 9 8 5 5
|
|7 7 4 3 4 4 7
|-
|8 5 5 9 8 8 5
|7 4 4 3 7 7 4
|-
|-
|
| colspan="2" |1 1 1 1 1 2 1 1 1 2
|
| colspan="4" |Too many (356 perfect circles, 356 plagal circles)
|-
|-
| rowspan="9" |1 1 1 1 3 1 1 3
| rowspan="9" |1 1 1 1 3 1 1 3
Line 1,011: Line 1,426:
|3 8 9 5 5 8 5 5
|3 8 9 5 5 8 5 5
|7 7 4 7 7 3 4 9
|7 7 4 7 7 3 4 9
|-
| rowspan="12" |1 1 1 2 2 1 2 2
| rowspan="12" |3<sup>-4</sup> • 5<sup>1</sup>
|5 4 9 8 9 8 9 8
| rowspan="12" |[[81/80]]
|4 3 4 3 4 3 8 7
| rowspan="12" |3<sup>4</sup> • 5<sup>-1</sup>
|-
|4 5 8 9 8 9 8 9
|3 4 3 4 3 4 7 8
|-
|3 5 5 4 5 4 5 5
|7 7 8 7 8 7 7 9
|-
|3 5 5 9 7 9 5 5
|7 7 3 5 3 7 7 9
|-
|3 5 5 4 9 8 9 5
|7 3 4 3 8 7 7 9
|-
|3 5 9 8 9 4 5 5
|7 7 8 3 4 3 7 9
|-
|3 5 9 4 5 8 9 5
|7 3 4 7 8 3 7 9
|-
|3 5 9 8 5 4 9 5
|7 3 8 7 4 3 7 9
|-
|7 9 5 8 9 8 5 9
|3 7 4 3 4 7 3 5
|-
|9 8 5 4 5 4 5 8
|4 7 8 7 8 7 4 3
|-
|9 4 5 8 5 4 5 8
|4 7 8 7 4 7 8 3
|-
|9 8 5 4 5 8 5 4
|8 7 4 7 8 7 4 3
|-
|1 1 1 4 1 4
| colspan="5" |None
|}
|}

Latest revision as of 21:41, 17 July 2024

Commas

Distinctly tempered out commas

12edo is distinctly consistent in the 5-odd-limit. It distinctly tempers out precisely 14 commas—ratios that vanish through a series of intervals in the distinct consistency odd-limit where each note is distinct, with the sole exception of the last note, which matches the first.

All commas vanishing in 12-tet throughout series of 5-odd-limit intervals with all notes distinct
Ratio Factorization Cents Limit - Cents 1 / Factorization 1 / Ratio
81/80 2-4 • 34 • 5-1 21.506 5 -21.506 24 • 3-4 • 51 80/81
128/125 27 • 5-3 41.059 5 -41.059 2-7 • 53 125/128
648/625 23 • 34 • 5-4 62.565 5 -62.565 2-3 • 3-4 • 54 625/648
2048/2025 211 • 3-4 • 5-2 19.553 5 -19.553 2-11 • 34 • 52 2025/2048
6561/6250 2-1 • 38 • 5-5 84.071 5 -84.071 21 • 3-8 • 55 6250/6561
32805/32768 2-15 • 38 • 51 1.954 5 -1.954 215 • 3-8 • 5-1 32768/32805
82944/78125 210 • 34 • 5-7 103.624 5 -103.624 2-10 • 3-4 • 57 78125/82944
262144/253125 218 • 3-4 • 5-5 60.611 5 -60.611 2-18 • 34 • 55 253125/262144
531441/500000 2-5 • 312 • 5-6 105.578 5 -105.578 25 • 3-12 • 56 500000/531441
531441/524288 2-19 • 312 23.460 3 -23.460 219 • 3-12 524288/531441
2125764/1953125 22 • 312 • 5-9 146.637 5 -146.637 2-2 • 3-12 • 59 1953125/2125764
10616832/9765625 217 • 34 • 5-10 144.683 5 -144.683 2-17 • 3-4 • 510 9765625/10616832
33554432/31640625 225 • 3-4 • 5-8 101.670 5 -101.670 2-25 • 34 • 58 31640625/33554432
53747712/48828125 213 • 38 • 5-11 166.189 5 -166.189 2-13 • 3-8 • 511 48828125/53747712

Other tempered out commas

12edo remains consistent within the 9-odd-limit. Therefore, it's worthwhile to explore ratios tempered out in the 7-limit, particularly those with simple factorizations that facilitate quick harmonic operations.

7-limit commas vanishing in 12-tet within three 9-odd-limit intervals
Ratio Factorization Cents Limit - Cents 1 / Factorization 1 / Ratio
36/35 22 • 32 • 5-1 • 7-1 48.770 7 -48.770 2-2 • 3-2 • 51 • 71 35/36
50/49 21 • 52 • 7-2 34.976 7 -34.976 2-1 • 5-2 • 72 49/50
64/63 26 • 3-2 • 7-1 27.264 7 -27.264 2-6 • 32 • 71 63/64
126/125 21 • 32 • 5-3 • 71 13.795 7 -13.795 2-1 • 3-2 • 53 • 7-1 125/126
225/224 2-5 • 32 • 52 • 7-1 7.712 7 -7.712 25 • 3-2 • 5-2 • 71 224/225
256/245 28 • 5-1 • 7-2 76.034 7 -76.034 2-8 • 51 • 72 245/256
360/343 23 • 32 • 51 • 7-3 83.746 7 -83.746 2-3 • 3-2 • 5-1 • 73 343/360
405/392 2-3 • 34 • 51 • 7-2 56.482 7 -56.482 23 • 3-4 • 5-1 • 72 392/405
729/686 2-1 • 36 • 7-3 105.252 7 -105.252 21 • 3-6 • 73 686/729
729/700 2-2 • 36 • 5-2 • 7-1 70.277 7 -70.277 22 • 3-6 • 52 • 71 700/729

MOS series

Due to the octave equivalence principle inherent in odd-limits, the 5-odd-limit contains only two primes: 3 and 5. As a result, every ratio distinctly tempered in 12-tet possess at least one rank-2 MOS series of 5-odd-limit intervals that tempers them out.

All MOS series of 5-odd-limit intervals tempering out ratios in 12-tet
Perfect circle Ratio Plagal circle
3-4 • 51 9 5 5 5 81/80 7 7 7 3 34 • 5-1
5 5 4 5 5 7 7 8 7 7
9 8 9 8 9 8 9 3 4 3 4 3 4 3
5-3 8 8 8 128/125 4 4 4 53
3 5 3 5 3 5 7 9 7 9 7 9
3-4 • 54 9 9 9 9 648/625 3 3 3 3 34 • 5-4
4 5 4 5 4 5 4 5 7 8 7 8 7 8 7 8
3-4 • 5-2 5 8 5 5 8 5 2048/2025 7 4 7 7 4 7 34 • 52
3 5 5 5 3 5 5 5 7 7 7 9 7 7 7 9
8 9 8 9 8 8 9 8 9 8 4 3 4 3 4 4 3 4 3 4
3-8 • 55 9 9 5 9 9 5 9 5 6561/6250 7 3 7 3 3 7 3 3 38 • 5-5
9 8 9 9 9 8 9 9 9 8 9 3 4 3 3 3 4 3 3 3 4 3
3-8 • 5-1 5 5 5 5 8 5 5 5 5 32805/32768 7 7 7 7 4 7 7 7 7 38 • 51
3 5 5 5 5 5 5 5 5 5 7 7 7 7 7 7 7 7 7 9
3-4 • 57 9 4 9 4 9 4 9 82944/78125 3 8 3 8 3 8 3 34 • 5-7
9 9 9 7 9 9 7 9 9 7 5 3 3 5 3 3 5 3 3 3
4 5 4 4 5 4 5 4 4 5 4 8 7 8 8 7 8 7 8 8 7 8
3-4 • 5-5 8 5 8 5 8 5 8 5 8 262144/253125 4 7 4 7 4 7 4 7 4 34 • 55
3-12 • 56 9 5 9 5 9 5 9 5 9 5 9 5 531441/500000 7 3 7 3 7 3 7 3 7 3 7 3 312 • 5-6
3-12 5 5 5 5 5 5 5 5 5 5 5 5 531441/524288 7 7 7 7 7 7 7 7 7 7 7 7 312
3-12 • 59 9 9 9 5 9 9 9 5 9 9 9 5 2125764/1953125 7 3 3 3 7 3 3 3 7 3 3 3 312 • 5-9
3-4 • 510 4 9 4 9 4 4 9 4 9 4 10616832/9765625 8 3 8 3 8 8 3 8 3 8 34 • 5-10
3-4 • 5-8 8 8 5 8 8 5 8 8 5 8 8 5 33554432/31640625 7 4 4 7 4 4 7 4 4 7 4 4 34 • 58
3-8 • 511 9 4 9 9 9 4 9 9 9 4 9 53747712/48828125 3 8 3 3 3 8 3 3 3 8 3 38 • 5-11

Modes

Modes of limited transposition

Modes of limited transposition with at least 6 notes
Period Mode Distinctly tempered commas
1\12 1 1 1 1 1 1 1 1 1 1 1 1 All commas (see above)
2\12 2 2 2 2 2 2 None (128/125 for its truncation)
3\12 1 2 1 2 1 2 1 2 648/625
4\12 3 1 3 1 3 1 128/125
2 1 1 2 1 1 2 1 1
6\12 1 2 3 1 2 3 648/625
1 3 2 1 3 2
1 4 1 1 4 1 2048/2025
1 1 2 2 1 1 2 2 81/80, 648/625, 2048/2025
1 3 1 1 1 3 1 1 81/80, 128/125, 2048/2025
1 1 2 1 1 1 1 2 1 1 81/80, 128/125, 648/625, 2048/2025, 6561/6250, 82944/78125, 10616832/9765625

Modes based on the circle of 3-odd-limit

Circle of fourths and fifths with altered notes
Alteration Modes Distinctly tempered commas
Penta MOS 2 2 3 2 3 81/80
Penta dom 3 3 2 2 2 None
Penta app 2 2 2 4 2 None
Ion MOS 2 2 1 2 2 2 1 81/80
Ion b3 2 1 2 2 2 2 1 81/80, 648/625
Ion b6 2 2 1 2 1 3 1 81/80, 128/125, 648/625
Ion b3 b6 2 1 2 2 1 3 1 81/80, 128/125, 648/625
Ion b2 1 3 1 2 2 2 1 128/125
Ion b2 b3 1 2 2 2 2 2 1 None
Ion b2 b6 1 3 1 2 1 3 1 128/125
Ion b2 b3 b6 1 2 2 2 1 3 1 128/125
Diaschisma 2 2 1 2 1 1 2 1 81/80, 128/125, 648/625, 2048/2025
Schisma 2 2 1 1 1 2 1 1 1 81/80, 128/125, 648/625, 2048/2025, 6561/6250, 32805/32768

Pajara and Blues scales

Pajara
Truncation Modes Distinctly tempered commas
Pajara[10] 1 1 1 1 1 2 1 1 1 2 81/80, 128/125, 648/625, 2048/2025, 6561/6250, 32805/32768, 82944/78125, 262144/253125
Pajara[8] 1 1 1 1 3 1 1 3 81/80, 648/625, 2048/2025
Pajara[8] mod 1 1 1 2 2 1 2 2 81/80
Pajara[6] 1 1 1 4 1 4 None
Penta MOS with added notes
Added notes Modes Distinctly tempered commas
None 2 2 3 2 3 81/80
b3 2 1 1 3 2 3 81/80
#1 1 1 2 3 2 3
#5/b6 2 2 3 1 1 3 648/625, 2048/2025
#1 b3 1 1 1 1 3 2 3 81/80
b3 #5/b6 2 1 1 3 1 1 3 81/80, 648/625, 2048/2025
#1 #5/b6 1 1 2 3 1 1 3
#1 b3 #5/b6 1 1 1 1 3 1 1 3 81/80, 648/625, 2048/2025

Scales notes

Diatonic modes & alterations
Ion MOS Ion b3 Ion b6 Ion b3 b6 Ion b2 Ion b2 b3 Ion b2 b6 Ion b2 b3 b6
Db Eb F Gb Ab Bb C Db Eb Fb Gb Ab Bb C C# D# E F# G# A# B# C# D# E F# G# A B# C# D E# F# G# A# B# C# D E F# G# A# B# C# D E# F# G# A B# C# D E F# G# A B#
F# G# A# B C# D# E# F# G# A B C# D# E# F# G# A# B C# D E# F# G# A B C# D E# F# G A# B C# D# E# F# G A B C# D# E# F# G A# B C# D E# F# G A B C# D E#
B C# D# E F# G# A# B C# D E F# G# A# B C# D# E F# G A# B C# D E F# G A# B C D# E F# G# A# B C D E F# G# A# B C D# E F# G A# B C D E F# G A#
E F# G# A B C# D# E F# G A B C# D# E F# G# A B C D# E F# G A B C D# E F G# A B C# D# E F G A B C# D# E F G# A B C D# E F G A B C D#
A B C# D E F# G# A B C D E F# G# A B C# D E F G# A B C D E F G# A Bb C# D E F# G# A Bb C D E F# G# A Bb C# D E F G# A Bb C D E F G#
D E F# G A B C# D E F G A B C# D E F# G A Bb C# D E F G A Bb C# D Eb F# G A B C# D Eb F G A B C# D Eb F# G A Bb C# D Eb F G A Bb C#
G A B C D E F# G A Bb C D E F# G A B C D Eb F# G A Bb C D Eb F# G Ab B C D E F# G Ab Bb C D E F# G Ab B C D Eb F# G Ab Bb C D Eb F#
C D E F G A B C D Eb F G A B C D E F G Ab B C D Eb F G Ab B C Db E F G A B C Db Eb F G A B C Db E F G Ab B C Db Eb F G Ab B
F G A Bb C D E F G Ab Bb C D E F G A Bb C Db E F G Ab Bb C Db E F Gb A Bb C D E F Gb Ab Bb C D E F Gb A Bb C Db E F Gb Ab Bb C Db E
Bb C D Eb F G A Bb C Db Eb F G A Bb C D Eb F Gb A Bb C Db Eb F Gb A Bb Cb D Eb F G A Bb Cb Db Eb F G A Bb Cb D Eb F Gb A Bb Cb Db Eb F Gb A
Eb F G Ab Bb C D Eb F Gb Ab Bb C D Eb F G Ab Bb Cb D Eb F Gb Ab Bb Cb D Eb Fb G Ab Bb C D Eb Fb Gb Ab Bb C D Eb Fb G Ab Bb Cb D Eb Fb Gb Ab Bb Cb D
Ab Bb C Db Eb F G Ab Bb Cb Db Eb F G Ab Bb C Db Eb Fb G G# A# B C# D# E Fx Ab Bbb C Db Eb F G Ab Bbb Cb Db Eb F G Ab Bbb C Db Eb Fb G G# A B C# D# E Fx
Pentatonic modes & diatonic extended
Penta MOS Penta dom Penta app 2 2 1 1 1 2 1 1 1 2 2 1 2 1 1 2 1
F# G# A# C# D# E# G# B C# D# F# G# A# B# E F# G# A# B B# C# D# E E# F# G# A# B C# D D# E#
B C# D# F# G# A# C# E F# G# B C# D# E# A B C# D# E E# F# G# A A# B C# D# E F# G G# A#
E F# G# B C# D# F# A B C# E F# G# A# D E F# G# A A# B C# D D# E F# G# A B C C# D#
A B C# E F# G# B D E F# A B C# D# G A B C# D D# E F# G G# A B C# D E F F# G#
D E F# A B C# E G A B D E F# G# C D E F# G G# A B C C# D E F# G A Bb B C#
G A B D E F# A C D E G A B C# F G A B C C# D E F F# G A B C D Eb E F#
C D E G A B D F G A C D E F# Bb C D E F F# G A Bb B C D E F G Ab A B
F G A C D E G Bb C D F G A B Eb F G A Bb B C D Eb E F G A Bb C Db D E
Bb C D F G A C Eb F G Bb C D E Ab Bb C D Eb E F G Ab A Bb C D Eb F Gb G A
Eb F G Bb C D F Ab Bb C Eb F G A Db Eb F G Ab A Bb C Db D Eb F G Ab Bb Cb C D
Ab Bb C Eb F G Bb Db Eb F Ab Bb C D Gb Ab Bb C Db D Eb F Gb G Ab Bb C Db Eb Fb F G
Db Eb F Ab Bb C Eb Gb Ab Bb Db Eb F G Cb Db Eb F Gb G Ab Bb Cb C C# D# E# F# G# A A# B#
Modes of limited transposition
2 2 2 2 2 2 1 2 1 2 1 2 1 2 1 2 3 1 2 3 1 3 2 1 3 2 2 1 1 2 1 1 2 1 1 3 1 3 1 3 1 1 1 2 1 1 1 1 2 1 1 1 3 1 1 1 3 1 1 1 1 2 2 1 1 2 2 1 4 1 1 4 1 1 1 1 1 1 1 1 1 1 1 1 1
C D E F# G# Bb C Db Eb E F# G A Bb C Db Eb F# G A C Db E F# G Bb C D Eb E F# G Ab Bb B C Eb E G Ab B C Db D E F F# G Ab Bb B C Db E F F# G Bb B C Db D E F# G Ab Bb C Db F F# G B C Db D Eb E F F# G Ab A Bb B
C# Eb F G A B C# D E F G Ab Bb B C# D E G Ab Bb C# D F G Ab B C# D# E F G G# A B C C# E F G# A C C# D Eb F F# G Ab A B C C# D F F# G Ab B C C# D Eb F G Ab A B C# D F# G Ab C
D Eb F F# G# A B C D Eb F G# A B D Eb F# G# A C D E F F# G# A Bb C C# D F F# A Bb C# D Eb E F# G G# A Bb C C# D Eb F# G G# A C C# D Eb E F# G# A Bb C D Eb G G# A C#
D# E F# A Bb C D# E G A Bb C# Eb F F# G A Bb B C# D Eb F# G Bb B D D# E F G G# A Bb B C# D D# E G G# A Bb C# D D# E F G A Bb B C# D# E G# A Bb D
E F G A# B C# E F G# A# B D E F F# G# A A# B C D D# E F G# A A# B D D# E F F# G# A# B C D E F A A# B D#
F Gb Ab B C D F Gb A B C Eb F Gb G A Bb B C Db Eb E F Gb A Bb B C Eb E F Gb G A B C Db Eb F Gb Bb B C E
Pajara modes
1 1 1 1 1 2 1 1 1 2 1 1 1 1 3 1 1 3 1 1 1 2 2 1 2 2 1 1 1 4 1 4
F# G G# A A# B C# D D# E F# G G# A A# C# D D# F# G G# A B C# D E F# G G# A C# D
B C C# D D# E F# G G# A B C C# D D# F# G G# B C C# D E F# G A B C C# D F# G
E F F# G G# A B C C# D E F F# G G# B C C# E F F# G A B C D E F F# G B C
A Bb B C C# D E F F# G A Bb B C C# E F F# A Bb B C D E F G A Bb B C E F
D Eb E F F# G A Bb B C D Eb E F F# A Bb B D Eb E F G A Bb C D Eb E F A Bb
G Ab A Bb B C D Eb E F G Ab A Bb B D Eb E G Ab A Bb C D Eb F G Ab A Bb D Eb
C Db D Eb E F G Ab A Bb C Db D Eb E G Ab A C Db D Eb F G Ab Bb C Db D Eb G Ab
F Gb G Ab A Bb C Db D Eb F Gb G Ab A C Db D F Gb G Ab Bb C Db Eb F Gb G Ab C Db
Bb Cb C Db D Eb F Gb G Ab Bb Cb C Db D F Gb G Bb Cb C Db Eb F Gb Ab Bb Cb C Db F Gb
Eb Fb F Gb G Ab Bb Cb C Db Eb Fb F Gb G Bb Cb C Eb Fb F Gb Ab Bb Cb Db Eb Fb F Gb Bb Cb
G# A A# B B# C# D# E E# F# G# A A# B B# D# E E# G# A A# B C# D# E F# G# A A# B D# E
C# D D# E E# F# G# A A# B C# D D# E E# G# A A# C# D D# E F# G# A B C# D D# E G# A

File:12edo modes.pdf

Modes series

Modes of limited transposition

Period Mode Perfect circle Ratio Plagal circle
1\12 1 Too many (967 perfect circles, 967 plagal circles)
2\12 2 None
3\12 1 2 3-4 • 54 4 5 4 5 4 5 4 5 648/625 7 8 7 8 7 8 7 8 34 • 5-4
9 7 9 5 9 7 9 5 7 3 5 3 7 3 5 3
4 9 8 9 4 9 8 9 3 4 3 8 3 4 3 8
4 5 4 5 9 7 9 5 7 3 5 3 7 8 7 8
4 9 8 9 4 5 4 5 7 8 7 8 3 4 3 8
4 9 8 9 9 7 9 5 7 3 5 3 3 4 3 8
5 9 7 9 9 8 9 4 8 3 4 3 3 5 3 7
4\12 3 1 5-3 3 5 3 5 3 5 128/125 7 9 7 9 7 9 53
8 7 8 8 9 8 4 3 4 4 5 4
3 5 3 8 9 8 4 3 4 9 7 9
8 7 8 5 3 5 7 9 7 4 5 4
2 1 1 3 8 8 9 4 9 8 8 3 9 4 4 3 8 3 4 4 9
7 8 8 5 4 5 8 8 7 5 4 4 7 8 7 4 4 5
3 8 8 9 4 5 8 8 7 5 4 4 7 8 3 4 4 9
7 8 8 5 4 9 8 8 3 9 4 4 3 8 7 4 4 5
5 8 8 7 4 7 8 8 5 7 4 4 5 8 5 4 4 7
9 8 8 3 4 3 8 8 9 3 4 4 9 8 9 4 4 3
5 8 8 7 4 3 8 8 9 3 4 4 9 8 5 4 4 7
9 8 8 3 4 7 8 8 5 7 4 4 5 8 9 4 4 3
7 8 8 9 8 9 8 8 7 5 4 4 3 4 3 4 4 5
9 8 8 7 8 7 8 8 9 3 4 4 5 4 5 4 4 3
7 8 8 9 5 3 5 8 7 5 4 7 9 7 3 4 4 5
9 8 8 7 3 5 3 8 9 3 4 9 7 9 5 4 4 3
7 8 5 3 5 9 8 8 7 5 4 4 3 7 9 7 4 5
9 8 3 5 3 7 8 8 9 3 4 4 5 9 7 9 4 3
6\12 1 2 3 3-4 • 54 9 4 5 9 4 5 648/625 7 8 3 7 8 3 34 • 5-4
1 3 2 4 9 5 4 9 5 7 3 8 7 3 8
1 4 1 3-4 • 5-2 5 8 5 5 8 5 2048/2025 7 4 7 7 4 7 34 • 52
1 3 1 1 3-4 • 51 8 5 8 5 4 9 4 5 81/80 7 8 3 8 7 4 7 4 34 • 5-1
5-3 8 7 3 7 8 5 5 5 128/125 7 7 7 4 5 9 5 4 53
3-4 • 5-2 5 3 5 5 5 3 5 5 2048/2025 7 7 9 7 7 7 9 7 34 • 52
8 9 8 5 8 9 8 5 7 4 3 4 7 4 3 4
8 9 8 5 5 3 5 5 7 7 9 7 7 4 3 4
1 1 2 2 3-4 • 51 5 8 8 9 5 4 4 5 81/80 7 8 8 7 3 4 4 7 34 • 5-1
5 4 4 5 9 8 8 5 7 4 4 3 7 8 8 7
3-4 • 54 5 4 4 5 5 4 4 5 648/625 7 8 8 7 7 8 8 7 34 • 5-4
3-4 • 5-2 5 8 8 9 9 8 8 5 2048/2025 7 4 4 3 3 4 4 7 34 • 52
5 8 8 9 5 8 8 9 3 4 4 7 3 4 4 7
9 8 8 5 9 8 8 5 7 4 4 3 7 4 4 3
1 1 2 1 1 Too many (130 perfect circles, 130 plagal circles)

Modes based on the circle of 3-odd-limit

Mode Perfect circle Ratio Plagal circle
2 2 3 2 3 3-4 • 51 5 5 4 5 5 81/80 7 7 8 7 7 34 • 5-1
5 9 8 9 5 7 3 4 3 7
2 2 3 3 2  ; 2 2 2 4 2 None
2 2 1 2 2 2 1 3-4 • 51 5 4 5 8 5 4 5 81/80 7 8 7 4 7 8 7 34 • 5-1
5 4 5 9 5 3 5 7 9 7 3 7 8 7
5 3 5 9 5 4 5 7 8 7 3 7 9 7
9 8 9 8 9 8 9 3 4 3 4 3 4 3
9 8 5 4 5 8 9 3 4 7 8 7 4 3
9 8 9 5 3 5 9 3 7 9 7 3 4 3
9 5 3 5 9 8 9 3 4 3 7 9 7 3
9 8 5 4 5 9 8 4 3 7 8 7 4 3
8 9 5 4 5 8 9 3 4 7 8 7 3 4
2 1 2 2 2 2 1 3-4 • 51 8 9 9 8 9 9 8 81/80 4 3 3 4 3 3 4 34 • 5-1
8 9 5 4 5 9 8 4 3 7 8 7 3 4
3-4 • 54 4 4 9 5 4 5 5 648/625 7 7 8 7 3 8 8 34 • 5-4
5 5 4 5 9 4 4 8 8 3 7 8 7 7
4 4 5 9 8 9 9 3 3 4 3 7 8 8
9 9 8 9 5 4 4 8 8 7 3 4 3 3
2 2 1 2 1 3 1 3-4 • 51 8 8 9 9 9 8 9 81/80 3 4 3 3 3 4 4 34 • 5-1
5 5 9 9 9 8 3 9 4 3 3 3 7 7
5 8 9 9 9 5 3 9 7 3 3 3 4 7
9 9 5 4 5 8 8 4 4 7 8 7 3 3
8 8 5 9 5 4 9 3 8 7 3 7 4 4
5-3 7 3 5 5 5 3 8 128/125 4 9 7 7 7 9 5 53
8 5 3 3 3 5 9 3 7 9 9 9 7 4
3-4 • 54 4 4 9 9 9 8 5 648/625 7 4 3 3 3 8 8 34 • 5-4
4 7 9 9 9 5 5 7 7 3 3 3 5 8
5 9 5 4 5 4 4 8 8 7 8 7 3 7
5 9 5 9 7 9 4 8 3 5 3 7 3 7
4 5 9 9 8 9 4 8 3 4 3 3 7 8
2 1 2 2 1 3 1 3-4 • 51 9 8 9 9 9 8 8 81/80 4 4 3 3 3 4 3 34 • 5-1
3 8 9 9 9 5 5 7 7 3 3 3 4 9
3 5 9 9 9 8 5 7 4 3 3 3 7 9
8 8 5 4 5 9 9 3 3 7 8 7 4 4
9 4 5 9 5 8 8 4 4 7 3 7 8 3
5-3 8 3 5 5 5 3 7 128/125 5 9 7 7 7 9 4 53
9 5 3 3 3 5 8 4 7 9 9 9 7 3
3-4 • 54 5 8 9 9 9 4 4 648/625 8 8 3 3 3 4 7 34 • 5-4
5 5 9 9 9 7 4 8 5 3 3 3 7 7
4 4 5 4 5 9 5 7 3 7 8 7 8 8
4 9 7 9 5 9 5 7 3 7 3 5 3 8
4 9 8 9 9 5 4 8 7 3 3 4 3 8
1 3 1 2 2 2 1 5-3 8 7 7 8 5 5 8 128/125 4 7 7 4 5 5 4 53
8 3 7 8 5 9 8 4 3 7 4 5 9 4
1 2 2 2 2 2 1 None
1 3 1 2 1 3 1 5-3 5 8 7 8 7 8 5 128/125 7 4 5 4 5 4 7 53
7 8 5 8 9 8 3 9 4 3 4 7 4 5
3 8 9 8 5 8 7 5 4 7 4 3 4 9
5 8 3 4 3 8 5 7 4 9 8 9 4 7
5 5 8 7 3 5 3 9 7 9 5 4 7 7
3 5 3 7 8 5 5 7 7 4 5 9 7 9
1 2 2 2 1 3 1 5-3 8 5 5 8 7 7 8 128/125 4 5 5 4 7 7 4 53
8 9 5 8 7 3 8 4 9 5 4 7 3 4
2 2 1 2 1 1 2 1 Too many (41 perfect circles, 41 plagal circles)
2 2 1 1 1 2 1 1 1 Too many (70 perfect circles, 70 plagal circles)

Pajara and Blues scales

Mode Perfect circle Ratio Plagal circle
2 2 3 2 3 3-4 • 51 5 5 4 5 5 81/80 7 7 8 7 7 34 • 5-1
5 9 8 9 5 7 3 4 3 7
2 1 1 3 2 3 3-4 • 51 5 5 5 8 9 4 81/80 8 3 4 7 7 7 34 • 5-1
1 1 2 3 2 3 4 9 8 5 5 5 7 7 7 4 3 8
2 2 3 1 1 3 3-4 • 54 5 9 4 4 9 5 648/625 7 3 8 8 3 7 34 • 5-4
3-4 • 5-2 5 5 8 8 5 5 2048/2025 7 7 4 4 7 7 34 • 52
1 1 1 1 3 2 3 3-4 • 51 5 8 9 4 9 8 5 81/80 7 4 3 8 3 4 7 34 • 5-1
2 1 1 3 1 1 3 3-4 • 51 5 5 5 8 5 4 4 81/80 8 8 7 4 7 7 7 34 • 5-1
3-4 • 54 5 9 4 7 9 9 5 648/625 7 3 3 5 8 3 7 34 • 5-4
3-4 • 5-2 5 5 3 5 8 5 5 2048/2025 7 7 4 7 9 7 7 34 • 52
5 5 8 9 8 8 5 7 4 4 3 4 7 7
5 8 8 9 5 5 8 4 7 7 3 4 4 7
1 1 2 3 1 1 3 3-4 • 51 4 4 5 8 5 5 5 81/80 7 7 7 4 7 8 8 34 • 5-1
3-4 • 54 5 9 9 7 4 9 5 648/625 7 3 8 5 3 3 7 34 • 5-4
3-4 • 5-2 5 5 8 5 3 5 5 2048/2025 7 7 9 7 4 7 7 34 • 52
5 8 8 9 8 5 5 7 7 4 3 4 4 7
8 5 5 9 8 8 5 7 4 4 3 7 7 4
1 1 1 1 1 2 1 1 1 2 Too many (356 perfect circles, 356 plagal circles)
1 1 1 1 3 1 1 3 3-4 • 51 5 8 9 4 4 5 8 5 81/80 7 4 7 8 8 3 4 7 34 • 5-1
5 8 5 4 4 9 8 5 7 4 3 8 8 7 4 7
3-4 • 54 5 9 9 7 7 9 9 5 648/625 7 3 3 5 5 3 3 7 34 • 5-4
3-4 • 5-2 5 3 5 5 5 5 3 5 2048/2025 7 9 7 7 7 7 9 7 34 • 52
8 9 8 5 5 8 9 8 4 3 4 7 7 4 3 4
5 3 5 5 5 8 9 8 4 3 4 7 7 7 9 7
8 9 8 5 5 5 3 5 7 9 7 7 7 4 3 4
5 5 8 5 5 9 8 3 9 4 3 7 7 4 7 7
3 8 9 5 5 8 5 5 7 7 4 7 7 3 4 9
1 1 1 2 2 1 2 2 3-4 • 51 5 4 9 8 9 8 9 8 81/80 4 3 4 3 4 3 8 7 34 • 5-1
4 5 8 9 8 9 8 9 3 4 3 4 3 4 7 8
3 5 5 4 5 4 5 5 7 7 8 7 8 7 7 9
3 5 5 9 7 9 5 5 7 7 3 5 3 7 7 9
3 5 5 4 9 8 9 5 7 3 4 3 8 7 7 9
3 5 9 8 9 4 5 5 7 7 8 3 4 3 7 9
3 5 9 4 5 8 9 5 7 3 4 7 8 3 7 9
3 5 9 8 5 4 9 5 7 3 8 7 4 3 7 9
7 9 5 8 9 8 5 9 3 7 4 3 4 7 3 5
9 8 5 4 5 4 5 8 4 7 8 7 8 7 4 3
9 4 5 8 5 4 5 8 4 7 8 7 4 7 8 3
9 8 5 4 5 8 5 4 8 7 4 7 8 7 4 3
1 1 1 4 1 4 None
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