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. There are precisely 14 distinct tempered out commas, which are ratios formed by sequences of intervals where every note is unique, except for the starting and ending notes, which remain identical.
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 with Benedetti height < 2**16
|+style=white-space:nowrap|7-limit commas vanishing in 12-tet within three 9-odd-limit intervals
! Ratio
! Ratio
! Factorization
! Factorization
Line 187: Line 187:
| 2<sup>-8</sup> • 5<sup>1</sup> • 7<sup>2</sup>
| 2<sup>-8</sup> • 5<sup>1</sup> • 7<sup>2</sup>
| [[245/256]]
| [[245/256]]
|-
| [[360/343]]
| 2<sup>3</sup> • 3<sup>2</sup> • 5<sup>1</sup> • 7<sup>-3</sup>
| 83.746
| 7
| -83.746
| 2<sup>-3</sup> • 3<sup>-2</sup> • 5<sup>-1</sup> • 7<sup>3</sup>
| [[343/360]]
|-
| [[405/392]]
| 2<sup>-3</sup> • 3<sup>4</sup> • 5<sup>1</sup> • 7<sup>-2</sup>
| 56.482
| 7
| -56.482
| 2<sup>3</sup> • 3<sup>-4</sup> • 5<sup>-1</sup> • 7<sup>2</sup>
| [[392/405]]
|-
| [[729/686]]
| 2<sup>-1</sup> • 3<sup>6</sup> • 7<sup>-3</sup>
| 105.252
| 7
| -105.252
| 2<sup>1</sup> • 3<sup>-6</sup> • 7<sup>3</sup>
| [[686/729]]
|-
| [[729/700]]
| 2<sup>-2</sup> • 3<sup>6</sup> • 5<sup>-2</sup> • 7<sup>-1</sup>
| 70.277
| 7
| -70.277
| 2<sup>2</sup> • 3<sup>-6</sup> • 5<sup>2</sup> • 7<sup>1</sup>
| [[700/729]]
|}
|}


== Modes ==
== MOS series ==


=== Modes of limited transposition ===
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"
{| class="wikitable center-3"
|+
|+
!Period
All MOS series of 5-odd-limit intervals tempering out ratios in 12-tet
!Modes
! colspan="2" |Perfect circle
!Ratio
! colspan="2" |Plagal circle
|-
| rowspan="3" |3<sup>-4</sup> • 5<sup>1</sup>
|9 5 5 5
| rowspan="3" |[[81/80]]
|7 7 7 3
| rowspan="3" |3<sup>4</sup> • 5<sup>-1</sup>
|-
|5 5 4 5 5
|7 7 8 7 7
|-
|9 8 9 8 9 8 9
|3 4 3 4 3 4 3
|-
| rowspan="2" |5<sup>-3</sup>
|8 8 8
| rowspan="2" |[[128/125]]
|4 4 4
| rowspan="2" |5<sup>3</sup>
|-
|3 5 3 5 3 5
|7 9 7 9 7 9
|-
| rowspan="2" |3<sup>-4</sup> • 5<sup>4</sup>
|9 9 9 9
| rowspan="2" |[[648/625]]
|3 3 3 3
| rowspan="2" |3<sup>4</sup> • 5<sup>-4</sup>
|-
|4 5 4 5 4 5 4 5
|7 8 7 8 7 8 7 8
|-
| rowspan="3" |3<sup>-4</sup> • 5<sup>-2</sup>
|5 8 5 5 8 5
| rowspan="3" |[[2048/2025]]
|7 4 7 7 4 7
| rowspan="3" |3<sup>4</sup> • 5<sup>2</sup>
|-
|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
|-
| rowspan="2" |3<sup>-8</sup> • 5<sup>5</sup>
|9 9 5 9 9 5 9 5
| rowspan="2" |[[6561/6250]]
|7 3 7 3 3 7 3 3
| rowspan="2" |3<sup>8</sup> • 5<sup>-5</sup>
|-
|9 8 9 9 9 8 9 9 9 8 9
|3 4 3 3 3 4 3 3 3 4 3
|-
| rowspan="2" |3<sup>-8</sup> • 5<sup>-1</sup>
|5 5 5 5 8 5 5 5 5
| rowspan="2" |[[32805/32768]]
|7 7 7 7 4 7 7 7 7
| rowspan="2" |3<sup>8</sup> • 5<sup>1</sup>
|-
|3 5 5 5 5 5 5 5 5 5
|7 7 7 7 7 7 7 7 7 9
|-
| rowspan="3" |3<sup>-4</sup> • 5<sup>7</sup>
|9 4 9 4 9 4 9
| rowspan="3" |[[82944/78125]]
|3 8 3 8 3 8 3
| rowspan="3" |3<sup>4</sup> • 5<sup>-7</sup>
|-
|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<sup>-4</sup> • 5<sup>-5</sup>
|8 5 8 5 8 5 8 5 8
|[[262144/253125]]
|4 7 4 7 4 7 4 7 4
|3<sup>4</sup> • 5<sup>5</sup>
|-
|-
|1\12
|3<sup>-12</sup> • 5<sup>6</sup>
|'''1'''
|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
|3<sup>12</sup> • 5<sup>-6</sup>
|-
|-
|2\12
|3<sup>-12</sup>
|'''2'''
|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
|3<sup>12</sup>
|-
|-
|3\12
|3<sup>-12</sup> • 5<sup>9</sup>
|3 ; '''1 2'''
|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
|3<sup>12</sup> • 5<sup>-9</sup>
|-
|-
|4\12
|3<sup>-4</sup> • 5<sup>10</sup>
|4 ; '''1 3''' ; '''1 1 2'''
|4 9 4 9 4 4 9 4 9 4
|[[10616832/9765625]]
|8 3 8 3 8 8 3 8 3 8
|3<sup>4</sup> • 5<sup>-10</sup>
|-
|-
|6\12
|3<sup>-4</sup> • 5<sup>-8</sup>
|6 ; 1 5 ; 2 4 ; '''1 1 4''' ; '''1 2 3''' ; '''1 3 2''' ; '''1 1 1 3''' ; '''1 1 2 2''' ; '''1 1 1 1 2'''
|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
|3<sup>4</sup> • 5<sup>8</sup>
|-
|-
|12\12
|3<sup>-8</sup> • 5<sup>11</sup>
|12
|9 4 9 9 9 4 9 9 9 4 9
|[[53747712/48828125]]
|3 8 3 3 3 8 3 3 3 8 3
|3<sup>8</sup> • 5<sup>-11</sup>
|}
|}
== Modes ==
=== Modes of limited transposition ===


{| class="wikitable"
{| class="wikitable"
|+
|+
All commas tempered out throughout series of 5-odd-limit intervals with all notes distinct and played
Modes of limited transposition with at least 6 notes
!Period
!Period
!Mode
!Mode
!5-limit commas tempered out
!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#Tempered commas|see below]])
|All commas ([[User:Contribution/Exploring Selected Modes in 12-EDO#Distinctly tempered out commas|see above]])
|-
|-
|2\12
|2\12
Line 242: 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 250: 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
|81/80, 128/125, 648/625, 2048/2025, 6561/6250, 82944/78125, 10616832/9765625
|81/80, 128/125, 648/625, 2048/2025, 6561/6250, 82944/78125, 10616832/9765625
|}
|}
=== Modes based on the circle of 3-odd-limit ===
{| class="wikitable"
|+
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 ===
{| 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
!Modes
! Distinctly tempered commas
|-
|None
|2 2 3 2 3
|81/80
|-
|b3
|2 1 1 3 2 3
| rowspan="2" |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
| rowspan="2" |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 ==
{| 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 of limited transposition ===
{| class="wikitable"
{| class="wikitable"
|+
|+
Line 269: Line 899:
|1\12
|1\12
|1
|1
| colspan="5" |
| colspan="5" |Too many (967 perfect circles, 967 plagal circles)
|-
|-
|2\12
|2\12
Line 302: 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 318: 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 362: 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 380: 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 433: 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" |130 perfect circles, 130 plagal circles
| colspan="4" |Too many (130 perfect circles, 130 plagal circles)
|}
|}


=== Modes based on the circle of 3-odd-limit ===
=== Modes based on the circle of 3-odd-limit ===
{| class="wikitable"
|+
!Alteration
!Modes
!5-limit commas tempered out
|-
|Penta MOS
|2 2 3 2 3
|81/80
|-
|Penta b7
|2 2 3 3 2
|None
|-
|Penta #4 b7
|2 2 2 4 2
|None
|-
|Ion
|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
|-
|Schisma
|2 2 1 1 1 2 1 1 1
|81/80, 128/125, 648/625, 2048/2025, 6561/6250, 32805/32768
|}
{| class="wikitable"
{| class="wikitable"
|+
|+
Line 509: 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" rowspan="2" |None
| colspan="4" |None
|-
|2 2 2 4 2
|-
|-
| rowspan="9" |2 2 1 2 2 2 1
| rowspan="9" |2 2 1 2 2 2 1
Line 706: 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
| colspan="4" |70 perfect circles, 70 plagal circles
| colspan="4" |Too many (70 perfect circles, 70 plagal circles)
|}
|}


=== Blues scales ===
=== Pajara and Blues scales ===
{| class="wikitable"
{| class="wikitable"
|+
|+
 
!Mode
!Added notes
! colspan="2" |Perfect circle
!Modes
!Ratio
!5-limit commas tempered out
! colspan="2" |Plagal circle
|-
|-
|None
| rowspan="2" |2 2 3 2 3
|2 2 3 2 3
| rowspan="2" |3<sup>-4</sup> • 5<sup>1</sup>
|81/80
|5 5 4 5 5
| rowspan="2" |[[81/80]]
|7 7 8 7 7
| rowspan="2" |3<sup>4</sup> • 5<sup>-1</sup>
|-
|-
|#1
|5 9 8 9 5
|1 1 2 3 2 3
|7 3 4 3 7
|81/80
|-
|-
|b3
|2 1 1 3 2 3
|2 1 1 3 2 3
|81/80
| rowspan="2" |3<sup>-4</sup> • 5<sup>1</sup>
|5 5 5 8 9 4
| rowspan="2" |[[81/80]]
|8 3 4 7 7 7
| rowspan="2" |3<sup>4</sup> • 5<sup>-1</sup>
|-
|-
|#5/b6
|1 1 2 3 2 3
|2 2 3 1 1 3
|4 9 8 5 5 5
|648/625, 2048/2025
|7 7 7 4 3 8
|-
|#1 b3
|1 1 1 1 3 2 3
|81/80
|-
|#1 b3 #5/b6
|1 1 1 1 3 1 1 3
|81/80, 648/625, 2048/2025
|}
{| class="wikitable"
|+
!Mode
! colspan="2" |Perfect circle
!Ratio
! colspan="2" |Plagal circle
|-
|-
| rowspan="2" |2 2 3 1 1 3
| rowspan="2" |2 2 3 1 1 3
Line 762: Line 1,329:
|7 7 4 4 7 7
|7 7 4 4 7 7
|3<sup>4</sup> • 5<sup>2</sup>
|3<sup>4</sup> • 5<sup>2</sup>
|-
|1 1 2 3 2 3
| rowspan="5" |3<sup>-4</sup> • 5<sup>1</sup>
|4 9 8 5 5 5
| rowspan="5" |[[81/80]]
|7 7 7 4 3 8
| rowspan="5" |3<sup>4</sup> • 5<sup>-1</sup>
|-
|2 1 1 3 2 3
|5 5 5 8 9 4
|8 3 4 7 7 7
|-
|-
|1 1 1 1 3 2 3
|1 1 1 1 3 2 3
|3<sup>-4</sup> • 5<sup>1</sup>
|5 8 9 4 9 8 5
|5 8 9 4 9 8 5
|[[81/80]]
|7 4 3 8 3 4 7
|7 4 3 8 3 4 7
|3<sup>4</sup> • 5<sup>-1</sup>
|-
| rowspan="5" |2 1 1 3 1 1 3
|3<sup>-4</sup> • 5<sup>1</sup>
|5 5 5 8 5 4 4
|[[81/80]]
|8 8 7 4 7 7 7
|3<sup>4</sup> • 5<sup>-1</sup>
|-
|3<sup>-4</sup> • 5<sup>4</sup>
|5 9 4 7 9 9 5
|[[648/625]]
|7 3 3 5 8 3 7
|3<sup>4</sup> • 5<sup>-4</sup>
|-
| rowspan="3" |3<sup>-4</sup> • 5<sup>-2</sup>
|5 5 3 5 8 5 5
| rowspan="3" |[[2048/2025]]
|7 7 4 7 9 7 7
| 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
|3<sup>-4</sup> • 5<sup>1</sup>
|4 4 5 8 5 5 5
|[[81/80]]
|7 7 7 4 7 8 8
|3<sup>4</sup> • 5<sup>-1</sup>
|-
|3<sup>-4</sup> • 5<sup>4</sup>
|5 9 9 7 4 9 5
|[[648/625]]
|7 3 8 5 3 3 7
|3<sup>4</sup> • 5<sup>-4</sup>
|-
| rowspan="3" |3<sup>-4</sup> • 5<sup>-2</sup>
|5 5 8 5 3 5 5
| rowspan="3" |[[2048/2025]]
|7 7 9 7 4 7 7
| 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
| rowspan="2" |3<sup>-4</sup> • 5<sup>1</sup>
|5 8 9 4 4 5 8 5
|5 8 9 4 4 5 8 5
| rowspan="2" |[[81/80]]
|7 4 7 8 8 3 4 7
|7 4 7 8 8 3 4 7
| rowspan="2" |3<sup>4</sup> • 5<sup>-1</sup>
|-
|-
|5 8 5 4 4 9 8 5
|5 8 5 4 4 9 8 5
Line 811: 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
|}
== MOS series of 5-odd-limit intervals tempering out 5-limit commas ==
{| class="wikitable center-3"
|+
! colspan="2" |Perfect circle
!Ratio
! colspan="2" |Plagal circle
|-
|-
| rowspan="3" |3<sup>-4</sup> • 5<sup>1</sup>
| rowspan="12" |1 1 1 2 2 1 2 2
|9 5 5 5
| rowspan="12" |3<sup>-4</sup> • 5<sup>1</sup>
| rowspan="3" |[[81/80]]
|5 4 9 8 9 8 9 8
|7 7 7 3
| rowspan="12" |[[81/80]]
| rowspan="3" |3<sup>4</sup> • 5<sup>-1</sup>
|4 3 4 3 4 3 8 7
| rowspan="12" |3<sup>4</sup> • 5<sup>-1</sup>
|-
|-
|5 5 4 5 5
|4 5 8 9 8 9 8 9
|7 7 8 7 7
|3 4 3 4 3 4 7 8
|-
|-
|9 8 9 8 9 8 9
|3 5 5 4 5 4 5 5
|3 4 3 4 3 4 3
|7 7 8 7 8 7 7 9
|-
|-
| rowspan="2" |5<sup>-3</sup>
|3 5 5 9 7 9 5 5
|8 8 8
|7 7 3 5 3 7 7 9
| rowspan="2" |[[128/125]]
|4 4 4
| rowspan="2" |5<sup>3</sup>
|-
|-
|3 5 3 5 3 5
|3 5 5 4 9 8 9 5
|7 9 7 9 7 9
|7 3 4 3 8 7 7 9
|-
|-
| rowspan="2" |3<sup>-4</sup> • 5<sup>4</sup>
|3 5 9 8 9 4 5 5
|9 9 9 9
|7 7 8 3 4 3 7 9
| rowspan="2" |[[648/625]]
|3 3 3 3
| rowspan="2" |3<sup>4</sup> • 5<sup>-4</sup>
|-
|-
|4 5 4 5 4 5 4 5
|3 5 9 4 5 8 9 5
|7 8 7 8 7 8 7 8
|7 3 4 7 8 3 7 9
|-
|-
| rowspan="3" |3<sup>-4</sup> • 5<sup>-2</sup>
|3 5 9 8 5 4 9 5
|5 8 5 5 8 5
|7 3 8 7 4 3 7 9
| rowspan="3" |[[2048/2025]]
|7 4 7 7 4 7
| rowspan="3" |3<sup>4</sup> • 5<sup>2</sup>
|-
|-
|3 5 5 5 3 5 5 5
|7 9 5 8 9 8 5 9
|7 7 7 9 7 7 7 9
|3 7 4 3 4 7 3 5
|-
|-
|8 9 8 9 8 8 9 8 9 8
|9 8 5 4 5 4 5 8
|4 3 4 3 4 4 3 4 3 4
|4 7 8 7 8 7 4 3
|-
|-
| rowspan="2" |3<sup>-8</sup> • 5<sup>5</sup>
|9 4 5 8 5 4 5 8
|9 9 5 9 9 5 9 5
|4 7 8 7 4 7 8 3
| rowspan="2" |[[6561/6250]]
|7 3 7 3 3 7 3 3
| rowspan="2" |3<sup>8</sup> • 5<sup>-5</sup>
|-
|-
|9 8 9 9 9 8 9 9 9 8 9
|9 8 5 4 5 8 5 4
|3 4 3 3 3 4 3 3 3 4 3
|8 7 4 7 8 7 4 3
|-
|-
| rowspan="2" |3<sup>-8</sup> • 5<sup>-1</sup>
|1 1 1 4 1 4
|5 5 5 5 8 5 5 5 5
| colspan="5" |None
| rowspan="2" |[[32805/32768]]
|7 7 7 7 4 7 7 7 7
| rowspan="2" |3<sup>8</sup> • 5<sup>1</sup>
|-
|3 5 5 5 5 5 5 5 5 5
|7 7 7 7 7 7 7 7 7 9
|-
| rowspan="3" |3<sup>-4</sup> • 5<sup>7</sup>
|9 4 9 4 9 4 9
| rowspan="3" |[[82944/78125]]
|3 8 3 8 3 8 3
| rowspan="3" |3<sup>4</sup> • 5<sup>-7</sup>
|-
|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<sup>-4</sup> • 5<sup>-5</sup>
|8 5 8 5 8 5 8 5 8
|[[262144/253125]]
|4 7 4 7 4 7 4 7 4
|3<sup>4</sup> • 5<sup>5</sup>
|-
|3<sup>-12</sup> • 5<sup>6</sup>
|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
|3<sup>12</sup> • 5<sup>-6</sup>
|-
|3<sup>-12</sup>
|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
|3<sup>12</sup>
|-
|3<sup>-12</sup> • 5<sup>9</sup>
|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
|3<sup>12</sup> • 5<sup>-9</sup>
|-
|3<sup>-4</sup> • 5<sup>10</sup>
|4 9 4 9 4 4 9 4 9 4
|[[10616832/9765625]]
|8 3 8 3 8 8 3 8 3 8
|3<sup>4</sup> • 5<sup>-10</sup>
|-
|3<sup>-4</sup> • 5<sup>-8</sup>
|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
|3<sup>4</sup> • 5<sup>8</sup>
|-
|3<sup>-8</sup> • 5<sup>11</sup>
|9 4 9 9 9 4 9 9 9 4 9
|[[53747712/48828125]]
|3 8 3 3 3 8 3 3 3 8 3
|3<sup>8</sup> • 5<sup>-11</sup>
|}
|}

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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