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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 the same.
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>
|}
|}
Retrieved from "https://en.xen.wiki/w/User:Contribution/Exploring_Selected_Modes_in_12-EDO"