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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 commas [[distinctly tempered out]], 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 < 10<sup>6</sup>
|+style=white-space:nowrap|7-limit commas vanishing in 12-tet within three 9-odd-limit intervals
! Ratio
! Ratio
! Factorization
! Factorization
Line 221: Line 221:
|}
|}


== 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
|-
|-
|1\12
|4 5 4 4 5 4 5 4 4 5 4
|'''1'''
|8 7 8 8 7 8 7 8 8 7 8
|-
|-
|2\12
|3<sup>-4</sup> • 5<sup>-5</sup>
|'''2'''
|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\12
|3<sup>-12</sup> • 5<sup>6</sup>
|3 ; '''1 2'''
|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>
|-
|-
|4\12
|3<sup>-12</sup>
|4 ; '''1 3''' ; '''1 1 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>
|-
|-
|6\12
|3<sup>-12</sup> • 5<sup>9</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'''
|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>
|-
|-
|12\12
|3<sup>-4</sup> • 5<sup>10</sup>
|12
|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>
|}
|}
== 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 274: 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 282: 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 334: 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 350: 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 394: 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 412: 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 465: 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)
|}
|}


=== 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 541: 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 738: 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 743: Line 1,288:
|}
|}


=== 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
|-
|-
|#5/b6
| rowspan="2" |2 2 3 2 3
|2 2 3 1 1 3
| rowspan="2" |3<sup>-4</sup> • 5<sup>1</sup>
|648/625, 2048/2025
|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>
|-
|-
|#1 b3
|1 1 2 3 2 3
|1 1 1 1 3 2 3
|4 9 8 5 5 5
|81/80
|7 7 7 4 3 8
|-
|#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 790: 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 839: 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>
|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
| rowspan="12" |1 1 1 2 2 1 2 2
|7 7 7 9 7 7 7 9
| 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>
|-
|-
|8 9 8 9 8 8 9 8 9 8
|4 5 8 9 8 9 8 9
|4 3 4 3 4 4 3 4 3 4
|3 4 3 4 3 4 7 8
|-
|-
| rowspan="2" |3<sup>-8</sup> • 5<sup>5</sup>
|3 5 5 4 5 4 5 5
|9 9 5 9 9 5 9 5
|7 7 8 7 8 7 7 9
| 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 5 5 9 7 9 5 5
|3 4 3 3 3 4 3 3 3 4 3
|7 7 3 5 3 7 7 9
|-
|-
| rowspan="2" |3<sup>-8</sup> • 5<sup>-1</sup>
|3 5 5 4 9 8 9 5
|5 5 5 5 8 5 5 5 5
|7 3 4 3 8 7 7 9
| 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
|3 5 9 8 9 4 5 5
|7 7 7 7 7 7 7 7 7 9
|7 7 8 3 4 3 7 9
|-
|-
| rowspan="3" |3<sup>-4</sup> • 5<sup>7</sup>
|3 5 9 4 5 8 9 5
|9 4 9 4 9 4 9
|7 3 4 7 8 3 7 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
|3 5 9 8 5 4 9 5
|5 3 3 5 3 3 5 3 3 3
|7 3 8 7 4 3 7 9
|-
|-
|4 5 4 4 5 4 5 4 4 5 4
|7 9 5 8 9 8 5 9
|8 7 8 8 7 8 7 8 8 7 8
|3 7 4 3 4 7 3 5
|-
|-
|3<sup>-4</sup> • 5<sup>-5</sup>
|9 8 5 4 5 4 5 8
|8 5 8 5 8 5 8 5 8
|4 7 8 7 8 7 4 3
|[[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 4 5 8 5 4 5 8
|9 5 9 5 9 5 9 5 9 5 9 5
|4 7 8 7 4 7 8 3
|[[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>
|9 8 5 4 5 8 5 4
|5 5 5 5 5 5 5 5 5 5 5 5
|8 7 4 7 8 7 4 3
|[[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>
|1 1 1 4 1 4
|9 9 9 5 9 9 9 5 9 9 9 5
| colspan="5" |None
|[[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>
|}
|}
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