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

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== Modes ==
== Commas ==


=== Modes of limited transposition ===
=== Distinctly tempered out commas ===
{| class="wikitable"
|+
!Period
!Modes
|-
|1\12
|'''1'''
|-
|2\12
|'''2'''
|-
|3\12
|3 ; '''1 2'''
|-
|4\12
|4 ; '''1 3''' ; '''1 1 2'''
|-
|6\12
|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'''
|-
|12\12
|12
|}


=== Pentatonic modes ===
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"
|+
!MOS
!Modes
|-
|2L 3s
|2 2 3 2 3 (2 1 1 3 2 3) ; 2 2 2 3 3
|-
|1L 4s
|2 2 2 2 4
|}
 
=== Heptatonic modes ===
{| class="wikitable"
|+
!Alteration
!Modes
|-
|Ion
|2 2 1 2 2 2 1
|-
|Ion b3
|2 1 2 2 2 2 1
|-
|Ion b6
|2 2 1 2 1 3 1
|-
|Ion b3 b6
|2 1 2 2 1 3 1
|-
|Ion b2
|1 3 1 2 2 2 1
|-
|Ion b2 b3
|1 2 2 2 2 2 1
|-
|Ion b2 b6
|1 3 1 2 1 3 1
|-
|Ion b2 b3 b6
|1 2 2 2 1 3 1
|}
 
== Tempered commas ==


{| 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| Selected 7-limit commas tempered out in 12-tet
|+style=white-space:nowrap|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
|-
| [[531441/524288]]
| 2<sup>-19</sup> • 3<sup>12</sup>
| 23.460
| 3
| -23.460
| 2<sup>19</sup> • 3<sup>-12</sup>
| [[524288/531441]]
|-
| [[81/80]]
| 2<sup>-4</sup> • 3<sup>4</sup> • 5<sup>-1</sup>
| 21.506
| 5
| -21.506
| 2<sup>4</sup> • 3<sup>-4</sup> • 5<sup>1</sup>
| [[80/81]]
|-
| [[32805/32768]]
| 2<sup>-15</sup> • 3<sup>8</sup> • 5<sup>1</sup>
| 1.954
| 5
| -1.954
| 2<sup>15</sup> • 3<sup>-8</sup> • 5<sup>-1</sup>
| [[32768/32805]]
|-
| [[2048/2025]]
| 2<sup>11</sup> • 3<sup>-4</sup> • 5<sup>-2</sup>
| 19.553
| 5
| -19.553
| 2<sup>-11</sup> • 3<sup>4</sup> • 5<sup>2</sup>
| [[2025/2048]]
|-
| [[6561/6400]]
| 2<sup>-8</sup> • 3<sup>8</sup> • 5<sup>-2</sup>
| 43.013
| 5
| -43.013
| 2<sup>8</sup> • 3<sup>-8</sup> • 5<sup>2</sup>
| [[6400/6561]]
|-
| [[128/125]]
| 2<sup>7</sup> • 5<sup>-3</sup>
| 41.059
| 5
| -41.059
| 2<sup>-7</sup> • 5<sup>3</sup>
| [[125/128]]
|-
| [[648/625]]
| 2<sup>3</sup> • 3<sup>4</sup> • 5<sup>-4</sup>
| 62.565
| 5
| -62.565
| 2<sup>-3</sup> • 3<sup>-4</sup> • 5<sup>4</sup>
| [[625/648]]
|-
| [[36/35]]
| 2<sup>2</sup> • 3<sup>2</sup> • 5<sup>-1</sup> • 7<sup>-1</sup>
| 48.770
| 7
| -48.770
| 2<sup>-2</sup> • 3<sup>-2</sup> • 5<sup>1</sup> • 7<sup>1</sup>
| [[35/36]]
|-
| [[50/49]]
| 2<sup>1</sup> • 5<sup>2</sup> • 7<sup>-2</sup>
| 34.976
| 7
| -34.976
| 2<sup>-1</sup> • 5<sup>-2</sup> • 7<sup>2</sup>
| [[49/50]]
|-
| [[64/63]]
| 2<sup>6</sup> • 3<sup>-2</sup> • 7<sup>-1</sup>
| 27.264
| 7
| -27.264
| 2<sup>-6</sup> • 3<sup>2</sup> • 7<sup>1</sup>
| [[63/64]]
|-
| [[126/125]]
| 2<sup>1</sup> • 3<sup>2</sup> • 5<sup>-3</sup> • 7<sup>1</sup>
| 13.795
| 7
| -13.795
| 2<sup>-1</sup> • 3<sup>-2</sup> • 5<sup>3</sup> • 7<sup>-1</sup>
| [[125/126]]
|-
| [[225/224]]
| 2<sup>-5</sup> • 3<sup>2</sup> • 5<sup>2</sup> • 7<sup>-1</sup>
| 7.712
| 7
| -7.712
| 2<sup>5</sup> • 3<sup>-2</sup> • 5<sup>-2</sup> • 7<sup>1</sup>
| [[224/225]]
|-
| [[256/245]]
| 2<sup>8</sup> • 5<sup>-1</sup> • 7<sup>-2</sup>
| 76.034
| 7
| -76.034
| 2<sup>-8</sup> • 5<sup>1</sup> • 7<sup>2</sup>
| [[245/256]]
|}
 
{| class="wikitable center-all left-2 right-3 right-5 left-6 mw-collapsible mw-collapsed"
|+style=white-space:nowrap| 7-limit ratios tempered out in 12-tet, maximal benedetti height = 3<sup>24</sup>
! Ratio
! Ratio
! Factorization
! Factorization
Line 197: Line 15:
! 1 / Ratio
! 1 / Ratio
|-
|-
| [[531441/524288]]
|[[81/80]]
| 2<sup>-19</sup> • 3<sup>12</sup>
| 23.460
| 3
| -23.460
| 2<sup>19</sup> • 3<sup>-12</sup>
| [[524288/531441]]
|-
| [[81/80]]
| 2<sup>-4</sup> • 3<sup>4</sup> • 5<sup>-1</sup>
| 2<sup>-4</sup> • 3<sup>4</sup> • 5<sup>-1</sup>
| 21.506
| 21.506
| 5
| 5
| -21.506
| -21.506
| 2<sup>4</sup> • 3<sup>-4</sup> • 5<sup>1</sup>
| 2<sup>4</sup> • 3<sup>-4</sup> • 5<sup>1</sup>
| [[80/81]]
|[[80/81]]
|-
|-
| [[128/125]]
|[[128/125]]
| 2<sup>7</sup> • 5<sup>-3</sup>
| 2<sup>7</sup> • 5<sup>-3</sup>
| 41.059
| 41.059
| 5
| 5
| -41.059
| -41.059
| 2<sup>-7</sup> • 5<sup>3</sup>
| 2<sup>-7</sup> • 5<sup>3</sup>
| [[125/128]]
|[[125/128]]
|-
|-
| [[648/625]]
|[[648/625]]
| 2<sup>3</sup> • 3<sup>4</sup> • 5<sup>-4</sup>
| 2<sup>3</sup> • 3<sup>4</sup> • 5<sup>-4</sup>
| 62.565
| 62.565
| 5
| 5
| -62.565
| -62.565
| 2<sup>-3</sup> • 3<sup>-4</sup> • 5<sup>4</sup>
| 2<sup>-3</sup> • 3<sup>-4</sup> • 5<sup>4</sup>
| [[625/648]]
|[[625/648]]
|-
|-
| [[2048/2025]]
|[[2048/2025]]
| 2<sup>11</sup> • 3<sup>-4</sup> • 5<sup>-2</sup>
| 2<sup>11</sup> • 3<sup>-4</sup> • 5<sup>-2</sup>
| 19.553
| 19.553
| 5
| 5
| -19.553
| -19.553
| 2<sup>-11</sup> • 3<sup>4</sup> • 5<sup>2</sup>
| 2<sup>-11</sup> • 3<sup>4</sup> • 5<sup>2</sup>
| [[2025/2048]]
|[[2025/2048]]
|-
|-
| [[6561/6250]]
|[[6561/6250]]
| 2<sup>-1</sup> • 3<sup>8</sup> • 5<sup>-5</sup>
| 2<sup>-1</sup> • 3<sup>8</sup> • 5<sup>-5</sup>
| 84.071
| 84.071
| 5
| 5
| -84.071
| -84.071
| 2<sup>1</sup> • 3<sup>-8</sup> • 5<sup>5</sup>
| 2<sup>1</sup> • 3<sup>-8</sup> • 5<sup>5</sup>
| [[6250/6561]]
|[[6250/6561]]
|-
| [[6561/6400]]
| 2<sup>-8</sup> • 3<sup>8</sup> • 5<sup>-2</sup>
| 43.013
| 5
| -43.013
| 2<sup>8</sup> • 3<sup>-8</sup> • 5<sup>2</sup>
| [[6400/6561]]
|-
| [[16384/15625]]
| 2<sup>14</sup> • 5<sup>-6</sup>
| 82.118
| 5
| -82.118
| 2<sup>-14</sup> • 5<sup>6</sup>
| [[15625/16384]]
|-
|-
| [[32805/32768]]
|[[32805/32768]]
| 2<sup>-15</sup> • 3<sup>8</sup> • 5<sup>1</sup>
| 2<sup>-15</sup> • 3<sup>8</sup> • 5<sup>1</sup>
| 1.954
| 1.954
| 5
| 5
| -1.954
| -1.954
| 2<sup>15</sup> • 3<sup>-8</sup> • 5<sup>-1</sup>
| 2<sup>15</sup> • 3<sup>-8</sup> • 5<sup>-1</sup>
| [[32768/32805]]
|[[32768/32805]]
|-
|-
| [[82944/78125]]
|[[82944/78125]]
| 2<sup>10</sup> • 3<sup>4</sup> • 5<sup>-7</sup>
| 2<sup>10</sup> • 3<sup>4</sup> • 5<sup>-7</sup>
| 103.624
| 103.624
| 5
| 5
| -103.624
| -103.624
| 2<sup>-10</sup> • 3<sup>-4</sup> • 5<sup>7</sup>
| 2<sup>-10</sup> • 3<sup>-4</sup> • 5<sup>7</sup>
| [[78125/82944]]
|[[78125/82944]]
|-
|-
| [[262144/253125]]
|[[262144/253125]]
| 2<sup>18</sup> • 3<sup>-4</sup> • 5<sup>-5</sup>
| 2<sup>18</sup> • 3<sup>-4</sup> • 5<sup>-5</sup>
| 60.611
| 60.611
| 5
| 5
| -60.611
| -60.611
| 2<sup>-18</sup> • 3<sup>4</sup> • 5<sup>5</sup>
| 2<sup>-18</sup> • 3<sup>4</sup> • 5<sup>5</sup>
| [[253125/262144]]
|[[253125/262144]]
|-
| [[419904/390625]]
| 2<sup>6</sup> • 3<sup>8</sup> • 5<sup>-8</sup>
| 125.130
| 5
| -125.130
| 2<sup>-6</sup> • 3<sup>-8</sup> • 5<sup>8</sup>
| [[390625/419904]]
|-
|-
| [[531441/500000]]
|[[531441/500000]]
| 2<sup>-5</sup> • 3<sup>12</sup> • 5<sup>-6</sup>
| 2<sup>-5</sup> • 3<sup>12</sup> • 5<sup>-6</sup>
| 105.578
| 105.578
| 5
| 5
| -105.578
| -105.578
| 2<sup>5</sup> • 3<sup>-12</sup> • 5<sup>6</sup>
| 2<sup>5</sup> • 3<sup>-12</sup> • 5<sup>6</sup>
| [[500000/531441]]
|[[500000/531441]]
|-
|-
| [[531441/512000]]
| [[531441/524288]]
| 2<sup>-12</sup> • 3<sup>12</sup> • 5<sup>-3</sup>
| 2<sup>-19</sup> • 3<sup>12</sup>
| 64.519
| 23.460
| 3
| -23.460
| 2<sup>19</sup> • 3<sup>-12</sup>
| [[524288/531441]]
|-
| [[2125764/1953125]]
| 2<sup>2</sup> • 3<sup>12</sup> • 5<sup>-9</sup>
| 146.637
| 5
| -146.637
| 2<sup>-2</sup> • 3<sup>-12</sup> • 5<sup>9</sup>
| [[1953125/2125764]]
|-
| [[10616832/9765625]]
| 2<sup>17</sup> • 3<sup>4</sup> • 5<sup>-10</sup>
| 144.683
| 5
| -144.683
| 2<sup>-17</sup> • 3<sup>-4</sup> • 5<sup>10</sup>
| [[9765625/10616832]]
|-
| [[33554432/31640625]]
| 2<sup>25</sup> • 3<sup>-4</sup> • 5<sup>-8</sup>
| 101.670
| 5
| 5
| -64.519
| -101.670
| 2<sup>12</sup> • 3<sup>-12</sup> • 5<sup>3</sup>
| 2<sup>-25</sup> • 3<sup>4</sup> • 5<sup>8</sup>
| [[512000/531441]]
| [[31640625/33554432]]
|-
| [[53747712/48828125]]
| 2<sup>13</sup> • 3<sup>8</sup> • 5<sup>-11</sup>
| 166.189
| 5
| -166.189
| 2<sup>-13</sup> • 3<sup>-8</sup> • 5<sup>11</sup>
| [[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.
{| class="wikitable center-all left-2 right-3 right-5 left-6"
|+style=white-space:nowrap|7-limit commas vanishing in 12-tet within three 9-odd-limit intervals
! Ratio
! Factorization
! Cents
! Limit
! - Cents
! 1 / Factorization
! 1 / Ratio
|-
|-
| [[36/35]]
| [[36/35]]
Line 388: Line 219:
| 2<sup>2</sup> • 3<sup>-6</sup> • 5<sup>2</sup> • 7<sup>1</sup>
| 2<sup>2</sup> • 3<sup>-6</sup> • 5<sup>2</sup> • 7<sup>1</sup>
| [[700/729]]
| [[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.
{| class="wikitable center-3"
|+
All MOS series of 5-odd-limit intervals tempering out ratios in 12-tet
! 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>
|-
|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>
|}
== Modes ==
=== Modes of limited transposition ===
{| class="wikitable"
|+
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 ([[User:Contribution/Exploring Selected Modes in 12-EDO#Distinctly tempered out commas|see above]])
|-
|-
| [[1296/1225]]
|2\12
| 2<sup>4</sup> • 3<sup>4</sup> • 5<sup>-2</sup> • 7<sup>-2</sup>
|2 2 2 2 2 2
| 97.541
|None (128/125 for its truncation)
| 7
|-
| -97.541
|3\12
| 2<sup>-4</sup> • 3<sup>-4</sup> • 5<sup>2</sup> • 7<sup>2</sup>
|1 2 1 2 1 2 1 2
| [[1225/1296]]
|648/625
|-
| rowspan="2" |4\12
|3 1 3 1 3 1
| rowspan="2" |128/125
|-
|2 1 1 2 1 1 2 1 1
|-
| rowspan="6" |6\12
|1 2 3 1 2 3
| rowspan="2" |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 ===
 
{| 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"
|+
!Period
!Mode
! colspan="2" |Perfect circle
!Ratio
! colspan="2" |Plagal circle
|-
|1\12
|1
| colspan="5" |Too many (967 perfect circles, 967 plagal circles)
|-
|2\12
|2
| colspan="5" |None
|-
| rowspan="7" |3\12
| rowspan="7" |1 2
| rowspan="7" |3<sup>-4</sup> • 5<sup>4</sup>
|4 5 4 5 4 5 4 5
| rowspan="7" |[[648/625]]
|7 8 7 8 7 8 7 8
| rowspan="7" |3<sup>4</sup> • 5<sup>-4</sup>
|-
|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
|-
| rowspan="18" |4\12
| rowspan="4" |3 1
| rowspan="18" |5<sup>-3</sup>
|3 5 3 5 3 5
| rowspan="18" |[[128/125]]
|7 9 7 9 7 9
| rowspan="18" |5<sup>3</sup>
|-
|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
|-
| rowspan="14" |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
|-
| rowspan="15" |6\12
|1 2 3
| rowspan="2" |3<sup>-4</sup> • 5<sup>4</sup>
|9 4 5 9 4 5
| rowspan="2" |[[648/625]]
|7 8 3 7 8 3
| rowspan="2" |3<sup>4</sup> • 5<sup>-4</sup>
|-
|1 3 2
|4 9 5 4 9 5
|7 3 8 7 3 8
|-
|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>
|8 5 8 5 4 9 4 5
|[[81/80]]
|7 8 3 8 7 4 7 4
|3<sup>4</sup> • 5<sup>-1</sup>
|-
|5<sup>-3</sup>
|8 7 3 7 8 5 5 5
|[[128/125]]
|7 7 7 4 5 9 5 4
|5<sup>3</sup>
|-
| rowspan="3" |3<sup>-4</sup> • 5<sup>-2</sup>
|5 3 5 5 5 3 5 5
| rowspan="3" |[[2048/2025]]
|7 7 9 7 7 7 9 7
| rowspan="3" |3<sup>4</sup> • 5<sup>2</sup>
|-
|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
|-
| rowspan="6" |1 1 2 2
| rowspan="2" |3<sup>-4</sup> • 5<sup>1</sup>
|5 8 8 9 5 4 4 5
| rowspan="2" |[[81/80]]
|7 8 8 7 3 4 4 7
| rowspan="2" |3<sup>4</sup> • 5<sup>-1</sup>
|-
|5 4 4 5 9 8 8 5
|7 4 4 3 7 8 8 7
|-
|3<sup>-4</sup> • 5<sup>4</sup>
|5 4 4 5 5 4 4 5
|[[648/625]]
|7 8 8 7 7 8 8 7
|3<sup>4</sup> • 5<sup>-4</sup>
|-
| rowspan="3" |3<sup>-4</sup> • 5<sup>-2</sup>
|5 8 8 9 9 8 8 5
| rowspan="3" |[[2048/2025]]
|7 4 4 3 3 4 4 7
| rowspan="3" |3<sup>4</sup> • 5<sup>2</sup>
|-
|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
|-
| colspan="2" |1 1 2 1 1
| colspan="4" |Too many (130 perfect circles, 130 plagal circles)
|}
 
=== Modes based on the circle of 3-odd-limit ===
{| class="wikitable"
|+
!Mode
! colspan="2" |Perfect circle
!Ratio
! 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
|-
| colspan="2" |2 2 3 3 2  ;  2 2 2 4 2
| colspan="4" |None
|-
| rowspan="9" |2 2 1 2 2 2 1
| rowspan="9" |3<sup>-4</sup> • 5<sup>1</sup>
|5 4 5 8 5 4 5
| rowspan="9" |[[81/80]]
|7 8 7 4 7 8 7
| rowspan="9" |3<sup>4</sup> • 5<sup>-1</sup>
|-
|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
|-
| rowspan="6" |2 1 2 2 2 2 1
| rowspan="2" |3<sup>-4</sup> • 5<sup>1</sup>
|8 9 9 8 9 9 8
| rowspan="2" |[[81/80]]
|4 3 3 4 3 3 4
| rowspan="2" |3<sup>4</sup> • 5<sup>-1</sup>
|-
|8 9 5 4 5 9 8
|4 3 7 8 7 3 4
|-
| rowspan="4" |3<sup>-4</sup> • 5<sup>4</sup>
|4 4 9 5 4 5 5
| rowspan="4" |[[648/625]]
|7 7 8 7 3 8 8
| rowspan="4" |3<sup>4</sup> • 5<sup>-4</sup>
|-
|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
|-
| rowspan="12" |2 2 1 2 1 3 1
| rowspan="5" |3<sup>-4</sup> • 5<sup>1</sup>
|8 8 9 9 9 8 9
| rowspan="5" |[[81/80]]
|3 4 3 3 3 4 4
| rowspan="5" |3<sup>4</sup> • 5<sup>-1</sup>
|-
|-
| [[2500/2401]]
|5 5 9 9 9 8 3
| 2<sup>2</sup> • 5<sup>4</sup> • 7<sup>-4</sup>
|9 4 3 3 3 7 7
| 69.951
| 7
| -69.951
| 2<sup>-2</sup> • 5<sup>-4</sup> • 7<sup>4</sup>
| [[2401/2500]]
|-
|-
| [[2560/2401]]
|5 8 9 9 9 5 3
| 2<sup>9</sup> • 5<sup>1</sup> • 7<sup>-4</sup>
|9 7 3 3 3 4 7
| 111.010
| 7
| -111.010
| 2<sup>-9</sup> • 5<sup>-1</sup> • 7<sup>4</sup>
| [[2401/2560]]
|-
|-
| [[2592/2401]]
|9 9 5 4 5 8 8
| 2<sup>5</sup> • 3<sup>4</sup> • 7<sup>-4</sup>
|4 4 7 8 7 3 3
| 132.516
| 7
| -132.516
| 2<sup>-5</sup> • 3<sup>-4</sup> • 7<sup>4</sup>
| [[2401/2592]]
|-
|-
| [[3125/3087]]
|8 8 5 9 5 4 9
| 3<sup>-2</sup> • 5<sup>5</sup> • 7<sup>-3</sup>
|3 8 7 3 7 4 4
| 21.181
| 7
| -21.181
| 3<sup>2</sup> • 5<sup>-5</sup> • 7<sup>3</sup>
| [[3087/3125]]
|-
|-
| [[3136/3125]]
| rowspan="2" |5<sup>-3</sup>
| 2<sup>6</sup> • 5<sup>-5</sup> • 7<sup>2</sup>
|7 3 5 5 5 3 8
| 6.083
| rowspan="2" |[[128/125]]
| 7
|4 9 7 7 7 9 5
| -6.083
| rowspan="2" |5<sup>3</sup>
| 2<sup>-6</sup> • 5<sup>5</sup> • 7<sup>-2</sup>
| [[3125/3136]]
|-
|-
| [[3200/3087]]
|8 5 3 3 3 5 9
| 2<sup>7</sup> • 3<sup>-2</sup> • 5<sup>2</sup> • 7<sup>-3</sup>
|3 7 9 9 9 7 4
| 62.240
| 7
| -62.240
| 2<sup>-7</sup> • 3<sup>2</sup> • 5<sup>-2</sup> • 7<sup>3</sup>
| [[3087/3200]]
|-
|-
| [[3645/3584]]
| rowspan="5" |3<sup>-4</sup> • 5<sup>4</sup>
| 2<sup>-9</sup> • 3<sup>6</sup> • 5<sup>1</sup> • 7<sup>-1</sup>
|4 4 9 9 9 8 5
| 29.218
| rowspan="5" |[[648/625]]
| 7
|7 4 3 3 3 8 8
| -29.218
| rowspan="5" |3<sup>4</sup> • 5<sup>-4</sup>
| 2<sup>9</sup> • 3<sup>-6</sup> • 5<sup>-1</sup> • 7<sup>1</sup>
| [[3584/3645]]
|-
|-
| [[4000/3969]]
|4 7 9 9 9 5 5
| 2<sup>5</sup> • 3<sup>-4</sup> • 5<sup>3</sup> • 7<sup>-2</sup>
|7 7 3 3 3 5 8
| 13.469
| 7
| -13.469
| 2<sup>-5</sup> • 3<sup>4</sup> • 5<sup>-3</sup> • 7<sup>2</sup>
| [[3969/4000]]
|-
|-
| [[4096/3969]]
|5 9 5 4 5 4 4
| 2<sup>12</sup> • 3<sup>-4</sup> • 7<sup>-2</sup>
|8 8 7 8 7 3 7
| 54.528
| 7
| -54.528
| 2<sup>-12</sup> • 3<sup>4</sup> • 7<sup>2</sup>
| [[3969/4096]]
|-
|-
| [[4608/4375]]
|5 9 5 9 7 9 4
| 2<sup>9</sup> • 3<sup>2</sup> • 5<sup>-4</sup> • 7<sup>-1</sup>
|8 3 5 3 7 3 7
| 89.829
| 7
| -89.829
| 2<sup>-9</sup> • 3<sup>-2</sup> • 5<sup>4</sup> • 7<sup>1</sup>
| [[4375/4608]]
|-
|-
| [[5103/5000]]
|4 5 9 9 8 9 4
| 2<sup>-3</sup> • 3<sup>6</sup> • 5<sup>-4</sup> • 7<sup>1</sup>
|8 3 4 3 3 7 8
| 35.301
| 7
| -35.301
| 2<sup>3</sup> • 3<sup>-6</sup> • 5<sup>4</sup> • 7<sup>-1</sup>
| [[5000/5103]]
|-
|-
| [[5120/5103]]
| rowspan="12" |2 1 2 2 1 3 1
| 2<sup>10</sup> • 3<sup>-6</sup> • 5<sup>1</sup> • 7<sup>-1</sup>
| rowspan="5" |3<sup>-4</sup> • 5<sup>1</sup>
| 5.758
|9 8 9 9 9 8 8
| 7
| rowspan="5" |[[81/80]]
| -5.758
|4 4 3 3 3 4 3
| 2<sup>-10</sup> • 3<sup>6</sup> • 5<sup>-1</sup> • 7<sup>1</sup>
| rowspan="5" |3<sup>4</sup> • 5<sup>-1</sup>
| [[5103/5120]]
|-
|-
| [[5625/5488]]
|3 8 9 9 9 5 5
| 2<sup>-4</sup> • 3<sup>2</sup> • 5<sup>4</sup> • 7<sup>-3</sup>
|7 7 3 3 3 4 9
| 42.687
| 7
| -42.687
| 2<sup>4</sup> • 3<sup>-2</sup> • 5<sup>-4</sup> • 7<sup>3</sup>
| [[5488/5625]]
|-
|-
| [[6561/6125]]
|3 5 9 9 9 8 5
| 3<sup>8</sup> • 5<sup>-3</sup> • 7<sup>-2</sup>
|7 4 3 3 3 7 9
| 119.047
| 7
| -119.047
| 3<sup>-8</sup> • 5<sup>3</sup> • 7<sup>2</sup>
| [[6125/6561]]
|-
|-
| [[6561/6272]]
|8 8 5 4 5 9 9
| 2<sup>-7</sup> • 3<sup>8</sup> • 7<sup>-2</sup>
|3 3 7 8 7 4 4
| 77.988
| 7
| -77.988
| 2<sup>7</sup> • 3<sup>-8</sup> • 7<sup>2</sup>
| [[6272/6561]]
|-
|-
| [[8192/7875]]
|9 4 5 9 5 8 8
| 2<sup>13</sup> • 3<sup>-2</sup> • 5<sup>-3</sup> • 7<sup>-1</sup>
|4 4 7 3 7 8 3
| 68.323
| 7
| -68.323
| 2<sup>-13</sup> • 3<sup>2</sup> • 5<sup>3</sup> • 7<sup>1</sup>
| [[7875/8192]]
|-
|-
| [[9216/8575]]
| rowspan="2" |5<sup>-3</sup>
| 2<sup>10</sup> • 3<sup>2</sup> • 5<sup>-2</sup> • 7<sup>-3</sup>
|8 3 5 5 5 3 7
| 124.805
| rowspan="2" |[[128/125]]
| 7
|5 9 7 7 7 9 4
| -124.805
| rowspan="2" |5<sup>3</sup>
| 2<sup>-10</sup> • 3<sup>-2</sup> • 5<sup>2</sup> • 7<sup>3</sup>
| [[8575/9216]]
|-
|-
| [[10125/9604]]
|9 5 3 3 3 5 8
| 2<sup>-2</sup> • 3<sup>4</sup> • 5<sup>3</sup> • 7<sup>-4</sup>
|4 7 9 9 9 7 3
| 91.458
| 7
| -91.458
| 2<sup>2</sup> • 3<sup>-4</sup> • 5<sup>-3</sup> • 7<sup>4</sup>
| [[9604/10125]]
|-
|-
| [[13122/12005]]
| rowspan="5" |3<sup>-4</sup> • 5<sup>4</sup>
| 2<sup>1</sup> • 3<sup>8</sup> • 5<sup>-1</sup> • 7<sup>-4</sup>
|5 8 9 9 9 4 4
| 154.023
| rowspan="5" |[[648/625]]
| 7
|8 8 3 3 3 4 7
| -154.023
| rowspan="5" |3<sup>4</sup> • 5<sup>-4</sup>
| 2<sup>-1</sup> • 3<sup>-8</sup> • 5<sup>1</sup> • 7<sup>4</sup>
| [[12005/13122]]
|-
|-
| [[15876/15625]]
|5 5 9 9 9 7 4
| 2<sup>2</sup> • 3<sup>4</sup> • 5<sup>-6</sup> • 7<sup>2</sup>
|8 5 3 3 3 7 7
| 27.590
| 7
| -27.590
| 2<sup>-2</sup> • 3<sup>-4</sup> • 5<sup>6</sup> • 7<sup>-2</sup>
| [[15625/15876]]
|-
|-
| [[16128/15625]]
|4 4 5 4 5 9 5
| 2<sup>8</sup> • 3<sup>2</sup> • 5<sup>-6</sup> • 7<sup>1</sup>
|7 3 7 8 7 8 8
| 54.854
| 7
| -54.854
| 2<sup>-8</sup> • 3<sup>-2</sup> • 5<sup>6</sup> • 7<sup>-1</sup>
| [[15625/16128]]
|-
|-
| [[16384/15435]]
|4 9 7 9 5 9 5
| 2<sup>14</sup> • 3<sup>-2</sup> • 5<sup>-1</sup> • 7<sup>-3</sup>
|7 3 7 3 5 3 8
| 103.299
| 7
| -103.299
| 2<sup>-14</sup> • 3<sup>2</sup> • 5<sup>1</sup> • 7<sup>3</sup>
| [[15435/16384]]
|-
|-
| [[18000/16807]]
|4 9 8 9 9 5 4
| 2<sup>4</sup> • 3<sup>2</sup> • 5<sup>3</sup> • 7<sup>-5</sup>
|8 7 3 3 4 3 8
| 118.722
| 7
| -118.722
| 2<sup>-4</sup> • 3<sup>-2</sup> • 5<sup>-3</sup> • 7<sup>5</sup>
| [[16807/18000]]
|-
|-
| [[18225/16807]]
| rowspan="2" |1 3 1 2 2 2 1
| 3<sup>6</sup> • 5<sup>2</sup> • 7<sup>-5</sup>
| rowspan="2" |5<sup>-3</sup>
| 140.228
|8 7 7 8 5 5 8
| 7
| rowspan="2" |[[128/125]]
| -140.228
|4 7 7 4 5 5 4
| 3<sup>-6</sup> • 5<sup>-2</sup> • 7<sup>5</sup>
| rowspan="2" |5<sup>3</sup>
| [[16807/18225]]
|-
|-
| [[18432/16807]]
|8 3 7 8 5 9 8
| 2<sup>11</sup> • 3<sup>2</sup> • 7<sup>-5</sup>
|4 3 7 4 5 9 4
| 159.780
| 7
| -159.780
| 2<sup>-11</sup> • 3<sup>-2</sup> • 7<sup>5</sup>
| [[16807/18432]]
|-
|-
| [[23328/21875]]
|1 2 2 2 2 2 1
| 2<sup>5</sup> • 3<sup>6</sup> • 5<sup>-5</sup> • 7<sup>-1</sup>
| colspan="5" |None
| 111.336
| 7
| -111.336
| 2<sup>-5</sup> • 3<sup>-6</sup> • 5<sup>5</sup> • 7<sup>1</sup>
| [[21875/23328]]
|-
|-
| [[28672/28125]]
| rowspan="6" |1 3 1 2 1 3 1
| 2<sup>12</sup> • 3<sup>-2</sup> • 5<sup>-5</sup> • 7<sup>1</sup>
| rowspan="6" |5<sup>-3</sup>
| 33.347
|5 8 7 8 7 8 5
| 7
| rowspan="6" |[[128/125]]
| -33.347
|7 4 5 4 5 4 7
| 2<sup>-12</sup> • 3<sup>2</sup> • 5<sup>5</sup> • 7<sup>-1</sup>
| rowspan="6" |5<sup>3</sup>
| [[28125/28672]]
|-
|-
| [[32768/30625]]
|7 8 5 8 9 8 3
| 2<sup>15</sup> • 5<sup>-4</sup> • 7<sup>-2</sup>
|9 4 3 4 7 4 5
| 117.093
| 7
| -117.093
| 2<sup>-15</sup> • 5<sup>4</sup> • 7<sup>2</sup>
| [[30625/32768]]
|-
|-
| [[46656/42875]]
|3 8 9 8 5 8 7
| 2<sup>6</sup> • 3<sup>6</sup> • 5<sup>-3</sup> • 7<sup>-3</sup>
|5 4 7 4 3 4 9
| 146.311
| 7
| -146.311
| 2<sup>-6</sup> • 3<sup>-6</sup> • 5<sup>3</sup> • 7<sup>3</sup>
| [[42875/46656]]
|-
|-
| [[50625/50176]]
|5 8 3 4 3 8 5
| 2<sup>-10</sup> • 3<sup>4</sup> • 5<sup>4</sup> • 7<sup>-2</sup>
|7 4 9 8 9 4 7
| 15.423
| 7
| -15.423
| 2<sup>10</sup> • 3<sup>-4</sup> • 5<sup>-4</sup> • 7<sup>2</sup>
| [[50176/50625]]
|-
|-
| [[59049/54880]]
|5 5 8 7 3 5 3
| 2<sup>-5</sup> • 3<sup>10</sup> • 5<sup>-1</sup> • 7<sup>-3</sup>
|9 7 9 5 4 7 7
| 126.759
| 7
| -126.759
| 2<sup>5</sup> • 3<sup>-10</sup> • 5<sup>1</sup> • 7<sup>3</sup>
| [[54880/59049]]
|-
|-
| [[59049/56000]]
|3 5 3 7 8 5 5
| 2<sup>-6</sup> • 3<sup>10</sup> • 5<sup>-3</sup> • 7<sup>-1</sup>
|7 7 4 5 9 7 9
| 91.783
| 7
| -91.783
| 2<sup>6</sup> • 3<sup>-10</sup> • 5<sup>3</sup> • 7<sup>1</sup>
| [[56000/59049]]
|-
|-
| [[59049/57344]]
| rowspan="2" |1 2 2 2 1 3 1
| 2<sup>-13</sup> • 3<sup>10</sup> • 7<sup>-1</sup>
| rowspan="2" |5<sup>-3</sup>
| 50.724
|8 5 5 8 7 7 8
| 7
| rowspan="2" |[[128/125]]
| -50.724
|4 5 5 4 7 7 4
| 2<sup>13</sup> • 3<sup>-10</sup> • 7<sup>1</sup>
| rowspan="2" |5<sup>3</sup>
| [[57344/59049]]
|-
|-
| [[65536/60025]]
|8 9 5 8 7 3 8
| 2<sup>16</sup> • 5<sup>-2</sup> • 7<sup>-4</sup>
|4 9 5 4 7 3 4
| 152.069
| 7
| -152.069
| 2<sup>-16</sup> • 5<sup>2</sup> • 7<sup>4</sup>
| [[60025/65536]]
|-
|-
| [[78125/76832]]
| colspan="2" |2 2 1 2 1 1 2 1
| 2<sup>-5</sup> • 5<sup>7</sup> • 7<sup>-4</sup>
| colspan="4" |Too many (41 perfect circles, 41 plagal circles)
| 28.892
| 7
| -28.892
| 2<sup>5</sup> • 5<sup>-7</sup> • 7<sup>4</sup>
| [[76832/78125]]
|-
|-
| [[81648/78125]]
| colspan="2" |2 2 1 1 1 2 1 1 1
| 2<sup>4</sup> • 3<sup>6</sup> • 5<sup>-7</sup> • 7<sup>1</sup>
| colspan="4" |Too many (70 perfect circles, 70 plagal circles)
| 76.360
|}
| 7
 
| -76.360
=== Pajara and Blues scales ===
| 2<sup>-4</sup> • 3<sup>-6</sup> • 5<sup>7</sup> • 7<sup>-1</sup>
{| class="wikitable"
| [[78125/81648]]
|+
!Mode
! colspan="2" |Perfect circle
!Ratio
! colspan="2" |Plagal circle
|-
|-
| [[93312/84035]]
| rowspan="2" |2 2 3 2 3
| 2<sup>7</sup> • 3<sup>6</sup> • 5<sup>-1</sup> • 7<sup>-5</sup>
| rowspan="2" |3<sup>-4</sup> • 5<sup>1</sup>
| 181.287
|5 5 4 5 5
| 7
| rowspan="2" |[[81/80]]
| -181.287
|7 7 8 7 7
| 2<sup>-7</sup> • 3<sup>-6</sup> • 5<sup>1</sup> • 7<sup>5</sup>
| rowspan="2" |3<sup>4</sup> • 5<sup>-1</sup>
| [[84035/93312]]
|-
|-
| [[91125/87808]]
|5 9 8 9 5
| 2<sup>-8</sup> • 3<sup>6</sup> • 5<sup>3</sup> • 7<sup>-3</sup>
|7 3 4 3 7
| 64.193
| 7
| -64.193
| 2<sup>8</sup> • 3<sup>-6</sup> • 5<sup>-3</sup> • 7<sup>3</sup>
| [[87808/91125]]
|-
|-
| [[118098/109375]]
|2 1 1 3 2 3
| 2<sup>1</sup> • 3<sup>10</sup> • 5<sup>-6</sup> • 7<sup>-1</sup>
| rowspan="2" |3<sup>-4</sup> • 5<sup>1</sup>
| 132.842
|5 5 5 8 9 4
| 7
| rowspan="2" |[[81/80]]
| -132.842
|8 3 4 7 7 7
| 2<sup>-1</sup> • 3<sup>-10</sup> • 5<sup>6</sup> • 7<sup>1</sup>
| rowspan="2" |3<sup>4</sup> • 5<sup>-1</sup>
| [[109375/118098]]
|-
|-
| [[125000/117649]]
|1 1 2 3 2 3
| 2<sup>3</sup> • 5<sup>6</sup> • 7<sup>-6</sup>
|4 9 8 5 5 5
| 104.927
|7 7 7 4 3 8
| 7
| -104.927
| 2<sup>-3</sup> • 5<sup>-6</sup> • 7<sup>6</sup>
| [[117649/125000]]
|-
|-
| [[128000/117649]]
| rowspan="2" |2 2 3 1 1 3
| 2<sup>10</sup> • 5<sup>3</sup> • 7<sup>-6</sup>
|3<sup>-4</sup> • 5<sup>4</sup>
| 145.986
|5 9 4 4 9 5
| 7
|[[648/625]]
| -145.986
|7 3 8 8 3 7
| 2<sup>-10</sup> • 5<sup>-3</sup> • 7<sup>6</sup>
|3<sup>4</sup> • 5<sup>-4</sup>
| [[117649/128000]]
|-
|-
| [[129600/117649]]
|3<sup>-4</sup> • 5<sup>-2</sup>
| 2<sup>6</sup> • 3<sup>4</sup> • 5<sup>2</sup> • 7<sup>-6</sup>
|5 5 8 8 5 5
| 167.492
|[[2048/2025]]
| 7
|7 7 4 4 7 7
| -167.492
|3<sup>4</sup> • 5<sup>2</sup>
| 2<sup>-6</sup> • 3<sup>-4</sup> • 5<sup>-2</sup> • 7<sup>6</sup>
| [[117649/129600]]
|-
|-
| [[131072/117649]]
|1 1 1 1 3 2 3
| 2<sup>17</sup> • 7<sup>-6</sup>
|3<sup>-4</sup> • 5<sup>1</sup>
| 187.045
|5 8 9 4 9 8 5
| 7
|[[81/80]]
| -187.045
|7 4 3 8 3 4 7
| 2<sup>-17</sup> • 7<sup>6</sup>
|3<sup>4</sup> • 5<sup>-1</sup>
| [[117649/131072]]
|-
|-
| [[131220/117649]]
| rowspan="5" |2 1 1 3 1 1 3
| 2<sup>2</sup> • 3<sup>8</sup> • 5<sup>1</sup> • 7<sup>-6</sup>
|3<sup>-4</sup> • 5<sup>1</sup>
| 188.998
|5 5 5 8 5 4 4
| 7
|[[81/80]]
| -188.998
|8 8 7 4 7 7 7
| 2<sup>-2</sup> • 3<sup>-8</sup> • 5<sup>-1</sup> • 7<sup>6</sup>
|3<sup>4</sup> • 5<sup>-1</sup>
| [[117649/131220]]
|-
|-
| [[131072/127575]]
|3<sup>-4</sup> • 5<sup>4</sup>
| 2<sup>17</sup> • 3<sup>-6</sup> • 5<sup>-2</sup> • 7<sup>-1</sup>
|5 9 4 7 9 9 5
| 46.817
|[[648/625]]
| 7
|7 3 3 5 8 3 7
| -46.817
|3<sup>4</sup> • 5<sup>-4</sup>
| 2<sup>-17</sup> • 3<sup>6</sup> • 5<sup>2</sup> • 7<sup>1</sup>
| [[127575/131072]]
|-
|-
| [[140625/134456]]
| rowspan="3" |3<sup>-4</sup> • 5<sup>-2</sup>
| 2<sup>-3</sup> • 3<sup>2</sup> • 5<sup>6</sup> • 7<sup>-5</sup>
|5 5 3 5 8 5 5
| 77.663
| rowspan="3" |[[2048/2025]]
| 7
|7 7 4 7 9 7 7
| -77.663
| rowspan="3" |3<sup>4</sup> • 5<sup>2</sup>
| 2<sup>3</sup> • 3<sup>-2</sup> • 5<sup>-6</sup> • 7<sup>5</sup>
| [[134456/140625]]
|-
|-
| [[156250/151263]]
|5 5 8 9 8 8 5
| 2<sup>1</sup> • 3<sup>-2</sup> • 5<sup>7</sup> • 7<sup>-5</sup>
|7 4 4 3 4 7 7
| 56.156
| 7
| -56.156
| 2<sup>-1</sup> • 3<sup>2</sup> • 5<sup>-7</sup> • 7<sup>5</sup>
| [[151263/156250]]
|-
|-
| [[160000/151263]]
|5 8 8 9 5 5 8
| 2<sup>8</sup> • 3<sup>-2</sup> • 5<sup>4</sup> • 7<sup>-5</sup>
|4 7 7 3 4 4 7
| 97.215
| 7
| -97.215
| 2<sup>-8</sup> • 3<sup>2</sup> • 5<sup>-4</sup> • 7<sup>5</sup>
| [[151263/160000]]
|-
|-
| [[163840/151263]]
| rowspan="5" |1 1 2 3 1 1 3
| 2<sup>15</sup> • 3<sup>-2</sup> • 5<sup>1</sup> • 7<sup>-5</sup>
|3<sup>-4</sup> • 5<sup>1</sup>
| 138.274
|4 4 5 8 5 5 5
| 7
|[[81/80]]
| -138.274
|7 7 7 4 7 8 8
| 2<sup>-15</sup> • 3<sup>2</sup> • 5<sup>-1</sup> • 7<sup>5</sup>
|3<sup>4</sup> • 5<sup>-1</sup>
| [[151263/163840]]
|-
|-
| [[164025/153664]]
|3<sup>-4</sup> • 5<sup>4</sup>
| 2<sup>-6</sup> • 3<sup>8</sup> • 5<sup>2</sup> • 7<sup>-4</sup>
|5 9 9 7 4 9 5
| 112.964
|[[648/625]]
| 7
|7 3 8 5 3 3 7
| -112.964
|3<sup>4</sup> • 5<sup>-4</sup>
| 2<sup>6</sup> • 3<sup>-8</sup> • 5<sup>-2</sup> • 7<sup>4</sup>
| [[153664/164025]]
|-
|-
| [[165888/153125]]
| rowspan="3" |3<sup>-4</sup> • 5<sup>-2</sup>
| 2<sup>11</sup> • 3<sup>4</sup> • 5<sup>-5</sup> • 7<sup>-2</sup>
|5 5 8 5 3 5 5
| 138.600
| rowspan="3" |[[2048/2025]]
| 7
|7 7 9 7 4 7 7
| -138.600
| rowspan="3" |3<sup>4</sup> • 5<sup>2</sup>
| 2<sup>-11</sup> • 3<sup>-4</sup> • 5<sup>5</sup> • 7<sup>2</sup>
| [[153125/165888]]
|-
|-
| [[200000/194481]]
|5 8 8 9 8 5 5
| 2<sup>6</sup> • 3<sup>-4</sup> • 5<sup>5</sup> • 7<sup>-4</sup>
|7 7 4 3 4 4 7
| 48.445
| 7
| -48.445
| 2<sup>-6</sup> • 3<sup>4</sup> • 5<sup>-5</sup> • 7<sup>4</sup>
| [[194481/200000]]
|-
|-
| [[204800/194481]]
|8 5 5 9 8 8 5
| 2<sup>13</sup> • 3<sup>-4</sup> • 5<sup>2</sup> • 7<sup>-4</sup>
|7 4 4 3 7 7 4
| 89.504
| 7
| -89.504
| 2<sup>-13</sup> • 3<sup>4</sup> • 5<sup>-2</sup> • 7<sup>4</sup>
| [[194481/204800]]
|-
|-
| [[236196/214375]]
| colspan="2" |1 1 1 1 1 2 1 1 1 2
| 2<sup>2</sup> • 3<sup>10</sup> • 5<sup>-4</sup> • 7<sup>-3</sup>
| colspan="4" |Too many (356 perfect circles, 356 plagal circles)
| 167.817
| 7
| -167.817
| 2<sup>-2</sup> • 3<sup>-10</sup> • 5<sup>4</sup> • 7<sup>3</sup>
| [[214375/236196]]
|-
|-
| [[253125/235298]]
| rowspan="9" |1 1 1 1 3 1 1 3
| 2<sup>-1</sup> • 3<sup>4</sup> • 5<sup>5</sup> • 7<sup>-6</sup>
| rowspan="2" |3<sup>-4</sup> • 5<sup>1</sup>
| 126.433
|5 8 9 4 4 5 8 5
| 7
| rowspan="2" |[[81/80]]
| -126.433
|7 4 7 8 8 3 4 7
| 2<sup>1</sup> • 3<sup>-4</sup> • 5<sup>-5</sup> • 7<sup>6</sup>
| rowspan="2" |3<sup>4</sup> • 5<sup>-1</sup>
| [[235298/253125]]
|-
|-
| [[250047/250000]]
|5 8 5 4 4 9 8 5
| 2<sup>-4</sup> • 3<sup>6</sup> • 5<sup>-6</sup> • 7<sup>3</sup>
|7 4 3 8 8 7 4 7
| 0.325
| 7
| -0.325
| 2<sup>4</sup> • 3<sup>-6</sup> • 5<sup>6</sup> • 7<sup>-3</sup>
| [[250000/250047]]
|-
|-
| [[256000/250047]]
|3<sup>-4</sup> • 5<sup>4</sup>
| 2<sup>11</sup> • 3<sup>-6</sup> • 5<sup>3</sup> • 7<sup>-3</sup>
|5 9 9 7 7 9 9 5
| 40.733
|[[648/625]]
| 7
|7 3 3 5 5 3 3 7
| -40.733
|3<sup>4</sup> • 5<sup>-4</sup>
| 2<sup>-11</sup> • 3<sup>6</sup> • 5<sup>-3</sup> • 7<sup>3</sup>
| [[250047/256000]]
|-
|-
| [[262144/250047]]
| rowspan="6" |3<sup>-4</sup> • 5<sup>-2</sup>
| 2<sup>18</sup> • 3<sup>-6</sup> • 7<sup>-3</sup>
|5 3 5 5 5 5 3 5
| 81.792
| rowspan="6" |[[2048/2025]]
| 7
|7 9 7 7 7 7 9 7
| -81.792
| rowspan="6" |3<sup>4</sup> • 5<sup>2</sup>
| 2<sup>-18</sup> • 3<sup>6</sup> • 7<sup>3</sup>
| [[250047/262144]]
|-
|-
| [[295245/268912]]
|8 9 8 5 5 8 9 8
| 2<sup>-4</sup> • 3<sup>10</sup> • 5<sup>1</sup> • 7<sup>-5</sup>
|4 3 4 7 7 4 3 4
| 161.734
| 7
| -161.734
| 2<sup>4</sup> • 3<sup>-10</sup> • 5<sup>-1</sup> • 7<sup>5</sup>
| [[268912/295245]]
|-
|-
| [[331776/300125]]
|5 3 5 5 5 8 9 8
| 2<sup>12</sup> • 3<sup>4</sup> • 5<sup>-3</sup> • 7<sup>-4</sup>
|4 3 4 7 7 7 9 7
| 173.575
| 7
| -173.575
| 2<sup>-12</sup> • 3<sup>-4</sup> • 5<sup>3</sup> • 7<sup>4</sup>
| [[300125/331776]]
|-
|-
| [[321489/312500]]
|8 9 8 5 5 5 3 5
| 2<sup>-2</sup> • 3<sup>8</sup> • 5<sup>-7</sup> • 7<sup>2</sup>
|7 9 7 7 7 4 3 4
| 49.096
| 7
| -49.096
| 2<sup>2</sup> • 3<sup>-8</sup> • 5<sup>7</sup> • 7<sup>-2</sup>
| [[312500/321489]]
|-
|-
| [[321489/320000]]
|5 5 8 5 5 9 8 3
| 2<sup>-9</sup> • 3<sup>8</sup> • 5<sup>-4</sup> • 7<sup>2</sup>
|9 4 3 7 7 4 7 7
| 8.037
| 7
| -8.037
| 2<sup>9</sup> • 3<sup>-8</sup> • 5<sup>4</sup> • 7<sup>-2</sup>
| [[320000/321489]]
|-
|-
| [[327680/321489]]
|3 8 9 5 5 8 5 5
| 2<sup>16</sup> • 3<sup>-8</sup> • 5<sup>1</sup> • 7<sup>-2</sup>
|7 7 4 7 7 3 4 9
| 33.022
| 7
| -33.022
| 2<sup>-16</sup> • 3<sup>8</sup> • 5<sup>-1</sup> • 7<sup>2</sup>
| [[321489/327680]]
|-
|-
| [[390625/388962]]
| rowspan="12" |1 1 1 2 2 1 2 2
| 2<sup>-1</sup> • 3<sup>-4</sup> • 5<sup>8</sup> • 7<sup>-4</sup>
| rowspan="12" |3<sup>-4</sup> • 5<sup>1</sup>
| 7.386
|5 4 9 8 9 8 9 8
| 7
| rowspan="12" |[[81/80]]
| -7.386
|4 3 4 3 4 3 8 7
| 2<sup>1</sup> • 3<sup>4</sup> • 5<sup>-8</sup> • 7<sup>4</sup>
| rowspan="12" |3<sup>4</sup> • 5<sup>-1</sup>
| [[388962/390625]]
|-
|-
| [[395136/390625]]
|4 5 8 9 8 9 8 9
| 2<sup>7</sup> • 3<sup>2</sup> • 5<sup>-8</sup> • 7<sup>3</sup>
|3 4 3 4 3 4 7 8
| 19.878
| 7
| -19.878
| 2<sup>-7</sup> • 3<sup>-2</sup> • 5<sup>8</sup> • 7<sup>-3</sup>
| [[390625/395136]]
|-
|-
| [[401408/390625]]
|3 5 5 4 5 4 5 5
| 2<sup>13</sup> • 5<sup>-8</sup> • 7<sup>2</sup>
|7 7 8 7 8 7 7 9
| 47.142
| 7
| -47.142
| 2<sup>-13</sup> • 5<sup>8</sup> • 7<sup>-2</sup>
| [[390625/401408]]
|-
|-
| [[413343/390625]]
|3 5 5 9 7 9 5 5
| 3<sup>10</sup> • 5<sup>-8</sup> • 7<sup>1</sup>
|7 7 3 5 3 7 7 9
| 97.866
| 7
| -97.866
| 3<sup>-10</sup> • 5<sup>8</sup> • 7<sup>-1</sup>
| [[390625/413343]]
|-
|-
| [[413343/400000]]
|3 5 5 4 9 8 9 5
| 2<sup>-7</sup> • 3<sup>10</sup> • 5<sup>-5</sup> • 7<sup>1</sup>
|7 3 4 3 8 7 7 9
| 56.807
| 7
| -56.807
| 2<sup>7</sup> • 3<sup>-10</sup> • 5<sup>5</sup> • 7<sup>-1</sup>
| [[400000/413343]]
|-
|-
| [[413343/409600]]
|3 5 9 8 9 4 5 5
| 2<sup>-14</sup> • 3<sup>10</sup> • 5<sup>-2</sup> • 7<sup>1</sup>
|7 7 8 3 4 3 7 9
| 15.748
| 7
| -15.748
| 2<sup>14</sup> • 3<sup>-10</sup> • 5<sup>2</sup> • 7<sup>-1</sup>
| [[409600/413343]]
|-
|-
| [[472392/420175]]
|3 5 9 4 5 8 9 5
| 2<sup>3</sup> • 3<sup>10</sup> • 5<sup>-2</sup> • 7<sup>-5</sup>
|7 3 4 7 8 3 7 9
| 202.793
| 7
| -202.793
| 2<sup>-3</sup> • 3<sup>-10</sup> • 5<sup>2</sup> • 7<sup>5</sup>
| [[420175/472392]]
|-
|-
| [[458752/455625]]
|3 5 9 8 5 4 9 5
| 2<sup>16</sup> • 3<sup>-6</sup> • 5<sup>-4</sup> • 7<sup>1</sup>
|7 3 8 7 4 3 7 9
| 11.841
| 7
| -11.841
| 2<sup>-16</sup> • 3<sup>6</sup> • 5<sup>4</sup> • 7<sup>-1</sup>
| [[455625/458752]]
|-
|-
| [[531441/470596]]
|7 9 5 8 9 8 5 9
| 2<sup>-2</sup> • 3<sup>12</sup> • 7<sup>-6</sup>
|3 7 4 3 4 7 3 5
| 210.505
| 7
| -210.505
| 2<sup>2</sup> • 3<sup>-12</sup> • 7<sup>6</sup>
| [[470596/531441]]
|-
|-
| [[531441/480200]]
|9 8 5 4 5 4 5 8
| 2<sup>-3</sup> • 3<sup>12</sup> • 5<sup>-2</sup> • 7<sup>-4</sup>
|4 7 8 7 8 7 4 3
| 175.529
| 7
| -175.529
| 2<sup>3</sup> • 3<sup>-12</sup> • 5<sup>2</sup> • 7<sup>4</sup>
| [[480200/531441]]
|-
|-
| [[524288/496125]]
|9 4 5 8 5 4 5 8
| 2<sup>19</sup> • 3<sup>-4</sup> • 5<sup>-3</sup> • 7<sup>-2</sup>
|4 7 8 7 4 7 8 3
| 95.587
| 7
| -95.587
| 2<sup>-19</sup> • 3<sup>4</sup> • 5<sup>3</sup> • 7<sup>2</sup>
| [[496125/524288]]
|-
|-
| [[531441/490000]]
|9 8 5 4 5 8 5 4
| 2<sup>-4</sup> • 3<sup>12</sup> • 5<sup>-4</sup> • 7<sup>-2</sup>
|8 7 4 7 8 7 4 3
| 140.553
| 7
| -140.553
| 2<sup>4</sup> • 3<sup>-12</sup> • 5<sup>4</sup> • 7<sup>2</sup>
| [[490000/531441]]
|-
|-
| [[531441/501760]]
|1 1 1 4 1 4
| 2<sup>-11</sup> • 3<sup>12</sup> • 5<sup>-1</sup> • 7<sup>-2</sup>
| colspan="5" |None
| 99.494
| 7
| -99.494
| 2<sup>11</sup> • 3<sup>-12</sup> • 5<sup>1</sup> • 7<sup>2</sup>
| [[501760/531441]]
|}
|}

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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