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Diaschismic family: Difference between revisions

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→Five-limit srutal (aka diaschismic): update keys; +srutal archagall
Update keys and normalize mappings and generators
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[[Mapping]]: [{{val| 2 0 11 }}, {{val| 0 1 -2 }}]
[[Mapping]]: [{{val| 2 0 11 }}, {{val| 0 1 -2 }}]
Mapping generators: ~45/32, ~3


[[Optimal tuning]] ([[POTE]]): ~45/32 = 1\2, ~3/2 = 704.898
[[Optimal tuning]] ([[POTE]]): ~45/32 = 1\2, ~3/2 = 704.898
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Sval mapping: [{{val| 2 0 11 5 }}], {{val| 0 1 -2 1 }}]
Sval mapping: [{{val| 2 0 11 5 }}], {{val| 0 1 -2 1 }}]
Sval mapping generators: ~17/12, ~3


Optimal tuning (CTE): ~17/12 = 1\2, ~3/2 = 705.1272
Optimal tuning (CTE): ~17/12 = 1\2, ~3/2 = 705.1272
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== Srutal ==
== Srutal ==
{{see also| Srutal vs diaschismic }}
{{See also| Srutal vs diaschismic }}


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 2048/2025, 4375/4374
[[Comma list]]: 2048/2025, 4375/4374


[[Mapping]]: [{{val|2 0 11 -42}}, {{val|0 1 -2 15}}]
[[Mapping]]: [{{val| 2 0 11 -42 }}, {{val| 0 1 -2 15 }}]


{{Multival|legend=1| 2 -4 30 -11 42 81 }}
{{Multival|legend=1| 2 -4 30 -11 42 81 }}


[[POTE generator]]: ~3/2 = 704.814
[[Optimal tuning]] ([[POTE]]): ~45/32 = 1\2, ~3/2 = 704.814


[[Tuning ranges]]:  
[[Tuning ranges]]:  
Line 80: Line 80:
Mapping: [{{val| 2 0 11 -42 -28 }}, {{val| 0 1 -2 15 11 }}]
Mapping: [{{val| 2 0 11 -42 -28 }}, {{val| 0 1 -2 15 11 }}]


POTE generator: ~3/2 = 704.856
Optimal tuning (POTE): ~45/32 = 1\2, ~3/2 = 704.856


Tuning ranges:  
Tuning ranges:  
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Mapping: [{{val| 2 0 11 -42 -28 -18 }}, {{val| 0 1 -2 15 11 8 }}]
Mapping: [{{val| 2 0 11 -42 -28 -18 }}, {{val| 0 1 -2 15 11 8 }}]


POTE generator: ~3/2 = 704.881
Optimal tuning (POTE): ~45/32 = 1\2, ~3/2 = 704.881


Tuning ranges:  
Tuning ranges:  
Line 117: Line 117:
Mapping: [{{val| 2 0 11 -42 -28 -18 5 }}, {{val| 0 1 -2 15 11 8 1 }}]
Mapping: [{{val| 2 0 11 -42 -28 -18 5 }}, {{val| 0 1 -2 15 11 8 1 }}]


POTE generator: ~3/2 = 704.840
Optimal tuning (POTE): ~17/12 = 1\2, ~3/2 = 704.840


Tuning ranges:  
Tuning ranges:  
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Mapping: [{{val| 2 0 11 -42 -28 -18 5 -55 }}, {{val| 0 1 -2 15 11 8 1 20 }}]
Mapping: [{{val| 2 0 11 -42 -28 -18 5 -55 }}, {{val| 0 1 -2 15 11 8 1 20 }}]


POTE generator: ~3/2 = 704.905
Optimal tuning (POTE): ~17/12 = 1\2, ~3/2 = 704.905


Optimal GPV sequence: {{Val list| 34dh, 46, 80, 206cd }}
Optimal GPV sequence: {{Val list| 34dh, 46, 80, 206cd }}
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Mapping: [{{val| 2 0 11 -42 -28 -18 5 -55 -10 }}, {{val| 0 1 -2 15 11 8 1 20 6 }}]
Mapping: [{{val| 2 0 11 -42 -28 -18 5 -55 -10 }}, {{val| 0 1 -2 15 11 8 1 20 6 }}]


POTE generator: ~3/2 = 704.899
Optimal tuning (POTE): ~17/12 = 1\2, ~3/2 = 704.899


Optimal GPV sequence: {{Val list| 34dh, 46, 80, 206cd }}
Optimal GPV sequence: {{Val list| 34dh, 46, 80, 206cd }}
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Mapping: [{{val| 2 0 11 -42 -28 -18 5 -55 -10 -76 }}, {{val| 0 1 -2 15 11 8 1 20 6 27 }}]
Mapping: [{{val| 2 0 11 -42 -28 -18 5 -55 -10 -76 }}, {{val| 0 1 -2 15 11 8 1 20 6 27 }}]


POTE generator: ~3/2 = 704.906
Optimal tuning (POTE): ~17/12 = 1\2, ~3/2 = 704.906


Optimal GPV sequence: {{Val list| 34dhj, 46, 80, 206cd }}
Optimal GPV sequence: {{Val list| 34dhj, 46, 80, 206cd }}
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== Pajara ==
== Pajara ==
{{main| Pajara }}
{{Main| Pajara }}


Pajara is closely associated with 22edo (not to mention [[Paul Erlich]]) but other tunings are possible. The 1/2 octave period serves as both a [[10/7]] and a [[7/5]]. Aside from 22edo, 34 with the val {{val| 34 54 79 96 }} and 56 with the val {{val| 56 89 130 158 }} are are interesting alternatives, with more accpetable fifths, and a tetrad which is more clearly a dominant seventh. As such, they are closer to the tuning of 12edo and of common practice Western music in general, while retaining the distictiveness of a sharp fifth.
Pajara is closely associated with 22edo (not to mention [[Paul Erlich]]) but other tunings are possible. The 1/2 octave period serves as both a [[10/7]] and a [[7/5]]. Aside from 22edo, 34 with the val {{val| 34 54 79 96 }} and 56 with the val {{val| 56 89 130 158 }} are are interesting alternatives, with more accpetable fifths, and a tetrad which is more clearly a dominant seventh. As such, they are closer to the tuning of 12edo and of common practice Western music in general, while retaining the distictiveness of a sharp fifth.
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Pajara extends nicely to an 11-limit version, for which the 56 tuning can be used, but a good alternative is to make the major thirds pure by setting the fifth to be 706.843 cents. Now 99/98, 100/99, 176/175 and 896/891 are being tempered out.
Pajara extends nicely to an 11-limit version, for which the 56 tuning can be used, but a good alternative is to make the major thirds pure by setting the fifth to be 706.843 cents. Now 99/98, 100/99, 176/175 and 896/891 are being tempered out.


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 50/49, 64/63
[[Comma list]]: 50/49, 64/63
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{{Multival|legend=1| 2 -4 -4 -11 -12 2 }}
{{Multival|legend=1| 2 -4 -4 -11 -12 2 }}


[[POTE generator]]: ~3/2 = 707.048
[[Optimal tuning]] ([[POTE]]): ~7/5 = 1\2, ~3/2 = 707.048


[[Tuning ranges]]:
[[Tuning ranges]]:
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Mapping: [{{val| 2 0 11 12 26 }}, {{val| 0 1 -2 -2 -6 }}]
Mapping: [{{val| 2 0 11 12 26 }}, {{val| 0 1 -2 -2 -6 }}]


POTE generator: ~3/2 = 706.885
Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 706.885


Tuning ranges:
Tuning ranges:
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Mapping: [{{val| 2 0 11 12 26 1 }}, {{val| 0 1 -2 -2 -6 2 }}]
Mapping: [{{val| 2 0 11 12 26 1 }}, {{val| 0 1 -2 -2 -6 2 }}]


POTE generator: ~3/2 = 708.919
Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 708.919


Optimal GPV sequence: {{Val list| 10e, 12, 22 }}
Optimal GPV sequence: {{Val list| 10e, 12, 22 }}
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Mapping: [{{val| 2 0 11 12 26 1 5 }}, {{val| 0 1 -2 -2 -6 2 1 }}]
Mapping: [{{val| 2 0 11 12 26 1 5 }}, {{val| 0 1 -2 -2 -6 2 1 }}]


POTE generator: ~3/2 = 708.806
Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 708.806


Optimal GPV sequence: {{Val list| 10e, 12, 22 }}
Optimal GPV sequence: {{Val list| 10e, 12, 22 }}
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Mapping: [{{val| 2 0 11 12 26 36 }}, {{val| 0 1 -2 -2 -6 -9 }}]
Mapping: [{{val| 2 0 11 12 26 36 }}, {{val| 0 1 -2 -2 -6 -9 }}]


POTE generator: ~3/2 = 706.133
Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 706.133


Optimal GPV sequence: {{Val list| 12f, 22, 34d }}
Optimal GPV sequence: {{Val list| 12f, 22, 34d }}
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Mapping: [{{val| 2 0 11 12 26 36 5 }}, {{val| 0 1 -2 -2 -6 -9 1 }}]
Mapping: [{{val| 2 0 11 12 26 36 5 }}, {{val| 0 1 -2 -2 -6 -9 1 }}]


POTE generator: ~3/2 = 706.410
Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 706.410


Optimal GPV sequence: {{Val list| 12f, 22, 34d }}
Optimal GPV sequence: {{Val list| 12f, 22, 34d }}
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Mapping: [{{val|2 0 11 12 26 17}}, {{val|0 1 -2 -2 -6 -3}}]
Mapping: [{{val|2 0 11 12 26 17}}, {{val|0 1 -2 -2 -6 -3}}]


POTE generator: ~3/2 = 707.450
Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 707.450


Optimal GPV sequence: {{Val list| 10e, 12f, 22f }}
Optimal GPV sequence: {{Val list| 10e, 12f, 22f }}
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Mapping: [{{val|2 0 11 12 26 17 5}}, {{val|0 1 -2 -2 -6 -3 1}}]
Mapping: [{{val|2 0 11 12 26 17 5}}, {{val|0 1 -2 -2 -6 -3 1}}]


POTE generator: ~3/2 = 707.947
Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 707.947


Optimal GPV sequence: {{Val list| 10e, 12f, 22f }}
Optimal GPV sequence: {{Val list| 10e, 12f, 22f }}
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Mapping: [{{val| 2 0 11 12 -9 }}, {{val| 0 1 -2 -2 5 }}]
Mapping: [{{val| 2 0 11 12 -9 }}, {{val| 0 1 -2 -2 5 }}]


POTE generator: ~3/2 = 709.578
Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 709.578


Tuning ranges:
Tuning ranges:
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Mapping: [{{val| 2 0 11 12 -9 1 }}, {{val| 0 1 -2 -2 5 2 }}]
Mapping: [{{val| 2 0 11 12 -9 1 }}, {{val| 0 1 -2 -2 5 2 }}]


POTE generator: ~3/2 = 710.240
Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 710.240


Optimal GPV sequence: {{Val list| 10, 22, 54f, 76bdff }}
Optimal GPV sequence: {{Val list| 10, 22, 54f, 76bdff }}
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Mapping: [{{val| 2 0 11 12 -9 1 5 }}, {{val| 0 1 -2 -2 5 2 1 }}]
Mapping: [{{val| 2 0 11 12 -9 1 5 }}, {{val| 0 1 -2 -2 5 2 1 }}]


POTE generator: ~3/2 = 710.221
Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 710.221


Optimal GPV sequence: {{Val list| 10, 22, 54f, 76bdff }}
Optimal GPV sequence: {{Val list| 10, 22, 54f, 76bdff }}
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Mapping: [{{val| 2 0 11 12 -9 17 }}, {{val| 0 1 -2 -2 5 -3 }}]
Mapping: [{{val| 2 0 11 12 -9 17 }}, {{val| 0 1 -2 -2 5 -3 }}]


POTE generator ~3/2 = 710.818
Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 710.818


Optimal GPV sequence: {{Val list| 10, 22f, 32f, 54ff }}
Optimal GPV sequence: {{Val list| 10, 22f, 32f, 54ff }}
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Mapping: [{{val| 2 0 11 12 -9 17 5 }}, {{val| 0 1 -2 -2 5 -3 1 }}]
Mapping: [{{val| 2 0 11 12 -9 17 5 }}, {{val| 0 1 -2 -2 5 -3 1 }}]


POTE generator ~3/2 = 710.866
Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 710.866


Optimal GPV sequence: {{Val list| 10, 22f, 32f, 54ff }}
Optimal GPV sequence: {{Val list| 10, 22f, 32f, 54ff }}
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Mapping: [{{val| 2 0 11 12 7 }}, {{val| 0 1 -2 -2 0 }}]
Mapping: [{{val| 2 0 11 12 7 }}, {{val| 0 1 -2 -2 0 }}]


POTE generator: ~3/2 = 705.524
Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 705.524


Optimal GPV sequence: {{Val list| 10, 12, 22e, 34dee }}
Optimal GPV sequence: {{Val list| 10, 12, 22e, 34dee }}
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Mapping: [{{val| 2 0 11 12 7 17 }}, {{val| 0 1 -2 -2 0 -3 }}]
Mapping: [{{val| 2 0 11 12 7 17 }}, {{val| 0 1 -2 -2 0 -3 }}]


POTE generator: ~3/2 = 707.442
Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 707.442


Optimal GPV sequence: {{Val list| 10, 12f, 22ef }}
Optimal GPV sequence: {{Val list| 10, 12f, 22ef }}
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Mapping: [{{val| 2 0 11 12 7 17 5 }}, {{val| 0 1 -2 -2 0 -3 1 }}]
Mapping: [{{val| 2 0 11 12 7 17 5 }}, {{val| 0 1 -2 -2 0 -3 1 }}]


POTE generator: ~3/2 = 708.544
Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 708.544


Optimal GPV sequence: {{Val list| 10, 12f, 22ef }}
Optimal GPV sequence: {{Val list| 10, 12f, 22ef }}
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Mapping: [{{val| 2 1 9 10 8 }}, {{val| 0 2 -4 -4 -1 }}]
Mapping: [{{val| 2 1 9 10 8 }}, {{val| 0 2 -4 -4 -1 }}]


POTE generator: ~11/8 = 546.383
Optimal tuning (POTE): ~7/5 = 1\2, ~11/8 = 546.383


Optimal GPV sequence: {{Val list| 20, 22, 68d, 90d }}
Optimal GPV sequence: {{Val list| 20, 22, 68d, 90d }}
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Mapping: [{{val| 2 0 11 12 -1 }}, {{val| 0 2 -4 -4 5 }}]
Mapping: [{{val| 2 0 11 12 -1 }}, {{val| 0 2 -4 -4 5 }}]


POTE generator: ~64/55 = 246.907
Optimal tuning (POTE): ~7/5 = 1\2, ~55/32 = 953.093


Optimal GPV sequence: {{Val list| 10, 24d, 34d }}
Optimal GPV sequence: {{Val list| 10, 24d, 34d }}
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Mapping: [{{val| 2 0 11 12 -1 9 }}, {{val| 0 2 -4 -4 5 -1 }}]
Mapping: [{{val| 2 0 11 12 -1 9 }}, {{val| 0 2 -4 -4 5 -1 }}]


POTE generator: ~15/13 = 246.926
Optimal tuning (POTE): ~7/5 = 1\2, ~26/15 = 953.074


Optimal GPV sequence: {{Val list| 10, 24d, 34d }}
Optimal GPV sequence: {{Val list| 10, 24d, 34d }}
Line 448: Line 448:
Mapping: [{{val| 2 0 11 12 -1 9 5 }}, {{val| 0 2 -4 -4 5 -1 2 }}]
Mapping: [{{val| 2 0 11 12 -1 9 5 }}, {{val| 0 2 -4 -4 5 -1 2 }}]


POTE generator: ~15/13 = 246.790
Optimal tuning (POTE): ~7/5 = 1\2, ~26/15 = 953.210


Optimal GPV sequence: {{Val list| 10, 24d, 34d }}
Optimal GPV sequence: {{Val list| 10, 24d, 34d }}
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Diaschismic extends naturally to the 17-limit, for which the same tunings may be used, making it one of the most important of the higher limit rank two temperaments. Adding the 11-limit adds the commas 176/175, 896/891 and 441/440. The 13-limit yields 196/195, 351/350, and 364/363; the 17-limit adds 136/135, 221/220, and 442/441. If you want to explore higher limit harmonies, diaschismic is certainly one excellent way to do it; MOS of 34 notes and even more the 46 note MOS will encompass very great deal of it. Of course 46 or 58 equal provide alternatives which in many ways are similar, particularly in the case of 58.
Diaschismic extends naturally to the 17-limit, for which the same tunings may be used, making it one of the most important of the higher limit rank two temperaments. Adding the 11-limit adds the commas 176/175, 896/891 and 441/440. The 13-limit yields 196/195, 351/350, and 364/363; the 17-limit adds 136/135, 221/220, and 442/441. If you want to explore higher limit harmonies, diaschismic is certainly one excellent way to do it; MOS of 34 notes and even more the 46 note MOS will encompass very great deal of it. Of course 46 or 58 equal provide alternatives which in many ways are similar, particularly in the case of 58.


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 126/125, 2048/2025
[[Comma list]]: 126/125, 2048/2025
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{{Multival|legend=1| 2 -4 -16 -11 -31 -26 }}
{{Multival|legend=1| 2 -4 -16 -11 -31 -26 }}


[[POTE generator]]: ~3/2 = 703.681
[[Optimal tuning]] ([[POTE]]): ~45/32 = 1\2, ~3/2 = 703.681


[[Tuning ranges]]:  
[[Tuning ranges]]:  
Line 487: Line 487:
Mapping: [{{val| 2 0 11 31 45 }}, {{val| 0 1 -2 -8 -12 }}]
Mapping: [{{val| 2 0 11 31 45 }}, {{val| 0 1 -2 -8 -12 }}]


POTE generator: ~3/2 = 703.714
Optimal tuning (POTE): ~45/32 = 1\2, ~3/2 = 703.714


Tuning ranges:  
Tuning ranges:  
Line 505: Line 505:
Mapping: [{{val| 2 0 11 31 45 55 }}, {{val| 0 1 -2 -8 -12 -15 }}]
Mapping: [{{val| 2 0 11 31 45 55 }}, {{val| 0 1 -2 -8 -12 -15 }}]


POTE generator: ~3/2 = 703.704
Optimal tuning (POTE): ~45/32 = 1\2, ~3/2 = 703.704


Tuning ranges:  
Tuning ranges:  
Line 524: Line 524:
Mapping: [{{val| 2 0 11 31 45 55 5 }}, {{val| 0 1 -2 -8 -12 -15 1 }}]
Mapping: [{{val| 2 0 11 31 45 55 5 }}, {{val| 0 1 -2 -8 -12 -15 1 }}]


POTE generator: ~3/2 = 703.812
Optimal tuning (POTE): ~17/12 = 1\2, ~3/2 = 703.812


Tuning ranges:  
Tuning ranges:  
Line 540: Line 540:
Subgroup: 2.3.5.7.11.13.17.23
Subgroup: 2.3.5.7.11.13.17.23


[[Comma list]]: 126/125, 136/135, 176/175, 196/195, 231/230, 256/255
Comma list: 126/125, 136/135, 176/175, 196/195, 231/230, 256/255


[[Mapping|Sval mapping]]: [{{val| 2 0 11 31 45 55 5 63 }}, {{val| 0 1 -2 -8 -12 -15 1 -17 }}]
Sval mapping: [{{val| 2 0 11 31 45 55 5 63 }}, {{val| 0 1 -2 -8 -12 -15 1 -17 }}]


[[POTE generator|POL2 generator]]: ~3/2 = 703.870
Optimal tuning (POTE): ~17/12 = 1\2, ~3/2 = 703.870


{{Val list|legend=1| 46, 58i, 104ci }}
{{Val list|legend=1| 46, 58i, 104ci }}
Line 551: Line 551:
Keen adds 875/864 as well as 2240/2187 to the set of commas. It may also be described as the 22&56 temperament. [[78edo]] is a good tuning choice, and remains a good one in the 11-limit, where keen, {{multival| 2 -4 18 -12 … }}, is really more interesting, adding 100/99 and 385/384 to the commas.
Keen adds 875/864 as well as 2240/2187 to the set of commas. It may also be described as the 22&56 temperament. [[78edo]] is a good tuning choice, and remains a good one in the 11-limit, where keen, {{multival| 2 -4 18 -12 … }}, is really more interesting, adding 100/99 and 385/384 to the commas.


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 875/864, 2048/2025
[[Comma list]]: 875/864, 2048/2025


[[Mapping]]: [{{val|2 0 11 -23}}, {{val|0 1 -2 9}}]
[[Mapping]]: [{{val| 2 0 11 -23 }}, {{val| 0 1 -2 9 }}]


{{Multival|legend=1| 2 -4 18 -11 23 53 }}
{{Multival|legend=1| 2 -4 18 -11 23 53 }}


[[POTE generator]]: ~3/2 = 707.571
[[Optimal tuning]] ([[POTE]]): ~45/32 = 1\2, ~3/2 = 707.571


{{Val list|legend=1| 22, 56, 78, 134b, 212b, 290bb }}
{{Val list|legend=1| 22, 56, 78, 134b, 212b, 290bb }}
Line 572: Line 572:
Mapping: [{{val|2 0 11 -23 26}}, {{val|0 1 -2 9 -6}}]
Mapping: [{{val|2 0 11 -23 26}}, {{val|0 1 -2 9 -6}}]


POTE generator: ~3/2 = 707.609
Optimal tuning (POTE): ~45/32 = 1\2, ~3/2 = 707.609


Optimal GPV sequence: {{Val list| 22, 56, 78, 212be, 290bbe }}
Optimal GPV sequence: {{Val list| 22, 56, 78, 212be, 290bbe }}
Line 585: Line 585:
Mapping: [{{val|2 0 11 -23 26 -18}}, {{val|0 1 -2 9 -6 8}}]
Mapping: [{{val|2 0 11 -23 26 -18}}, {{val|0 1 -2 9 -6 8}}]


POTE generator: ~3/2 = 707.167
Optimal tuning (POTE): ~45/32 = 1\2, ~3/2 = 707.167


Optimal GPV sequence: {{Val list| 22f, 34, 56f }}
Optimal GPV sequence: {{Val list| 22f, 34, 56f }}
Line 598: Line 598:
Mapping: [{{val|2 0 11 -23 26 -18 5}}, {{val|0 1 -2 9 -6 8 1}}]
Mapping: [{{val|2 0 11 -23 26 -18 5}}, {{val|0 1 -2 9 -6 8 1}}]


POTE generator: ~3/2 = 707.155
Optimal tuning (POTE): ~17/12 = 1\2, ~3/2 = 707.155


Optimal GPV sequence: {{Val list| 22f, 34, 56f }}
Optimal GPV sequence: {{Val list| 22f, 34, 56f }}
Line 611: Line 611:
Mapping: [{{val|2 0 11 -23 26 36}}, {{val|0 1 -2 9 -6 -9}}]
Mapping: [{{val|2 0 11 -23 26 36}}, {{val|0 1 -2 9 -6 -9}}]


POTE generator: ~3/2 = 707.257
Optimal tuning (POTE): ~45/32 = 1\2, ~3/2 = 707.257


Optimal GPV sequence: {{Val list| 22, 34, 56 }}
Optimal GPV sequence: {{Val list| 22, 34, 56 }}
Line 624: Line 624:
Mapping: [{{val|2 0 11 -23 26 36 5}}, {{val|0 1 -2 9 -6 -9 1}}]
Mapping: [{{val|2 0 11 -23 26 36 5}}, {{val|0 1 -2 9 -6 -9 1}}]


POTE generator: ~3/2 = 707.252
Optimal tuning (POTE): ~17/12 = 1\2, ~3/2 = 707.252


Optimal GPV sequence: {{Val list| 22, 34, 56 }}
Optimal GPV sequence: {{Val list| 22, 34, 56 }}
Line 633: Line 633:
Bidia adds [[3136/3125]] to the commas, splitting the period into 1/4 octave. It may be called the 12&56 temperament.
Bidia adds [[3136/3125]] to the commas, splitting the period into 1/4 octave. It may be called the 12&56 temperament.


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 2048/2025, 3136/3125
[[Comma list]]: 2048/2025, 3136/3125


[[Mapping]]: [{{val|4 0 22 43}}, {{val|0 1 -2 -5}}]
[[Mapping]]: [{{val| 4 0 22 43 }}, {{val| 0 1 -2 -5 }}]


{{Multival|legend=1|4 -8 -20 -22 -43 -24}}
{{Multival|legend=1| 4 -8 -20 -22 -43 -24 }}


[[POTE generator]]: ~3/2 = 705.364
[[Optimal tuning]] ([[POTE]]): ~25/21 = 1\4, ~3/2 = 705.364


{{Val list|legend=1| 12, 56, 68, 80, 148d }}
{{Val list|legend=1| 12, 56, 68, 80, 148d }}
Line 652: Line 652:
Comma list: 176/175, 896/891, 1375/1372
Comma list: 176/175, 896/891, 1375/1372


Mapping: [{{val|4 0 22 43 71}}, {{val|0 1 -2 -5 -9}}]
Mapping: [{{val| 4 0 22 43 71 }}, {{val| 0 1 -2 -5 -9 }}]


POTE generator: ~3/2 = 705.087
Optimal tuning (POTE): ~25/21 = 1\4, ~3/2 = 705.087


Optimal GPV sequence: {{Val list| 12, 68, 80 }}
Optimal GPV sequence: {{Val list| 12, 68, 80 }}
Line 665: Line 665:
Comma list: 176/175, 325/324, 640/637, 896/891
Comma list: 176/175, 325/324, 640/637, 896/891


Mapping: [{{val|4 0 22 43 71 -36}}, {{val|0 1 -2 -5 -9 8 }}]
Mapping: [{{val| 4 0 22 43 71 -36 }}, {{val| 0 1 -2 -5 -9 8 }}]


POTE generator: ~3/2 = 705.301
Optimal tuning (POTE): ~25/21 = 1\4, ~3/2 = 705.301


Optimal GPV sequence: {{Val list| 12, 68, 80, 148d, 228bcd, 376bbcddf }}
Optimal GPV sequence: {{Val list| 12, 68, 80, 148d, 228bcd, 376bbcddf }}
Line 678: Line 678:
Comma list: 136/135, 176/175, 256/255, 325/324, 640/637
Comma list: 136/135, 176/175, 256/255, 325/324, 640/637


Mapping: [{{val|4 0 22 43 71 -36 10 }}, {{val|0 1 -2 -5 -9 8 1 }}]
Mapping: [{{val| 4 0 22 43 71 -36 10 }}, {{val| 0 1 -2 -5 -9 8 1 }}]


POTE generator: ~3/2 = 705.334
Optimal tuning (POTE): ~25/21 = 1\4, ~3/2 = 705.334


Optimal GPV sequence: {{Val list| 12, 68, 80, 148d, 228bcd, 376bbcddf }}
Optimal GPV sequence: {{Val list| 12, 68, 80, 148d, 228bcd, 376bbcddf }}
Line 691: Line 691:
Comma list: 136/135, 176/175, 190/189, 256/255, 325/324, 640/637
Comma list: 136/135, 176/175, 190/189, 256/255, 325/324, 640/637


Mapping: [{{val|4 0 22 43 71 -36 10 17 }}, {{val|0 1 -2 -5 -9 8 1 0 }}]
Mapping: [{{val| 4 0 22 43 71 -36 10 17 }}, {{val| 0 1 -2 -5 -9 8 1 0 }}]


POTE generator: ~3/2 = 705.339
Optimal tuning (POTE): ~19/16 = 1\4, ~3/2 = 705.339


Optimal GPV sequence: {{Val list| 12, 68, 80, 148d, 376bbcddfh }}
Optimal GPV sequence: {{Val list| 12, 68, 80, 148d, 376bbcddfh }}
Line 704: Line 704:
Echidna becomes more interesting when extended to be an 11-limit temperament by adding 176/175, 896/891 or 540/539 to the commas, where the same tunings can be used as before. It then is able to represent the entire 11-limit diamond to within about six cents of error, within a compass of 24 notes. The 28 note 2MOS gives scope for this, and the 36 note MOS much more.
Echidna becomes more interesting when extended to be an 11-limit temperament by adding 176/175, 896/891 or 540/539 to the commas, where the same tunings can be used as before. It then is able to represent the entire 11-limit diamond to within about six cents of error, within a compass of 24 notes. The 28 note 2MOS gives scope for this, and the 36 note MOS much more.


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 1728/1715, 2048/2025
[[Comma list]]: 1728/1715, 2048/2025


[[Mapping]]: [{{val|2 1 9 2}}, {{val|0 3 -6 5}}]
[[Mapping]]: [{{val| 2 1 9 2 }}, {{val| 0 3 -6 5 }}]


{{Multival|legend=1| 6 -12 10 -33 -1 57 }}
{{Multival|legend=1| 6 -12 10 -33 -1 57 }}


[[POTE generator]]: ~9/7 = 434.856
[[Optimal tuning]] ([[POTE]]): ~45/32 = 1\2, ~9/7 = 434.856


{{Val list|legend=1| 22, 58, 80, 138cd, 218cd }}
{{Val list|legend=1| 22, 58, 80, 138cd, 218cd }}
Line 723: Line 723:
Comma list: 176/175, 540/539, 896/891
Comma list: 176/175, 540/539, 896/891


Mapping: [{{val|2 1 9 2 12}}, {{val|0 3 -6 5 -7}}]
Mapping: [{{val| 2 1 9 2 12 }}, {{val| 0 3 -6 5 -7 }}]


POTE generator: ~9/7 = 434.852
Optimal tuning (POTE): ~45/32 = 1\2, ~9/7 = 434.852


Minimax tuning:  
Minimax tuning:  
* 11-odd-limit: ~9/7 = {{monzo|5/12 0 0 1/12 -1/12}}
* 11-odd-limit: ~9/7 = {{monzo| 5/12 0 0 1/12 -1/12 }}
: [{{monzo|1 0 0 0 0}}, {{monzo|7/4 0 0 1/4 -1/4}}, {{monzo|2 0 0 -1/2 1/2}}, {{monzo|37/12 0 0 5/12 -5/12}}, {{monzo|37/12 0 0 -7/12 7/12}}]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 7/4 0 0 1/4 -1/4 }}, {{monzo| 2 0 0 -1/2 1/2 }}, {{monzo| 37/12 0 0 5/12 -5/12 }}, {{monzo| 37/12 0 0 -7/12 7/12 }}]
: Eigenmonzos: 2, 11/7
: Eigenmonzo subgroup: 2.11/7


Optimal GPV sequence: {{Val list| 22, 58, 80, 138cde, 218cde }}
Optimal GPV sequence: {{Val list| 22, 58, 80, 138cde, 218cde }}
Line 741: Line 741:
Comma list: 176/175, 351/350, 364/363, 540/539
Comma list: 176/175, 351/350, 364/363, 540/539


Mapping: [{{val|2 1 9 2 12 19}}, {{val|0 3 -6 5 -7 -16}}]
Mapping: [{{val| 2 1 9 2 12 19 }}, {{val| 0 3 -6 5 -7 -16 }}]


POTE generator: ~9/7 = 434.756
Optimal tuning (POTE): ~45/32 = 1\2, ~9/7 = 434.756


Optimal GPV sequence: {{Val list| 22, 58, 80, 138cde }}
Optimal GPV sequence: {{Val list| 22, 58, 80, 138cde }}
Line 754: Line 754:
Comma list: 136/135, 176/175, 221/220, 256/255, 540/539
Comma list: 136/135, 176/175, 221/220, 256/255, 540/539


Mapping: [{{val|2 1 9 2 12 19 6}}, {{val|0 3 -6 5 -7 -16 3}}]
Mapping: [{{val| 2 1 9 2 12 19 6 }}, {{val| 0 3 -6 5 -7 -16 3 }}]


POTE generator: ~9/7 = 434.816
Optimal tuning (POTE): ~17/12 = 1\2, ~9/7 = 434.816


Optimal GPV sequence: {{Val list| 22, 58, 80, 138cde }}
Optimal GPV sequence: {{Val list| 22, 58, 80, 138cde }}
Line 763: Line 763:


== Echidnic ==
== Echidnic ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 686/675, 1029/1024
[[Comma list]]: 686/675, 1029/1024


[[Mapping]]: [{{val|2 2 7 6}}, {{val|0 3 -6 -1}}]
[[Mapping]]: [{{val| 2 2 7 6 }}, {{val| 0 3 -6 -1 }}]


{{Multival|legend=1| 6 -12 -2 -33 -20 29 }}
{{Multival|legend=1| 6 -12 -2 -33 -20 29 }}


[[POTE generator]]: ~8/7 = 234.492
[[Optimal tuning]] ([[POTE]]): ~45/32 = 1\2, ~8/7 = 234.492


{{Val list|legend=1| 10, 36, 46, 194bcd, 240bcd, 286bcd, 332bccdd }}
{{Val list|legend=1| 10, 36, 46, 194bcd, 240bcd, 286bcd, 332bccdd }}
Line 782: Line 782:
Comma list: 385/384, 441/440, 686/675
Comma list: 385/384, 441/440, 686/675


Mapping: [{{val|2 2 7 6 3}}, {{val|0 3 -6 -1 10}}]
Mapping: [{{val| 2 2 7 6 3 }}, {{val| 0 3 -6 -1 10 }}]


POTE generator: ~8/7 = 235.096
Optimal tuning (POTE): ~45/32 = 1\2, ~8/7 = 235.096


Optimal GPV sequence: {{Val list| 10, 36e, 46, 102, 148, 342bcdd }}
Optimal GPV sequence: {{Val list| 10, 36e, 46, 102, 148, 342bcdd }}
Line 795: Line 795:
Comma list: 91/90, 169/168, 385/384, 441/440
Comma list: 91/90, 169/168, 385/384, 441/440


Mapping: [{{val|2 2 7 6 3 7}}, {{val|0 3 -6 -1 10 1}}]
Mapping: [{{val| 2 2 7 6 3 7 }}, {{val| 0 3 -6 -1 10 1 }}]


POTE generator: ~8/7 = 235.088
Optimal tuning (POTE): ~45/32 = 1\2, ~8/7 = 235.088


Optimal GPV sequence: {{Val list| 10, 46, 102, 148f, 194bcdf }}
Optimal GPV sequence: {{Val list| 10, 46, 102, 148f, 194bcdf }}
Line 808: Line 808:
Comma list: 91/90, 136/135, 154/153, 169/168, 256/255
Comma list: 91/90, 136/135, 154/153, 169/168, 256/255


Mapping: [{{val|2 2 7 6 3 7 7}}, {{val|0 3 -6 -1 10 1 3}}]
Mapping: [{{val| 2 2 7 6 3 7 7 }}, {{val| 0 3 -6 -1 10 1 3 }}]


POTE generator: ~8/7 = 235.088
Optimal tuning (POTE): ~17/12 = 1\2, ~8/7 = 235.088


Optimal GPV sequence: {{Val list| 10, 46, 102, 148f, 194bcdf }}
Optimal GPV sequence: {{Val list| 10, 46, 102, 148f, 194bcdf }}
Line 820: Line 820:


== Shrutar ==
== Shrutar ==
Shrutar adds 245/243 to the commas, and also tempers out 6144/6125. It can also be described as 22&amp;46. Its generator can be taken as either 36/35 or 35/24; the latter is interesting since along with 15/14 and 21/20, it connects opposite sides of a hexany. [[68edo]] makes for a good tuning, but another and excellent choice is a generator of 14<sup>(1/7)</sup>, making 7s just.
Shrutar adds 245/243 to the commas, and also tempers out 6144/6125. It can also be described as 22&amp;46. Its generator can be taken as either 36/35 or 35/24; the latter is interesting since along with 15/14 and 21/20, it connects opposite sides of a hexany. [[68edo]] makes for a good tuning, but another and excellent choice is a generator of 14<sup>(1/7)</sup>, making 7's just.


By adding 121/120 or 176/175 to the commas, shrutar can be extended to the 11-limit, which loses a bit of accuracy, but picks up low-complexity 11-limit harmony, making shrutar quite an interesting 11-limit system. 68, 114 or a 14<sup>(1/7)</sup> generator can again be used as tunings.
By adding 121/120 or 176/175 to the commas, shrutar can be extended to the 11-limit, which loses a bit of accuracy, but picks up low-complexity 11-limit harmony, making shrutar quite an interesting 11-limit system. 68, 114 or a 14<sup>(1/7)</sup> generator can again be used as tunings.


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 245/243, 2048/2025
[[Comma list]]: 245/243, 2048/2025


[[Mapping]]: [{{val| 2 3 5 5 }}, {{val| 0 2 -4 7 }}]
[[Mapping]]: [{{val| 2 1 9 -2 }}, {{val| 0 2 -4 7 }}]


{{Multival|legend=1| 4 -8 14 -22 11 55 }}
{{Multival|legend=1| 4 -8 14 -22 11 55 }}


[[POTE generator]]: ~36/35 = 52.811
[[Optimal tuning]] ([[POTE]]): ~45/32 = 1\2, ~35/24 = 652.811


{{Val list|legend=1| 22, 46, 68, 182b, 250bc }}
{{Val list|legend=1| 22, 46, 68, 182b, 250bc }}
Line 843: Line 843:
Comma list: 121/120, 176/175, 245/243
Comma list: 121/120, 176/175, 245/243


Mapping: [{{val| 2 3 5 5 7 }}, {{val| 0 2 -4 7 -1 }}]
Mapping: [{{val| 2 1 9 -2 8 }}, {{val| 0 2 -4 7 -1 }}]


POTE generator: ~33/32 = 52.680
Optimal tuning (POTE): ~45/32 = 1\2, ~16/11 = 652.680


Optimal GPV sequence: {{Val list| 22, 46, 68, 114, 296bce, 410bce }}
Optimal GPV sequence: {{Val list| 22, 46, 68, 114, 296bce, 410bce }}
Line 856: Line 856:
Comma list: 121/120, 176/175, 196/195, 245/243
Comma list: 121/120, 176/175, 196/195, 245/243


Mapping: [{{val| 2 3 5 5 7 6 }}, {{val| 0 2 -4 7 -1 16 }}]
Mapping: [{{val| 2 1 9 -2 8 -10 }}, {{val| 0 2 -4 7 -1 16 }}]


POTE generator: ~33/32 = 52.654
Optimal tuning (POTE): ~45/32 = 1\2, ~16/11 = 652.654


Optimal GPV sequence: {{Val list| 22f, 24f, 46, 68, 114 }}
Optimal GPV sequence: {{Val list| 22f, 24f, 46, 68, 114 }}
Line 869: Line 869:
Comma list: 121/120, 136/135, 154/153, 176/175, 196/195
Comma list: 121/120, 136/135, 154/153, 176/175, 196/195


Mapping: [{{val| 2 3 5 5 7 6 8 }}, {{val| 0 2 -4 7 -1 16 2 }}]
Mapping: [{{val| 2 1 9 -2 8 -10 6 }}, {{val| 0 2 -4 7 -1 16 2 }}]


POTE generator: ~33/32 = 52.647
Optimal tuning (POTE): ~17/12 = 1\2, ~16/11 = 652.647


Optimal GPV sequence: {{Val list| 22f, 24f, 46, 68, 114 }}
Optimal GPV sequence: {{Val list| 22f, 24f, 46, 68, 114 }}
Line 882: Line 882:
Comma list: 121/120, 136/135, 154/153, 176/175, 196/195, 343/342
Comma list: 121/120, 136/135, 154/153, 176/175, 196/195, 343/342


Mapping: [{{val| 2 3 5 5 7 6 8 7 }}, {{val| 0 2 -4 7 -1 16 2 17 }}]
Mapping: [{{val| 2 1 9 -2 8 -10 6 -10 }}, {{val| 0 2 -4 7 -1 16 2 17 }}]


POTE generator: ~33/32 = 52.730
Optimal tuning (POTE): ~17/12 = 1\2, ~16/11 = 652.730


Optimal GPV sequence: {{Val list| 22fh, 24fh, 46, 68, 114, 182bef }}
Optimal GPV sequence: {{Val list| 22fh, 24fh, 46, 68, 114, 182bef }}
Line 891: Line 891:


== Sruti ==
== Sruti ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 2048/2025, 19683/19600
[[Comma list]]: 2048/2025, 19683/19600
Line 899: Line 899:
{{Multival|legend=1| 4 -8 26 -22 30 83 }}
{{Multival|legend=1| 4 -8 26 -22 30 83 }}


POTE generator: ~175/144 = 351.876
[[Optimal tuning]] ([[POTE]]): ~45/32 = 1\2, ~140/81 = 951.876


{{Val list|legend=1| 24, 34d, 58, 150cd, 208ccdd, 266ccdd }}
{{Val list|legend=1| 24, 34d, 58, 150cd, 208ccdd, 266ccdd }}
Line 912: Line 912:
Mapping: [{{val| 2 0 11 -15 -1 }}, {{val| 0 2 -4 13 5 }}]
Mapping: [{{val| 2 0 11 -15 -1 }}, {{val| 0 2 -4 13 5 }}]


POTE generator: ~11/9 = 351.863
Optimal tuning (POTE): ~45/32 = 1\2, ~121/70 = 951.863


Optimal GPV sequence: {{Val list| 24, 34d, 58 }}
Optimal GPV sequence: {{Val list| 24, 34d, 58 }}
Line 925: Line 925:
Mapping: [{{val| 2 0 11 -15 -1 9 }}, {{val| 0 2 -4 13 5 -1 }}]
Mapping: [{{val| 2 0 11 -15 -1 9 }}, {{val| 0 2 -4 13 5 -1 }}]


POTE generator: ~11/9 = 351.886
Optimal tuning (POTE): ~45/32 = 1\2, ~26/15 = 951.886


Optimal GPV sequence: {{Val list| 24, 34d, 58, 150cdeef, 208ccddeeff }}
Optimal GPV sequence: {{Val list| 24, 34d, 58, 150cdeef, 208ccddeeff }}
Line 938: Line 938:
Mapping: [{{val| 2 0 11 -15 -1 9 5 }}, {{val| 0 2 -4 13 5 -1 2 }}]
Mapping: [{{val| 2 0 11 -15 -1 9 5 }}, {{val| 0 2 -4 13 5 -1 2 }}]


POTE generator: ~11/9 = 351.857
Optimal tuning (POTE): ~17/12 = 1\2, ~26/15 = 951.857


Optimal GPV sequence: {{Val list| 24, 34d, 58 }}
Optimal GPV sequence: {{Val list| 24, 34d, 58 }}
Line 945: Line 945:


== Anguirus ==
== Anguirus ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 49/48, 2048/2025
[[Comma list]]: 49/48, 2048/2025
Line 953: Line 953:
{{Multival|legend=1| 4 -8 2 -22 -8 27 }}
{{Multival|legend=1| 4 -8 2 -22 -8 27 }}


[[POTE generator]]: ~8/7 = 246.979
[[Optimal tuning]] ([[POTE]]): ~45/32 = 1\2, ~7/4 = 953.021


{{Val list|legend=1| 10, 24, 34 }}
{{Val list|legend=1| 10, 24, 34 }}
Line 966: Line 966:
Mapping: [{{val| 2 0 11 4 -1 }}, {{val| 0 2 -4 1 5 }}]
Mapping: [{{val| 2 0 11 4 -1 }}, {{val| 0 2 -4 1 5 }}]


POTE generator: ~8/7 = 247.816
Optimal tuning (POTE): ~45/32 = 1\2, ~7/4 = 952.184


Optimal GPV sequence: {{Val list| 10, 24, 34, 58d, 92de }}
Optimal GPV sequence: {{Val list| 10, 24, 34, 58d, 92de }}
Line 979: Line 979:
Mapping: [{{val| 2 0 11 4 -1 9 }}, {{val| 0 2 -4 1 5 -1 }}]
Mapping: [{{val| 2 0 11 4 -1 9 }}, {{val| 0 2 -4 1 5 -1 }}]


POTE generator: ~8/7 = 247.691
Optimal tuning (POTE): ~45/32 = 1\2, ~7/4 = 952.309


Optimal GPV sequence: {{Val list| 10, 24, 34, 58d, 92ddef }}
Optimal GPV sequence: {{Val list| 10, 24, 34, 58d, 92ddef }}
Line 992: Line 992:
Mapping: [{{val| 2 0 11 4 -1 9 5 }}, {{val| 0 2 -4 1 5 -1 2 }}]
Mapping: [{{val| 2 0 11 4 -1 9 5 }}, {{val| 0 2 -4 1 5 -1 2 }}]


POTE generator: ~8/7 = 247.670
Optimal tuning (POTE): ~17/12 = 1\2, ~7/4 = 952.330


Optimal GPV sequence: {{Val list| 10, 24, 34, 58d, 92ddef }}
Optimal GPV sequence: {{Val list| 10, 24, 34, 58d, 92ddef }}
Line 999: Line 999:


== Shru ==
== Shru ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 392/375, 1323/1280
[[Comma list]]: 392/375, 1323/1280
Line 1,007: Line 1,007:
{{Multival|legend=1| 4 -8 -10 -22 -27 -1 }}
{{Multival|legend=1| 4 -8 -10 -22 -27 -1 }}


[[POTE generator]]: ~64/63 = 50.135
[[Optimal tuning]] ([[POTE]]): ~45/32 = 1\2, ~10/7 = 650.135


{{Val list|legend=1| 2, 22d, 24 }}
{{Val list|legend=1| 2, 22d, 24 }}
Line 1,020: Line 1,020:
Mapping: [{{val| 2 1 9 11 8 }}, {{val| 0 2 -4 -5 -1 }}]
Mapping: [{{val| 2 1 9 11 8 }}, {{val| 0 2 -4 -5 -1 }}]


POTE generator: ~33/32 = 50.130
Optimal tuning (POTE): ~45/32 = 1\2, ~10/7 = 650.130


Optimal GPV sequence: {{Val list| 2, 22d, 24 }}
Optimal GPV sequence: {{Val list| 2, 22d, 24 }}
Line 1,033: Line 1,033:
Mapping: [{{val| 2 1 9 11 8 15 }}, {{val| 0 2 -4 -5 -1 -7 }}]
Mapping: [{{val| 2 1 9 11 8 15 }}, {{val| 0 2 -4 -5 -1 -7 }}]


POTE generator: ~33/32 = 50.535
Optimal tuning (POTE): ~45/32 = 1\2, ~10/7 = 650.535


Optimal GPV sequence: {{Val list| 22df, 24 }}
Optimal GPV sequence: {{Val list| 22df, 24 }}
Line 1,040: Line 1,040:


== Quadrasruta ==
== Quadrasruta ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 2048/2025, 2401/2400
[[Comma list]]: 2048/2025, 2401/2400


[[Mapping]]: [{{val|2 0 11 8}}, {{val|0 4 -8 -3}}]
[[Mapping]]: [{{val| 2 0 11 8 }}, {{val| 0 4 -8 -3 }}]


{{Multival|legend=1| 8 -16 -6 -44 -32 31 }}
{{Multival|legend=1| 8 -16 -6 -44 -32 31 }}


[[POTE generator]]: ~15/14 = 123.784
[[Optimal tuning]] ([[POTE]]): ~45/32 = 1\2, ~21/16 = 476.216


{{Val list|legend=1| 10, 38c, 48c, 58, 68, 126 }}
{{Val list|legend=1| 10, 38c, 48c, 58, 68, 126 }}
Line 1,059: Line 1,059:
Comma list: 176/175, 896/891, 2401/2400
Comma list: 176/175, 896/891, 2401/2400


Mapping: [{{val|2 0 11 8 22}}, {{val|0 4 -8 -3 -19}}]
Mapping: [{{val| 2 0 11 8 22 }}, {{val| 0 4 -8 -3 -19 }}]


POTE generator: ~15/14 = 123.882
Optimal tuning (POTE): ~45/32 = 1\2, ~21/16 = 476.118


Optimal GPV sequence: {{Val list| 58, 126, 184c, 310bccde }}
Optimal GPV sequence: {{Val list| 58, 126, 184c, 310bccde }}
Line 1,072: Line 1,072:
Comma list: 176/175, 196/195, 512/507, 676/675
Comma list: 176/175, 196/195, 512/507, 676/675


Mapping: [{{val|2 0 11 8 22 9}}, {{val|0 4 -8 -3 -19 -2}}]
Mapping: [{{val| 2 0 11 8 22 9 }}, {{val| 0 4 -8 -3 -19 -2 }}]


POTE generator: ~14/13 = 123.901
Optimal tuning (POTE): ~45/32 = 1\2, ~21/16 = 476.099


Optimal GPV sequence: {{Val list| 58, 126f, 184cff }}
Optimal GPV sequence: {{Val list| 58, 126f, 184cff }}
Line 1,085: Line 1,085:
Comma list: 136/135, 170/169, 176/175, 196/195, 256/255
Comma list: 136/135, 170/169, 176/175, 196/195, 256/255


Mapping: [{{val|2 0 11 8 22 9 5}}, {{val|0 4 -8 -3 -19 -2 4}}]
Mapping: [{{val| 2 0 11 8 22 9 5 }}, {{val| 0 4 -8 -3 -19 -2 4 }}]


POTE generator: ~14/13 = 123.838
Optimal tuning (POTE): ~17/12 = 1\2, ~21/16 = 476.162


Optimal GPV sequence: {{Val list| 58, 126f }}
Optimal GPV sequence: {{Val list| 58, 126f }}
Line 1,098: Line 1,098:
Comma list: 243/242, 441/440, 2048/2025
Comma list: 243/242, 441/440, 2048/2025


Mapping: [{{val|2 0 11 8 -1}}, {{val|0 4 -8 -3 10}}]
Mapping: [{{val| 2 0 11 8 -1 }}, {{val| 0 4 -8 -3 10 }}]


POTE generator: ~15/14 = 123.983
Optimal tuning (POTE): ~45/32 = 1\2, ~21/16 = 476.017


Optimal GPV sequence: {{Val list| 10, 38c, 48c, 58 }}
Optimal GPV sequence: {{Val list| 10, 38c, 48c, 58 }}
Line 1,111: Line 1,111:
Comma list: 144/143, 196/195, 243/242, 676/675
Comma list: 144/143, 196/195, 243/242, 676/675


Mapping: [{{val|2 0 11 8 -1 9}}, {{val|0 4 -8 -3 10 -2}}]
Mapping: [{{val| 2 0 11 8 -1 9 }}, {{val| 0 4 -8 -3 10 -2 }}]


POTE generator: ~14/13 = 123.972
Optimal tuning (POTE): ~45/32 = 1\2, ~21/16 = 476.028


Optimal GPV sequence: {{Val list| 10, 38c, 48c, 58 }}
Optimal GPV sequence: {{Val list| 10, 38c, 48c, 58 }}
Line 1,124: Line 1,124:
Comma list: 136/135, 144/143, 170/169, 196/195, 221/220
Comma list: 136/135, 144/143, 170/169, 196/195, 221/220


Mapping: [{{val|2 0 11 8 -1 9 5}}, {{val|0 4 -8 -3 10 -2 4}}]
Mapping: [{{val| 2 0 11 8 -1 9 5 }}, {{val| 0 4 -8 -3 10 -2 4 }}]


POTE generator: ~14/13 = 123.923
Optimal tuning (POTE): ~17/12 = 1\2, ~21/16 = 476.077


Optimal GPV sequence: {{Val list| 10, 38c, 48c, 58, 126eef, 184ceeff }}
Optimal GPV sequence: {{Val list| 10, 38c, 48c, 58, 126eef, 184ceeff }}
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