No-fives subgroup temperaments: Difference between revisions

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"optimal GPV sequence" → "optimal ET sequence", per Talk:Optimal_ET_sequence
Line 14: Line 14:
[[Tp tuning|POL2 generator]]: ~7/6 = 250.385
[[Tp tuning|POL2 generator]]: ~7/6 = 250.385


{{Val list|legend=1| 5, 14, 19, 24, 67d, 91d }}
{{Optimal ET sequence|legend=1| 5, 14, 19, 24, 67d, 91d }}


[[Tp tuning #T2 tuning|RMS error]]: 2.523 cents
[[Tp tuning #T2 tuning|RMS error]]: 2.523 cents
Line 65: Line 65:
[[Tp tuning|POL2 generator]]: ~12/11 = 139.990
[[Tp tuning|POL2 generator]]: ~12/11 = 139.990


{{Val list|legend=1| 17, 43, 60d }}
{{Optimal ET sequence|legend=1| 17, 43, 60d }}


[[Tp tuning #T2 tuning|RMS error]]: 1.752 cents
[[Tp tuning #T2 tuning|RMS error]]: 1.752 cents
Line 84: Line 84:
[[Tp tuning|POL2 generator]]: ~3/2 = 709.321
[[Tp tuning|POL2 generator]]: ~3/2 = 709.321


{{Val list|legend=1| 5, 12, 17, 22, 27, 137bd }}
{{Optimal ET sequence|legend=1| 5, 12, 17, 22, 27, 137bd }}


[[Tp tuning #T2 tuning|RMS error]]: 1.856 cents
[[Tp tuning #T2 tuning|RMS error]]: 1.856 cents
Line 101: Line 101:
[[Tp tuning|POL2 generator]]: ~3/2 = 707.192
[[Tp tuning|POL2 generator]]: ~3/2 = 707.192


{{Val list|legend=1| 5, 12, 17, 39d, 56d }}
{{Optimal ET sequence|legend=1| 5, 12, 17, 39d, 56d }}


[[Tp tuning #T2 tuning|RMS error]]: 1.977 cents
[[Tp tuning #T2 tuning|RMS error]]: 1.977 cents
Line 118: Line 118:
[[Tp tuning|POL2 generator]]: ~3/2 = 706.137
[[Tp tuning|POL2 generator]]: ~3/2 = 706.137


{{Val list|legend=1| 5f, 12f, 17, 22, 39d, 56d }}
{{Optimal ET sequence|legend=1| 5f, 12f, 17, 22, 39d, 56d }}


[[Tp tuning #T2 tuning|RMS error]]: 2.095 cents
[[Tp tuning #T2 tuning|RMS error]]: 2.095 cents
Line 135: Line 135:
[[Tp tuning|POL2 generator]]: ~11/9 = 353.958
[[Tp tuning|POL2 generator]]: ~11/9 = 353.958


{{Val list|legend=1| 7, 10, 17, 44e, 61de, 78de }}
{{Optimal ET sequence|legend=1| 7, 10, 17, 44e, 61de, 78de }}


[[Tp tuning #T2 tuning|RMS error]]: 1.968 cents
[[Tp tuning #T2 tuning|RMS error]]: 1.968 cents
Line 152: Line 152:
[[Tp tuning|POL2 generator]]: ~11/9 = 353.775
[[Tp tuning|POL2 generator]]: ~11/9 = 353.775


{{Val list|legend=1| 7, 10, 17, 44e, 61de, 78de }}
{{Optimal ET sequence|legend=1| 7, 10, 17, 44e, 61de, 78de }}


[[Tp tuning #T2 tuning|RMS error]]: 1.953 cents
[[Tp tuning #T2 tuning|RMS error]]: 1.953 cents
Line 169: Line 169:
[[Tp tuning|POL2 generator]]: ~9/7 = 425.365
[[Tp tuning|POL2 generator]]: ~9/7 = 425.365


{{Val list|legend=1| 14, 17, 31, 48, 79, 189b, 268bd, 347bd }}
{{Optimal ET sequence|legend=1| 14, 17, 31, 48, 79, 189b, 268bd, 347bd }}


[[Tp tuning #T2 tuning|RMS error]]: 1.149 cents
[[Tp tuning #T2 tuning|RMS error]]: 1.149 cents
Line 188: Line 188:
[[Tp tuning|POL2 generator]]: ~9/7 = 425.244
[[Tp tuning|POL2 generator]]: ~9/7 = 425.244


{{Val list|legend=1| 14, 17, 31, 48, 79, 127, 206bde }}
{{Optimal ET sequence|legend=1| 14, 17, 31, 48, 79, 127, 206bde }}


[[Tp tuning #T2 tuning|RMS error]]: 1.099 cents
[[Tp tuning #T2 tuning|RMS error]]: 1.099 cents
Line 205: Line 205:
[[Tp tuning|POL2 generator]]: ~9/7 = 424.457
[[Tp tuning|POL2 generator]]: ~9/7 = 424.457


{{Val list|legend=1| 17, 48f, 65ef, 82d, 147df }}
{{Optimal ET sequence|legend=1| 17, 48f, 65ef, 82d, 147df }}


[[Tp tuning #T2 tuning|RMS error]]: 1.769 cents
[[Tp tuning #T2 tuning|RMS error]]: 1.769 cents
Line 222: Line 222:
[[Tp tuning|POL2 generator]]: ~9/7 = 424.702
[[Tp tuning|POL2 generator]]: ~9/7 = 424.702


{{Val list|legend=1| 14, 17, 31 }}
{{Optimal ET sequence|legend=1| 14, 17, 31 }}


[[Tp tuning #T2 tuning|RMS error]]: 1.290 cents
[[Tp tuning #T2 tuning|RMS error]]: 1.290 cents
Line 239: Line 239:
[[Tp tuning|POL2 generator]]: ~3/2 = 696.544
[[Tp tuning|POL2 generator]]: ~3/2 = 696.544


{{Val list|legend=1| 12, 19, 31, 112b, 143b, 174b }}
{{Optimal ET sequence|legend=1| 12, 19, 31, 112b, 143b, 174b }}


[[Tp tuning #T2 tuning|RMS error]]: 1.226 cents
[[Tp tuning #T2 tuning|RMS error]]: 1.226 cents
Line 260: Line 260:
[[Tp tuning|POL2 generator]]: ~3/2 = 704.721 cents
[[Tp tuning|POL2 generator]]: ~3/2 = 704.721 cents


{{Val list|legend=1| 17, 46, 63 }}
{{Optimal ET sequence|legend=1| 17, 46, 63 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.6202 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.6202 cents
Line 279: Line 279:
[[Tp tuning|POL2 generator]]: ~3/2 = 704.753 cents
[[Tp tuning|POL2 generator]]: ~3/2 = 704.753 cents


{{Val list|legend=1| 17, 46, 63 }}
{{Optimal ET sequence|legend=1| 17, 46, 63 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.6047 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.6047 cents
Line 296: Line 296:
[[Tp tuning|POL2 generator]]: ~3/2 = 704.745 cents
[[Tp tuning|POL2 generator]]: ~3/2 = 704.745 cents


{{Val list|legend=1| 17, 46, 63 }}
{{Optimal ET sequence|legend=1| 17, 46, 63 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.7541 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.7541 cents
Line 313: Line 313:
[[Tp tuning|POL2 generator]]: ~3/2 = 704.729 cents
[[Tp tuning|POL2 generator]]: ~3/2 = 704.729 cents


{{Val list|legend=1| 17, 46, 63 }}
{{Optimal ET sequence|legend=1| 17, 46, 63 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.6265 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.6265 cents
Line 330: Line 330:
[[Tp tuning|POL2 generator]]: ~3/2 = 704.729 cents
[[Tp tuning|POL2 generator]]: ~3/2 = 704.729 cents


{{Val list|legend=1| 17, 46, 63 }}
{{Optimal ET sequence|legend=1| 17, 46, 63 }}


; Music
; Music
Line 355: Line 355:
[[Tp tuning|POL2 generator]]: ~81/56 = 633.525
[[Tp tuning|POL2 generator]]: ~81/56 = 633.525


{{Val list|legend=1| 17, 36, 89, 125, 161, 358, 519b }}
{{Optimal ET sequence|legend=1| 17, 36, 89, 125, 161, 358, 519b }}


[[Tp tuning #T2 tuning|RMS error]]: 0.3519 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.3519 cents
Line 372: Line 372:
[[Tp tuning|POL2 generator]]: ~8/7 = 233.688
[[Tp tuning|POL2 generator]]: ~8/7 = 233.688


{{Val list|legend=1| 5, 21, 26, 31, 36, 77, 113, 190 }}
{{Optimal ET sequence|legend=1| 5, 21, 26, 31, 36, 77, 113, 190 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.3202 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.3202 cents
Line 387: Line 387:
[[Tp tuning|POL2 generator]]: ~8/7 = 233.6044
[[Tp tuning|POL2 generator]]: ~8/7 = 233.6044


{{Val list|legend=1| 10, 26, 36, 154…, 190…, 226…, 262… }}
{{Optimal ET sequence|legend=1| 10, 26, 36, 154…, 190…, 226…, 262… }}


[[Tp tuning #T2 tuning|RMS error]]: 0.5452 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.5452 cents
Line 402: Line 402:
[[Tp tuning|POL2 generator]]: ~8/7 = 233.6155
[[Tp tuning|POL2 generator]]: ~8/7 = 233.6155


{{Val list|legend=1| 10, 26, 36, 154…, 190…, 226… }}
{{Optimal ET sequence|legend=1| 10, 26, 36, 154…, 190…, 226… }}


[[Tp tuning #T2 tuning|RMS error]]: 0.5073 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.5073 cents
Line 419: Line 419:
[[Tp tuning|POL2 generator]]: ~2187/1792 = 351.485
[[Tp tuning|POL2 generator]]: ~2187/1792 = 351.485


{{Val list|legend=1| 7, 17, 41, 58, 99 }}
{{Optimal ET sequence|legend=1| 7, 17, 41, 58, 99 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.2344 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.2344 cents
Line 438: Line 438:
[[Tp tuning|POL2 generator]]: ~11/9 = 351.535
[[Tp tuning|POL2 generator]]: ~11/9 = 351.535


{{Val list|legend=1| 7, 17, 41, 58, 99e }}
{{Optimal ET sequence|legend=1| 7, 17, 41, 58, 99e }}


[[Tp tuning #T2 tuning|RMS error]]: 0.6108 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.6108 cents
Line 455: Line 455:
[[Tp tuning|POL2 generator]]: ~11/9 = 351.691
[[Tp tuning|POL2 generator]]: ~11/9 = 351.691


{{Val list|legend=1| 7, 10, 17, 24, 41, 58 }}
{{Optimal ET sequence|legend=1| 7, 10, 17, 24, 41, 58 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.7167 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.7167 cents
Line 472: Line 472:
[[Tp tuning|POL2 generator]]: ~99/98 = 17.6258
[[Tp tuning|POL2 generator]]: ~99/98 = 17.6258


{{Val list|legend=1| 54, 63, 72, 135, 342, 477, 1089, 1566 }}
{{Optimal ET sequence|legend=1| 54, 63, 72, 135, 342, 477, 1089, 1566 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.0383 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.0383 cents
Line 491: Line 491:
[[Tp tuning|POL2 tuning]]: ~3 = 1903.834, ~7 = 3369.872
[[Tp tuning|POL2 tuning]]: ~3 = 1903.834, ~7 = 3369.872


{{Val list|legend=1| 17, 36, 41, 58, 63, 104 }}
{{Optimal ET sequence|legend=1| 17, 36, 41, 58, 63, 104 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.4149 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.4149 cents
Line 510: Line 510:
[[Tp tuning|POL2 tuning]]: ~3 = 1903.856, ~7 = 3369.907
[[Tp tuning|POL2 tuning]]: ~3 = 1903.856, ~7 = 3369.907


{{Val list|legend=1| 17, 41, 46, 58, 87, 104 }}
{{Optimal ET sequence|legend=1| 17, 41, 46, 58, 87, 104 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.3789 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.3789 cents
Line 529: Line 529:
[[Tp tuning|POL2 generator]]: ~11/9 = 350.525
[[Tp tuning|POL2 generator]]: ~11/9 = 350.525


{{Val list|legend=1| 7, 10, 17, 24, 41, 65, 89, 202, 291, 380 }}
{{Optimal ET sequence|legend=1| 7, 10, 17, 24, 41, 65, 89, 202, 291, 380 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.3021 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.3021 cents
Line 551: Line 551:
[[Tp tuning|POL2 generator]]: ~11/9 = 351.488
[[Tp tuning|POL2 generator]]: ~11/9 = 351.488


{{Val list|legend=1| 7, 10, 17, 24, 41 }}
{{Optimal ET sequence|legend=1| 7, 10, 17, 24, 41 }}


[[Tp tuning #T2 tuning|RMS error]]: 0.7038 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.7038 cents
Line 568: Line 568:
[[Tp tuning|POL2 generator]]: ~4/3 = 494.509
[[Tp tuning|POL2 generator]]: ~4/3 = 494.509


{{Val list|legend=1| 5, 12, 17 }}
{{Optimal ET sequence|legend=1| 5, 12, 17 }}


=== 2.3.7.41 subgroup ===
=== 2.3.7.41 subgroup ===
Line 583: Line 583:
[[TOP tuning|TOP generator]]s: ~2 = 1197.2342, ~4/3 = 488.9029
[[TOP tuning|TOP generator]]s: ~2 = 1197.2342, ~4/3 = 488.9029


[[Optimal GPV sequence]]: {{Val list| 5, 12, 17, 22, 49 }}
{{Optimal ET sequence|legend=1| 5, 12, 17, 22, 49 }}


=== 2.3.7.11.41 subgroup ===
=== 2.3.7.11.41 subgroup ===
Line 598: Line 598:
[[TOP tuning|TOP generator]]s: ~2 = 1197.9683, ~4/3 = 491.3454
[[TOP tuning|TOP generator]]s: ~2 = 1197.9683, ~4/3 = 491.3454


[[Optimal GPV sequence]]: {{Val list| 5, 12, 17, 22, 39d }}
{{Optimal ET sequence|legend=1| 5, 12, 17, 22, 39d }}


== Magi ==
== Magi ==

Revision as of 18:00, 7 May 2023

This is a collection of subgroup temperaments which omit the prime harmonic of 5.

Semaphore

Subgroup: 2.3.7

Comma: 49/48

Gencom: [2 8/7; 49/48]

Gencom mapping: [⟨1 2 0 3], ⟨0 -2 0 -1]]

Sval mapping: [⟨1 2 3], ⟨0 -2 -1]]

POL2 generator: ~7/6 = 250.385

Optimal ET sequence: 5, 14, 19, 24, 67d, 91d

RMS error: 2.523 cents

Bleu

Subgroup: 2.3.7

Comma: 17496/16807

Gencom: [2 54/49; 17496/16807]

Gencom mapping: [⟨1 1 0 2], ⟨0 5 0 7]]

Sval mapping: [⟨1 1 2], ⟨0 5 7]]

POL2 generator: ~54/49 = 139.848

Template:Val list

RMS error: 1.917 cents

2.3.7.11

Subgroup: 2.3.7.11

Comma list: 99/98, 864/847

Gencom: [2 12/11; 99/98 864/847]

Gencom mapping: [⟨1 1 0 2 3], ⟨0 5 0 7 4]]

Sval mapping: [⟨1 1 2 3], ⟨0 5 7 4]]

POL2 generator: ~12/11 = 140.005

Template:Val list

RMS error: 1.829 cents

2.3.7.11.13

Subgroup: 2.3.7.11.13

Comma list: 78/77, 99/98, 144/143

Gencom: [2 12/11; 78/77 99/98 144/143]

Gencom mapping: [⟨1 1 0 2 3 3], ⟨0 5 0 7 4 6]]

Sval mapping: [⟨1 1 2 3 3], ⟨0 5 7 4 6]]

POL2 generator: ~12/11 = 139.990

Optimal ET sequence: 17, 43, 60d

RMS error: 1.752 cents

Archy

Archy (properly pronounced "arky", after the Greek theorist Archytas) can be thought of as "no-fives dominant" or "no-fives superpyth". The name comes from the fact that it tempers out 64/63, the Archytas comma.

Subgroup: 2.3.7

Comma: 64/63

Gencom: [2 3/2; 64/63]

Gencom mapping: [⟨1 1 0 4], ⟨0 1 0 -2]]

Sval mapping: [⟨1 2 2], ⟨0 -1 2]]

POL2 generator: ~3/2 = 709.321

Optimal ET sequence: 5, 12, 17, 22, 27, 137bd

RMS error: 1.856 cents

Supra

Subgroup: 2.3.7.11

Comma list: 64/63, 99/98

Gencom: [2 3/2; 64/63 99/98]

Gencom mapping: [⟨1 1 0 4 7], ⟨0 1 0 -2 -6]]

Sval mapping: [⟨1 0 6 13], ⟨0 1 -2 -6]]

POL2 generator: ~3/2 = 707.192

Optimal ET sequence: 5, 12, 17, 39d, 56d

RMS error: 1.977 cents

Supraphon

Subgroup: 2.3.7.11.13

Comma list: 64/63, 78/77, 99/98

Gencom: [2 3/2; 64/63 78/77 99/98]

Gencom mapping: [⟨1 1 0 4 7 9], ⟨0 1 0 -2 -6 -9]]

Sval mapping: [⟨1 0 6 13 18], ⟨0 1 -2 -6 -9]]

POL2 generator: ~3/2 = 706.137

Optimal ET sequence: 5f, 12f, 17, 22, 39d, 56d

RMS error: 2.095 cents

Suhajira

Subgroup: 2.3.7.11

Comma list: 64/63, 243/242

Gencom: [2 11/9; 64/63 243/242]

Gencom mapping: [⟨1 1 0 4 2], ⟨0 2 0 -4 5]]

Sval mapping: [⟨1 1 4 2], ⟨0 2 -4 5]]

POL2 generator: ~11/9 = 353.958

Optimal ET sequence: 7, 10, 17, 44e, 61de, 78de

RMS error: 1.968 cents

2.3.7.11.13

Subgroup: 2.3.7.11.13

Comma list: 64/63, 78/77, 144/143

Gencom: [2 11/9; 64/63 78/77 144/143]

Gencom mapping: [⟨1 1 0 4 2 4], ⟨0 2 0 -4 5 -1]]

Sval mapping: [⟨1 1 4 2 4], ⟨0 2 -4 5 -1]]

POL2 generator: ~11/9 = 353.775

Optimal ET sequence: 7, 10, 17, 44e, 61de, 78de

RMS error: 1.953 cents

Skwares

Subgroup: 2.3.7

Comma: 19683/19208

Gencom: [2 9/7; 19683/19208]

Gencom mapping: [⟨1 3 6], ⟨0 -4 -9]]

Sval mapping: [⟨1 3 6], ⟨0 -4 -9]]

POL2 generator: ~9/7 = 425.365

Optimal ET sequence: 14, 17, 31, 48, 79, 189b, 268bd, 347bd

RMS error: 1.149 cents

Related temperament: squares

2.3.7.11

Subgroup: 2.3.7.11

Comma list: 99/98, 243/242

Gencom: [2 9/7; 99/98 243/242]

Gencom mapping: [⟨1 3 0 6 7], ⟨0 -4 0 -9 -10]]

Sval mapping: [⟨1 3 6 7], ⟨0 -4 -9 -10]]

POL2 generator: ~9/7 = 425.244

Optimal ET sequence: 14, 17, 31, 48, 79, 127, 206bde

RMS error: 1.099 cents

2.3.7.11.13

Subgroup: 2.3.7.11.13

Comma list: 78/77, 99/98, 243/242

Gencom: [2 9/7; 78/77, 99/98, 243/242]

Gencom mapping: [⟨1 3 0 6 7 9], ⟨0 -4 0 -9 -10 -15]]

Sval mapping: [⟨1 3 6 7 9], ⟨0 -4 -9 -10 -15]]

POL2 generator: ~9/7 = 424.457

Optimal ET sequence: 17, 48f, 65ef, 82d, 147df

RMS error: 1.769 cents

Skwairs

Subgroup: 2.3.7.11.13

Comma list: 99/98, 144/143, 243/242

Gencom: [2 9/7; 99/98, 144/143, 243/242]

Gencom mapping: [⟨1 3 0 6 7 3], ⟨0 -4 0 -9 -10 2]]

Sval mapping: [⟨1 3 6 7 3], ⟨0 -4 -9 -10 2]]

POL2 generator: ~9/7 = 424.702

Optimal ET sequence: 14, 17, 31

RMS error: 1.290 cents

Harrison

Subgroup: 2.3.7

Comma: 59049/57344

Gencom: [2 3/2; 59049/57344]

Gencom mapping: [⟨1 1 0 -3], ⟨0 1 0 10]]

Sval mapping: [⟨1 1 -3], ⟨0 1 10]]

POL2 generator: ~3/2 = 696.544

Optimal ET sequence: 12, 19, 31, 112b, 143b, 174b

RMS error: 1.226 cents

Related temperament: meantone

Leapfrog

See also: Gentle region

Subgroup: 2.3.7

Comma list: 14680064/14348907

Gencom: [2 3/2; 14680064/14348907]

Gencom mapping: [⟨1 1 0 -6], ⟨0 1 0 15]]

Sval mapping: [⟨1 0 -21], ⟨0 1 15]]

POL2 generator: ~3/2 = 704.721 cents

Optimal ET sequence: 17, 46, 63

RMS error: 0.6202 cents

Related temperaments: leapday, leapweek, srutal

2.3.7.11

Subgroup: 2.3.7.11

Comma list: 896/891, 1331/1323

Gencom: [2 3/2; 896/891 1331/1323]

Gencom mapping: [⟨1 1 0 -6 -3], ⟨0 1 0 15 11]]

Sval mapping: [⟨1 0 -21 -14], ⟨0 1 15 11]]

POL2 generator: ~3/2 = 704.753 cents

Optimal ET sequence: 17, 46, 63

RMS error: 0.6047 cents

2.3.7.11.13

Subgroup: 2.3.7.11.13

Comma list: 169/168, 352/351, 364/363

Gencom: [2 3/2; 169/169 352/351 364/363]

Gencom mapping: [⟨1 1 0 -6 -3 -1], ⟨0 1 0 15 11 8]]

Sval mapping: [⟨1 0 -21 -14 -9], ⟨0 1 15 11 8]]

POL2 generator: ~3/2 = 704.745 cents

Optimal ET sequence: 17, 46, 63

RMS error: 0.7541 cents

Skidoo

Subgroup: 2.3.7.11.13.23

Comma list: 169/168, 208/207, 352/351, 364/363

Gencom: [2 3/2; 169/169 208/207 352/351 364/363]

Gencom mapping: [⟨1 1 0 -6 -3 -1 0 0 1], ⟨0 1 0 15 11 8 0 0 6]]

Sval mapping: [⟨1 0 -21 -14 -9 -5], ⟨0 1 15 11 8 6]]

POL2 generator: ~3/2 = 704.729 cents

Optimal ET sequence: 17, 46, 63

RMS error: 0.6265 cents

2.3.7.11.13.23.29

Subgroup: 2.3.7.11.13.23.29

Comma list: 169/168, 208/207, 232/231, 352/351, 364/363

Gencom: [2 3/2; 169/169 208/207 352/351 364/363]

Gencom mapping: [⟨1 1 0 -6 -3 -1 0 0 1 -11], ⟨0 1 0 15 11 8 0 0 6 27]]

Sval mapping: [⟨1 0 -21 -14 -9 -5 -38], ⟨0 1 15 11 8 6 27]]

POL2 generator: ~3/2 = 704.729 cents

Optimal ET sequence: 17, 46, 63

Music

Lee

Subgroup: 2.3.7

Comma: 177147/175616

Gencom: [2 81/56; 177147/175616]

Gencom mapping: [⟨1 0 0 -3], ⟨0 3 0 11]]

Sval mapping: [⟨1 0 -3], ⟨0 3 11]]

POL2 generator: ~81/56 = 633.525

Optimal ET sequence: 17, 36, 89, 125, 161, 358, 519b

RMS error: 0.3519 cents

Slendric

Subgroup: 2.3.7

Comma: 1029/1024

Gencom: [2 8/7; 1029/1024]

Gencom mapping: [⟨1 1 0 3], ⟨0 3 0 -1]]

Sval mapping: [⟨1 1 3], ⟨0 3 -1]]

POL2 generator: ~8/7 = 233.688

Optimal ET sequence: 5, 21, 26, 31, 36, 77, 113, 190

RMS error: 0.3202 cents

Baladic

Subgroup: 2.3.7.13

Comma list: 169/168, 1029/1024

Gencom: [91/64 8/7; 169/168 1029/1024]

Sval mapping: [⟨2 2 6 7], ⟨0 3 -1 1]]

POL2 generator: ~8/7 = 233.6044

Optimal ET sequence: 10, 26, 36, 154…, 190…, 226…, 262…

RMS error: 0.5452 cents

2.3.7.13.17

Subgroup: 2.3.7.13.17

Comma list: 169/168, 273/272, 289/288

Gencom: [17/12 8/7; 169/168 273/272 289/288]

Sval mapping: [⟨2 2 6 7 7], ⟨0 3 -1 1 3]]

POL2 generator: ~8/7 = 233.6155

Optimal ET sequence: 10, 26, 36, 154…, 190…, 226…

RMS error: 0.5073 cents

Hemif

Subgroup: 2.3.7

Comma: 1605632/1594323

Gencom: [2 2187/1792; 1605632/1594323]

Gencom mapping: [⟨1 1 0 -1], ⟨0 2 0 13]]

Sval mapping: [⟨1 1 -1], ⟨0 2 13]]

POL2 generator: ~2187/1792 = 351.485

Optimal ET sequence: 7, 17, 41, 58, 99

RMS error: 0.2344 cents

Related temperaments: hemififths, namo

2.3.7.11

Subgroup: 2.3.7.11

Comma list: 243/242, 896/891

Gencom: [2 11/9; 243/242 896/891]

Gencom mapping: [⟨1 1 0 -1 2], ⟨0 2 0 13 5]]

Sval mapping: [⟨1 1 -1 2], ⟨0 2 13 5]]

POL2 generator: ~11/9 = 351.535

Optimal ET sequence: 7, 17, 41, 58, 99e

RMS error: 0.6108 cents

2.3.7.11.13

Subgroup: 2.3.7.11.13

Comma list: 144/143, 243/242, 364/363

Gencom: [2 11/9; 144/143 243/242 364/363]

Gencom mapping: [⟨1 1 0 -1 2 4], ⟨0 2 0 13 5 -1]]

Sval mapping: [⟨1 1 -1 2 4], ⟨0 2 13 5 -1]]

POL2 generator: ~11/9 = 351.691

Optimal ET sequence: 7, 10, 17, 24, 41, 58

RMS error: 0.7167 cents

Ennea

Subgroup: 2.3.7.11

Comma list: 41503/41472, 43923/43904

Gencom: [2 99/98; 41503/41472, 43923/43904]

Gencom mapping: [⟨1 14/9 0 25/9 31/9], ⟨0 2 0 2 1]]

Sval mapping: [⟨9 0 11 24], ⟨0 2 2 1]]

POL2 generator: ~99/98 = 17.6258

Optimal ET sequence: 54, 63, 72, 135, 342, 477, 1089, 1566

RMS error: 0.0383 cents

Parapyth (rank 3)

See also: Pentacircle temperaments § Parapyth

Subgroup: 2.3.7.11

Comma list: 896/891

Gencom: [2 3/2 28/27; 896/891]

Gencom mapping: [⟨1 1 0 1 4], ⟨0 1 0 3 -1], ⟨0 0 0 1 1]]

Sval mapping: [⟨1 0 0 7], ⟨0 1 0 -4], ⟨0 0 1 1]]

POL2 tuning: ~3 = 1903.834, ~7 = 3369.872

Optimal ET sequence: 17, 36, 41, 58, 63, 104

RMS error: 0.4149 cents

2.3.7.11.13

Subgroup: 2.3.7.11.13

Comma list: 352/351, 364/363

The gencom below gives Margo Schulter's favored basis

Gencom: [2 3/2 28/27; 352/351 364/363]

Gencom mapping: [⟨1 1 0 1 4 6], ⟨0 1 0 3 -1 -4], ⟨0 0 0 1 1 1]]

Sval mapping: [⟨1 0 0 7 12], ⟨0 1 0 -4 -7], ⟨0 0 1 1 1]]

POL2 tuning: ~3 = 1903.856, ~7 = 3369.907

Optimal ET sequence: 17, 41, 46, 58, 87, 104

RMS error: 0.3789 cents

Neutral

Neutral can be thought of as the 2.3.11 version of either mohajira or maqamic, as well as suhajira and ringo. Among other things, it is the temperament optimizing the neutral tetrad.

Subgroup: 2.3.11

Comma: 243/242

Gencom: [2 11/9; 243/242]

Gencom mapping: [⟨1 1 0 0 2], ⟨0 2 0 0 5]]

Sval mapping: [⟨1 1 2], ⟨0 2 5]]

POL2 generator: ~11/9 = 350.525

Optimal ET sequence: 7, 10, 17, 24, 41, 65, 89, 202, 291, 380

RMS error: 0.3021 cents

Scales

Namo

Subgroup: 2.3.11.13

Comma list: 144/143, 243/242

Gencom: [2 11/9; 144/143 243/242]

Gencom mapping: [⟨1 1 0 0 2 4], ⟨0 2 0 0 5 -1]]

Sval mapping: [⟨1 1 2 4], ⟨0 2 5 -1]]

POL2 generator: ~11/9 = 351.488

Optimal ET sequence: 7, 10, 17, 24, 41

RMS error: 0.7038 cents

Reversed meantone

Main article: Reversed meantone

Subgroup: 2.3.41

Comma list: 82/81

Gencom: [2 4/3; 82/81]

Sval mapping: [⟨1 2 7], ⟨0 -1 -4]]

POL2 generator: ~4/3 = 494.509

Optimal ET sequence: 5, 12, 17

2.3.7.41 subgroup

Subgroup: 2.3.7.41

Comma list: 64/63, 82/81

Gencom: [2 4/3; 64/63 82/81]

Sval mapping: [⟨1 2 2 7], ⟨0 -1 2 -4]]

POTE generator: ~4/3 = 490.0323

TOP generators: ~2 = 1197.2342, ~4/3 = 488.9029

Optimal ET sequence: 5, 12, 17, 22, 49

2.3.7.11.41 subgroup

Subgroup: 2.3.7.11.41

Comma list: 64/63, 82/81, 99/98

Gencom: [2 4/3; 64/63 82/81 99/98]

Sval mapping: [⟨1 2 2 1 7], ⟨0 -1 2 6 -4]]

POTE generator: ~4/3 = 492.1787

TOP generators: ~2 = 1197.9683, ~4/3 = 491.3454

Optimal ET sequence: 5, 12, 17, 22, 39d

Magi

Subgroup: 2.3.7.11

Comma list: 896/891, 537824/531441

Gencom: [2 96/77; 896/891, 537824/531441]

Balthazar

Subgroup: 2.3.7.11.13

Comma list: 896/891, 537824/531441, 169/168

Caspar

Subgroup: 2.3.7.11.13

Comma list: 896/891, 537824/531441, 144/143

Melchior

Subgroup: 2.3.7.11.13

Comma list: 896/891, 537824/531441, 364/363

Hogwarts

Subgroup: 2.3.7.29

Comma list: 537824/531441, 784/783

Gencom: [2 36/29; 537824/531441, 784/783]

Twenothology

Subgroup: 2.3.7.11.13.29

Comma list: 537824/531441, 896/891, 144/143, 784/783

Hectosaros leap week

Defined as the 320 & 1803 temperament, in the 2.3.7.13.17.19 on the basis of the fact that 1803 tropical years make up almost exactly 100 saros cycles.

Subgroup: 2.3.7

Comma list: [-50 -746 439⟩

Mapping: [⟨1 313 532], ⟨0 -439 -746]]

Optimal tuning (CTE): ~[17 343 143⟩ = 851.248

2.3.7.13 subgroup

Subgroup: 2.3.7.13

Comma list: [-42  -2 -5 16⟩, [10 -46 29 -5⟩

Mapping: [⟨1 313 532 208], ⟨0 -439 -746 -288]]

Optimal tuning (CTE): ~1235079060111/755603996672 = 851.248

2.3.7.13.17 subgroup

Subgroup: 2.3.7.13.17

Comma list: 39337984/39328497, [0 -14  7  4 -3⟩, [-18 -24 14 -1  5⟩

Mapping: [⟨1 313 532 208 58], ⟨0 -439 -746 -288 -76]]

Optimal tuning (CTE): ~6144/3757 = 851.248

2.3.7.13.17.19 subgroup

Subgroup: 2.3.7.13.17

Comma list: 10081799/10077696, 39337984/39328497, 10754912/10744731, 480024727/480020256

Mapping: [⟨1 313 532 208 58 432], ⟨0 -439 -746 -288 -76 -603]]

Optimal tuning (CTE): ~6144/3757 = 851.248

Sematology

This temperament tempers out 4107/4096 and thus equates 2 37/32's with 4/3.

Subgroup: 2.3.37

Comma list: 4107/4096

Gencom: [2 37/32; 4107/4096]

Mapping: [⟨1 1 5], ⟨0 -2 1]]

POTE generator: ~37/32 = 249.075

2.3.7.37 subgroup

Subgroup: 2.3.7.37

Comma list: 4107/4096, 259/256

Gencom: [2 37/32; 4107/4096 259/256]

Mapping: [⟨1 1 1 5], ⟨0 -2 -1 1]]

POTE generator: ~37/32 = 247.782

2.3.5.37 subgroup

It is difficult to extend sematology to include 5, due the 5th harmonic being quite high-complexity.

Subgroup: 2.3.5.37

Comma list: 4107/4096, 17592186044416/17562397269605

Gencom: [2 37/32; 4107/4096 17592186044416/17562397269605]

Mapping: [⟨1 1 4 5], ⟨0 -2 -8 1]]

POTE generator: ~37/32 = 251.393

2.3.5.7.37 subgroup

Subgroup: 2.3.5.7.37

Comma list: 4107/4096, 17592186044416/17562397269605, 259/256

Gencom: [2 37/32; 4107/4096 17592186044416/17562397269605 259/256]

Mapping: [⟨1 1 4 1 5], ⟨0 -2 -8 -1 1]]

POTE generator: ~37/32 = 251.204

Reversed mavila

Subgroup: 2.3.37

Comma list: 81/74

Gencom: [2 4/3; 81/74]

Mapping: [⟨1 1 0], ⟨0 -1 12]]

POTE generator: ~4/3 = 521.397

Aerophore

Subgroup: 2.3.11.19

Comma list: 363/361, 729/704

Mapping: [⟨1 0 -6 -6], ⟨0 2 12 13]]

POTE generator: ~19/11 = 945.4

Vals: 14, 5[+11, +19], 19, 33, 9[-11, -19], 47[-11, -19], 24[+11, +19], 52, 23[--11, --19], 61[-3, -11, -19], 43[+11, +19], 71[-3], 37[-3, --11, --19], 80[-3, -11, -19]

Semaerophore

Subgroup: 2.3.7.11.19

Comma list: 49/48, 77/76, 729/704

Mapping: [⟨1 0 2 -6 -6], ⟨0 2 1 12 13]]

POTE generator: ~7/4 = 944.667

Vals: 14, 5[+11, +19], 19, 9[-11, -19], 33[-7], 47[-7, -11, -19], 23[-7, --11, --19], 61[-3, -7, -11, -19], 37[-3, -7, --11, --19], 75[-3, --7, -11, --19], 89[-3, --7, --11, --19]

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