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No-fives subgroup temperaments: Difference between revisions

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Comma list: 64/63, 364/363, 1078/1053
Comma list: 64/63, 364/363, 1078/1053


Mapping: {{mapping| 1 0 6 21 34 | 0 1 -2 -11 -19 }}
{{Mapping|legend=0| 1 0 6 21 34 | 0 1 -2 -11 -19 }}


Optimal tunings:  
Optimal tunings:  
Line 31: Line 31:
Comma list: 64/63, 209/208, 343/342, 364/363
Comma list: 64/63, 209/208, 343/342, 364/363


Mapping: {{mapping| 1 0 6 21 34 17 | 0 1 -2 -11 -19 -8 }}
{{Mapping|legend=0| 1 0 6 21 34 17 | 0 1 -2 -11 -19 -8 }}


Optimal tunings:  
Optimal tunings:  
Line 57: Line 57:
[[Comma list]]: 282429536481/281974669312
[[Comma list]]: 282429536481/281974669312


{{Mapping|legend=2| 1 1 0| 0 5 24 }}
{{Mapping|legend=3| 1 1 0| 0 5 24 }}
: mapping generators: ~2, ~243/224
: mapping generators: ~2, ~243/224


Line 75: Line 75:
Comma list: 1331/1323, 19712/19683
Comma list: 1331/1323, 19712/19683


Subgroup-val mapping: {{mapping| 1 1 0 1 | 0 5 24 21 }}
{{Mapping|legend=2| 1 1 0 1 | 0 5 24 21 }}


Optimal tunings:  
Optimal tunings:  
Line 90: Line 90:
Comma list: 352/351, 729/728, 1331/1323
Comma list: 352/351, 729/728, 1331/1323


Subgroup-val mapping: {{mapping| 1 1 0 1 3 | 0 5 24 21 6 }}
{{Mapping|legend=2| 1 1 0 1 3 | 0 5 24 21 6 }}


Optimal tunings:  
Optimal tunings:  
Line 105: Line 105:
[[Comma list]]: 177147/175616
[[Comma list]]: 177147/175616


{{Mapping|legend=2| 1 0 -3 | 0 3 11 }}
{{Mapping|legend=3| 1 0 -3 | 0 3 11 }}


{{Mapping|legend=3| 1 0 0 -3 | 0 3 0 11 }}
{{Mapping|legend=5| 1 0 0 -3 | 0 3 0 11 }}
: mapping generators: ~2, ~81/56
: mapping generators: ~2, ~81/56


Line 124: Line 124:


=== Hemif ===
=== Hemif ===
Hemif is the no-5 [[restriction]] of [[hemififths]], and the add-7 [[extension]] of [[namo]].  
Hemif tempers out the [[parahemif comma]], equating a [[Pythagorean apotome]] with a stack of two septimal third-tones of [[28/27]], and splits the fifth in two. In this temperament, the interval class of [[7/1|7]] is found at +13 steps, as a semi-augmented sixth (C–A{{demisharp2}}). It also equates the large septimal diesis of [[49/48]] with the [[Pythagorean comma]]. This means the fifth should be tuned sharp and the septimal third-tone should be flattened to a somewhat large quartertone which can be used as the undecimal quartertone of [[33/32]], leading to a natural 11-limit extension where [[243/242]] and [[896/891]] are tempered out.
 
Hemif is the common [[restriction]] of [[hemififths]] (41 & 58) and [[salsa]] (24 & 41), and the add-7 [[extension]] of [[namo]]. A [[weak extension]] of it is [[sruti]] (24 & 34d).  


[[Subgroup]]: 2.3.7
[[Subgroup]]: 2.3.7
Line 130: Line 132:
[[Comma list]]: 1605632/1594323
[[Comma list]]: 1605632/1594323


{{Mapping|legend=2| 1 1 -1 | 0 2 13 }}
{{Mapping|legend=3| 1 1 -1 | 0 2 13 }}


{{Mapping|legend=3| 1 1 0 -1 | 0 2 0 13 }}
{{Mapping|legend=5| 1 1 0 -1 | 0 2 0 13 }}
: mapping generators: ~2, ~2187/1792
: mapping generators: ~2, ~2187/1792


Line 150: Line 152:
Comma list: 243/242, 896/891
Comma list: 243/242, 896/891


Subgroup-val mapping: {{mapping| 1 1 -1 2 | 0 2 13 5 }}
{{Mapping|legend=2| 1 1 -1 2 | 0 2 13 5 }}


Gencom mapping: {{mapping| 1 1 0 -1 2 | 0 2 0 13 5 }}
{{Mapping|legend=4| 1 1 0 -1 2 | 0 2 0 13 5 }}


Optimal tunings:  
Optimal tunings:  
Line 169: Line 171:
Sval mapping: {{mapping| 1 1 -1 2 4 | 0 2 13 5 -1 }}
Sval mapping: {{mapping| 1 1 -1 2 4 | 0 2 13 5 -1 }}


Gencom mapping: {{mapping| 1 1 0 -1 2 4 | 0 2 0 13 5 -1 }}
{{Mapping|legend=4| 1 1 0 -1 2 4 | 0 2 0 13 5 -1 }}


Optimal tunings:  
Optimal tunings:  
Line 188: Line 190:
Comma list: 243/242, 896/891, 1083/1078
Comma list: 243/242, 896/891, 1083/1078


Subgroup-val mapping: {{mapping| 1 1 -1 2 0 | 0 4 26 10 29 }}
{{Mapping|legend=2| 1 1 -1 2 0 | 0 4 26 10 29 }}
: mapping generators: ~2, ~21/19
: mapping generators: ~2, ~21/19


Line 208: Line 210:
[[Comma list]]: 34451725707/34359738368
[[Comma list]]: 34451725707/34359738368


{{Mapping|legend=2| 1 1 5 | 0 4 -15 }}
{{Mapping|legend=3| 1 1 5 | 0 4 -15 }}
: mapping generators: ~2, ~567/512
: mapping generators: ~2, ~567/512


Line 226: Line 228:
Comma list: 243/242, 65536/65219
Comma list: 243/242, 65536/65219


Subgroup-val mapping: {{mapping| 1 1 5 2 | 0 4 -15 10 }}
{{Mapping|legend=2| 1 1 5 2 | 0 4 -15 10 }}


Optimal tunings:  
Optimal tunings:  
Line 241: Line 243:
Comma list: 243/242, 513/512, 1083/1078
Comma list: 243/242, 513/512, 1083/1078


Subgroup-val mapping: {{mapping| 1 1 5 2 6 | 0 4 -15 10 -12 }}
{{Mapping|legend=2| 1 1 5 2 6 | 0 4 -15 10 -12 }}


Optimal tunings:  
Optimal tunings:  
Line 258: Line 260:
[[Comma list]]: 537824/531441
[[Comma list]]: 537824/531441


{{Mapping|legend=2| 1 0 -1 | 0 5 12 }}
{{Mapping|legend=3| 1 0 -1 | 0 5 12 }}
: mapping generators: ~2, ~243/196
: mapping generators: ~2, ~243/196


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Comma list: 896/891, 26411/26244
Comma list: 896/891, 26411/26244


Subgroup-val mapping: {{mapping| 1 0 -1 6 | 0 5 12 -8 }}
{{Mapping|legend=2| 1 0 -1 6 | 0 5 12 -8 }}


Optimal tunings:  
Optimal tunings:  
Line 291: Line 293:
Comma list: 144/143, 343/338, 729/728
Comma list: 144/143, 343/338, 729/728


Subgroup-val mapping: {{mapping| 1 0 -1 6 -2 | 0 5 12 -8 18 }}
{{Mapping|legend=2| 1 0 -1 6 -2 | 0 5 12 -8 18 }}


Optimal tunings:  
Optimal tunings:  
Line 306: Line 308:
Comma list: 144/143, 232/231, 343/338, 729/728
Comma list: 144/143, 232/231, 343/338, 729/728


Subgroup-val mapping: {{mapping| 1 0 -1 6 -2 2 | 0 5 12 -8 18 9 }}
{{Mapping|legend=2| 1 0 -1 6 -2 2 | 0 5 12 -8 18 9 }}


Optimal tunings:  
Optimal tunings:  
Line 321: Line 323:
Comma list: 352/351, 364/363, 26411/26244
Comma list: 352/351, 364/363, 26411/26244


Subgroup-val mapping: {{mapping| 1 0 -1 6 11 | 0 5 12 -8 -23 }}
{{Mapping|legend=2| 1 0 -1 6 11 | 0 5 12 -8 -23 }}


Optimal tunings:  
Optimal tunings:  
Line 336: Line 338:
Comma list: 169/168, 896/891, 26411/26244
Comma list: 169/168, 896/891, 26411/26244


Subgroup-val mapping: {{mapping| 1 0 -1 6 1 | 0 10 24 -16 17 }}
{{Mapping|legend=2| 1 0 -1 6 1 | 0 10 24 -16 17 }}
: mapping generators: ~2, ~143/128
: mapping generators: ~2, ~143/128


Line 352: Line 354:
Comma list: 784/783, 5887/5832
Comma list: 784/783, 5887/5832


Subgroup-val mapping: {{mapping| 1 0 -1 2 | 0 5 12 9 }}
{{Mapping|legend=2| 1 0 -1 2 | 0 5 12 9 }}


Optimal tunings:  
Optimal tunings:  
Line 371: Line 373:
[[Comma list]]: 19683/19208
[[Comma list]]: 19683/19208


{{Mapping|legend=2| 1 -1 -3 | 0 4 9 }}
{{Mapping|legend=3| 1 -1 -3 | 0 4 9 }}


{{Mapping|legend=3| 1 -1 0 -3 | 0 4 0 9 }}
{{Mapping|legend=5| 1 -1 0 -3 | 0 4 0 9 }}
: mapping generators: ~2, ~14/9
: mapping generators: ~2, ~14/9


Line 391: Line 393:
Comma list: 99/98, 243/242
Comma list: 99/98, 243/242


Subgroup-val mapping: {{mapping| 1 -1 -3 -3 | 0 4 9 10 }}
{{Mapping|legend=2| 1 -1 -3 -3 | 0 4 9 10 }}


Gencom mapping: {{mapping| 1 -1 0 -3 -3 | 0 4 0 9 10 }}
{{Mapping|legend=4| 1 -1 0 -3 -3 | 0 4 0 9 10 }}


Optimal tunings:  
Optimal tunings:  
Line 408: Line 410:
Comma list: 78/77, 99/98, 243/242
Comma list: 78/77, 99/98, 243/242


Subgroup-val mapping: {{mapping| 1 -1 -3 -3 -6 | 0 4 9 10 15 }}
{{Mapping|legend=2| 1 -1 -3 -3 -6 | 0 4 9 10 15 }}


Gencom mapping: {{mapping| 1 -1 0 -3 -3 -6 | 0 4 0 9 10 15}}
{{Mapping|legend=4| 1 -1 0 -3 -3 -6 | 0 4 0 9 10 15}}


Optimal tunings:  
Optimal tunings:  
Line 425: Line 427:
Comma list: 99/98, 144/143, 243/242
Comma list: 99/98, 144/143, 243/242


Subgroup-val mapping: {{mapping| 1 -1 -3 -3 5 | 0 4 9 10 -2 }}
{{Mapping|legend=2| 1 -1 -3 -3 5 | 0 4 9 10 -2 }}


Gencom mapping: {{mapping| 1 -1 0 -3 -3 5 | 0 4 0 9 10 -2 }}
{{Mapping|legend=4| 1 -1 0 -3 -3 5 | 0 4 0 9 10 -2 }}


Optimal tunings:  
Optimal tunings:  
Line 444: Line 446:
Comma list: 99/98, 243/242, 363/361
Comma list: 99/98, 243/242, 363/361


Subgroup-val mapping: {{mapping| 2 2 3 4 5 | 0 4 9 10 12 }}
{{Mapping|legend=2| 2 2 3 4 5 | 0 4 9 10 12 }}
: mapping generators: ~209/147, ~21/19
: mapping generators: ~209/147, ~21/19


Line 462: Line 464:
[[Comma list]]: [[59049/57344]]
[[Comma list]]: [[59049/57344]]


{{Mapping|legend=2| 1 0 -13 | 0 1 10 }}
{{Mapping|legend=3| 1 0 -13 | 0 1 10 }}


{{Mapping|legend=3| 1 0 0 -13 | 0 1 0 10 }}
{{Mapping|legend=5| 1 0 0 -13 | 0 1 0 10 }}
: mapping generators: ~2, ~3
: mapping generators: ~2, ~3


Line 478: Line 480:


=== Bleu ===
=== Bleu ===
Bleu can be described as the {{nowrap| 8d & 9 }} temperament in the no-5 13-limit, and is the common [[restriction]] of [[progression]] and [[jerome]].  
Bleu can be described as the {{nowrap| 8d & 9 }} temperament in the no-5 13-limit, and is the common [[restriction]] of [[progression]], [[secund]] and [[jerome]].  


[[Subgroup]]: 2.3.7
[[Subgroup]]: 2.3.7
Line 484: Line 486:
[[Comma list]]: 17496/16807
[[Comma list]]: 17496/16807


{{Mapping|legend=2| 1 1 2 | 0 5 7 }}
{{Mapping|legend=3| 1 1 2 | 0 5 7 }}


{{Mapping|legend=3| 1 1 0 2 | 0 5 0 7 }}
{{Mapping|legend=5| 1 1 0 2 | 0 5 0 7 }}
: mapping generators: ~2, ~54/49
: mapping generators: ~2, ~54/49


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Comma list: 99/98, 864/847
Comma list: 99/98, 864/847


Subgroup-val mapping: {{mapping| 1 1 2 3 | 0 5 7 4 }}
{{Mapping|legend=2| 1 1 2 3 | 0 5 7 4 }}


Gencom mapping: {{mapping| 1 1 0 2 3 | 0 5 0 7 4 }}
{{Mapping|legend=4| 1 1 0 2 3 | 0 5 0 7 4 }}


Optimal tunings:  
Optimal tunings:  
Line 521: Line 523:
Comma list: 78/77, 99/98, 144/143
Comma list: 78/77, 99/98, 144/143


Subgroup-val mapping: {{mapping| 1 1 2 3 3 | 0 5 7 4 6 }}
{{Mapping|legend=2| 1 1 2 3 3 | 0 5 7 4 6 }}


Gencom mapping: {{mapping| 1 1 0 2 3 3 | 0 5 0 7 4 6 }}
{{Mapping|legend=4| 1 1 0 2 3 3 | 0 5 0 7 4 6 }}


Optimal tunings:  
Optimal tunings:  
Line 545: Line 547:
[[Comma list]]: 5764801/5668704
[[Comma list]]: 5764801/5668704


{{Mapping|legend=2| 1 1 2 | 0 8 11 }}
{{Mapping|legend=3| 1 1 2 | 0 8 11 }}
: mapping generators: ~2, ~343/342
: mapping generators: ~2, ~343/342


Line 563: Line 565:
Comma list: 243/242, 2401/2376
Comma list: 243/242, 2401/2376


Subgroup-val mapping: {{mapping| 1 1 2 2 | 0 8 11 20 }}
{{Mapping|legend=2| 1 1 2 2 | 0 8 11 20 }}


Optimal tunings:  
Optimal tunings:  
Line 578: Line 580:
Comma list: 133/132, 243/242, 343/342
Comma list: 133/132, 243/242, 343/342


Subgroup-val mapping: {{mapping| 1 1 2 2 3 | 0 8 11 20 17 }}
{{Mapping|legend=2| 1 1 2 2 3 | 0 8 11 20 17 }}


Optimal tunings:  
Optimal tunings:  
Line 595: Line 597:
[[Comma list]]: 2187/2048
[[Comma list]]: 2187/2048


{{Mapping|legend=2| 7 11 0 | 0 0 1 }}
{{Mapping|legend=3| 7 11 0 | 0 0 1 }}
: mapping generators: ~9/8, ~7
: mapping generators: ~9/8, ~7


Line 615: Line 617:
[[Comma list]]: {{monzo| 83 -1 -29 }}
[[Comma list]]: {{monzo| 83 -1 -29 }}


{{Mapping|legend=2| 1 -4 3 | 0 29 -1 }}
{{Mapping|legend=3| 1 -4 3 | 0 29 -1 }}
: mapping generators: ~2, ~8/7
: mapping generators: ~2, ~8/7


Line 633: Line 635:
Comma list: 43923/43904, 5767168/5764801
Comma list: 43923/43904, 5767168/5764801


Subgroup-val mapping: {{mapping| 1 -4 3 5 | 0 29 -1 -8 }}
{{Mapping|legend=2| 1 -4 3 5 | 0 29 -1 -8 }}


Optimal tunings:  
Optimal tunings:  
Line 656: Line 658:
[[Comma list]]: 14680064/14348907
[[Comma list]]: 14680064/14348907


{{Mapping|legend=2| 1 0 -21 | 0 1 15 }}
{{Mapping|legend=3| 1 0 -21 | 0 1 15 }}


{{Mapping|legend=3| 1 0 0 -21 | 0 1 0 15 }}
{{Mapping|legend=5| 1 0 0 -21 | 0 1 0 15 }}
: mapping generators: ~2, ~3
: mapping generators: ~2, ~3


Line 676: Line 678:
Comma list: 896/891, 1331/1323
Comma list: 896/891, 1331/1323


Subgroup-val mapping: {{mapping| 1 0 -21 -14 | 0 1 15 11 }}
{{Mapping|legend=2| 1 0 -21 -14 | 0 1 15 11 }}


Gencom mapping: {{mapping| 1 0 0 -21 -14 | 0 1 0 15 11 }}
{{Mapping|legend=4| 1 0 0 -21 -14 | 0 1 0 15 11 }}


Optimal tunings:  
Optimal tunings:  
Line 693: Line 695:
Comma list: 169/168, 352/351, 364/363
Comma list: 169/168, 352/351, 364/363


Subgroup-val mapping: {{mapping| 1 0 -21 -14 -9 | 0 1 15 11 8 }}
{{Mapping|legend=2| 1 0 -21 -14 -9 | 0 1 15 11 8 }}


Gencom mapping: {{mapping| 1 0 0 -21 -14 -9 | 0 1 0 15 11 8 }}
{{Mapping|legend=4| 1 0 0 -21 -14 -9 | 0 1 0 15 11 8 }}


Optimal tunings:  
Optimal tunings:  
Line 710: Line 712:
Comma list: 169/168, 208/207, 352/351, 364/363
Comma list: 169/168, 208/207, 352/351, 364/363


Subgroup-val mapping: {{mapping| 1 0 -21 -14 -9 -5 | 0 1 15 11 8 6 }}
{{Mapping|legend=2| 1 0 -21 -14 -9 -5 | 0 1 15 11 8 6 }}


Gencom mapping: {{mapping| 1 0 0 -21 -14 -9 0 0 -5 | 0 1 0 15 11 8 0 0 6 }}
{{Mapping|legend=4| 1 0 0 -21 -14 -9 0 0 -5 | 0 1 0 15 11 8 0 0 6 }}


Optimal tunings:  
Optimal tunings:  
Line 727: Line 729:
Comma list: 169/168, 208/207, 232/231, 352/351, 364/363
Comma list: 169/168, 208/207, 232/231, 352/351, 364/363


Subgroup-val mapping: {{mapping| 1 0 -21 -14 -9 -5 -38 | 0 1 15 11 8 6 27 }}
{{Mapping|legend=2| 1 0 -21 -14 -9 -5 -38 | 0 1 15 11 8 6 27 }}


Gencom mapping: {{mapping| 1 0 0 -21 -14 -9 -5 0 0 -38 | 0 1 0 15 11 8 0 0 6 27 }}
{{Mapping|legend=4| 1 0 0 -21 -14 -9 -5 0 0 -38 | 0 1 0 15 11 8 0 0 6 27 }}


Optimal tunings:  
Optimal tunings:  
Line 756: Line 758:
[[Comma list]]: 17294403/16777216
[[Comma list]]: 17294403/16777216


{{Mapping|legend=2| 1 0 3 | 0 8 -1 }}
{{Mapping|legend=3| 1 0 3 | 0 8 -1 }}
: mapping generators: ~2, ~8/7
: mapping generators: ~2, ~8/7


Line 770: Line 772:


=== Hectosaros leap week ===
=== Hectosaros leap week ===
This temperament may be described as the {{nowrap| 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.
This temperament may be described as the {{nowrap| 320 & 1803 }} temperament, in the 2.3.7.13.17.19. It is named after the fact that 1803 tropical years make up almost exactly 100 saros cycles.


[[Subgroup]]: 2.3.7
[[Subgroup]]: 2.3.7
Line 776: Line 778:
[[Comma list]]: {{monzo| -50 -746 439 }}
[[Comma list]]: {{monzo| -50 -746 439 }}


{{Mapping|legend=2| 1 -126 -214 | 0 439 746 }}
{{Mapping|legend=3| 1 -126 -214 | 0 439 746 }}
: mapping generators: ~2, ~{{monzo| -16 -243 143 }}
: mapping generators: ~2, ~{{monzo| -16 -243 143 }}


Line 794: Line 796:
Comma list: {{monzo| -42 -2 -5 16 }}, {{monzo| 10 -46 29 -5 }}
Comma list: {{monzo| -42 -2 -5 16 }}, {{monzo| 10 -46 29 -5 }}


Subgroup-val mapping: {{mapping| 1 -126 -214 -80 | 0 439 746 288 }}
{{Mapping|legend=2| 1 -126 -214 -80 | 0 439 746 288 }}


Optimal tunings:  
Optimal tunings:  
Line 809: Line 811:
Comma list: 39337984/39328497, {{monzo| 0 -14 7 4 -3 }}, {{monzo| -18 -24 14 -1 5 }}
Comma list: 39337984/39328497, {{monzo| 0 -14 7 4 -3 }}, {{monzo| -18 -24 14 -1 5 }}


Subgroup-val mapping: {{mapping| 1 -126 -214 -80 -18 | 0 439 746 288 76 }}
{{Mapping|legend=2| 1 -126 -214 -80 -18 | 0 439 746 288 76 }}


Optimal tunings:  
Optimal tunings:  
Line 824: Line 826:
Comma list: 10081799/10077696, 10754912/10744731, 39337984/39328497, 480024727/480020256
Comma list: 10081799/10077696, 10754912/10744731, 39337984/39328497, 480024727/480020256


Subgroup-val mapping: {{mapping| 1 -126 -214 -80 -18 -171 | 0 439 746 288 76 603 }}
{{Mapping|legend=2| 1 -126 -214 -80 -18 -171 | 0 439 746 288 76 603 }}


Optimal tunings:  
Optimal tunings:  
Line 843: Line 845:
[[Comma list]]: 243/242, 1083/1078
[[Comma list]]: 243/242, 1083/1078


{{Mapping|legend=2| 1 1 0 2 1 | 0 4 0 10 3 | 0 0 1 0 1 }}
{{Mapping|legend=3| 1 1 0 2 1 | 0 4 0 10 3 | 0 0 1 0 1 }}
: mapping generators: ~2, ~21/19, ~7
: mapping generators: ~2, ~21/19, ~7


Line 861: Line 863:
Io is a very low-complexity temperament which tempers out the undecimal quartertone [[33/32]]. This equates very different intervals (for example, the generator itself represents both [[3/2]] and [[16/11]]), and as such some consider it to be an [[exotemperament]]. It has an extremely wide generator range, but the most accurate tunings are generally inside the range of [[flattone]] temperament.
Io is a very low-complexity temperament which tempers out the undecimal quartertone [[33/32]]. This equates very different intervals (for example, the generator itself represents both [[3/2]] and [[16/11]]), and as such some consider it to be an [[exotemperament]]. It has an extremely wide generator range, but the most accurate tunings are generally inside the range of [[flattone]] temperament.


The name ''io'' was coined by [[User:CompactStar|CompactStar]] in 2024 based on the [[Kite's color notation|color name]] ''ilo'', prior to which it could only be termed as "undecimal temperament" with 33/32 being known as the undecimal comma.
The name ''io'' was coined by {{u|CompactStar}} in 2024 based on the [[Kite's color notation|color name]] ''ilo'', prior to which it could only be termed as "undecimal temperament" with 33/32 being known as the undecimal comma.


[[Subgroup]]: 2.3.11
[[Subgroup]]: 2.3.11
Line 867: Line 869:
[[Comma list]]: 33/32
[[Comma list]]: 33/32


{{Mapping|legend=2| 1 0 5 | 0 1 -1 }}
{{Mapping|legend=3| 1 0 5 | 0 1 -1 }}
: mapping generators: ~2, ~3
: mapping generators: ~2, ~3


Line 881: Line 883:


=== Alphaxenean ===
=== Alphaxenean ===
Alphaxenean tempers out the [[Alpharabian comma]] and equates a stack of four undecimal quartertones with the [[9/8|Pythagorean whole tone]]. It also divides the [[2/1|octave]] into two.  
Alphaxenean tempers out the [[Alpharabian comma]], and equates a stack of four undecimal quartertones with the [[9/8|Pythagorean whole tone]]. It has a [[semi-octave]] period, and the generator can be taken to be the aforementioned undecimal quartertone or that plus a period, which represents ~[[16/11]]. The interval class of a perfect fifth is found by two generator steps and a period.  


[[Subgroup]]: 2.3.11
[[Subgroup]]: 2.3.11
Line 887: Line 889:
[[Comma list]]: 131769/131072
[[Comma list]]: 131769/131072


{{Mapping|legend=2| 2 1 8 | 0 2 -1 }}
{{Mapping|legend=3| 2 1 8 | 0 2 -1 }}
: mapping generators: ~363/256, ~16/11
: mapping generators: ~363/256, ~16/11


Line 899: Line 901:


[[Badness]] (Sintel): 0.395
[[Badness]] (Sintel): 0.395
=== Octatonic ===
[[12/11]] is very close to one step of 8edo, and hence this temperament tempers out the [[undecimal octatonic comma]], the difference between a stack of eight 12/11's and the octave.
[[Subgroup]]: 2.3.11
[[Comma list]]: 214990848/214358881
{{Mapping|legend=3| 8 0 15 | 0 1 1 }}
: mapping generators: ~12/11, ~3
[[Optimal tuning]]s:
* [[WE]]: ~12/11 = 149.9917{{c}}, ~3/2 = 701.9324{{c}}
: [[error map]]: {{val| -0.066 -0.089 +0.424 }}
* [[CWE]]: ~12/11 = 150.0000{{c}}, ~3/2 = 701.9147{{c}}
: error map: {{val| 0.000 -0.040 +0.597 }}
{{Optimal ET sequence|legend=1| 8, 16, 24, 104, 128, 152, 176, 200, 1024e, 1224e, 1424e, 1624e }}
[[Badness]] (Sintel): 0.515
=== Betaxenean ===
Betaxenean tempers out the [[Betarabian comma]]. It splits a [[3/1|perfect twelfth]] into two halves of [[512/297]]. [[Baragon]] is a full 11-limit extension that combines this temperament with [[godzilla]], albeit not as accurate as normal betaxenean.
[[Subgroup]]: 2.3.11
[[Comma list]]: 264627/262144
{{Mapping|legend=3| 1 0 9 | 0 2 -7 }}
: mapping generators: ~2, ~512/297
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.5931{{c}}, ~512/297 = 950.6872{{c}}
: [[error map]]: {{val| +0.593 -0.581 -0.791 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~512/297 = 950.1972{{c}}
: error map: {{val| 0.000 -1.561 -2.699 }}
{{Optimal ET sequence|legend=1| 5, 14e, 19, 24, 125, 149, 173, 197e, 221e, 466bee, 687beee }}
[[Badness]] (Sintel): 0.709


=== Infraug ===
=== Infraug ===
Line 905: Line 947:
[[Comma list]]: 729/704
[[Comma list]]: 729/704


{{Mapping|legend=2| 1 0 -6 | 0 1 6 }}
{{Mapping|legend=3| 1 0 -6 | 0 1 6 }}
: mapping generators: ~2, ~3
: mapping generators: ~2, ~3


Line 923: Line 965:
Comma list: 144/143, 729/704
Comma list: 144/143, 729/704


Subgroup-val mapping: {{mapping| 1 0 -6 10 | 0 1 6 -4 }}
{{Mapping|legend=2| 1 0 -6 10 | 0 1 6 -4 }}


Optimal tunings:  
Optimal tunings:  
Line 938: Line 980:
Comma list: 363/361, 729/704
Comma list: 363/361, 729/704


Subgroup-val mapping: {{mapping| 1 0 -6 -6 | 0 2 12 13 }}
{{Mapping|legend=2| 1 0 -6 -6 | 0 2 12 13 }}
: mapping generators: ~2, ~19/11
: mapping generators: ~2, ~19/11


Line 945: Line 987:
* CWE: ~2 = 1200.0000{{c}}, ~19/11 = 945.7779{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~19/11 = 945.7779{{c}}


{{Optimal ET sequence|legend=1| 14, 19, 33 }}
{{Optimal ET sequence|legend=0| 14, 19, 33 }}


Badness (Sintel): 1.59
Badness (Sintel): 1.59
Line 954: Line 996:
[[Comma list]]: 4096/3993
[[Comma list]]: 4096/3993


{{Mapping|legend=2| 1 0 4 | 0 3 -1 }}
{{Mapping|legend=3| 1 0 4 | 0 3 -1 }}
: mapping generators: ~2, ~16/11
: mapping generators: ~2, ~16/11


Line 976: Line 1,018:
Comma list: 512/507, 1352/1331
Comma list: 512/507, 1352/1331


Subgroup-val mapping: {{mapping| 1 -3 5 6 | 0 6 -2 -3 }}
{{Mapping|legend=2| 1 -3 5 6 | 0 6 -2 -3 }}
: mapping generators: ~2, ~22/13
: mapping generators: ~2, ~22/13


Line 994: Line 1,036:
[[Comma list]]: 10554638336/10460353203
[[Comma list]]: 10554638336/10460353203


{{Mapping|legend=2| 1 1 0 | 0 5 21 }}
{{Mapping|legend=3| 1 1 0 | 0 5 21 }}
: mapping generators: ~2, ~88/81
: mapping generators: ~2, ~88/81


Line 1,012: Line 1,054:
Comma list: 352/351, 531674/531441
Comma list: 352/351, 531674/531441


Subgroup-val mapping: {{mapping| 1 1 0 3 | 0 5 21 6 }}
{{Mapping|legend=2| 1 1 0 3 | 0 5 21 6 }}


Optimal tunings:  
Optimal tunings:  
Line 1,029: Line 1,071:
[[Comma list]]: 19487171/19131876
[[Comma list]]: 19487171/19131876


{{Mapping|legend=2| 7 0 2 | 0 1 2 }}
{{Mapping|legend=3| 7 0 2 | 0 1 2 }}
: mapping generators: ~1458/1331, ~3
: mapping generators: ~1458/1331, ~3


Line 1,048: Line 1,090:
[[Comma list]]: 2197/2187
[[Comma list]]: 2197/2187


{{Mapping|legend=2| 1 0 0 | 0 3 7 }}
{{Mapping|legend=3| 1 0 0 | 0 3 7 }}
: mapping generators: ~2, ~13/9
: mapping generators: ~2, ~13/9


Line 1,068: Line 1,110:
[[Comma list]]: 27/26
[[Comma list]]: 27/26


{{Mapping|legend=2| 1 0 -1 | 0 1 3 }}
{{Mapping|legend=3| 1 0 -1 | 0 1 3 }}
: mapping generators: ~2, ~3
: mapping generators: ~2, ~3


Line 1,082: Line 1,124:


=== Tridecapyth ===
=== Tridecapyth ===
Tridecapyth sets a stack of ten [[9/8]]'s equal to [[13/4]]. It can be seen as a way of giving a mapping of prime 13 to any temperament with an extremely accurate tuning of its fifth (like the [[53edo]] tuning of [[schismic]]). Interestingly, the mapping is ''so'' accurate that more optimized tunings of schismic that use a flatter fifth are not accurate enough to preserve the mapping; for 118edo we get the 118f [[val]] that takes the second-best, flat mapping of prime 13, and the same is true for [[171edo]] where we get the 171f val. However, due to its small note count, [[41edo]] technically uses this mapping too, so that [[94edo]], the val sum of 41 and 53, also uses this mapping, suggesting it is of interest to flatter tunings of [[garibaldi]] with fifths tending close to pure; this corresponds to the extension of garibaldi called [[cassandra]]. If we add this mapping to 5-limit schismic instead we get [[tridecaschismic]].
Tridecapyth sets a stack of ten [[9/8]]'s equal to [[13/4]]. It shows a way of giving a mapping of prime 13 to any temperament with an extremely accurate tuning of its fifth (like the [[53edo]] tuning of [[schismic]]). Interestingly, the mapping is ''so'' accurate that more optimized tunings of schismic that use a flatter fifth are not accurate enough to preserve the mapping – for 118edo we get the 118f [[val]] that takes the second-closest, flat mapping of prime 13, and the same is true for [[171edo]] where we get the 171f val. However, [[41edo]] uses this mapping, and so does [[94edo]], the val sum of 41 and 53. Thus it is of interest to flatter tunings of [[garibaldi]]/[[cassandra]] with fifths tending close to pure. If we add this mapping to 5-limit schismic instead we get [[tridecaschismic]].


[[Subgroup]]: 2.3.13
[[Subgroup]]: 2.3.13
Line 1,108: Line 1,150:
[[Comma list]]: 1053/1024
[[Comma list]]: 1053/1024


{{Mapping|legend=2| 1 0 10 | 0 1 -4 }}
{{Mapping|legend=3| 1 0 10 | 0 1 -4 }}
: mapping generators: ~2, ~3
: mapping generators: ~2, ~3


Line 1,115: Line 1,157:
: [[error map]]: {{val| +3.129 -3.177 -4.349 }}
: [[error map]]: {{val| +3.129 -3.177 -4.349 }}
* [[CWE]]: ~2 = 1200.0000{{c}} ~3/2 = 693.6081{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}} ~3/2 = 693.6081{{c}}
: error map: {{val| 0.000 -8.347 +14.960 }}
: error map: {{val| 0.000 -8.347 -14.960 }}


{{Optimal ET sequence|legend=1| 5f, 7, 12, 19, 45f, 64f, 147bfff }}
{{Optimal ET sequence|legend=1| 5f, 7, 12, 19, 45f, 64f, 147bfff }}
Line 1,128: Line 1,170:
[[Comma list]]: 816293376/815730721
[[Comma list]]: 816293376/815730721


{{Mapping|legend=2| 1 -5 -7 | 0 8 13 }}
{{Mapping|legend=3| 1 -5 -7 | 0 8 13 }}
: mapping generators: ~2, ~3888/2197
: mapping generators: ~2, ~3888/2197


Line 1,146: Line 1,188:
Comma list: 3888/3887, 12168/12167
Comma list: 3888/3887, 12168/12167


Subgroup-val mapping: {{mapping| 1 -5 -7 -7 | 0 8 13 14 }}
{{Mapping|legend=2| 1 -5 -7 -7 | 0 8 13 14 }}


Optimal tunings:  
Optimal tunings:  
Line 1,169: Line 1,211:
[[Comma list]]: 373248/371293
[[Comma list]]: 373248/371293


{{Mapping|legend=2| 1 1 3 | 0 5 6 }}
{{Mapping|legend=3| 1 1 3 | 0 5 6 }}
: mapping generators: ~2, ~13/12
: mapping generators: ~2, ~13/12


Line 1,184: Line 1,226:
== Temperaments with a higher-limit gene ==
== Temperaments with a higher-limit gene ==
=== Semitonic ===
=== Semitonic ===
Note that [[34edo]] and [[46edo]] also support this temperament but do not appear in the optimal ET sequence due to their sharp fifths, but they are on opposite sides of the just-17's tuning such that 34 + 46 = [[80edo]] is effectively a pure [[consistent circle]] of 80 [[17/16]]'s; all of these are good tunings for the related temperament [[srutal]] in 2.3.5.17 (sometimes called [[srutal archagall]] to distinguish it from diaschismic, which prefers tunings with a more just fifth), but the accuracy of primes 7, 11 and 13 varies depending on the tuning; especially, 34edo's 7 is extremely sharp and its 11 not much better.
[[Subgroup]]: 2.3.17
[[Subgroup]]: 2.3.17


[[Comma list]]: [[289/288]]
[[Comma list]]: [[289/288]]


{{Mapping|legend=2| 2 0 5 | 0 1 1 }}
{{Mapping|legend=3| 2 0 5 | 0 1 1 }}
: mapping generators: ~17/12, ~3
: mapping generators: ~17/12, ~3


Line 1,208: Line 1,252:
[[Comma list]]: 513/512
[[Comma list]]: 513/512


{{Mapping|legend=2| 1 0 9 | 0 1 -3 }}
{{Mapping|legend=3| 1 0 9 | 0 1 -3 }}
: mapping generators: ~2, ~3
: mapping generators: ~2, ~3


Line 1,220: Line 1,264:


[[Badness]] (Sintel): 0.0294
[[Badness]] (Sintel): 0.0294
=== Deviaug ===
Deviaug tempers out [[6912/6859]] in the 2.3.19 subgroup, setting [[24/19]] to 1/3 of an octave.
[[Subgroup]]: 2.3.19
[[Comma list]]: 6912/6859
{{Mapping|legend=3| 3 0 8 | 0 1 1 }}
: mapping generators: ~24/19, ~3
[[Optimal tuning]]s:
* [[WE]]: ~24/19 = 399.8571{{c}}, ~3/2 = 701.9795{{c}}
: [[error map]]: {{val| -0.429 -0.404 +2.894 }}
* [[CWE]]: ~24/19 = 400.0000{{c}}, ~3/2 = 701.8810{{c}}
: error map: {{val| 0.000 -0.074 +4.368 }}
{{Optimal ET sequence|legend=1| 9, 12, 75, 87, 99, 111, 123h, 135h }}
[[Badness]] (Sintel): 0.237


=== Lipsett ===
=== Lipsett ===
Line 1,228: Line 1,292:
[[Comma list]]: 2187/2116
[[Comma list]]: 2187/2116


{{Mapping|legend=2| 1 0 -1 | 0 2 7 }}
{{Mapping|legend=3| 1 0 -1 | 0 2 7 }}
: mapping generators: ~2, ~46/27
: mapping generators: ~2, ~46/27


Line 1,246: Line 1,310:
[[Comma list]]: 24576/24389
[[Comma list]]: 24576/24389


{{Mapping|legend=2| 1 2 5 | 0 -3 -1 }}
{{Mapping|legend=3| 1 2 5 | 0 -3 -1 }}
: mapping generators: ~2, ~32/29
: mapping generators: ~2, ~32/29


Line 1,258: Line 1,322:


[[Badness]] (Sintel): 0.317
[[Badness]] (Sintel): 0.317
=== Jackpot ===
Named by {{u|CompactStar}} in 2024, jackpot tempers out the [[7-29-comma]] in 2.3.29, setting 32/29 to one step of [[7edo]] with an independent generator for the [[3/1|3rd harmonic]]. The name refers to the [[Temperament merging|EDO join]] of 7 & 77.
[[Subgroup]]: 2.3.29
[[Comma list]]: 17249876309/17179869184
{{Mapping|legend=3| 7 0 34 | 0 1 0 }}
: mapping generators: ~32/29, ~3
[[Optimal tuning]]s:
* [[WE]]: ~32/29 = 171.4434{{c}}, ~3/2 = 701.8515{{c}}
: [[error map]]: {{val| +0.103 -0.000 -0.503 }}
* [[CWE]]: ~32/29 = 171.4286{{c}}, ~3/2 = 701.8456{{c}}
: error map: {{val| 0.000 -0.109 -1.006 }}
{{Optimal ET sequence|legend=1| 7, 42, 49, 56, 63, 70, 77, 147, 224 }}
[[Badness]] (Sintel): 1.02


=== Sematology ===
=== Sematology ===
Line 1,266: Line 1,350:
[[Comma list]]: 4107/4096
[[Comma list]]: 4107/4096


{{Mapping|legend=2| 1 0 6 | 0 2 -1 }}
{{Mapping|legend=3| 1 0 6 | 0 2 -1 }}
: mapping generators: ~2, ~64/37
: mapping generators: ~2, ~64/37


Line 1,284: Line 1,368:
[[Comma list]]: 81/74
[[Comma list]]: 81/74


{{Mapping|legend=2| 1 0 -1 | 0 1 4 }}
{{Mapping|legend=3| 1 0 -1 | 0 1 4 }}
: mapping generators: ~2, ~3
: mapping generators: ~2, ~3


Line 1,304: Line 1,388:
[[Comma list]]: 82/81
[[Comma list]]: 82/81


{{Mapping|legend=2| 1 0 -1 | 0 1 4 }}
{{Mapping|legend=3| 1 0 -1 | 0 1 4 }}
: mapping generators: ~2, ~3
: mapping generators: ~2, ~3


Line 1,317: Line 1,401:
[[Badness]] (Sintel): 0.0841
[[Badness]] (Sintel): 0.0841


[[Category:Temperament collections]]
[[Category:Catalogs of subgroup temperaments]]
[[Category:Subgroup temperaments]]
[[Category:Catalogs of rank-2 temperaments]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/No-fives_subgroup_temperaments"