Septiennealimmal clan: Difference between revisions

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→Ennealimnic: + distinguish (very similar name; just letter "m" and "n" difference)
m Units & misc. cleanup. + category
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{{Mapping|legend=2| 9 0 11 | 0 1 1 }}
{{Mapping|legend=2| 9 0 11 | 0 1 1 }}
 
: mapping generators: ~2592/2401, ~3
: sval mapping generators: ~2592/2401, ~3


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2592/2401 = 133.3333, ~3/2 = 702.0044
* [[CTE]]: ~2592/2401 = 133.3333{{c}}, ~3/2 = 702.0044{{c}}
: [[error map]]: {{val| 0.0000 +0.0494 -0.1549 }}
: [[error map]]: {{val| 0.0000 +0.0494 -0.1549 }}
* [[POTE]]: ~2592/2401 = 133.3333, ~3/2 = 701.9649
* [[POTE]]: ~2592/2401 = 133.3333{{c}}, ~3/2 = 701.9649{{c}}
: error map: {{val| 0.0000 +0.0099 -0.1944 }}
: error map: {{val| 0.0000 +0.0099 -0.1944 }}


Line 40: Line 39:
{{Mapping|legend=2| 9 0 11 19 | 0 1 1 1 }}
{{Mapping|legend=2| 9 0 11 19 | 0 1 1 1 }}


[[Optimal tuning]] ([[CTE]]): ~13/12 = 133.333, ~3/2 = 702.638
[[Optimal tuning]] ([[CTE]]): ~13/12 = 133.333{{c}}, ~3/2 = 702.638{{c}}


{{Optimal ET sequence|legend=1| 27, 36, 99, 135f, 171f }}
{{Optimal ET sequence|legend=1| 27, 36, 99, 135f, 171f }}
Line 58: Line 57:


Optimal tunings:  
Optimal tunings:  
* CTE: ~121/112 = 133.3333, 343/198 = 951.0143 (~99/98 = 17.6810)
* CTE: ~121/112 = 133.3333{{c}}, 343/198 = 951.0143{{c}} (~99/98 = 17.6810{{c}})
* POTE: ~121/112 = 133.3333, 343/198 = 950.9591 (~99/98 = 17.6258)
* POTE: ~121/112 = 133.3333{{c}}, 343/198 = 950.9591{{c}} (~99/98 = 17.6258{{c}})


{{Optimal ET sequence|legend=0| 54, 63, 72, 135, 342, 477, 1089, 1566 }}
{{Optimal ET sequence|legend=0| 54, 63, 72, 135, 342, 477, 1089, 1566 }}
Line 84: Line 83:


{{Mapping|legend=1| 9 1 1 12 | 0 2 3 2 }}
{{Mapping|legend=1| 9 1 1 12 | 0 2 3 2 }}
: mapping generators: ~27/25, ~5/3
: mapping generators: ~27/25, ~5/3


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~27/25 = 133.3333, ~5/3 = 884.3317 (~36/35 = 49.0016)
* [[CTE]]: ~27/25 = 133.3333{{c}}, ~5/3 = 884.3317{{c}} (~36/35 = 49.0016{{c}})
: [[error map]]: {{val| 0.0000 +0.0416 +0.0146 -0.1626 }}
: [[error map]]: {{val| 0.0000 +0.0416 +0.0146 -0.1626 }}
* [[POTE]]: ~27/25 = 133.3333, ~5/3 = 884.3129 (~36/35 = 49.0205)
* [[POTE]]: ~27/25 = 133.3333{{c}}, ~5/3 = 884.3129{{c}} (~36/35 = 49.0205{{c}})
: error map: {{val| 0.0000 +0.0040 -0.0418 -0.2002 }}
: error map: {{val| 0.0000 +0.0040 -0.0418 -0.2002 }}


Line 112: Line 110:


Optimal tunings:  
Optimal tunings:  
* CTE: ~27/25 = 133.3333, ~5/3 = 884.4385 (~36/35 = 48.8948)
* CTE: ~27/25 = 133.3333{{c}}, ~5/3 = 884.4385 (~36/35 = 48.8948{{c}})
* POTE: ~27/25 = 133.3333, ~5/3 = 884.4679 (~36/35 = 48.8654)
* POTE: ~27/25 = 133.3333{{c}}, ~5/3 = 884.4679 (~36/35 = 48.8654{{c}})


{{Optimal ET sequence|legend=0| 99e, 171e, 270, 909, 1179, 1449c, 1719c }}
{{Optimal ET sequence|legend=0| 99e, 171e, 270, 909, 1179, 1449c, 1719c }}
Line 127: Line 125:


Optimal tunings:  
Optimal tunings:  
* CTE: ~27/25 = 133.3333, ~5/3 = 884.4285 (~36/35 = 48.9048)
* CTE: ~27/25 = 133.3333{{c}}, ~5/3 = 884.4285{{c}} (~36/35 = 48.9048{{c}})
* POTE: ~27/25 = 133.3333, ~5/3 = 884.4304 (~36/35 = 48.9030)
* POTE: ~27/25 = 133.3333{{c}}, ~5/3 = 884.4304{{c}} (~36/35 = 48.9030{{c}})


{{Optimal ET sequence|legend=0| 99e, 171e, 270 }}
{{Optimal ET sequence|legend=0| 99e, 171e, 270 }}
Line 142: Line 140:


Optimal tunings:  
Optimal tunings:  
* CTE: ~27/25 = 133.3333, ~5/3 = 884.4110 (~36/35 = 48.9223)
* CTE: ~27/25 = 133.3333{{c}}, ~5/3 = 884.4110{{c}} (~36/35 = 48.9223{{c}})
* POTE: ~27/25 = 133.3333, ~5/3 = 884.4234 (~36/35 = 48.9099)
* POTE: ~27/25 = 133.3333{{c}}, ~5/3 = 884.4234{{c}} (~36/35 = 48.9099{{c}})


{{Optimal ET sequence|legend=0| 99e, 171e, 270 }}
{{Optimal ET sequence|legend=0| 99e, 171e, 270 }}
Line 155: Line 153:


Optimal tunings:  
Optimal tunings:  
* CTE: ~27/25 = 133.3333, ~5/3 = 884.4139 (~36/35 = 48.9194)
* CTE: ~27/25 = 133.3333{{c}}, ~5/3 = 884.4139{{c}} (~36/35 = 48.9194{{c}})
* POTE: ~27/25 = 133.3333, ~5/3 = 884.4270 (~36/35 = 48.9063)
* POTE: ~27/25 = 133.3333{{c}}, ~5/3 = 884.4270{{c}} (~36/35 = 48.9063{{c}})


{{Optimal ET sequence|legend=0| 99e, 171e, 270 }}
{{Optimal ET sequence|legend=0| 99e, 171e, 270 }}
Line 168: Line 166:


Optimal tunings:  
Optimal tunings:  
* CTE: ~27/25 = 133.3333, ~5/3 = 884.4560 (~36/35 = 48.8773)
* CTE: ~27/25 = 133.3333{{c}}, ~5/3 = 884.4560{{c}} (~36/35 = 48.8773{{c}})
* CWE: ~27/25 = 133.3333, ~5/3 = 884.4745 (~36/35 = 48.8588)
* CWE: ~27/25 = 133.3333{{c}}, ~5/3 = 884.4745{{c}} (~36/35 = 48.8588{{c}})


{{Optimal ET sequence|legend=0| 99ef, 171ef, 270, 639, 909, 1179, 2088bce }}
{{Optimal ET sequence|legend=0| 99ef, 171ef, 270, 639, 909, 1179, 2088bce }}
Line 185: Line 183:


Optimal tunings:  
Optimal tunings:  
* CTE: ~27/25 = 133.3333, ~5/3 = 884.4113 (~36/35 = 48.9220)
* CTE: ~27/25 = 133.3333{{c}}, ~5/3 = 884.4113{{c}} (~36/35 = 48.9220{{c}})
* POTE: ~27/25 = 133.3333, ~5/3 = 884.4089 (~36/35 = 48.9244)
* POTE: ~27/25 = 133.3333{{c}}, ~5/3 = 884.4089{{c}} (~36/35 = 48.9244{{c}})


{{Optimal ET sequence|legend=0| 99, 171, 270, 711, 981, 1251, 2232e }}
{{Optimal ET sequence|legend=0| 99, 171, 270, 711, 981, 1251, 2232e }}
Line 200: Line 198:


Optimal tunings:  
Optimal tunings:  
* CTE: ~27/25 = 133.3333, ~5/3 = 884.4055 (~36/35 = 48.9278)
* CTE: ~27/25 = 133.3333{{c}}, ~5/3 = 884.4055{{c}} (~36/35 = 48.9278{{c}})
* POTE: ~27/25 = 133.3333, ~5/3 = 884.3997 (~36/35 = 48.9336)
* POTE: ~27/25 = 133.3333{{c}}, ~5/3 = 884.3997{{c}} (~36/35 = 48.9336{{c}})


{{Optimal ET sequence|legend=0| 99, 171, 270, 711, 981, 1692e }}
{{Optimal ET sequence|legend=0| 99, 171, 270, 711, 981, 1692e }}
Line 215: Line 213:


Optimal tunings:  
Optimal tunings:  
* CTE: ~27/25 = 133.3333, ~5/3 = 884.3867 (~36/35 = 48.9466)
* CTE: ~27/25 = 133.3333{{c}}, ~5/3 = 884.3867{{c}} (~36/35 = 48.9466{{c}})
* POTE: ~27/25 = 133.3333, ~5/3 = 884.3808 (~36/35 = 48.9525)
* POTE: ~27/25 = 133.3333{{c}}, ~5/3 = 884.3808{{c}} (~36/35 = 48.9525{{c}})


{{Optimal ET sequence|legend=0| 99, 171, 270 }}
{{Optimal ET sequence|legend=0| 99, 171, 270 }}
Line 228: Line 226:


Optimal tunings:  
Optimal tunings:  
* CTE: ~27/25 = 133.3333, ~5/3 = 884.3960 (~36/35 = 48.9373)
* CTE: ~27/25 = 133.3333{{c}}, ~5/3 = 884.3960{{c}} (~36/35 = 48.9373{{c}})
* POTE: ~27/25 = 133.3333, ~5/3 = 884.3985 (~36/35 = 48.9348)
* POTE: ~27/25 = 133.3333{{c}}, ~5/3 = 884.3985{{c}} (~36/35 = 48.9348{{c}})


{{Optimal ET sequence|legend=0| 99, 171, 270 }}
{{Optimal ET sequence|legend=0| 99, 171, 270 }}


=== Ennealimnic ===
=== Ennealimnic ===
{{Distinguish|Ennealimmic}}
{{Distinguish| Ennealimmic }}
 
Ennealimnic ({{nowrap| 72 & 171 }}) equates 11/9 with 27/22, 49/40, and 60/49 as a neutral third interval.
Ennealimnic ({{nowrap| 72 & 171 }}) equates 11/9 with 27/22, 49/40, and 60/49 as a neutral third interval.


Line 244: Line 243:


Optimal tunings:  
Optimal tunings:  
* CTE: ~27/25 = 133.3333, ~5/3 = 884.0697 (~36/35 = 49.2636)
* CTE: ~27/25 = 133.3333{{c}}, ~5/3 = 884.0697{{c}} (~36/35 = 49.2636{{c}})
* POTE: ~27/25 = 133.3333, ~5/3 = 883.9386 (~36/35 = 49.3948)
* POTE: ~27/25 = 133.3333{{c}}, ~5/3 = 883.9386{{c}} (~36/35 = 49.3948{{c}})


Tuning ranges:  
Tuning ranges:  
Line 265: Line 264:


Optimal tunings:  
Optimal tunings:  
* CTE: ~27/25 = 133.3333, ~5/3 = 884.0603 (~36/35 = 49.2730)
* CTE: ~27/25 = 133.3333{{c}}, ~5/3 = 884.0603{{c}} (~36/35 = 49.2730{{c}})
* POTE: ~27/25 = 133.3333, ~5/3 = 883.9920 (~36/35 = 49.3414)
* POTE: ~27/25 = 133.3333{{c}}, ~5/3 = 883.9920{{c}} (~36/35 = 49.3414{{c}})


Tuning ranges:  
Tuning ranges:  
Line 284: Line 283:


Optimal tunings:  
Optimal tunings:  
* CTE: ~27/25 = 133.3333, ~5/3 = 884.0742 (~36/35 = 49.2591)
* CTE: ~27/25 = 133.3333{{c}}, ~5/3 = 884.0742{{c}} (~36/35 = 49.2591{{c}})
* POTE: ~27/25 = 133.3333, ~5/3 = 883.9981 (~36/35 = 49.3353)
* POTE: ~27/25 = 133.3333{{c}}, ~5/3 = 883.9981{{c}} (~36/35 = 49.3353{{c}})


Tuning ranges:  
Tuning ranges:  
Line 303: Line 302:


Optimal tunings:  
Optimal tunings:  
* CTE: ~27/25 = 133.3333, ~5/3 = 884.0366 (~36/35 = 49.2967)
* CTE: ~27/25 = 133.3333{{c}}, ~5/3 = 884.0366{{c}} (~36/35 = 49.2967{{c}})
* CWE: ~27/25 = 133.3333, ~5/3 = 883.9630 (~36/35 = 49.3703)
* CWE: ~27/25 = 133.3333{{c}}, ~5/3 = 883.9630{{c}} (~36/35 = 49.3703{{c}})


{{Optimal ET sequence|legend=0| 72, 171, 243 }}
{{Optimal ET sequence|legend=0| 72, 171, 243 }}
Line 316: Line 315:


Optimal tunings:  
Optimal tunings:  
* CTE: ~13/12 = 133.3333, ~5/3 = 884.2055 (~36/35 = 49.1278)
* CTE: ~13/12 = 133.3333{{c}}, ~5/3 = 884.2055{{c}} (~36/35 = 49.1278{{c}})
* POTE: ~13/12 = 133.3333, ~5/3 = 883.6257 (~36/35 = 49.7076)
* POTE: ~13/12 = 133.3333{{c}}, ~5/3 = 883.6257{{c}} (~36/35 = 49.7076{{c}})


{{Optimal ET sequence|legend=0| 27e, 45ef, 72 }}
{{Optimal ET sequence|legend=0| 27e, 45ef, 72 }}
Line 331: Line 330:


Optimal tunings:  
Optimal tunings:  
* CTE: ~13/12 = 133.3333, ~5/3 = 884.1935 (~36/35 = 49.1398)
* CTE: ~13/12 = 133.3333{{c}}, ~5/3 = 884.1935{{c}} (~36/35 = 49.1398{{c}})
* POTE: ~13/12 = 133.3333, ~5/3 = 883.6257 (~36/35 = 49.7076)
* POTE: ~13/12 = 133.3333{{c}}, ~5/3 = 883.6257{{c}} (~36/35 = 49.7076{{c}})


{{Optimal ET sequence|legend=0| 27eg, 45efg, 72 }}
{{Optimal ET sequence|legend=0| 27eg, 45efg, 72 }}
Line 344: Line 343:


Optimal tunings:  
Optimal tunings:  
* CTE: ~13/12 = 133.3333, ~5/3 = 884.1388 (~36/35 = 49.1945)
* CTE: ~13/12 = 133.3333{{c}}, ~5/3 = 884.1388{{c}} (~36/35 = 49.1945{{c}})
* POTE: ~13/12 = 133.3333, ~5/3 = 883.6257 (~36/35 = 49.7076)
* POTE: ~13/12 = 133.3333{{c}}, ~5/3 = 883.6257{{c}} (~36/35 = 49.7076{{c}})


{{Optimal ET sequence|legend=0| 27eg, 45efg, 72 }}
{{Optimal ET sequence|legend=0| 27eg, 45efg, 72 }}
Line 357: Line 356:


Optimal tunings:  
Optimal tunings:  
* CTE: ~27/25 = 133.3333, ~5/3 = 884.0925 (~36/35 = 49.2408)
* CTE: ~27/25 = 133.3333{{c}}, ~5/3 = 884.0925{{c}} (~36/35 = 49.2408{{c}})
* POTE: ~27/25 = 133.3333, ~5/3 = 883.8298 (~36/35 = 49.5036)
* POTE: ~27/25 = 133.3333{{c}}, ~5/3 = 883.8298{{c}} (~36/35 = 49.5036{{c}})


{{Optimal ET sequence|legend=0| 27, 45, 72, 171e, 243e, 315e }}
{{Optimal ET sequence|legend=0| 27, 45, 72, 171e, 243e, 315e }}
Line 372: Line 371:


Optimal tunings:  
Optimal tunings:  
* CTE: ~13/12 = 133.3333, ~5/3 = 884.2648 (~36/35 = 49.0685)
* CTE: ~13/12 = 133.3333{{c}}, ~5/3 = 884.2648{{c}} (~36/35 = 49.0685{{c}})
* POTE: ~13/12 = 133.3333, ~5/3 = 883.8476 (~36/35 = 49.4857)
* POTE: ~13/12 = 133.3333{{c}}, ~5/3 = 883.8476{{c}} (~36/35 = 49.4857{{c}})


{{Optimal ET sequence|legend=0| 27, 45f, 72, 171ef, 243eff }}
{{Optimal ET sequence|legend=0| 27, 45f, 72, 171ef, 243eff }}
Line 387: Line 386:


Optimal tunings:  
Optimal tunings:  
* CTE: ~13/12 = 133.3333, ~5/3 = 884.1032 (~36/35 = 49.2301)
* CTE: ~13/12 = 133.3333{{c}}, ~5/3 = 884.1032{{c}} (~36/35 = 49.2301{{c}})
* POTE: ~13/12 = 133.3333, ~5/3 = 883.8476 (~36/35 = 49.4857)
* POTE: ~13/12 = 133.3333{{c}}, ~5/3 = 883.8476{{c}} (~36/35 = 49.4857{{c}})


{{Optimal ET sequence|legend=0| 27, 45f, 72 }}
{{Optimal ET sequence|legend=0| 27, 45f, 72 }}
Line 400: Line 399:


Optimal tunings:  
Optimal tunings:  
* CTE: ~13/12 = 133.3333, ~5/3 = 884.0186 (~36/35 = 49.3147)
* CTE: ~13/12 = 133.3333{{c}}, ~5/3 = 884.0186{{c}} (~36/35 = 49.3147{{c}})
* POTE: ~13/12 = 133.3333, ~5/3 = 883.8476 (~36/35 = 49.4857)
* POTE: ~13/12 = 133.3333{{c}}, ~5/3 = 883.8476{{c}} (~36/35 = 49.4857{{c}})


{{Optimal ET sequence|legend=0| 27, 45f, 72 }}
{{Optimal ET sequence|legend=0| 27, 45f, 72 }}
Line 415: Line 414:


Mapping: {{mapping| 18 0 -1 22 48 | 0 2 3 2 1 }}
Mapping: {{mapping| 18 0 -1 22 48 | 0 2 3 2 1 }}
: mapping generators: ~80/77, ~400/231
: mapping generators: ~80/77, ~400/231


Optimal tunings:  
Optimal tunings:  
* POTE: ~80/77 = 66.6667, ~400/231 = 950.9553
* POTE: ~80/77 = 66.6667{{c}}, ~400/231 = 950.9553{{c}}


Tuning ranges:  
Tuning ranges:  
Line 437: Line 435:


Optimal tunings:  
Optimal tunings:  
* POTE: ~27/26 = 66.6667, ~26/15 = 951.0837
* POTE: ~27/26 = 66.6667{{c}}, ~26/15 = 951.0837{{c}}


Tuning ranges:  
Tuning ranges:  
Line 457: Line 455:


Optimal tunings:  
Optimal tunings:  
* POTE: ~27/26 = 66.6667, ~26/15 = 951.0837
* POTE: ~27/26 = 66.6667{{c}}, ~26/15 = 951.0837{{c}}


{{Optimal ET sequence|legend=0| 72, 198g, 270 }}
{{Optimal ET sequence|legend=0| 72, 198g, 270 }}
Line 469: Line 467:


Optimal tunings:  
Optimal tunings:  
* POTE: ~27/26 = 66.6667, ~26/15 = 951.0837
* POTE: ~27/26 = 66.6667{{c}}, ~26/15 = 951.0837{{c}}


{{Optimal ET sequence|legend=1| 72, 198g, 270 }}
{{Optimal ET sequence|legend=1| 72, 198g, 270 }}
Line 483: Line 481:


Optimal tunings:  
Optimal tunings:  
* POTE: ~80/77 = 66.6667, ~1053/800 = 475.4727
* POTE: ~80/77 = 66.6667{{c}}, ~1053/800 = 475.4727{{c}}


{{Optimal ET sequence|legend=0| 126, 144, 270, 684, 954 }}
{{Optimal ET sequence|legend=0| 126, 144, 270, 684, 954 }}
Line 495: Line 493:


Mapping: {{mapping| 18 0 -1 22 48 88 -119 | 0 4 6 4 2 -3 27 }}
Mapping: {{mapping| 18 0 -1 22 48 88 -119 | 0 4 6 4 2 -3 27 }}
: mapping generators: ~80/77, ~1053/800


Optimal tunings:  
Optimal tunings:  
* POTE: ~80/77 = 66.6667, ~1053/800 = 475.4727
* POTE: ~80/77 = 66.6667{{c}}, ~1053/800 = 475.4727{{c}}


{{Optimal ET sequence|legend=0| 270, 684, 954 }}
{{Optimal ET sequence|legend=0| 270, 684, 954 }}
Line 511: Line 507:


Mapping: {{mapping| 18 0 -1 22 48 88 -119 -2 | 0 4 6 4 2 -3 27 11 }}
Mapping: {{mapping| 18 0 -1 22 48 88 -119 -2 | 0 4 6 4 2 -3 27 11 }}
: mapping generators: ~80/77, ~1053/800


Optimal tunings:  
Optimal tunings:  
* POTE: ~80/77 = 66.6667, ~1053/800 = 475.4727
* POTE: ~80/77 = 66.6667{{c}}, ~1053/800 = 475.4727{{c}}


{{Optimal ET sequence|legend=0| 270, 684h, 954h, 1224 }}
{{Optimal ET sequence|legend=0| 270, 684h, 954h, 1224 }}
Line 529: Line 523:


Mapping: {{mapping| 9 3 4 14 18 | 0 6 9 6 7 }}
Mapping: {{mapping| 9 3 4 14 18 | 0 6 9 6 7 }}
: mapping generators: ~27/25, ~140/121
: mapping generators: ~27/25, ~140/121


Optimal tunings:  
Optimal tunings:  
* POTE: ~27/25 = 133.3333, ~140/121 = 250.3367
* POTE: ~27/25 = 133.3333{{c}}, ~140/121 = 250.3367{{c}}


{{Optimal ET sequence|legend=0| 72, 369, 441 }}
{{Optimal ET sequence|legend=0| 72, 369, 441 }}
Line 547: Line 540:


Optimal tunings:  
Optimal tunings:  
* POTE: ~27/25 = 133.3333, ~140/121 = 250.3375
* POTE: ~27/25 = 133.3333{{c}}, ~140/121 = 250.3375{{c}}


{{Optimal ET sequence|legend=0| 72, 297ef, 369f, 441 }}
{{Optimal ET sequence|legend=0| 72, 297ef, 369f, 441 }}
Line 559: Line 552:


Mapping: {{mapping| 9 1 1 12 -7 | 0 8 12 8 23 }}
Mapping: {{mapping| 9 1 1 12 -7 | 0 8 12 8 23 }}
: mapping generators: ~27/25, ~25/22
: mapping generators: ~27/25, ~25/22


Optimal tunings:  
Optimal tunings:  
* POTE: ~27/25 = 133.3333, ~25/22 = 221.0717
* POTE: ~27/25 = 133.3333{{c}}, ~25/22 = 221.0717{{c}}


{{Optimal ET sequence|legend=0| 342, 1053, 1395, 1737, 4869dd, 6606cdd }}
{{Optimal ET sequence|legend=0| 342, 1053, 1395, 1737, 4869dd, 6606cdd }}
Line 575: Line 567:


Mapping: {{mapping| 27 1 0 34 177 | 0 2 3 2 -4 }}
Mapping: {{mapping| 27 1 0 34 177 | 0 2 3 2 -4 }}
: mapping generators: ~2744/2673, ~2352/1375
: mapping generators: ~2744/2673, ~2352/1375


Optimal tunings:  
Optimal tunings:  
* POTE: ~2744/2673 = 44.4444, ~2352/1375 = 928.8000
* POTE: ~2744/2673 = 44.4444{{c}}, ~2352/1375 = 928.8000{{c}}


{{Optimal ET sequence|legend=0| 27, 243, 270, 783, 1053, 1323 }}
{{Optimal ET sequence|legend=0| 27, 243, 270, 783, 1053, 1323 }}
Line 595: Line 586:


Mapping: {{mapping| 45 1 -1 56 226 | 0 2 3 2 -2 }}
Mapping: {{mapping| 45 1 -1 56 226 | 0 2 3 2 -2 }}
: mapping generators: ~3072/3025, ~55/32
: mapping generators: ~3072/3025, ~55/32


Optimal tunings:  
Optimal tunings:  
* CTE: ~3072/3025 = 26.6667, ~55/32 = 937.6658 (~385/384 = 4.3325)
* CTE: ~3072/3025 = 26.6667{{c}}, ~55/32 = 937.6658{{c}} (~385/384 = 4.3325{{c}})
* CWE: ~3072/3025 = 26.6667, ~55/32 = 937.6630 (~385/384 = 4.3397)
* CWE: ~3072/3025 = 26.6667{{c}}, ~55/32 = 937.6630{{c}} (~385/384 = 4.3397{{c}})


{{Optimal ET sequence|legend=0| 45, 225c, 270, 1125, 1395, 1665, 5265d }}
{{Optimal ET sequence|legend=0| 45, 225c, 270, 1125, 1395, 1665, 5265d }}
Line 614: Line 604:


Optimal tunings:  
Optimal tunings:  
* CTE: ~66/65 = 26.6667, ~55/32 = 937.6569 (~385/384 = 4.3236)
* CTE: ~66/65 = 26.6667{{c}}, ~55/32 = 937.6569{{c}} (~385/384 = 4.3236{{c}})
* CWE: ~66/65 = 26.6667, ~55/32 = 937.6515 (~385/384 = 4.3182)
* CWE: ~66/65 = 26.6667{{c}}, ~55/32 = 937.6515{{c}} (~385/384 = 4.3182{{c}})


{{Optimal ET sequence|legend=0| 45, 270, 855, 1125, 1395, 1665, 3060d, 4725df }}
{{Optimal ET sequence|legend=0| 45, 270, 855, 1125, 1395, 1665, 3060d, 4725df }}
Line 631: Line 621:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[POTE]]: ~126/125 = 12.121, ~11/8 = 552.756
* [[POTE]]: ~126/125 = 12.121{{c}}, ~11/8 = 552.756{{c}}


{{Optimal ET sequence|legend=1| 99e, 198, 297e, 495ce }}
{{Optimal ET sequence|legend=1| 99e, 198, 297e, 495ce }}
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Optimal tunings:  
Optimal tunings:  
* POTE: ~144/143 = 12.121, ~11/8 = 552.024
* POTE: ~144/143 = 12.121{{c}}, ~11/8 = 552.024{{c}}


{{Optimal ET sequence|legend=0| 99ef, 198 }}
{{Optimal ET sequence|legend=0| 99ef, 198 }}
Line 661: Line 651:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[POTE]]: ~225/224 = 7.018, ~11/8 = 550.954
* [[POTE]]: ~225/224 = 7.018{{c}}, ~11/8 = 550.954{{c}}


{{Optimal ET sequence|legend=1| 171, 342 }}
{{Optimal ET sequence|legend=1| 171, 342 }}
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Optimal tunings:  
Optimal tunings:  
* POTE: ~225/224 = 7.018, ~11/8 = 551.322
* POTE: ~225/224 = 7.018{{c}}, ~11/8 = 551.322{{c}}


{{Optimal ET sequence|legend=0| 171, 342, 855ff, 1197fff }}
{{Optimal ET sequence|legend=0| 171, 342, 855ff, 1197fff }}
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Optimal tunings:  
Optimal tunings:  
* POTE: ~225/224 = 7.018, ~11/8 = 551.365
* POTE: ~225/224 = 7.018{{c}}, ~11/8 = 551.365{{c}}


{{Optimal ET sequence|legend=0| 171, 342, 855ff, 1197fff }}
{{Optimal ET sequence|legend=0| 171, 342, 855ff, 1197fff }}
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Optimal tunings:  
Optimal tunings:  
* POTE: ~225/224 = 7.018, ~11/8 = 551.625
* POTE: ~225/224 = 7.018{{c}}, ~11/8 = 551.625{{c}}


{{Optimal ET sequence|legend=1| 171, 342f, 513, 855f }}
{{Optimal ET sequence|legend=1| 171, 342f, 513, 855f }}
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Optimal tunings:  
Optimal tunings:  
* POTE: ~225/224 = 7.018, ~11/8 = 551.756
* POTE: ~225/224 = 7.018{{c}}, ~11/8 = 551.756{{c}}


{{Optimal ET sequence|legend=0| 171, 342f, 513, 855f }}
{{Optimal ET sequence|legend=0| 171, 342f, 513, 855f }}
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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[POTE]]: ~32805/32768 = 2.7211, ~11/8 = 551.3584
* [[POTE]]: ~32805/32768 = 2.7211{{c}}, ~11/8 = 551.3584{{c}}


{{Optimal ET sequence|legend=1| 441, 882, 1323, 2205, 3528 }}
{{Optimal ET sequence|legend=1| 441, 882, 1323, 2205, 3528 }}
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Optimal tunings:  
Optimal tunings:  
* POTE: ~729/728 = 2.7211, ~11/8 = 551.4043
* POTE: ~729/728 = 2.7211{{c}}, ~11/8 = 551.4043{{c}}


{{Optimal ET sequence|legend=0| 441, 882, 1323, 3528f, 4851ff, 6174dff }}
{{Optimal ET sequence|legend=0| 441, 882, 1323, 3528f, 4851ff, 6174dff }}
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Optimal tunings:  
Optimal tunings:  
* POTE: ~729/728 = 2.7211, ~11/8 = 551.3688
* POTE: ~729/728 = 2.7211{{c}}, ~11/8 = 551.3688{{c}}


{{Optimal ET sequence|legend=0| 441, 882, 1323, 2205f, 3528f }}
{{Optimal ET sequence|legend=0| 441, 882, 1323, 2205f, 3528f }}
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Badness (Smith): 0.029334
Badness (Smith): 0.029334


[[Category:Septiennealimmal clan| ]] <!-- main article -->
[[Category:Temperament clans]]
[[Category:Temperament clans]]

Revision as of 07:54, 20 March 2026

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The septiennealimmal clan of temperaments tempers out the septimal ennealimma (monzo: [-11 -9 0 9⟩, ratio: 40353607/40310784). Primarily, this clan includes the 7-limit ennealimmal temperament and extensions of it.

Temperaments discussed elsewhere are:

No-five septiennealimmal

This rank-2 temperament is of interest to anyone who wants a different generator for the ennealimmal-like structure because it represents the part of ennealimmal supported by non-ennealimmal equal temperaments of interest that do well in the 2.3.7 subgroup, such as 36edo, which adds the gamelisma, or 63edo, which in the 7-limit can be used for magic and in higher limits for parapyth among other things.

Subgroup: 2.3.7

Comma list: 40353607/40310784

Subgroup-val mapping: [⟨9 0 11], ⟨0 1 1]]

mapping generators: ~2592/2401, ~3

Optimal tunings:

  • CTE: ~2592/2401 = 133.3333 ¢, ~3/2 = 702.0044 ¢
error map: ⟨0.0000 +0.0494 -0.1549]
  • POTE: ~2592/2401 = 133.3333 ¢, ~3/2 = 701.9649 ¢
error map: ⟨0.0000 +0.0099 -0.1944]

Optimal ET sequence: 27, 36, 99, 135, 171, 306, 4419d, 4725d, …, 8397dd, 8703dd

Septiennealic

Septiennealic finds a somewhat high-damage but very simple and intuitive mapping of prime 13 by fixing 13/12~14/13 at 1\9.

A notable tuning of septiennealic not appearing in the optimal ET sequence is 63edo. If we include a somewhat more complex mapping for 11 via 36e & 63, it will become the optimal patent val and largest in the sequence.

Subgroup: 2.3.7.13

Comma list: 169/168, 31213/31104

Subgroup-val mapping: [⟨9 0 11 19], ⟨0 1 1 1]]

Optimal tuning (CTE): ~13/12 = 133.333 ¢, ~3/2 = 702.638 ¢

Optimal ET sequence: 27, 36, 99, 135f, 171f

Badness:

  • Smith: 0.0134
  • Dirichlet: 0.540

Ennea

See also: No-fives subgroup temperaments

Subgroup: 2.3.7.11

Comma list: 41503/41472, 43923/43904

Mapping: [⟨9 0 11 24], ⟨0 2 2 1]]

Optimal tunings:

  • CTE: ~121/112 = 133.3333 ¢, 343/198 = 951.0143 ¢ (~99/98 = 17.6810 ¢)
  • POTE: ~121/112 = 133.3333 ¢, 343/198 = 950.9591 ¢ (~99/98 = 17.6258 ¢)

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

Ennealimmal

Main article: Ennealimmal
For the 5-limit version, see Ennealimma #Ennealimmal.

Ennealimmal tempers out the two smallest 7-limit superparticular commas, 2401/2400 and 4375/4374, leading to a temperament of unusual efficiency. It also tempers out the landscape comma, which is (2401/2400)/(4375/4374), and the wizma, which is (2401/2400)⋅(4375/4374).

In the 5-limit, it tempers out the ennealimma, [1 -27 18⟩, which leads to the identification of (27/25)9 with the octave, and gives ennealimmal a period of 1/9 octave. Its pergen is (P8/9, P5/2), and ploidacot enneaploid dicot. While 27/25 is a 5-limit interval, a stack of two periods equates to 7/6 because of identification by 4375/4374, and this represents 7/6 with such accuracy (a fifth of a cent flat) that there is no realistic possibility of treating ennealimmal as anything other than 7-limit.

Aside from 10/9 which has already been mentioned, possible generators include 36/35, 21/20, 6/5, 7/5 and the neutral thirds pair 49/40~60/49, all of which have their own interesting advantages. Possible tunings are 441-, 612-, or 3600edo, though its hardly likely anyone could tell the difference.

If 1/9 of an octave is too small of a period for you, you could try generator-period pairs of [3, 5], [5/3, 3], [6/5, 4/3], [4/3, 8/5] or [10/9, 4/3] (for example). In particular, people fond of the idea of "tritaves" as analogous to octaves might consider the 28- or 43-note mos with generator an approximate 5/3 within 3; for instance as given by 451/970 of a "tritave". Tetrads have a low enough complexity that (for example) there are nine 1–3/2–7/4–5/2 tetrads in the 28 notes to the tritave mos, which is equivalent in average step size to a 172⁄3 to the octave mos.

Ennealimmal extensions discussed elsewhere include omicronbeta.

7-limit ennealimmal's S-expression-based comma list is {S25/S27, S49}.

Subgroup: 2.3.5.7

Comma list: 2401/2400, 4375/4374

Mapping: [⟨9 1 1 12], ⟨0 2 3 2]]

mapping generators: ~27/25, ~5/3

Optimal tunings:

  • CTE: ~27/25 = 133.3333 ¢, ~5/3 = 884.3317 ¢ (~36/35 = 49.0016 ¢)
error map: ⟨0.0000 +0.0416 +0.0146 -0.1626]
  • POTE: ~27/25 = 133.3333 ¢, ~5/3 = 884.3129 ¢ (~36/35 = 49.0205 ¢)
error map: ⟨0.0000 +0.0040 -0.0418 -0.2002]

Tuning ranges:

  • 7-odd-limit diamond monotone: ~36/35 = [26.667, 66.667] (1\45 to 1\18)
  • 9-odd-limit diamond monotone: ~36/35 = [44.444, 53.333] (1\27 to 2\45)
  • 7- and 9-odd-limit diamond tradeoff: ~36/35 = [48.920, 49.179]

Optimal ET sequence: 27, 45, 72, 99, 171, 441, 612

Badness (Smith): 0.003610

11-limit

The ennealimmal temperament can be described as 99e & 171e, which tempers out 5632/5625 (vishdel comma) and 19712/19683 (symbiotic comma).

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 4375/4374, 5632/5625

Mapping: [⟨9 1 1 12 -75], ⟨0 2 3 2 16]]

Optimal tunings:

  • CTE: ~27/25 = 133.3333 ¢, ~5/3 = 884.4385 (~36/35 = 48.8948 ¢)
  • POTE: ~27/25 = 133.3333 ¢, ~5/3 = 884.4679 (~36/35 = 48.8654 ¢)

Optimal ET sequence: 99e, 171e, 270, 909, 1179, 1449c, 1719c

Badness (Smith): 0.027332

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1001/1000, 1716/1715, 4096/4095, 4375/4374

Mapping: [⟨9 1 1 12 -75 93], ⟨0 2 3 2 16 -9]]

Optimal tunings:

  • CTE: ~27/25 = 133.3333 ¢, ~5/3 = 884.4285 ¢ (~36/35 = 48.9048 ¢)
  • POTE: ~27/25 = 133.3333 ¢, ~5/3 = 884.4304 ¢ (~36/35 = 48.9030 ¢)

Optimal ET sequence: 99e, 171e, 270

Badness (Smith): 0.029404

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 715/714, 1001/1000, 1716/1715, 4096/4095, 4375/4374

Mapping: [⟨9 1 1 12 -75 93 -3], ⟨0 2 3 2 16 -9 6]]

Optimal tunings:

  • CTE: ~27/25 = 133.3333 ¢, ~5/3 = 884.4110 ¢ (~36/35 = 48.9223 ¢)
  • POTE: ~27/25 = 133.3333 ¢, ~5/3 = 884.4234 ¢ (~36/35 = 48.9099 ¢)

Optimal ET sequence: 99e, 171e, 270

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 715/714, 1001/1000, 1216/1215, 1716/1715, 4096/4095, 4375/4374

Mapping: [⟨9 1 1 12 -75 93 -3 -48], ⟨0 2 3 2 16 -9 6 13]]

Optimal tunings:

  • CTE: ~27/25 = 133.3333 ¢, ~5/3 = 884.4139 ¢ (~36/35 = 48.9194 ¢)
  • POTE: ~27/25 = 133.3333 ¢, ~5/3 = 884.4270 ¢ (~36/35 = 48.9063 ¢)

Optimal ET sequence: 99e, 171e, 270

Ennealimmalis

Subgroup: 2.3.5.7.11.13

Comma list: 2080/2079, 2401/2400, 4375/4374, 5632/5625

Mapping: [⟨9 1 1 12 -75 -106], ⟨0 2 3 2 16 21]]

Optimal tunings:

  • CTE: ~27/25 = 133.3333 ¢, ~5/3 = 884.4560 ¢ (~36/35 = 48.8773 ¢)
  • CWE: ~27/25 = 133.3333 ¢, ~5/3 = 884.4745 ¢ (~36/35 = 48.8588 ¢)

Optimal ET sequence: 99ef, 171ef, 270, 639, 909, 1179, 2088bce

Badness (Smith): 0.022068

Ennealimmia

The ennealimmia temperament is an alternative extension and can be described as 99 & 171, which tempers out 131072/130977 (olympia).

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 4375/4374, 131072/130977

Mapping: [⟨9 1 1 12 124], ⟨0 2 3 2 -14]]

Optimal tunings:

  • CTE: ~27/25 = 133.3333 ¢, ~5/3 = 884.4113 ¢ (~36/35 = 48.9220 ¢)
  • POTE: ~27/25 = 133.3333 ¢, ~5/3 = 884.4089 ¢ (~36/35 = 48.9244 ¢)

Optimal ET sequence: 99, 171, 270, 711, 981, 1251, 2232e

Badness (Smith): 0.026463

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 2080/2079, 2401/2400, 4096/4095, 4375/4374

Mapping: [⟨9 1 1 12 124 93], ⟨0 2 3 2 -14 -9]]

Optimal tunings:

  • CTE: ~27/25 = 133.3333 ¢, ~5/3 = 884.4055 ¢ (~36/35 = 48.9278 ¢)
  • POTE: ~27/25 = 133.3333 ¢, ~5/3 = 884.3997 ¢ (~36/35 = 48.9336 ¢)

Optimal ET sequence: 99, 171, 270, 711, 981, 1692e

Badness (Smith): 0.016607

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 936/935, 2080/2079, 2401/2400, 4096/4095, 4375/4374

Mapping: [⟨9 1 1 12 124 93 -3], ⟨0 2 3 2 -14 -9 6]]

Optimal tunings:

  • CTE: ~27/25 = 133.3333 ¢, ~5/3 = 884.3867 ¢ (~36/35 = 48.9466 ¢)
  • POTE: ~27/25 = 133.3333 ¢, ~5/3 = 884.3808 ¢ (~36/35 = 48.9525 ¢)

Optimal ET sequence: 99, 171, 270

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 936/935, 1216/1215, 2080/2079, 2401/2400, 4096/4095, 4375/4374

Mapping: [⟨9 1 1 12 124 93 -3 -48], ⟨0 2 3 2 -14 -9 6 13]]

Optimal tunings:

  • CTE: ~27/25 = 133.3333 ¢, ~5/3 = 884.3960 ¢ (~36/35 = 48.9373 ¢)
  • POTE: ~27/25 = 133.3333 ¢, ~5/3 = 884.3985 ¢ (~36/35 = 48.9348 ¢)

Optimal ET sequence: 99, 171, 270

Ennealimnic

Not to be confused with Ennealimmic.

Ennealimnic (72 & 171) equates 11/9 with 27/22, 49/40, and 60/49 as a neutral third interval.

Subgroup: 2.3.5.7.11

Comma list: 243/242, 441/440, 4375/4356

Mapping: [⟨9 1 1 12 -2], ⟨0 2 3 2 5]]

Optimal tunings:

  • CTE: ~27/25 = 133.3333 ¢, ~5/3 = 884.0697 ¢ (~36/35 = 49.2636 ¢)
  • POTE: ~27/25 = 133.3333 ¢, ~5/3 = 883.9386 ¢ (~36/35 = 49.3948 ¢)

Tuning ranges:

  • 11-odd-limit diamond monotone: ~36/35 = [44.444, 53.333] (1\27 to 2\45)
  • 11-odd-limit diamond tradeoff: ~36/35 = [48.920, 52.592]

Optimal ET sequence: 72, 171, 243

Badness (Smith): 0.020347

See also: Chords of ennealimnic

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 243/242, 364/363, 441/440, 625/624

Mapping: [⟨9 1 1 12 -2 -33], ⟨0 2 3 2 5 10]]

Optimal tunings:

  • CTE: ~27/25 = 133.3333 ¢, ~5/3 = 884.0603 ¢ (~36/35 = 49.2730 ¢)
  • POTE: ~27/25 = 133.3333 ¢, ~5/3 = 883.9920 ¢ (~36/35 = 49.3414 ¢)

Tuning ranges:

  • 13- and 15-odd-limit diamond monotone: ~36/35 = [48.485, 50.000] (4\99 to 3\72)
  • 13- and 15-odd-limit diamond tradeoff: ~36/35 = [48.825, 52.592]

Optimal ET sequence: 72, 171, 243

Badness (Smith): 0.023250

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 243/242, 364/363, 375/374, 441/440, 595/594

Mapping: [⟨9 1 1 12 -2 -33 -3], ⟨0 2 3 2 5 10 6]]

Optimal tunings:

  • CTE: ~27/25 = 133.3333 ¢, ~5/3 = 884.0742 ¢ (~36/35 = 49.2591 ¢)
  • POTE: ~27/25 = 133.3333 ¢, ~5/3 = 883.9981 ¢ (~36/35 = 49.3353 ¢)

Tuning ranges:

  • 17-odd-limit diamond monotone: ~36/35 = [48.485, 50.000] (4\99 to 3\72)
  • 17-odd-limit diamond tradeoff: ~36/35 = [46.363, 52.592]

Optimal ET sequence: 72, 171, 243

Badness (Smith): 0.014602

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 243/242, 364/363, 375/374, 441/440, 513/512, 595/594

Mapping: [⟨9 1 1 12 -2 -33 -3 78], ⟨0 2 3 2 5 10 6 -6]]

Optimal tunings:

  • CTE: ~27/25 = 133.3333 ¢, ~5/3 = 884.0366 ¢ (~36/35 = 49.2967 ¢)
  • CWE: ~27/25 = 133.3333 ¢, ~5/3 = 883.9630 ¢ (~36/35 = 49.3703 ¢)

Optimal ET sequence: 72, 171, 243

Ennealim

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 243/242, 325/324, 441/440

Mapping: [⟨9 1 1 12 -2 20], ⟨0 2 3 2 5 2]]

Optimal tunings:

  • CTE: ~13/12 = 133.3333 ¢, ~5/3 = 884.2055 ¢ (~36/35 = 49.1278 ¢)
  • POTE: ~13/12 = 133.3333 ¢, ~5/3 = 883.6257 ¢ (~36/35 = 49.7076 ¢)

Optimal ET sequence: 27e, 45ef, 72

Badness (Smith): 0.020697

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 169/168, 221/220, 243/242, 325/324, 441/440

Mapping: [⟨9 1 1 12 -2 20 -3], ⟨0 2 3 2 5 2 6]]

Optimal tunings:

  • CTE: ~13/12 = 133.3333 ¢, ~5/3 = 884.1935 ¢ (~36/35 = 49.1398 ¢)
  • POTE: ~13/12 = 133.3333 ¢, ~5/3 = 883.6257 ¢ (~36/35 = 49.7076 ¢)

Optimal ET sequence: 27eg, 45efg, 72

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 169/168, 221/220, 243/242, 325/324, 441/440

Mapping: [⟨9 1 1 12 -2 20 -3 25], ⟨0 2 3 2 5 2 6 2]]

Optimal tunings:

  • CTE: ~13/12 = 133.3333 ¢, ~5/3 = 884.1388 ¢ (~36/35 = 49.1945 ¢)
  • POTE: ~13/12 = 133.3333 ¢, ~5/3 = 883.6257 ¢ (~36/35 = 49.7076 ¢)

Optimal ET sequence: 27eg, 45efg, 72

Ennealiminal

Subgroup: 2.3.5.7.11

Comma list: 385/384, 1375/1372, 4375/4374

Mapping: [⟨9 1 1 12 51], ⟨0 2 3 2 -3]]

Optimal tunings:

  • CTE: ~27/25 = 133.3333 ¢, ~5/3 = 884.0925 ¢ (~36/35 = 49.2408 ¢)
  • POTE: ~27/25 = 133.3333 ¢, ~5/3 = 883.8298 ¢ (~36/35 = 49.5036 ¢)

Optimal ET sequence: 27, 45, 72, 171e, 243e, 315e

Badness (Smith): 0.031123

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 325/324, 385/384, 1375/1372

Mapping: [⟨9 1 1 12 51 20], ⟨0 2 3 2 -3 2]]

Optimal tunings:

  • CTE: ~13/12 = 133.3333 ¢, ~5/3 = 884.2648 ¢ (~36/35 = 49.0685 ¢)
  • POTE: ~13/12 = 133.3333 ¢, ~5/3 = 883.8476 ¢ (~36/35 = 49.4857 ¢)

Optimal ET sequence: 27, 45f, 72, 171ef, 243eff

Badness (Smith): 0.030325

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 169/168, 221/220, 325/324, 385/384, 1375/1372

Mapping: [⟨9 1 1 12 51 20 50], ⟨0 2 3 2 -3 2 -2]]

Optimal tunings:

  • CTE: ~13/12 = 133.3333 ¢, ~5/3 = 884.1032 ¢ (~36/35 = 49.2301 ¢)
  • POTE: ~13/12 = 133.3333 ¢, ~5/3 = 883.8476 ¢ (~36/35 = 49.4857 ¢)

Optimal ET sequence: 27, 45f, 72

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 153/152, 169/168, 221/220, 325/324, 385/384, 1375/1372

Mapping: [⟨9 1 1 12 51 20 50 25], ⟨0 2 3 2 -3 2 -2 2]]

Optimal tunings:

  • CTE: ~13/12 = 133.3333 ¢, ~5/3 = 884.0186 ¢ (~36/35 = 49.3147 ¢)
  • POTE: ~13/12 = 133.3333 ¢, ~5/3 = 883.8476 ¢ (~36/35 = 49.4857 ¢)

Optimal ET sequence: 27, 45f, 72

Hemiennealimmal

Hemiennealimmal (72 & 198) has a period of 1/18 octave and tempers out the four smallest superparticular commas of the 11-limit JI, 2401/2400, 3025/3024, 4375/4374, and 9801/9800. Tempering out 9801/9800 leads to an octave split into two equal parts. Notably, every one of these commas is part of one or more known infinite comma families; see directly below.

Its S-expression-based comma list is {(S22/S24 = S55 = S25/S27 × S99,) S25/S27, S49, S33/S35 = S99}.

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 3025/3024, 4375/4374

Mapping: [⟨18 0 -1 22 48], ⟨0 2 3 2 1]]

mapping generators: ~80/77, ~400/231

Optimal tunings:

  • POTE: ~80/77 = 66.6667 ¢, ~400/231 = 950.9553 ¢

Tuning ranges:

  • 11-odd-limit diamond monotone: ~99/98 = [13.333, 22.222] (1\90 to 1\54)
  • 11-odd-limit diamond tradeoff: ~99/98 = [17.304, 17.985]

Optimal ET sequence: 72, 198, 270, 342, 612, 954, 1566

Badness (Smith): 0.006283

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 676/675, 1001/1000, 1716/1715, 3025/3024

Mapping: [⟨18 0 -1 22 48 -19], ⟨0 2 3 2 1 6]]

Optimal tunings:

  • POTE: ~27/26 = 66.6667 ¢, ~26/15 = 951.0837 ¢

Tuning ranges:

  • 13-odd-limit diamond monotone: ~99/98 = [16.667, 22.222] (1\72 to 1\54)
  • 15-odd-limit diamond monotone: ~99/98 = [16.667, 19.048] (1\72 to 2\126)
  • 13-odd-limit diamond tradeoff: ~99/98 = [17.304, 18.309]
  • 15-odd-limit diamond tradeoff: ~99/98 = [17.304, 18.926]

Optimal ET sequence: 72, 198, 270

Badness (Smith): 0.012505

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 676/675, 715/714, 1001/1000, 1716/1715, 3025/3024

Mapping: [⟨18 0 -1 22 48 -19 -12], ⟨0 2 3 2 1 6 6]]

Optimal tunings:

  • POTE: ~27/26 = 66.6667 ¢, ~26/15 = 951.0837 ¢

Optimal ET sequence: 72, 198g, 270

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 676/675, 715/714, 1001/1000, 1331/1330, 1716/1715, 3025/3024

Mapping: [⟨18 0 -1 22 48 -19 -12 48 105], ⟨0 2 3 2 1 6 6 -2]]

Optimal tunings:

  • POTE: ~27/26 = 66.6667 ¢, ~26/15 = 951.0837 ¢

Optimal ET sequence: 72, 198g, 270

Semihemiennealimmal

Subgroup: 2.3.5.7.11.13

Comma list: 2401/2400, 3025/3024, 4225/4224, 4375/4374

Mapping: [⟨18 0 -1 22 48 88], ⟨0 4 6 4 2 -3]]

mapping generators: ~80/77, ~1053/800

Optimal tunings:

  • POTE: ~80/77 = 66.6667 ¢, ~1053/800 = 475.4727 ¢

Optimal ET sequence: 126, 144, 270, 684, 954

Badness (Smith): 0.013104

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 2401/2400, 2431/2430, 3025/3024, 4225/4224, 4375/4374

Mapping: [⟨18 0 -1 22 48 88 -119], ⟨0 4 6 4 2 -3 27]]

Optimal tunings:

  • POTE: ~80/77 = 66.6667 ¢, ~1053/800 = 475.4727 ¢

Optimal ET sequence: 270, 684, 954

Badness (Smith): 0.013104

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 2401/2400, 2431/2430, 2926/2925, 3025/3024, 4225/4224, 4375/4374

Mapping: [⟨18 0 -1 22 48 88 -119 -2], ⟨0 4 6 4 2 -3 27 11]]

Optimal tunings:

  • POTE: ~80/77 = 66.6667 ¢, ~1053/800 = 475.4727 ¢

Optimal ET sequence: 270, 684h, 954h, 1224

Badness (Smith): 0.013104

Semiennealimmal

Semiennealimmal tempers out 4000/3993, and uses a ~140/121 semifourth generator. Notably, however, two generator steps do not reach ~4/3, despite that the name may suggest so. In fact, it splits the generator of ennealimmal into three.

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 4000/3993, 4375/4374

Mapping: [⟨9 3 4 14 18], ⟨0 6 9 6 7]]

mapping generators: ~27/25, ~140/121

Optimal tunings:

  • POTE: ~27/25 = 133.3333 ¢, ~140/121 = 250.3367 ¢

Optimal ET sequence: 72, 369, 441

Badness (Smith): 0.034196

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1575/1573, 2080/2079, 2401/2400, 4375/4374

Mapping: [⟨9 3 4 14 18 -8], ⟨0 6 9 6 7 22]]

Optimal tunings:

  • POTE: ~27/25 = 133.3333 ¢, ~140/121 = 250.3375 ¢

Optimal ET sequence: 72, 297ef, 369f, 441

Badness (Smith): 0.026122

Quadraennealimmal

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 4375/4374, 234375/234256

Mapping: [⟨9 1 1 12 -7], ⟨0 8 12 8 23]]

mapping generators: ~27/25, ~25/22

Optimal tunings:

  • POTE: ~27/25 = 133.3333 ¢, ~25/22 = 221.0717 ¢

Optimal ET sequence: 342, 1053, 1395, 1737, 4869dd, 6606cdd

Badness (Smith): 0.021320

Trinealimmal

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 4375/4374, 2097152/2096325

Mapping: [⟨27 1 0 34 177], ⟨0 2 3 2 -4]]

mapping generators: ~2744/2673, ~2352/1375

Optimal tunings:

  • POTE: ~2744/2673 = 44.4444 ¢, ~2352/1375 = 928.8000 ¢

Optimal ET sequence: 27, 243, 270, 783, 1053, 1323

Badness (Smith): 0.029812

Rhodium

Main article: Rhodium

Rhodium splits the ennealimmal period in five parts and thereby features a period of 9 × 5 = 45. Thus the name is given after the 45th element.

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 4375/4374, 117440512/117406179

Mapping: [⟨45 1 -1 56 226], ⟨0 2 3 2 -2]]

mapping generators: ~3072/3025, ~55/32

Optimal tunings:

  • CTE: ~3072/3025 = 26.6667 ¢, ~55/32 = 937.6658 ¢ (~385/384 = 4.3325 ¢)
  • CWE: ~3072/3025 = 26.6667 ¢, ~55/32 = 937.6630 ¢ (~385/384 = 4.3397 ¢)

Optimal ET sequence: 45, 225c, 270, 1125, 1395, 1665, 5265d

Badness (Smith): 0.0381

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 2401/2400, 4225/4224, 4375/4374, 6656/6655

Mapping: [⟨45 1 -1 56 226 272], ⟨0 2 3 2 -2 -3]]

Optimal tunings:

  • CTE: ~66/65 = 26.6667 ¢, ~55/32 = 937.6569 ¢ (~385/384 = 4.3236 ¢)
  • CWE: ~66/65 = 26.6667 ¢, ~55/32 = 937.6515 ¢ (~385/384 = 4.3182 ¢)

Optimal ET sequence: 45, 270, 855, 1125, 1395, 1665, 3060d, 4725df

Badness (Smith): 0.0226

Undecentic

Undecentic (99 & 198) has a period of 1/99 octave.

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 3136/3125, 4375/4374

Mapping: [⟨99 157 230 278 0], ⟨0 0 0 0 1]]

Optimal tunings:

  • POTE: ~126/125 = 12.121 ¢, ~11/8 = 552.756 ¢

Optimal ET sequence: 99e, 198, 297e, 495ce

Badness (Smith): 0.058801

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 847/845, 2401/2400, 3136/3125

Mapping: [⟨99 157 230 278 0 24], ⟨0 0 0 0 1 1]]

Optimal tunings:

  • POTE: ~144/143 = 12.121 ¢, ~11/8 = 552.024 ¢

Optimal ET sequence: 99ef, 198

Badness (Smith): 0.042547

Schisennealimmal

Schisennealimmal (171 & 342) has a period of 1/171 octave. 171edo and its multiples are members of both schismic and ennealimmal, and from this it derives its name.

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 4375/4374, 32805/32768

Mapping: [⟨171 271 397 480 0], ⟨0 0 0 0 1]]

Optimal tunings:

  • POTE: ~225/224 = 7.018 ¢, ~11/8 = 550.954 ¢

Optimal ET sequence: 171, 342

Badness (Smith): 0.031739

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 625/624, 729/728, 2205/2197, 2401/2400

Mapping: [⟨171 271 397 480 0 633], ⟨0 0 0 0 1 0]]

Optimal tunings:

  • POTE: ~225/224 = 7.018 ¢, ~11/8 = 551.322 ¢

Optimal ET sequence: 171, 342, 855ff, 1197fff

Badness (Smith): 0.054029

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 625/624, 729/728, 833/832, 1225/1224, 2205/2197

Mapping: [⟨171 271 397 480 0 633 699], ⟨0 0 0 0 1 0 0]]

Optimal tunings:

  • POTE: ~225/224 = 7.018 ¢, ~11/8 = 551.365 ¢

Optimal ET sequence: 171, 342, 855ff, 1197fff

Badness (Smith): 0.031323

Schisennealimmic

Subgroup: 2.3.5.7.11.13

Comma list: 2080/2079, 2401/2400, 4375/4374, 32805/32768

Mapping: [⟨171 271 397 480 0 41], ⟨0 0 0 0 1 1]]

Optimal tunings:

  • POTE: ~225/224 = 7.018 ¢, ~11/8 = 551.625 ¢

Optimal ET sequence: 171, 342f, 513, 855f

Badness (Smith): 0.046843

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 936/935, 1225/1224, 1701/1700, 2025/2023, 11271/11264

Mapping: [⟨171 271 397 480 0 41 699], ⟨0 0 0 0 1 1 0]]

Optimal tunings:

  • POTE: ~225/224 = 7.018 ¢, ~11/8 = 551.756 ¢

Optimal ET sequence: 171, 342f, 513, 855f

Badness (Smith): 0.030622

Lunennealimmal

Lunennealimmal (441 & 882) has has a period of 1/441 octave. 441edo and its multiples are members of both luna and ennealimmal, and from this it derives its name.

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 4375/4374, 274877906944/274658203125

Mapping: [⟨441 699 1024 1238 1526], ⟨0 0 0 0 -1]]

Optimal tunings:

  • POTE: ~32805/32768 = 2.7211 ¢, ~11/8 = 551.3584 ¢

Optimal ET sequence: 441, 882, 1323, 2205, 3528

Badness (Smith): 0.091939

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 2401/2400, 4096/4095, 4375/4374, 85750/85683

Mapping: [⟨441 699 1024 1238 1526 1632], ⟨0 0 0 0 -1 0]]

Optimal tunings:

  • POTE: ~729/728 = 2.7211 ¢, ~11/8 = 551.4043 ¢

Optimal ET sequence: 441, 882, 1323, 3528f, 4851ff, 6174dff

Badness (Smith): 0.042975

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 2401/2400, 4096/4095, 4375/4374, 8624/8619, 14161/14157

Mapping: [⟨441 699 1024 1238 1526 1632 1803], ⟨0 0 0 0 -1 0 -1]]

Optimal tunings:

  • POTE: ~729/728 = 2.7211 ¢, ~11/8 = 551.3688 ¢

Optimal ET sequence: 441, 882, 1323, 2205f, 3528f

Badness (Smith): 0.029334

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