Compton family: Difference between revisions

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The '''Compton family''' tempers out the [[Pythagorean comma]], 531441/524288 = {{monzo| -19 12 }}, and hence the fifths form a closed 12-note circle of fifths, identical to [[12edo|12EDO]]. While the tuning of the fifth will be that of 12EDO, two cents flat, the tuning of the larger primes is not so constrained, and the point of these temperaments is to improve on it.
{{Technical data page}}
The '''compton family''', otherwise known as the '''aristoxenean family''', of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[Pythagorean comma]] ([[ratio]]: 531441/524288, {{monzo|legend=1| -19 12 }}, and hence the fifths form a closed 12-note [[circle of fifths]], identical to [[12edo]]. While the tuning of the fifth is fixed to 7 steps of 12edo, about 2{{cent}} flat of [[just]], these temperaments aim to add tunings for higher primes which are more in tune than in 12edo.


== Compton ==
== Compton ==
Subgroup: 2.3.5
{{Main| Compton }}


[[Comma]]: 531441/524288
5-limit compton tempers out the Pythagorean comma and has a period of 1\12, so it is the 12edo circle of fifths with an independent dimension for the harmonic 5. Equivalent generators are [[5/4]], [[6/5]], [[10/9]], [[16/15]] (the [[secor]]), [[45/32]], [[135/128]] and most importantly, [[81/80]]. In terms of [[equal temperament]]s, compton is the {{nowrap| 12 & 72 }} temperament; its [[ploidacot]] is dodecaploid acot. [[72edo]], [[84edo]] or [[240edo]] make for good tunings.


[[Mapping]]: [{{val| 12 19 0 }}, {{val| 0 0 1 }}
This temperament is documented as ''aristoxenean'' in [[Tonalsoft Encyclopedia]].


[[POTE generator]]: ~5/4 = 384.884 or ~81/80 = 15.116
[[Subgroup]]: 2.3.5


{{Val list|legend=1| 12, 72, 84, 156, 240, 396b }}
[[Comma list]]: 531441/524288


[[Badness]]: 0.094494
{{Mapping|legend=1| 12 19 0 | 0 0 1 }}
 
: mapping generators: ~256/243, ~5
 
[[Optimal tuning]]s:
* [[WE]]: ~256/243 = 100.0513{{c}}, ~5/4 = 385.0800{{c}} (~81/80 = 15.1253{{c}})
: [[error map]]: {{val| +0.616 -0.980 -0.001 }}
* [[CWE]]: ~256/243 = 100.0000{{c}}, ~5/4 = 385.3590{{c}} (~81/80 = 14.6410{{c}})
: error map: {{val| 0.000 -1.955 -0.955 }}
 
{{Optimal ET sequence|legend=1| 12, 48, 60, 72, 84, 156, 240, 396b, 636bbc }}
 
[[Badness]] (Sintel): 2.22


== Septimal compton ==
== Septimal compton ==
In terms of the normal list, compton adds 413343/409600 = {{monzo| -14 10 -2 1 }} to the Pythagorean comma; however it can also be characterized by saying it adds [[225/224]]. Compton, however, does not need to be used as a 7-limit temperament (also called as ''waage''); in the 5-limit it becomes the rank two 5-limit temperament tempering out the Pythagorean comma. In terms of equal temperaments, it is the 12&72 temperament, and [[72edo|72EDO]], [[84edo|84EDO]] or [[240edo|240EDO]] make for good tunings. Possible generators are 21/20, 10/9, the secor, 6/5, 5/4, 7/5 and most importantly, 81/80.
{{Main| Compton }}


In either the 5 or 7-limit, 240EDO is an excellent tuning, with 81/80 coming in at 15 cents exactly. In the 12EDO, the major third is sharp by 13.686 cents, and the minor third flat by 15.641 cents; adjusting these down and up by 15 cents puts them in excellent tune.
In terms of the [[normal forms #Normal forms for commas|normal comma list]], septimal compton adds 413343/409600 ({{monzo| -14 10 -2 1 }}) to the Pythagorean comma; however, it can also be characterized by saying it adds [[225/224]]. Other important commas of this temperament are 250047/250000, the [[landscape comma]], which sets [[63/50]] to 1/3 of an octave, and 390625/388962, the [[dimcomp comma]], which sets [[25/21]] to 1/4 of an octave.  


In terms of the normal comma list, we may add 8019/8000 to get to the 11-limit version of compton, which also adds [[441/440]]. For this 72EDO can be recommended as a tuning.
In either the 5- or 7-limit, 240edo is an excellent tuning, with [[81/80]] coming in at 15 cents exactly. In the 12edo, the major third is sharp by 13.686 cents, and the minor third flat by 15.641 cents; adjusting these down and up by 15 cents puts them in excellent tune.  


Subgroup: 2.3.5.7
Septimal compton is catalogued as ''waage'' in [[Graham Breed]]'s [https://x31eq.com/temper/ temperament finder].
 
In terms of the normal comma list, we may add [[8019/8000]] to get to the 11-limit version of compton, which also adds [[441/440]] and has a natural extension to the 13-limit. In 13-limit compton, intervals of 5, 7, 11, and 13 are off by one, two, three, and four generators, respectively. For these, 72edo can be recommended as a tuning.
 
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 225/224, 250047/250000
[[Comma list]]: 225/224, 250047/250000


[[Mapping]]: [{{val| 12 19 0 -22 }}, {{val| 0 0 1 2 }}]
{{Mapping|legend=1| 12 19 0 -22 | 0 0 1 2 }}


[[POTE generator]]: ~5/4 = 383.7752
[[Optimal tuning]]s:
* [[WE]]: ~256/243 = 100.0579{{c}}, ~5/4 = 383.9974{{c}} (~126/125 = 16.2342{{c}})
: [[error map]]: {{val| +0.695 -0.855 -0.927 +0.674 }}
* [[CWE]]: ~256/243 = 100.0000{{c}}, ~5/4 = 384.1429{{c}} (~126/125 = 15.8571{{c}})
: error map: {{val| 0.000 -1.955 -2.171 -0.540 }}


{{Val list|legend=1| 12, 60, 72, 228, 300c, 372bc, 444bc }}
{{Optimal ET sequence|legend=1| 12, …, 60, 72, 228, 300c, 372bc, 444bc }}


[[Badness]]: 0.035686
[[Badness]] (Sintel): 0.903


=== 11-limit ===
=== 11-limit ===
Line 38: Line 59:
Comma list: 225/224, 441/440, 4375/4356
Comma list: 225/224, 441/440, 4375/4356


Mapping: [{{val|12 19 0 -22 -42 }}, {{val| 0 0 1 2 3 }}]
Mapping: {{mapping| 12 19 0 -22 -42 | 0 0 1 2 3 }}


POTE generator: ~5/4 = 383.2660
Optimal tunings:
* WE: ~35/33 = 100.0633{{c}}, ~5/4 = 383.5087{{c}} (~100/99 = 16.7446{{c}})
* CWE: ~35/33 = 100.0000{{c}}, ~5/4 = 383.5958{{c}} (~100/99 = 16.4042{{c}})


Optimal GPV sequence: {{Val list| 12, 60e, 72 }}
{{Optimal ET sequence|legend=0| 12, …, 60e, 72 }}


Badness: 0.022235
Badness (Sintel): 0.735


==== 13-limit ====
==== 13-limit ====
Line 51: Line 74:
Comma list: 225/224, 351/350, 364/363, 441/440
Comma list: 225/224, 351/350, 364/363, 441/440


Mapping: [{{val| 12 19 0 -22 -42 -67 }}, {{val| 0 0 1 2 3 4 }}]
Mapping: {{mapping| 12 19 0 -22 -42 -67 | 0 0 1 2 3 4 }}


POTE generator: ~5/4 = 383.9628
Optimal tunings:  
* WE: ~35/33 = 100.0508{{c}}, ~5/4 = 384.1577{{c}} (~100/99 = 16.0454{{c}})
* CWE: ~35/33 = 100.0000{{c}}, ~5/4 = 384.1782{{c}} (~100/99 = 15.8218{{c}})


Optimal GPV sequence: {{Val list| 12f, 72, 84, 156, 228f, 300cf }}
{{Optimal ET sequence|legend=0| 12f, …, 60eff, 72, 228f }}


Badness: 0.021852
Badness (Sintel): 0.903


===== 17-limit =====
===== 17-limit =====
Line 64: Line 89:
Comma list: 221/220, 225/224, 289/288, 351/350, 441/440
Comma list: 221/220, 225/224, 289/288, 351/350, 441/440


Mapping: [{{val| 12 19 0 -22 -42 -67 49 }}, {{val| 0 0 1 2 3 4 0 }}]
Mapping: {{mapping| 12 19 0 -22 -42 -67 49 | 0 0 1 2 3 4 0 }}


POTE generator: ~5/4 = 383.7500
Optimal tunings:  
* WE: ~18/17 = 100.0658{{c}}, ~5/4 = 384.0024{{c}} (~100/99 = 16.2607{{c}})
* CWE: ~18/17 = 100.0000{{c}}, ~5/4 = 383.9647{{c}} (~100/99 = 16.0353{{c}})


Optimal GPV sequence: {{Val list| 12f, 72, 84, 156g, 228fg }}
{{Optimal ET sequence|legend=0| 12f, 60eff, 72 }}


Badness: 0.017131
Badness (Sintel): 0.873


==== Comptone ====
==== Comptone ====
Line 77: Line 104:
Comma list: 225/224, 325/324, 441/440, 1001/1000
Comma list: 225/224, 325/324, 441/440, 1001/1000


Mapping: [{{val| 12 19 0 -22 -42 100 }}, {{val| 0 0 1 2 3 -2 }}]
Mapping: {{mapping| 12 19 0 -22 -42 100 | 0 0 1 2 3 -2 }}


POTE generator: ~5/4 = 382.6116
Optimal tunings:
* WE: ~35/33 = 100.0926{{c}}, ~5/4 = 382.9660{{c}} (~100/99 = 17.4045{{c}})
* CWE: ~35/33 = 100.0000{{c}}, ~5/4 = 382.7748{{c}} (~100/99 = 17.2252{{c}})


Optimal GPV sequence: {{Val list| 12, 60e, 72, 204cdef, 276cdef }}
{{Optimal ET sequence|legend=0| 12, 60e, 72, 204cdef, 276cdeff }}


Badness: 0.025144
Badness (Sintel): 1.04


===== 17-limit =====
===== 17-limit =====
Line 90: Line 119:
Comma list: 225/224, 273/272, 289/288, 325/324, 441/440
Comma list: 225/224, 273/272, 289/288, 325/324, 441/440


Mapping: [{{val| 12 19 0 -22 -42 100 49 }}, {{val| 0 0 1 2 3 -2 0 }}]
Mapping: {{mapping| 12 19 0 -22 -42 100 49 | 0 0 1 2 3 -2 0 }}


POTE generator: ~5/4 = 382.5968
Optimal tunings:
* WE: ~18/17 = 100.0941{{c}}, ~5/4 = 382.9567{{c}} (~100/99 = 17.4796{{c}})
* CWE: ~18/17 = 100.0000{{c}}, ~5/4 = 382.7381{{c}} (~100/99 = 17.2619{{c}})


Optimal GPV sequence: {{Val list| 12, 60e, 72, 132deg, 204cdefg }}
{{Optimal ET sequence|legend=0| 12, 60e, 72, 204cdefg, 276cdeffgg }}


Badness: 0.016361
Badness (Sintel): 0.833


== Catler ==
== Catler ==
In terms of the normal comma list, catler is characterized by the addition of the [[schisma]], 32805/32768, to the Pythagorean comma, though it can also be characterized as adding [[81/80]], [[128/125]] or [[648/625]]. In any event, the 5-limit is exactly the same as the 5-limit of [[12edo|12EDO]]. Catler can also be characterized as the 12&24 temperament. [[36edo|36EDO]] or [[48edo|48EDO]] are possible tunings. Possible generators are 36/35, 21/20, 15/14, 8/7, 7/6, 9/7, 7/5, and most importantly, 64/63.   
In terms of the normal comma list, catler is characterized by the addition of the schisma, [[32805/32768]], to the Pythagorean comma, though it can also be characterized as adding [[81/80]], [[128/125]] or [[648/625]]. In any event, the 5-limit is exactly the same as the 5-limit of 12edo. Catler can also be characterized as the {{nowrap| 12 & 24 }} temperament. [[36edo]] or [[48edo]] are possible tunings. Possible generators are [[36/35]], [[21/20]], [[15/14]], [[8/7]], [[7/6]], [[9/7]], [[7/5]], and most importantly, [[64/63]].   


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 81/80, 128/125
[[Comma list]]: 81/80, 128/125


[[Mapping]]: [{{val| 12 19 28 0 }}, {{val| 0 0 0 1 }}]
{{Mapping|legend=1| 12 19 28 0 | 0 0 0 1 }}


[[POTE generator]]: ~64/63 = 26.790
: mapping generators: ~16/15, ~7


{{Val list|legend=1| 12, 24, 36, 48c }}
[[Optimal tuning]]s:
* [[WE]]: ~16/15 = 99.8680{{c}}, ~7/4 = 971.9257{{c}} (~64/63 = 26.7545{{c}})
: [[error map]]: {{val| -1.584 -4.463 +9.991 -0.068 }}
* [[CWE]]: ~16/15 = 100.0000{{c}}, ~7/4 = 972.0971{{c}} (~64/63 = 27.9029{{c}})
: error map: {{val| 0.000 -1.955 +13.686 +3.271 }}


[[Badness]]: 0.050297
{{Optimal ET sequence|legend=1| 12, 24, 36, 48c, 84c }}
 
[[Badness]] (Sintel): 1.27


=== 11-limit ===
=== 11-limit ===
Line 118: Line 155:
Comma list: 81/80, 99/98, 128/125
Comma list: 81/80, 99/98, 128/125


POTE generator: ~64/63 = 22.723
Mapping: {{mapping| 12 19 28 0 -26 | 0 0 0 1 2 }}


Mapping: [{{val| 12 19 28 0 -26 }}, {{val| 0 0 0 1 2 }}]
Optimal tunings:  
* WE: ~16/15 = 99.8542{{c}}, ~7/4 = 975.8519{{c}} (~64/63 = 22.6896{{c}})
* CWE: ~16/15 = 100.0000{{c}}, ~7/4 = 976.4125{{c}} (~64/63 = 23.5875{{c}})


Optimal GPV sequence: {{Val list| 12, 36e, 48c, 108ccd }}
{{Optimal ET sequence|legend=0| 12, 36e, 48c }}


Badness: 0.058213
Badness (Sintel): 1.92


=== Catlat ===
=== Catlat ===
Line 131: Line 170:
Comma list: 81/80, 128/125, 540/539
Comma list: 81/80, 128/125, 540/539


POTE generator: ~64/63 = 27.864
Mapping: {{mapping| 12 19 28 0 109 | 0 0 0 1 -2 }}


Mapping: [{{val| 12 19 28 0 109 }}, {{val| 0 0 0 1 -2 }}]
Optimal tunings:  
* WE: ~16/15 = 99.8791{{c}}, ~7/4 = 970.9614{{c}} (~64/63 = 27.8300{{c}})
* CWE: ~16/15 = 100.0000{{c}}, ~7/4 = 972.2549{{c}} (~64/63 = 27.7451{{c}})


Optimal GPV sequence: {{Val list| 36, 48c, 84c }}
{{Optimal ET sequence|legend=0| 12e, 36, 48c, 84c }}


Badness: 0.081909
Badness (Sintel): 2.71


=== Catcall ===
=== Catnip ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 56/55, 81/80, 128/125
Comma list: 56/55, 81/80, 128/125


POTE generator: ~36/35 = 32.776
Mapping: {{mapping| 12 19 28 0 8 | 0 0 0 1 1 }}


Mapping: [{{val| 12 19 28 0 8 }}, {{val| 0 0 0 1 1 }}]
Optimal tunings:  
* WE: ~16/15 = 99.8519{{c}}, ~7/4 = 965.7912{{c}} (~64/63 = 32.7275{{c}})
* CWE: ~16/15 = 100.0000{{c}}, ~7/4 = 965.8666{{c}} (~64/63 = 34.1334{{c}})


Optimal GPV sequence: {{Val list| 12, 24, 36, 72ce }}
{{Optimal ET sequence|legend=0| 12, 24, 36 }}


Badness: 0.034478
Badness (Sintel): 1.14


==== 13-limit ====
==== 13-limit ====
Line 157: Line 200:
Comma list: 56/55, 66/65, 81/80, 105/104
Comma list: 56/55, 66/65, 81/80, 105/104


POTE generator: ~36/35 = 37.232
Mapping: {{mapping| 12 19 28 0 8 11 | 0 0 0 1 1 1 }}


Mapping: [{{val| 12 19 28 0 8 11 }}, {{val| 0 0 0 1 1 1 }}]
Optimal tunings:  
* WE: ~16/15 = 99.8308{{c}}, ~7/4 = 961.1391{{c}} (~40/39 = 37.1694{{c}})
* CWE: ~16/15 = 100.0000{{c}}, ~7/4 = 961.1435{{c}} (~40/39 = 38.8565{{c}})


Optimal GPV sequence: {{Val list| 12f, 24, 36f, 60cf }}
{{Optimal ET sequence|legend=0| 12f, 24, 36f }}


Badness: 0.028363
Badness (Sintel): 1.18


===== 17-limit =====
===== 17-limit =====
Line 170: Line 215:
Comma list: 51/50, 56/55, 66/65, 81/80, 105/104
Comma list: 51/50, 56/55, 66/65, 81/80, 105/104


POTE generator: ~36/35 = 39.777
Mapping: {{mapping| 12 19 28 0 8 11 49 | 0 0 0 1 1 1 0 }}


Mapping: [{{val| 12 19 28 0 8 11 49 }}, {{val| 0 0 0 1 1 1 0 }}]
Optimal tunings:  
* WE: ~18/17 = 99.8958{{c}}, ~7/4 = 959.2226{{c}} (~40/39 = 39.7354{{c}})
* CWE: ~18/17 = 100.0000{{c}}, ~7/4 = 959.4216{{c}} (~40/39 = 40.5784{{c}})


Optimal GPV sequence: {{Val list| 12f, 24, 36f, 60cf }}
{{Optimal ET sequence|legend=0| 12f, 24, 36f }}


Badness: 0.023246
Badness (Sintel): 1.18


===== 19-limit =====
===== 19-limit =====
Line 183: Line 230:
Comma list: 51/50, 56/55, 66/65, 76/75, 81/80, 96/95
Comma list: 51/50, 56/55, 66/65, 76/75, 81/80, 96/95


POTE generator: ~36/35 = 40.165
Mapping: {{mapping| 12 19 28 0 8 11 49 51 | 0 0 0 1 1 1 0 0 }}


Mapping: [{{val| 12 19 28 0 8 11 49 51 }}, {{val| 0 0 0 1 1 1 0 0 }}]
Optimal tunings:  
* WE: ~18/17 = 99.9058{{c}}, ~7/4 = 958.9307{{c}} (~40/39 = 40.1270{{c}})
* CWE: ~18/17 = 100.0000{{c}}, ~7/4 = 959.2303{{c}} (~40/39 = 40.7697{{c}})


Optimal GPV sequence: {{Val list| 12f, 24, 36f, 60cf }}
{{Optimal ET sequence|legend=0| 12f, 24, 36f }}


Badness: 0.018985
Badness (Sintel): 1.15


==== Duodecic ====
==== Duodecic ====
Line 196: Line 245:
Comma list: 56/55, 81/80, 91/90, 128/125
Comma list: 56/55, 81/80, 91/90, 128/125


POTE generator: ~36/35 = 37.688
Mapping: {{mapping| 12 19 28 0 8 78 | 0 0 0 1 1 -1 }}


Mapping: [{{val| 12 19 28 0 8 78 }}, {{val| 0 0 0 1 1 -1 }}]
Optimal tunings:  
* WE: ~18/17 = 99.9301{{c}}, ~7/4 = 961.6396{{c}} (~64/63 = 37.6617{{c}})
* CWE: ~18/17 = 100.0000{{c}}, ~7/4 = 962.1413{{c}} (~64/63 = 37.8587{{c}})


Optimal GPV sequence: {{Val list| 12, 24, 36, 60c }}
{{Optimal ET sequence|legend=0| 12, 24, 36 }}


Badness: 0.038307
Badness (Sintel): 1.58


===== 17-limit =====
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17
Comma list: 51/50, 56/55, 81/80, 91/90, 128/125
Comma list: 51/50, 56/55, 81/80, 91/90, 128/125


POTE generator: ~36/35 = 38.097
Mapping:{{mapping| 12 19 28 0 8 78 49 | 0 0 0 1 1 -1 0 }}


Mapping: [{{val| 12 19 28 0 8 78 49 }}, {{val| 0 0 0 1 1 -1 0 }}]
Optimal tunings:  
* WE: ~18/17 = 99.9556{{c}}, ~7/4 = 961.4763{{c}} (~64/63 = 38.0796{{c}})
* CWE: ~18/17 = 100.0000{{c}}, ~7/4 = 961.8075{{c}} (~64/63 = 38.1925{{c}})


Optimal GPV sequence: {{Val list| 12, 24, 36, 60c }}
{{Optimal ET sequence|legend=0| 12, 24, 36, 60c }}


Badness: 0.027487
Badness (Sintel): 1.40


===== 19-limit =====
===== 19-limit =====
Line 221: Line 275:
Comma list: 51/50, 56/55, 76/75, 81/80, 91/90, 96/95
Comma list: 51/50, 56/55, 76/75, 81/80, 91/90, 96/95


POTE generator: ~36/35 = 38.080
Mapping: {{mapping| 12 19 28 0 8 78 49 51 | 0 0 0 1 1 -1 0 0 }}


Mapping: [{{val| 12 19 28 0 8 78 49 51 }}, {{val| 0 0 0 1 1 -1 0 0 }}]
Optimal tunings:  
* WE: ~18/17 = 99.9545{{c}}, ~7/4 = 961.4829{{c}} (~64/63 = 38.0624{{c}})
* CWE: ~18/17 = 100.0000{{c}}, ~7/4 = 961.8354{{c}} (~64/63 = 38.1646{{c}})


Optimal GPV sequence: {{Val list| 12, 24, 36, 60c }}
{{Optimal ET sequence|legend=0| 12, 24, 36, 60c }}


Badness: 0.020939
Badness (Sintel): 1.27


== Duodecim ==
== Duodecim ==
{{see also| Jubilismic clan #Duodecim }}
Duodecim uses exactly the same mapping as the 7-limit of 12edo, only correcting its poor approximation of prime 11.


Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 36/35, 50/49, 64/63
[[Comma list]]: 36/35, 50/49, 64/63


[[Mapping]]: [{{val| 12 19 28 34 0 }}, {{val| 0 0 0 0 1 }}]
{{Mapping|legend=1| 12 19 28 34 0 | 0 0 0 0 1 }}
 
: mapping genereators: ~16/15, ~11


[[POTE generator]]: ~45/44 = 34.977
[[Optimal tuning]]s:
* [[WE]]: ~16/15 = 99.6643{{c}}, ~11/8 = 563.1257{{c}} (~55/54 = 34.8599{{c}})
: [[error map]]: {{val| -4.029 -8.334 +4.285 +19.759 -0.279 }}
* [[CWE]]: ~16/15 = 100.0000{{c}}, ~11/8 = 562.2258{{c}} (~55/54 = 37.7742{{c}})
: error map: {{val| 0.000 -1.955 +13.686 +31.174 +10.908 }}


{{Val list|legend=1| 12, 24d }}
{{Optimal ET sequence|legend=1| 12, 24d, 36d }}


[[Badness]]: 0.030536
[[Badness]] (Sintel): 1.01


== Hours ==
== Hours ==
The hours temperament has a period of 1/24 octave and tempers out the cataharry comma (19683/19600) and the mirwomo comma (33075/32768). The name "hours" was so named for the following reasons - the period is 1/24 octave, and there are 24 hours per a day.
The hours temperament has a period of 1/24 octave and tempers out the [[cataharry comma]] (19683/19600) and the mirwomo comma (33075/32768). Its ploidacot is 24-ploid acot. The name ''hours'' was given for the reason that the period is 1/24 octave and there are 24 hours per day.


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 19683/19600, 33075/32768
[[Comma list]]: 19683/19600, 33075/32768


[[Mapping]]: [{{val| 24 38 0 123 }}, {{val| 0 0 1 -1 }}]
{{Mapping|legend=1| 24 38 0 123 | 0 0 1 -1 }}


{{Multival|legend=1| 0 24 -24 38 -38 -123 }}
: mapping generators: ~36/35, ~5


[[POTE generator]]: ~5/4 = 384.033
[[Optimal tuning]]s:
* [[WE]]: ~36/35 = 50.0337{{c}}, ~5/4 = 384.2919{{c}} (~81/80 = 15.9775{{c}})
: [[error map]]: {{val| +0.808 -0.675 -0.406 -0.592 }}
* [[CWE]]: ~36/35 = 50.0000{{c}}, ~5/4 = 384.0719{{c}} (~81/80 = 15.9281{{c}})
: error map: {{val| 0.000 -1.955 -2.242 -2.898 }}


{{Val list|legend=1| 24, 48, 72, 312bd, 384bcdd, 456bcdd, 528bcdd, 600bccdd }}
{{Optimal ET sequence|legend=1| 24, 48, 72, 312bd, 384bcdd, 456bcdd, 528bcdd, 600bccdd }}


[[Badness]]: 0.116091
[[Badness]] (Sintel): 2.94


=== 11-limit ===
=== 11-limit ===
Line 266: Line 332:
Comma list: 243/242, 385/384, 9801/9800
Comma list: 243/242, 385/384, 9801/9800


Mapping: [{{val| 24 38 0 123 83 }}, {{val| 0 0 1 -1 0 }}]
Mapping: {{mapping| 24 38 0 123 83 | 0 0 1 -1 0 }}


POTE generator: ~5/4 = 384.054
Optimal tunings:
* WE: ~36/35 = 50.0301{{c}}, ~5/4 = 384.2848{{c}} (~121/120 = 15.9559{{c}})
* CWE: ~36/35 = 50.0000{{c}}, ~5/4 = 384.0825{{c}} (~121/120 = 15.9175{{c}})


Optimal GPV sequence: {{Val list| 24, 48, 72, 312bd, 384bcdd, 456bcdde, 528bcdde }}
{{Optimal ET sequence|legend=0| 24, 48, 72, 312bd, 384bcdd, 456bcdde, 528bcdde }}


Badness: 0.036248
Badness (Sintel): 1.20


=== 13-limit ===
=== 13-limit ===
Line 279: Line 347:
Comma list: 243/242, 351/350, 364/363, 385/384
Comma list: 243/242, 351/350, 364/363, 385/384


Mapping: [{{val| 24 38 0 123 83 33 }}, {{val| 0 0 1 -1 0 1 }}]
Mapping: {{mapping| 24 38 0 123 83 33 | 0 0 1 -1 0 1 }}
 
Optimal tunings:
* WE: ~36/35 = 50.0358{{c}}, ~5/4 = 384.9267{{c}} (~121/120 = 15.3594{{c}})
* CWE: ~36/35 = 50.0000{{c}}, ~5/4 = 384.7662{{c}} (~121/120 = 15.2338{{c}})


POTE generator: ~5/4 = 384.652
{{Optimal ET sequence|legend=0| 24, 48f, 72, 168df, 240dff }}


Optimal GPV sequence: {{Val list| 24, 48f, 72, 168df, 240dff }}
Badness (Sintel): 1.11


Badness: 0.026931
== Gamelstearn ==
The gamelstearn temperament has a period of 1/36 octave and tempers out the [[gamelisma]] (1029/1024) and the [[stearnsma]] (118098/117649). Its ploidacot is 36-ploid acot.  


== Decades ==
It used to be called ''decades'', but was renamed in 2025 after the above two commas because the old name was deemed too confusing.
The decades temperament has a period of 1/36 octave and tempers out the [[1029/1024|gamelisma]] (1029/1024) and the stearnsma (118098/117649). The name "decades" was so named for the following reasons - the period is 1/36 octave, and there are 36 decades (''ten days'') per a year (12 months × 3 decades per a month).  


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 1029/1024, 118098/117649
[[Comma list]]: 1029/1024, 118098/117649


[[Mapping]]: [{{val| 36 57 0 101 }}, {{val| 0 0 1 0 }}]
{{Mapping|legend=1| 36 57 0 101 | 0 0 1 0 }}


{{Multival|legend=1| 0 36 0 57 0 -101 }}
: mapping generators: ~49/48, ~5


[[POTE generator]]: ~5/4 = 384.764
[[Optimal tuning]]s:
* [[WE]]: ~49/48 = 33.3519{{c}}, ~5/4 = 384.9781{{c}} (~81/80 = 15.2442{{c}})
: [[error map]]: {{val| +0.667 -0.899 -0.002 -0.288 }}
* [[CWE]]: ~49/48 = 33.3333{{c}}, ~5/4 = 385.1512{{c}} (~81/80 = 14.8488{{c}})
: error map: {{val| 0.000 -1.955 -1.162 -2.159 }}


{{Val list|legend=1| 36, 72, 252, 324bd, 396bd }}
{{Optimal ET sequence|legend=1| 36, 72, 252, 324bd, 396bd }}


[[Badness]]: 0.108016
[[Badness]] (Sintel): 2.73


=== 11-limit ===
=== 11-limit ===
Line 309: Line 385:
Comma list: 540/539, 1029/1024, 4000/3993
Comma list: 540/539, 1029/1024, 4000/3993


Mapping: [{{val| 36 57 0 101 41 }}, {{val| 0 0 1 0 1 }}]
Mapping: {{mapping| 36 57 0 101 41 | 0 0 1 0 1 }}


POTE generator: ~5/4 = 384.150
Optimal tunings:
* WE: ~49/48 = 33.3504{{c}}, ~5/4 = 384.3474{{c}} (~81/80 = 15.8576{{c}})
* CWE: ~49/48 = 33.333{{c}}, ~5/4 = 384.5541{{c}} (~81/80 = 15.4459{{c}})


Optimal GPV sequence: {{Val list| 36, 72, 396bd, 468bcd, 540bcd, 612bccdd, 684bbccdd, 756bbccdd }}
{{Optimal ET sequence|legend=0| 36, 72, 396bd }}


Badness: 0.043088
Badness (Sintel): 1.42


== Omicronbeta ==
== Omicronbeta ==
Subgroup: 2.3.5.7.11.13
[[Subgroup]]: 2.3.5.7.11.13
 
[[Comma list]]: 225/224, 243/242, 385/384, 4000/3993


[[Comma list]]: 225/224, 243/242, 441/440, 4375/4356
{{Mapping|legend=1| 72 114 167 202 249 0 | 0 0 0 0 0 1 }}


[[Mapping]]: [{{val| 72 114 167 202 249 266 }}, {{val| 0 0 0 0 0 1 }}]
: mapping generators: ~100/99, ~13


[[POTE generator]]: ~13/8 = 837.814
[[Optimal tuning]]s:  
* [[WE]]: ~100/99 = 16.6768{{c}}, ~13/8 = 838.3259{{c}} (~364/363 = 4.4838{{c}})
: [[error map]]: {{val| +0.733 -0.795 -1.281 -0.104 +1.216 -0.004 }}
* [[CWE]]: ~100/99 = 16.6667{{c}}, ~13/8 = 838.2660{{c}} (~364/363 = 4.9326{{c}})
: error map: {{val| 0.000 -1.955 -2.980 -2.159 -1.318 -2.262 }}


{{Val list|legend=1| 72, 144, 216c, 288cdf, 504bcdef }}
{{Optimal ET sequence|legend=1| 72, 144, 216c, 288cdf }}


[[Badness]]: 0.029956
[[Badness]] (Sintel): 1.24


[[Category:Regular temperament theory]]
[[Category:Temperament families]]
[[Category:Temperament family]]
[[Category:Compton]]
[[Category:Compton family| ]] <!-- main article -->
[[Category:Compton family| ]] <!-- main article -->
[[Category:Compton| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]

Latest revision as of 03:50, 4 May 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 compton family, otherwise known as the aristoxenean family, of temperaments tempers out the Pythagorean comma (ratio: 531441/524288, monzo: [-19 12⟩, and hence the fifths form a closed 12-note circle of fifths, identical to 12edo. While the tuning of the fifth is fixed to 7 steps of 12edo, about 2 ¢ flat of just, these temperaments aim to add tunings for higher primes which are more in tune than in 12edo.

Compton

Main article: Compton

5-limit compton tempers out the Pythagorean comma and has a period of 1\12, so it is the 12edo circle of fifths with an independent dimension for the harmonic 5. Equivalent generators are 5/4, 6/5, 10/9, 16/15 (the secor), 45/32, 135/128 and most importantly, 81/80. In terms of equal temperaments, compton is the 12 & 72 temperament; its ploidacot is dodecaploid acot. 72edo, 84edo or 240edo make for good tunings.

This temperament is documented as aristoxenean in Tonalsoft Encyclopedia.

Subgroup: 2.3.5

Comma list: 531441/524288

Mapping: [⟨12 19 0], ⟨0 0 1]]

mapping generators: ~256/243, ~5

Optimal tunings:

  • WE: ~256/243 = 100.0513 ¢, ~5/4 = 385.0800 ¢ (~81/80 = 15.1253 ¢)
error map: ⟨+0.616 -0.980 -0.001]
  • CWE: ~256/243 = 100.0000 ¢, ~5/4 = 385.3590 ¢ (~81/80 = 14.6410 ¢)
error map: ⟨0.000 -1.955 -0.955]

Optimal ET sequence: 12, 48, 60, 72, 84, 156, 240, 396b, 636bbc

Badness (Sintel): 2.22

Septimal compton

Main article: Compton

In terms of the normal comma list, septimal compton adds 413343/409600 ([-14 10 -2 1⟩) to the Pythagorean comma; however, it can also be characterized by saying it adds 225/224. Other important commas of this temperament are 250047/250000, the landscape comma, which sets 63/50 to 1/3 of an octave, and 390625/388962, the dimcomp comma, which sets 25/21 to 1/4 of an octave.

In either the 5- or 7-limit, 240edo is an excellent tuning, with 81/80 coming in at 15 cents exactly. In the 12edo, the major third is sharp by 13.686 cents, and the minor third flat by 15.641 cents; adjusting these down and up by 15 cents puts them in excellent tune.

Septimal compton is catalogued as waage in Graham Breed's temperament finder.

In terms of the normal comma list, we may add 8019/8000 to get to the 11-limit version of compton, which also adds 441/440 and has a natural extension to the 13-limit. In 13-limit compton, intervals of 5, 7, 11, and 13 are off by one, two, three, and four generators, respectively. For these, 72edo can be recommended as a tuning.

Subgroup: 2.3.5.7

Comma list: 225/224, 250047/250000

Mapping: [⟨12 19 0 -22], ⟨0 0 1 2]]

Optimal tunings:

  • WE: ~256/243 = 100.0579 ¢, ~5/4 = 383.9974 ¢ (~126/125 = 16.2342 ¢)
error map: ⟨+0.695 -0.855 -0.927 +0.674]
  • CWE: ~256/243 = 100.0000 ¢, ~5/4 = 384.1429 ¢ (~126/125 = 15.8571 ¢)
error map: ⟨0.000 -1.955 -2.171 -0.540]

Optimal ET sequence: 12, …, 60, 72, 228, 300c, 372bc, 444bc

Badness (Sintel): 0.903

11-limit

Subgroup: 2.3.5.7.11

Comma list: 225/224, 441/440, 4375/4356

Mapping: [⟨12 19 0 -22 -42], ⟨0 0 1 2 3]]

Optimal tunings:

  • WE: ~35/33 = 100.0633 ¢, ~5/4 = 383.5087 ¢ (~100/99 = 16.7446 ¢)
  • CWE: ~35/33 = 100.0000 ¢, ~5/4 = 383.5958 ¢ (~100/99 = 16.4042 ¢)

Optimal ET sequence: 12, …, 60e, 72

Badness (Sintel): 0.735

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 351/350, 364/363, 441/440

Mapping: [⟨12 19 0 -22 -42 -67], ⟨0 0 1 2 3 4]]

Optimal tunings:

  • WE: ~35/33 = 100.0508 ¢, ~5/4 = 384.1577 ¢ (~100/99 = 16.0454 ¢)
  • CWE: ~35/33 = 100.0000 ¢, ~5/4 = 384.1782 ¢ (~100/99 = 15.8218 ¢)

Optimal ET sequence: 12f, …, 60eff, 72, 228f

Badness (Sintel): 0.903

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 221/220, 225/224, 289/288, 351/350, 441/440

Mapping: [⟨12 19 0 -22 -42 -67 49], ⟨0 0 1 2 3 4 0]]

Optimal tunings:

  • WE: ~18/17 = 100.0658 ¢, ~5/4 = 384.0024 ¢ (~100/99 = 16.2607 ¢)
  • CWE: ~18/17 = 100.0000 ¢, ~5/4 = 383.9647 ¢ (~100/99 = 16.0353 ¢)

Optimal ET sequence: 12f, 60eff, 72

Badness (Sintel): 0.873

Comptone

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 325/324, 441/440, 1001/1000

Mapping: [⟨12 19 0 -22 -42 100], ⟨0 0 1 2 3 -2]]

Optimal tunings:

  • WE: ~35/33 = 100.0926 ¢, ~5/4 = 382.9660 ¢ (~100/99 = 17.4045 ¢)
  • CWE: ~35/33 = 100.0000 ¢, ~5/4 = 382.7748 ¢ (~100/99 = 17.2252 ¢)

Optimal ET sequence: 12, 60e, 72, 204cdef, 276cdeff

Badness (Sintel): 1.04

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 225/224, 273/272, 289/288, 325/324, 441/440

Mapping: [⟨12 19 0 -22 -42 100 49], ⟨0 0 1 2 3 -2 0]]

Optimal tunings:

  • WE: ~18/17 = 100.0941 ¢, ~5/4 = 382.9567 ¢ (~100/99 = 17.4796 ¢)
  • CWE: ~18/17 = 100.0000 ¢, ~5/4 = 382.7381 ¢ (~100/99 = 17.2619 ¢)

Optimal ET sequence: 12, 60e, 72, 204cdefg, 276cdeffgg

Badness (Sintel): 0.833

Catler

In terms of the normal comma list, catler is characterized by the addition of the schisma, 32805/32768, to the Pythagorean comma, though it can also be characterized as adding 81/80, 128/125 or 648/625. In any event, the 5-limit is exactly the same as the 5-limit of 12edo. Catler can also be characterized as the 12 & 24 temperament. 36edo or 48edo are possible tunings. Possible generators are 36/35, 21/20, 15/14, 8/7, 7/6, 9/7, 7/5, and most importantly, 64/63.

Subgroup: 2.3.5.7

Comma list: 81/80, 128/125

Mapping: [⟨12 19 28 0], ⟨0 0 0 1]]

mapping generators: ~16/15, ~7

Optimal tunings:

  • WE: ~16/15 = 99.8680 ¢, ~7/4 = 971.9257 ¢ (~64/63 = 26.7545 ¢)
error map: ⟨-1.584 -4.463 +9.991 -0.068]
  • CWE: ~16/15 = 100.0000 ¢, ~7/4 = 972.0971 ¢ (~64/63 = 27.9029 ¢)
error map: ⟨0.000 -1.955 +13.686 +3.271]

Optimal ET sequence: 12, 24, 36, 48c, 84c

Badness (Sintel): 1.27

11-limit

Subgroup: 2.3.5.7.11

Comma list: 81/80, 99/98, 128/125

Mapping: [⟨12 19 28 0 -26], ⟨0 0 0 1 2]]

Optimal tunings:

  • WE: ~16/15 = 99.8542 ¢, ~7/4 = 975.8519 ¢ (~64/63 = 22.6896 ¢)
  • CWE: ~16/15 = 100.0000 ¢, ~7/4 = 976.4125 ¢ (~64/63 = 23.5875 ¢)

Optimal ET sequence: 12, 36e, 48c

Badness (Sintel): 1.92

Catlat

Subgroup: 2.3.5.7.11

Comma list: 81/80, 128/125, 540/539

Mapping: [⟨12 19 28 0 109], ⟨0 0 0 1 -2]]

Optimal tunings:

  • WE: ~16/15 = 99.8791 ¢, ~7/4 = 970.9614 ¢ (~64/63 = 27.8300 ¢)
  • CWE: ~16/15 = 100.0000 ¢, ~7/4 = 972.2549 ¢ (~64/63 = 27.7451 ¢)

Optimal ET sequence: 12e, 36, 48c, 84c

Badness (Sintel): 2.71

Catnip

Subgroup: 2.3.5.7.11

Comma list: 56/55, 81/80, 128/125

Mapping: [⟨12 19 28 0 8], ⟨0 0 0 1 1]]

Optimal tunings:

  • WE: ~16/15 = 99.8519 ¢, ~7/4 = 965.7912 ¢ (~64/63 = 32.7275 ¢)
  • CWE: ~16/15 = 100.0000 ¢, ~7/4 = 965.8666 ¢ (~64/63 = 34.1334 ¢)

Optimal ET sequence: 12, 24, 36

Badness (Sintel): 1.14

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 56/55, 66/65, 81/80, 105/104

Mapping: [⟨12 19 28 0 8 11], ⟨0 0 0 1 1 1]]

Optimal tunings:

  • WE: ~16/15 = 99.8308 ¢, ~7/4 = 961.1391 ¢ (~40/39 = 37.1694 ¢)
  • CWE: ~16/15 = 100.0000 ¢, ~7/4 = 961.1435 ¢ (~40/39 = 38.8565 ¢)

Optimal ET sequence: 12f, 24, 36f

Badness (Sintel): 1.18

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 51/50, 56/55, 66/65, 81/80, 105/104

Mapping: [⟨12 19 28 0 8 11 49], ⟨0 0 0 1 1 1 0]]

Optimal tunings:

  • WE: ~18/17 = 99.8958 ¢, ~7/4 = 959.2226 ¢ (~40/39 = 39.7354 ¢)
  • CWE: ~18/17 = 100.0000 ¢, ~7/4 = 959.4216 ¢ (~40/39 = 40.5784 ¢)

Optimal ET sequence: 12f, 24, 36f

Badness (Sintel): 1.18

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 51/50, 56/55, 66/65, 76/75, 81/80, 96/95

Mapping: [⟨12 19 28 0 8 11 49 51], ⟨0 0 0 1 1 1 0 0]]

Optimal tunings:

  • WE: ~18/17 = 99.9058 ¢, ~7/4 = 958.9307 ¢ (~40/39 = 40.1270 ¢)
  • CWE: ~18/17 = 100.0000 ¢, ~7/4 = 959.2303 ¢ (~40/39 = 40.7697 ¢)

Optimal ET sequence: 12f, 24, 36f

Badness (Sintel): 1.15

Duodecic

Subgroup: 2.3.5.7.11.13

Comma list: 56/55, 81/80, 91/90, 128/125

Mapping: [⟨12 19 28 0 8 78], ⟨0 0 0 1 1 -1]]

Optimal tunings:

  • WE: ~18/17 = 99.9301 ¢, ~7/4 = 961.6396 ¢ (~64/63 = 37.6617 ¢)
  • CWE: ~18/17 = 100.0000 ¢, ~7/4 = 962.1413 ¢ (~64/63 = 37.8587 ¢)

Optimal ET sequence: 12, 24, 36

Badness (Sintel): 1.58

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 51/50, 56/55, 81/80, 91/90, 128/125

Mapping:[⟨12 19 28 0 8 78 49], ⟨0 0 0 1 1 -1 0]]

Optimal tunings:

  • WE: ~18/17 = 99.9556 ¢, ~7/4 = 961.4763 ¢ (~64/63 = 38.0796 ¢)
  • CWE: ~18/17 = 100.0000 ¢, ~7/4 = 961.8075 ¢ (~64/63 = 38.1925 ¢)

Optimal ET sequence: 12, 24, 36, 60c

Badness (Sintel): 1.40

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 51/50, 56/55, 76/75, 81/80, 91/90, 96/95

Mapping: [⟨12 19 28 0 8 78 49 51], ⟨0 0 0 1 1 -1 0 0]]

Optimal tunings:

  • WE: ~18/17 = 99.9545 ¢, ~7/4 = 961.4829 ¢ (~64/63 = 38.0624 ¢)
  • CWE: ~18/17 = 100.0000 ¢, ~7/4 = 961.8354 ¢ (~64/63 = 38.1646 ¢)

Optimal ET sequence: 12, 24, 36, 60c

Badness (Sintel): 1.27

Duodecim

Duodecim uses exactly the same mapping as the 7-limit of 12edo, only correcting its poor approximation of prime 11.

Subgroup: 2.3.5.7.11

Comma list: 36/35, 50/49, 64/63

Mapping: [⟨12 19 28 34 0], ⟨0 0 0 0 1]]

mapping genereators: ~16/15, ~11

Optimal tunings:

  • WE: ~16/15 = 99.6643 ¢, ~11/8 = 563.1257 ¢ (~55/54 = 34.8599 ¢)
error map: ⟨-4.029 -8.334 +4.285 +19.759 -0.279]
  • CWE: ~16/15 = 100.0000 ¢, ~11/8 = 562.2258 ¢ (~55/54 = 37.7742 ¢)
error map: ⟨0.000 -1.955 +13.686 +31.174 +10.908]

Optimal ET sequence: 12, 24d, 36d

Badness (Sintel): 1.01

Hours

The hours temperament has a period of 1/24 octave and tempers out the cataharry comma (19683/19600) and the mirwomo comma (33075/32768). Its ploidacot is 24-ploid acot. The name hours was given for the reason that the period is 1/24 octave and there are 24 hours per day.

Subgroup: 2.3.5.7

Comma list: 19683/19600, 33075/32768

Mapping: [⟨24 38 0 123], ⟨0 0 1 -1]]

mapping generators: ~36/35, ~5

Optimal tunings:

  • WE: ~36/35 = 50.0337 ¢, ~5/4 = 384.2919 ¢ (~81/80 = 15.9775 ¢)
error map: ⟨+0.808 -0.675 -0.406 -0.592]
  • CWE: ~36/35 = 50.0000 ¢, ~5/4 = 384.0719 ¢ (~81/80 = 15.9281 ¢)
error map: ⟨0.000 -1.955 -2.242 -2.898]

Optimal ET sequence: 24, 48, 72, 312bd, 384bcdd, 456bcdd, 528bcdd, 600bccdd

Badness (Sintel): 2.94

11-limit

Subgroup: 2.3.5.7.11

Comma list: 243/242, 385/384, 9801/9800

Mapping: [⟨24 38 0 123 83], ⟨0 0 1 -1 0]]

Optimal tunings:

  • WE: ~36/35 = 50.0301 ¢, ~5/4 = 384.2848 ¢ (~121/120 = 15.9559 ¢)
  • CWE: ~36/35 = 50.0000 ¢, ~5/4 = 384.0825 ¢ (~121/120 = 15.9175 ¢)

Optimal ET sequence: 24, 48, 72, 312bd, 384bcdd, 456bcdde, 528bcdde

Badness (Sintel): 1.20

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 243/242, 351/350, 364/363, 385/384

Mapping: [⟨24 38 0 123 83 33], ⟨0 0 1 -1 0 1]]

Optimal tunings:

  • WE: ~36/35 = 50.0358 ¢, ~5/4 = 384.9267 ¢ (~121/120 = 15.3594 ¢)
  • CWE: ~36/35 = 50.0000 ¢, ~5/4 = 384.7662 ¢ (~121/120 = 15.2338 ¢)

Optimal ET sequence: 24, 48f, 72, 168df, 240dff

Badness (Sintel): 1.11

Gamelstearn

The gamelstearn temperament has a period of 1/36 octave and tempers out the gamelisma (1029/1024) and the stearnsma (118098/117649). Its ploidacot is 36-ploid acot.

It used to be called decades, but was renamed in 2025 after the above two commas because the old name was deemed too confusing.

Subgroup: 2.3.5.7

Comma list: 1029/1024, 118098/117649

Mapping: [⟨36 57 0 101], ⟨0 0 1 0]]

mapping generators: ~49/48, ~5

Optimal tunings:

  • WE: ~49/48 = 33.3519 ¢, ~5/4 = 384.9781 ¢ (~81/80 = 15.2442 ¢)
error map: ⟨+0.667 -0.899 -0.002 -0.288]
  • CWE: ~49/48 = 33.3333 ¢, ~5/4 = 385.1512 ¢ (~81/80 = 14.8488 ¢)
error map: ⟨0.000 -1.955 -1.162 -2.159]

Optimal ET sequence: 36, 72, 252, 324bd, 396bd

Badness (Sintel): 2.73

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 1029/1024, 4000/3993

Mapping: [⟨36 57 0 101 41], ⟨0 0 1 0 1]]

Optimal tunings:

  • WE: ~49/48 = 33.3504 ¢, ~5/4 = 384.3474 ¢ (~81/80 = 15.8576 ¢)
  • CWE: ~49/48 = 33.333 ¢, ~5/4 = 384.5541 ¢ (~81/80 = 15.4459 ¢)

Optimal ET sequence: 36, 72, 396bd

Badness (Sintel): 1.42

Omicronbeta

Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 243/242, 385/384, 4000/3993

Mapping: [⟨72 114 167 202 249 0], ⟨0 0 0 0 0 1]]

mapping generators: ~100/99, ~13

Optimal tunings:

  • WE: ~100/99 = 16.6768 ¢, ~13/8 = 838.3259 ¢ (~364/363 = 4.4838 ¢)
error map: ⟨+0.733 -0.795 -1.281 -0.104 +1.216 -0.004]
  • CWE: ~100/99 = 16.6667 ¢, ~13/8 = 838.2660 ¢ (~364/363 = 4.9326 ¢)
error map: ⟨0.000 -1.955 -2.980 -2.159 -1.318 -2.262]

Optimal ET sequence: 72, 144, 216c, 288cdf

Badness (Sintel): 1.24

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