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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]]. 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 ==
{{Main| 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 temperament]]s, compton is the {{nowrap| 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
[[Subgroup]]: 2.3.5


[[Comma]]: 531441/524288
[[Comma list]]: 531441/524288


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


Mapping generators: ~256/243, ~5
: mapping generators: ~256/243, ~5


[[Optimal tuning]] ([[POTE]]): ~5/4 = 384.884 (~81/80 = 15.116)
[[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 }}


{{Val list|legend=1| 12, 48, 60, 72, 84, 156, 240, 396b, 636bbc }}
{{Optimal ET sequence|legend=1| 12, 48, 60, 72, 84, 156, 240, 396b, 636bbc }}


[[Badness]]: 0.094494
[[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 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 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 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 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.
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
[[Subgroup]]: 2.3.5.7
Line 27: Line 42:
[[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 }}


[[Optimal tuning]] ([[POTE]]): ~5/4 = 383.7752 (~126/125 = 16.2248)
[[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, 48d, 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 40: 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 }}


Optimal tuning (POTE): ~5/4 = 383.2660 (~100/99 = 16.7340)
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, 48dee, 60e, 72 }}
{{Optimal ET sequence|legend=0| 12, …, 60e, 72 }}


Badness: 0.022235
Badness (Sintel): 0.735


==== 13-limit ====
==== 13-limit ====
Line 53: 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 }}


Optimal tuning (POTE): ~5/4 = 383.9628 (~105/104 = 16.0372)
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, 48defff, 60eff, 72, 228f }}
{{Optimal ET sequence|legend=0| 12f, …, 60eff, 72, 228f }}


Badness: 0.021852
Badness (Sintel): 0.903


===== 17-limit =====
===== 17-limit =====
Line 66: 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 }}


Optimal tuning (POTE): ~5/4 = 383.7500 (~105/104 = 16.2500)
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, 60eff, 72 }}
{{Optimal ET sequence|legend=0| 12f, 60eff, 72 }}


Badness: 0.017131
Badness (Sintel): 0.873


==== Comptone ====
==== Comptone ====
Line 79: 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 }}


Optimal tuning (POTE): ~5/4 = 382.6116 (~100/99 = 17.3884)
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, 276cdeff }}
{{Optimal ET sequence|legend=0| 12, 60e, 72, 204cdef, 276cdeff }}


Badness: 0.025144
Badness (Sintel): 1.04


===== 17-limit =====
===== 17-limit =====
Line 92: 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 }}


Optimal tuning (POTE): ~5/4 = 382.5968 (~100/99 = 17.4032)
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, 204cdefg, 276cdeffgg }}
{{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]]. 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.   
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
Line 107: Line 136:
[[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 }}


Mapping generators: ~16/15, ~7
: mapping generators: ~16/15, ~7


[[Optimal tuning]] ([[POTE]]): ~64/63 = 26.790
[[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 }}


{{Val list|legend=1| 12, 24, 36, 48c }}
{{Optimal ET sequence|legend=1| 12, 24, 36, 48c, 84c }}


[[Badness]]: 0.050297
[[Badness]] (Sintel): 1.27


=== 11-limit ===
=== 11-limit ===
Line 122: Line 155:
Comma list: 81/80, 99/98, 128/125
Comma list: 81/80, 99/98, 128/125


Mapping: [{{val| 12 19 28 0 -26 }}, {{val| 0 0 0 1 2 }}]
Mapping: {{mapping| 12 19 28 0 -26 | 0 0 0 1 2 }}


Optimal tuning (POTE): ~64/63 = 22.723
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 135: Line 170:
Comma list: 81/80, 128/125, 540/539
Comma list: 81/80, 128/125, 540/539


Mapping: [{{val| 12 19 28 0 109 }}, {{val| 0 0 0 1 -2 }}]
Mapping: {{mapping| 12 19 28 0 109 | 0 0 0 1 -2 }}


Optimal tuning (POTE): ~64/63 = 27.864
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


Mapping: [{{val| 12 19 28 0 8 }}, {{val| 0 0 0 1 1 }}]
Mapping: {{mapping| 12 19 28 0 8 | 0 0 0 1 1 }}


Optimal tuning (POTE): ~36/35 = 32.776
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 161: Line 200:
Comma list: 56/55, 66/65, 81/80, 105/104
Comma list: 56/55, 66/65, 81/80, 105/104


Mapping: [{{val| 12 19 28 0 8 11 }}, {{val| 0 0 0 1 1 1 }}]
Mapping: {{mapping| 12 19 28 0 8 11 | 0 0 0 1 1 1 }}


Optimal tuning (POTE): ~36/35 = 37.232
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 174: 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


Mapping: [{{val| 12 19 28 0 8 11 49 }}, {{val| 0 0 0 1 1 1 0 }}]
Mapping: {{mapping| 12 19 28 0 8 11 49 | 0 0 0 1 1 1 0 }}


Optimal tuning (POTE): ~36/35 = 39.777
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 187: 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


Mapping: [{{val| 12 19 28 0 8 11 49 51 }}, {{val| 0 0 0 1 1 1 0 0 }}]
Mapping: {{mapping| 12 19 28 0 8 11 49 51 | 0 0 0 1 1 1 0 0 }}


Optimal tuning (POTE): ~36/35 = 40.165
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 ====
{{Distinguish|Duodecim}}
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


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


Mapping: [{{val| 12 19 28 0 8 78 }}, {{val| 0 0 0 1 1 -1 }}]
Mapping: {{mapping| 12 19 28 0 8 78 | 0 0 0 1 1 -1 }}


Optimal tuning (POTE): ~36/35 = 37.688
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 =====
Line 213: Line 262:
Comma list: 51/50, 56/55, 81/80, 91/90, 128/125
Comma list: 51/50, 56/55, 81/80, 91/90, 128/125


Mapping: [{{val| 12 19 28 0 8 78 49 }}, {{val| 0 0 0 1 1 -1 0 }}]
Mapping:{{mapping| 12 19 28 0 8 78 49 | 0 0 0 1 1 -1 0 }}


Optimal tuning (POTE): ~36/35 = 38.097
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 226: Line 277:
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


Mapping: [{{val| 12 19 28 0 8 78 49 51 }}, {{val| 0 0 0 1 1 -1 0 0 }}]
Mapping: {{mapping| 12 19 28 0 8 78 49 51 | 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 ET sequence|legend=0| 12, 24, 36, 60c }}
 
Badness (Sintel): 1.27
 
=== Paly ===
{{Todo|inline=1|complete section|unify precision|comment= Sort the comma basis. Normalize the mapping. Unify the cent values in optimal tunings to 4 decimal places. Add the optimal ET sequence. <br>Add the data for the 11-, 13-, and 17-limit versions. }}  
 
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 81/80, 91/90, 51/50, 128/125, 76/75, 121/120


Optimal tuning (POTE): ~36/35 = 38.080
{{Mapping| 24 38 56 48 83 108 98 102 | 0 0 0 1 0 -1 0 0 }}


Optimal GPV sequence: {{Val list| 12, 24, 36, 60c }}
Optimal tunings:
* WE: ~33/32 = 49.994{{c}}, ~7/4 = 965.340{{c}}
: error map: {{val| -0.144 -2.183 +13.350 -3.774 -1.816 -6.516 -5.543 +1.875 }}
* CWE: ~33/32 = 50.000{{c}}, ~7/4 = 965.449{{c}}
: error map: {{val| -0.000 -1.955 +13.686 -3.377 -1.318 -5.977 -4.955 +2.487 }}


Badness: 0.020939
Badness (Sintel): 2.383


== Duodecim ==
== Duodecim ==
{{See also| Jubilismic clan #Duodecim }}
{{Distinguish|Duodecic}}
 
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
Line 241: Line 314:
[[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 generators: ~16/15, ~11
: mapping genereators: ~16/15, ~11


[[Optimal tuning]] ([[POTE]]): ~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


Mapping generators: ~36/35, ~5
[[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 }}


[[Optimal tuning]] ([[POTE]]): ~5/4 = 384.033
{{Optimal ET sequence|legend=1| 24, 48, 72, 312bd, 384bcdd, 456bcdd, 528bcdd, 600bccdd }}


{{Val list|legend=1| 24, 48, 72, 312bd, 384bcdd, 456bcdd, 528bcdd, 600bccdd }}
[[Badness]] (Sintel): 2.94
 
[[Badness]]: 0.116091


=== 11-limit ===
=== 11-limit ===
Line 275: Line 354:
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 }}


Optimal tuning (POTE): ~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 288: Line 369:
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 tuning (POTE): ~5/4 = 384.652
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}})


Optimal GPV sequence: {{Val list| 24, 48f, 72, 168df, 240dff }}
{{Optimal ET sequence|legend=0| 24, 48f, 72, 168df, 240dff }}


Badness: 0.026931
Badness (Sintel): 1.11


== Decades ==
== Gamelstearn ==
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).  
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.  


Subgroup: 2.3.5.7
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
[[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 }}


Mapping generators: ~49/48, ~5
: mapping generators: ~49/48, ~5


{{Multival|legend=1| 0 36 0 57 0 -101 }}
[[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 }}


[[Optimal tuning]] ([[POTE]]): ~5/4 = 384.764
{{Optimal ET sequence|legend=1| 36, 72, 252, 324bd, 396bd }}


{{Val list|legend=1| 36, 72, 252, 324bd, 396bd }}
[[Badness]] (Sintel): 2.73
 
[[Badness]]: 0.108016


=== 11-limit ===
=== 11-limit ===
Line 320: Line 407:
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 }}


Optimal tuning (POTE): ~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, 441/440, 4375/4356
[[Comma list]]: 225/224, 243/242, 385/384, 4000/3993


[[Mapping]]: [{{val| 72 114 167 202 249 266 }}, {{val| 0 0 0 0 0 1 }}]
{{Mapping|legend=1| 72 114 167 202 249 0 | 0 0 0 0 0 1 }}


Mapping generators: ~100/99, ~13
: mapping generators: ~100/99, ~13


[[Optimal tuning]] ([[POTE]]): ~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:Compton family| ]] <!-- main article -->
[[Category:Compton| ]] <!-- key article -->
[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Compton family| ]] <!-- main article -->
[[Category:Catalogs of rank-2 temperaments]]
[[Category:Rank 2]]
[[Category:Compton]]
Retrieved from "https://en.xen.wiki/w/Compton_family"