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The '''compton family''', otherwise known as the '''aristoxenean 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 [[cent]]s 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 ==
5-limit compton is also known as ''aristoxenean''. It 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, it is the 12 & 72 temperament, and [[72edo]], [[84edo]] or [[240edo]] make for good tunings.  
{{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
Line 12: Line 17:
: mapping generators: ~256/243, ~5
: mapping generators: ~256/243, ~5


[[Optimal tuning]] ([[POTE]]): ~256/243 = 1\12, ~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 }}


{{Optimal ET sequence|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 ==
Septimal compton is also known as ''waage''. 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]].
{{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 11-limit compton, intervals of 5 are off by one generator, intervals of 7 are off by two generators, and intervals of 11 are off by 3 generators.  
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 [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 31: Line 44:
{{Mapping|legend=1| 12 19 0 -22 | 0 0 1 2 }}
{{Mapping|legend=1| 12 19 0 -22 | 0 0 1 2 }}


[[Optimal tuning]] ([[POTE]]): ~256/243 = 1\12, ~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 }}


{{Optimal ET sequence|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 44: Line 61:
Mapping: {{mapping| 12 19 0 -22 -42 | 0 0 1 2 3 }}
Mapping: {{mapping| 12 19 0 -22 -42 | 0 0 1 2 3 }}


Optimal tuning (POTE): ~256/243 = 1\12, ~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 ET sequence|legend=1| 12, 48dee, 60e, 72 }}
{{Optimal ET sequence|legend=0| 12, …, 60e, 72 }}


Badness: 0.022235
Badness (Sintel): 0.735


==== 13-limit ====
==== 13-limit ====
Line 57: Line 76:
Mapping: {{mapping| 12 19 0 -22 -42 -67 | 0 0 1 2 3 4 }}
Mapping: {{mapping| 12 19 0 -22 -42 -67 | 0 0 1 2 3 4 }}


Optimal tuning (POTE): ~256/243 = 1\12, ~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 ET sequence|legend=1| 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 70: Line 91:
Mapping: {{mapping| 12 19 0 -22 -42 -67 49 | 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): ~18/17 = 1\12, ~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 ET sequence|legend=1| 12f, 60eff, 72 }}
{{Optimal ET sequence|legend=0| 12f, 60eff, 72 }}


Badness: 0.017131
Badness (Sintel): 0.873


==== Comptone ====
==== Comptone ====
Line 83: Line 106:
Mapping: {{mapping| 12 19 0 -22 -42 100 | 0 0 1 2 3 -2 }}
Mapping: {{mapping| 12 19 0 -22 -42 100 | 0 0 1 2 3 -2 }}


Optimal tuning (POTE): ~256/243 = 1\12, ~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 ET sequence|legend=1| 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 96: Line 121:
Mapping: {{mapping| 12 19 0 -22 -42 100 49 | 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): ~18/17 = 1\12, ~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 ET sequence|legend=1| 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 113: Line 140:
: mapping generators: ~16/15, ~7
: mapping generators: ~16/15, ~7


{{Multival|legend=1| 0 0 12 0 19 28 }}
[[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 }}


[[Optimal tuning]] ([[POTE]]): ~16/15 = 1\12, ~7/4 = 973.210 (~64/63 = 26.790)
{{Optimal ET sequence|legend=1| 12, 24, 36, 48c, 84c }}


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


=== 11-limit ===
=== 11-limit ===
Line 128: Line 157:
Mapping: {{mapping| 12 19 28 0 -26 | 0 0 0 1 2 }}
Mapping: {{mapping| 12 19 28 0 -26 | 0 0 0 1 2 }}


Optimal tuning (POTE): ~16/15 = 1\12, ~7/4 = 977.277 (~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 ET sequence|legend=1| 12, 36e, 48c, 108ccd }}
{{Optimal ET sequence|legend=0| 12, 36e, 48c }}


Badness: 0.058213
Badness (Sintel): 1.92


=== Catlat ===
=== Catlat ===
Line 141: Line 172:
Mapping: {{mapping| 12 19 28 0 109 | 0 0 0 1 -2 }}
Mapping: {{mapping| 12 19 28 0 109 | 0 0 0 1 -2 }}


Optimal tuning (POTE): ~16/15 = 1\12, ~7/4 = 972.136 (~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 ET sequence|legend=1| 36, 48c, 84c }}
{{Optimal ET sequence|legend=0| 12e, 36, 48c, 84c }}


Badness: 0.081909
Badness (Sintel): 2.71


=== Catnip ===
=== Catnip ===
Line 154: Line 187:
Mapping: {{mapping| 12 19 28 0 8 | 0 0 0 1 1 }}
Mapping: {{mapping| 12 19 28 0 8 | 0 0 0 1 1 }}


Optimal tuning (POTE): ~16/15 = 1\12, ~7/4 = 967.224 (~64/63 = 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 ET sequence|legend=1| 12, 24, 36, 72ce }}
{{Optimal ET sequence|legend=0| 12, 24, 36 }}


Badness: 0.034478
Badness (Sintel): 1.14


==== 13-limit ====
==== 13-limit ====
Line 167: Line 202:
Mapping: {{mapping| 12 19 28 0 8 11 | 0 0 0 1 1 1 }}
Mapping: {{mapping| 12 19 28 0 8 11 | 0 0 0 1 1 1 }}


Optimal tuning (POTE): ~16/15 = 1\12, ~7/4 = 962.778 (~40/39 = 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 ET sequence|legend=1| 12f, 24, 36f, 60cf }}
{{Optimal ET sequence|legend=0| 12f, 24, 36f }}


Badness: 0.028363
Badness (Sintel): 1.18


===== 17-limit =====
===== 17-limit =====
Line 180: Line 217:
Mapping: {{mapping| 12 19 28 0 8 11 49 | 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): ~18/17 = 1\12, ~7/4 = 960.223 (~40/39 = 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 ET sequence|legend=1| 12f, 24, 36f, 60cf }}
{{Optimal ET sequence|legend=0| 12f, 24, 36f }}


Badness: 0.023246
Badness (Sintel): 1.18


===== 19-limit =====
===== 19-limit =====
Line 193: Line 232:
Mapping: {{mapping| 12 19 28 0 8 11 49 51 | 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): ~18/17 = 1\12, ~7/4 = 959.835 (~40/39 = 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 ET sequence|legend=1| 12f, 24, 36f, 60cf }}
{{Optimal ET sequence|legend=0| 12f, 24, 36f }}


Badness: 0.018985
Badness (Sintel): 1.15


==== Duodecic ====
==== Duodecic ====
Line 206: Line 247:
Mapping: {{mapping| 12 19 28 0 8 78 | 0 0 0 1 1 -1 }}
Mapping: {{mapping| 12 19 28 0 8 78 | 0 0 0 1 1 -1 }}


Optimal tuning (POTE): ~16/15 = 1\12, ~7/4 = 962.312 (~64/63 = 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 ET sequence|legend=1| 12, 24, 36, 60c }}
{{Optimal ET sequence|legend=0| 12, 24, 36 }}


Badness: 0.038307
Badness (Sintel): 1.58


===== 17-limit =====
===== 17-limit =====
Line 219: Line 262:
Mapping:{{mapping| 12 19 28 0 8 78 49 | 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): ~18/17 = 1\12, ~7/4 = 961.903 (~64/63 = 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 ET sequence|legend=1| 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 232: Line 277:
Mapping: {{mapping| 12 19 28 0 8 78 49 51 | 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 tuning (POTE): ~18/17 = 1\12, ~7/4 = 961.920 (~64/63 = 38.080)
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=1| 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
Line 247: Line 294:
{{Mapping|legend=1| 12 19 28 34 0 | 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]]): ~16/15 = 1\12, ~11/8 = 565.023 (~55/54 = 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 }}


{{Optimal ET sequence|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
Line 263: Line 314:


{{Mapping|legend=1| 24 38 0 123 | 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]] ([[POTE]]): ~36/35 = 1\24, ~5/4 = 384.033 (~81/80 = 15.967)
[[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 ET sequence|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 281: Line 334:
Mapping: {{mapping| 24 38 0 123 83 | 0 0 1 -1 0 }}
Mapping: {{mapping| 24 38 0 123 83 | 0 0 1 -1 0 }}


Optimal tuning (POTE): ~36/35 = 1\24, ~5/4 = 384.054 (~121/120 = 15.946)
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 ET sequence|legend=1| 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 294: Line 349:
Mapping: {{mapping| 24 38 0 123 83 33 | 0 0 1 -1 0 1 }}
Mapping: {{mapping| 24 38 0 123 83 33 | 0 0 1 -1 0 1 }}


Optimal tuning (POTE): ~36/35 = 1\24, ~5/4 = 384.652 (~121/120 = 15.348)
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 ET sequence|legend=0| 24, 48f, 72, 168df, 240dff }}


{{Optimal ET sequence|legend=1| 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 [[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
Line 311: Line 370:
: 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}})
[[Optimal tuning]] ([[POTE]]): ~49/48 = 1\36, ~5/4 = 384.764 (~81/80 = 15.236)
: [[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 ET sequence|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 326: Line 387:
Mapping: {{mapping| 36 57 0 101 41 | 0 0 1 0 1 }}
Mapping: {{mapping| 36 57 0 101 41 | 0 0 1 0 1 }}


Optimal tuning (POTE): ~49/48 = 1\36, ~5/4 = 384.150 (~81/80 = 15.850)
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 ET sequence|legend=1| 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, 4000/3993
[[Comma list]]: 225/224, 243/242, 385/384, 4000/3993


{{Mapping|legend=1| 72 114 167 202 249 0 | 0 0 0 0 0 1 }}
{{Mapping|legend=1| 72 114 167 202 249 0 | 0 0 0 0 0 1 }}
Line 341: Line 404:
: mapping generators: ~100/99, ~13
: mapping generators: ~100/99, ~13


{{Multival|legend=1| 0 0 0 0 72 0 0 0 114 0 0 167 0 202 249 }}
[[Optimal tuning]]s:
 
* [[WE]]: ~100/99 = 16.6768{{c}}, ~13/8 = 838.3259{{c}} (~364/363 = 4.4838{{c}})
[[Optimal tuning]] ([[POTE]]): ~100/99 = 1\72, ~13/8 = 837.814 (~364/363 = 4.481)
: [[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 }}


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


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


[[Category:Temperament families]]
[[Category:Temperament families]]
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