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{{Technical data page}}<br><br>
{{Technical data page}}
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.
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 - that is, it is equivalent to the root12(2).5 subgroup with 2^(7/12) mapped to 3/2. 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 &amp; 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 14: Line 18:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~256/243 = 100.000, ~5/4 = 386.314 (~81/80 = 13.686)
* [[WE]]: ~256/243 = 100.0513{{c}}, ~5/4 = 385.0800{{c}} (~81/80 = 15.1253{{c}})
: [[error map]]: {{val| 0.000 -1.955 0.000 }}
: [[error map]]: {{val| +0.616 -0.980 -0.001 }}
* [[POTE]]: ~256/243 = 100.000, ~5/4 = 384.884 (~81/80 = 15.116)
* [[CWE]]: ~256/243 = 100.0000{{c}}, ~5/4 = 385.3590{{c}} (~81/80 = 14.6410{{c}})
: error map: {{val| 0.000 -1.955 -1.431 }}
: 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]] (Smith): 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 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. 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.  
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 35: Line 43:


{{Mapping|legend=1| 12 19 0 -22 | 0 0 1 2 }}
{{Mapping|legend=1| 12 19 0 -22 | 0 0 1 2 }}
{{Multival|legend=1| 0 12 24 19 38 22 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~200/189 = 100.000, ~5/4 = 384.922 (~126/125 = 15.078)
* [[WE]]: ~256/243 = 100.0579{{c}}, ~5/4 = 383.9974{{c}} (~126/125 = 16.2342{{c}})
: [[error map]]: {{val| 0.000 -1.955 -1.392 -1.017 }}
: [[error map]]: {{val| +0.695 -0.855 -0.927 +0.674 }}
* [[POTE]]: ~200/189 = 100.000, ~5/4 = 383.775 (~126/125 = 16.225)
* [[CWE]]: ~256/243 = 100.0000{{c}}, ~5/4 = 384.1429{{c}} (~126/125 = 15.8571{{c}})
: error map: {{val| 0.000 -1.955 -2.538 -1.275 }}
: 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]] (Smith): 0.035686
[[Badness]] (Sintel): 0.903


=== 11-limit ===
=== 11-limit ===
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Mapping: {{mapping| 12 19 0 -22 -42 | 0 0 1 2 3 }}
Mapping: {{mapping| 12 19 0 -22 -42 | 0 0 1 2 3 }}
Wedgie: {{Multival| 0 12 24 36 19 38 57 22 42 18 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~35/33 = 100.000, ~5/4 = 384.324 (~100/99 = 15.676)
* WE: ~35/33 = 100.0633{{c}}, ~5/4 = 383.5087{{c}} (~100/99 = 16.7446{{c}})
* POTE: ~35/33 = 100.000, ~5/4 = 383.266 (~100/99 = 16.734)
* CWE: ~35/33 = 100.0000{{c}}, ~5/4 = 383.5958{{c}} (~100/99 = 16.4042{{c}})


{{Optimal ET sequence|legend=0| 12, 48dee, 60e, 72 }}
{{Optimal ET sequence|legend=0| 12, …, 60e, 72 }}


Badness (Smith): 0.022235
Badness (Sintel): 0.735


==== 13-limit ====
==== 13-limit ====
Line 71: Line 75:


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 }}
Wedgie: {{Multival| 0 12 24 36 48 19 38 57 76 22 42 67 18 46 33 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~35/33 = 100.000, ~5/4 = 384.685 (~105/104 = 15.315)
* WE: ~35/33 = 100.0508{{c}}, ~5/4 = 384.1577{{c}} (~100/99 = 16.0454{{c}})
* POTE: ~35/33 = 100.000, ~5/4 = 383.963 (~105/104 = 16.037)
* CWE: ~35/33 = 100.0000{{c}}, ~5/4 = 384.1782{{c}} (~100/99 = 15.8218{{c}})


{{Optimal ET sequence|legend=0| 12f, 48deefff, 60eff, 72, 228f }}
{{Optimal ET sequence|legend=0| 12f, …, 60eff, 72, 228f }}


Badness (Smith): 0.021852
Badness (Sintel): 0.903


===== 17-limit =====
===== 17-limit =====
Line 90: Line 92:


Optimal tunings:  
Optimal tunings:  
* CTE: ~18/17 = 100.000, ~5/4 = 384.685 (~105/104 = 15.315)
* WE: ~18/17 = 100.0658{{c}}, ~5/4 = 384.0024{{c}} (~100/99 = 16.2607{{c}})
* POTE: ~18/17 = 100.000, ~5/4 = 383.750 (~105/104 = 16.250)
* CWE: ~18/17 = 100.0000{{c}}, ~5/4 = 383.9647{{c}} (~100/99 = 16.0353{{c}})


{{Optimal ET sequence|legend=0| 12f, 60eff, 72 }}
{{Optimal ET sequence|legend=0| 12f, 60eff, 72 }}


Badness (Smith): 0.017131
Badness (Sintel): 0.873


==== Comptone ====
==== Comptone ====
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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 }}
Wedgie: {{Multival| 0 12 24 36 -24 19 38 57 -38 22 42 -100 18 -156 -216 }}


Optimal tunings:  
Optimal tunings:  
* CTE: ~35/33 = 100.000, ~5/4 = 383.552 (~100/99 = 16.448)
* WE: ~35/33 = 100.0926{{c}}, ~5/4 = 382.9660{{c}} (~100/99 = 17.4045{{c}})
* POTE: ~35/33 = 100.000, ~5/4 = 382.612 (~100/99 = 17.388)
* CWE: ~35/33 = 100.0000{{c}}, ~5/4 = 382.7748{{c}} (~100/99 = 17.2252{{c}})


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


Badness (Smith): 0.025144
Badness (Sintel): 1.04


===== 17-limit =====
===== 17-limit =====
Line 122: Line 122:


Optimal tunings:  
Optimal tunings:  
* CTE: ~18/17 = 100.000, ~5/4 = 383.552 (~100/99 = 16.448)
* WE: ~18/17 = 100.0941{{c}}, ~5/4 = 382.9567{{c}} (~100/99 = 17.4796{{c}})
* POTE: ~18/17 = 100.000, ~5/4 = 382.597 (~100/99 = 17.403)
* CWE: ~18/17 = 100.0000{{c}}, ~5/4 = 382.7381{{c}} (~100/99 = 17.2619{{c}})


{{Optimal ET sequence|legend=0| 12, 60e, 72, 204cdefg, 276cdeffgg }}
{{Optimal ET sequence|legend=0| 12, 60e, 72, 204cdefg, 276cdeffgg }}


Badness (Smith): 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 &amp; 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 139: Line 139:


: mapping generators: ~16/15, ~7
: mapping generators: ~16/15, ~7
{{Multival|legend=1| 0 0 12 0 19 28 }}


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~16/15 = 100.000, ~7/4 = 968.826 (~64/63 = 31.174)
* [[WE]]: ~16/15 = 99.8680{{c}}, ~7/4 = 971.9257{{c}} (~64/63 = 26.7545{{c}})
: [[error map]]: {{val| 0.000 -1.955 +13.686 0.000 }}
: [[error map]]: {{val| -1.584 -4.463 +9.991 -0.068 }}
* [[POTE]]: ~16/15 = 100.000, ~7/4 = 973.210 (~64/63 = 26.790)
* [[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 +4.384 }}
: error map: {{val| 0.000 -1.955 +13.686 +3.271 }}


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


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


=== 11-limit ===
=== 11-limit ===
Line 160: Line 158:


Optimal tunings:  
Optimal tunings:  
* CTE: ~16/15 = 100.000, ~7/4 = 973.779 (~64/63 = 26.221)
* WE: ~16/15 = 99.8542{{c}}, ~7/4 = 975.8519{{c}} (~64/63 = 22.6896{{c}})
* POTE: ~16/15 = 100.000, ~7/4 = 977.277 (~64/63 = 22.723)
* CWE: ~16/15 = 100.0000{{c}}, ~7/4 = 976.4125{{c}} (~64/63 = 23.5875{{c}})


{{Optimal ET sequence|legend=0| 12, 36e, 48c }}
{{Optimal ET sequence|legend=0| 12, 36e, 48c }}


Badness (Smith): 0.058213
Badness (Sintel): 1.92


=== Catlat ===
=== Catlat ===
Line 175: Line 173:


Optimal tunings:  
Optimal tunings:  
* CTE: ~16/15 = 100.000, ~7/4 = 972.823 (~64/63 = 27.177)
* WE: ~16/15 = 99.8791{{c}}, ~7/4 = 970.9614{{c}} (~64/63 = 27.8300{{c}})
* POTE: ~16/15 = 100.000, ~7/4 = 972.136 (~64/63 = 27.864)
* CWE: ~16/15 = 100.0000{{c}}, ~7/4 = 972.2549{{c}} (~64/63 = 27.7451{{c}})


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


Badness (Smith): 0.081909
Badness (Sintel): 2.71


=== Catnip ===
=== Catnip ===
Line 190: Line 188:


Optimal tunings:  
Optimal tunings:  
* CTE: ~16/15 = 100.000, ~7/4 = 961.874 (~64/63 = 38.126)
* WE: ~16/15 = 99.8519{{c}}, ~7/4 = 965.7912{{c}} (~64/63 = 32.7275{{c}})
* POTE: ~16/15 = 100.000, ~7/4 = 967.224 (~64/63 = 32.776)
* CWE: ~16/15 = 100.0000{{c}}, ~7/4 = 965.8666{{c}} (~64/63 = 34.1334{{c}})


{{Optimal ET sequence|legend=0| 12, 24, 36, 72ce }}
{{Optimal ET sequence|legend=0| 12, 24, 36 }}


Badness (Smith): 0.034478
Badness (Sintel): 1.14


==== 13-limit ====
==== 13-limit ====
Line 205: Line 203:


Optimal tunings:  
Optimal tunings:  
* CTE: ~16/15 = 100.000, ~7/4 = 956.375 (~40/39 = 43.625)
* WE: ~16/15 = 99.8308{{c}}, ~7/4 = 961.1391{{c}} (~40/39 = 37.1694{{c}})
* POTE: ~16/15 = 100.000, ~7/4 = 962.778 (~40/39 = 37.232)
* CWE: ~16/15 = 100.0000{{c}}, ~7/4 = 961.1435{{c}} (~40/39 = 38.8565{{c}})


{{Optimal ET sequence|legend=0| 12f, 24, 36f }}
{{Optimal ET sequence|legend=0| 12f, 24, 36f }}


Badness (Smith): 0.028363
Badness (Sintel): 1.18


===== 17-limit =====
===== 17-limit =====
Line 220: Line 218:


Optimal tunings:  
Optimal tunings:  
* CTE: ~18/17 = 100.000, ~7/4 = 956.375 (~40/39 = 43.625)
* WE: ~18/17 = 99.8958{{c}}, ~7/4 = 959.2226{{c}} (~40/39 = 39.7354{{c}})
* POTE: ~18/17 = 100.000, ~7/4 = 960.223 (~40/39 = 39.777)
* CWE: ~18/17 = 100.0000{{c}}, ~7/4 = 959.4216{{c}} (~40/39 = 40.5784{{c}})


{{Optimal ET sequence|legend=0| 12f, 24, 36f }}
{{Optimal ET sequence|legend=0| 12f, 24, 36f }}


Badness (Smith): 0.023246
Badness (Sintel): 1.18


===== 19-limit =====
===== 19-limit =====
Line 235: Line 233:


Optimal tunings:  
Optimal tunings:  
* CTE: ~18/17 = 100.000, ~7/4 = 956.375 (~40/39 = 43.625)
* WE: ~18/17 = 99.9058{{c}}, ~7/4 = 958.9307{{c}} (~40/39 = 40.1270{{c}})
* POTE: ~18/17 = 100.000, ~7/4 = 959.835 (~40/39 = 40.165)
* CWE: ~18/17 = 100.0000{{c}}, ~7/4 = 959.2303{{c}} (~40/39 = 40.7697{{c}})


{{Optimal ET sequence|legend=0| 12f, 24, 36f }}
{{Optimal ET sequence|legend=0| 12f, 24, 36f }}


Badness (Smith): 0.018985
Badness (Sintel): 1.15


==== Duodecic ====
==== Duodecic ====
{{Distinguish|Duodecim}}
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Line 250: Line 250:


Optimal tunings:  
Optimal tunings:  
* CTE: ~16/15 = 100.000, ~7/4 = 961.255 (~64/63 = 38.745)
* WE: ~18/17 = 99.9301{{c}}, ~7/4 = 961.6396{{c}} (~64/63 = 37.6617{{c}})
* POTE: ~16/15 = 100.000, ~7/4 = 962.312 (~64/63 = 37.688)
* CWE: ~18/17 = 100.0000{{c}}, ~7/4 = 962.1413{{c}} (~64/63 = 37.8587{{c}})


{{Optimal ET sequence|legend=0| 12, 24, 36 }}
{{Optimal ET sequence|legend=0| 12, 24, 36 }}


Badness (Smith): 0.038307
Badness (Sintel): 1.58


===== 17-limit =====
===== 17-limit =====
Line 265: Line 265:


Optimal tunings:  
Optimal tunings:  
* CTE: ~18/17 = 100.000, ~7/4 = 961.255 (~64/63 = 38.745)
* WE: ~18/17 = 99.9556{{c}}, ~7/4 = 961.4763{{c}} (~64/63 = 38.0796{{c}})
* POTE: ~18/17 = 100.000, ~7/4 = 961.903 (~64/63 = 38.097)
* CWE: ~18/17 = 100.0000{{c}}, ~7/4 = 961.8075{{c}} (~64/63 = 38.1925{{c}})


{{Optimal ET sequence|legend=0| 12, 24, 36, 60c }}
{{Optimal ET sequence|legend=0| 12, 24, 36, 60c }}


Badness (Smith): 0.027487
Badness (Sintel): 1.40


===== 19-limit =====
===== 19-limit =====
Line 280: Line 280:


Optimal tunings:  
Optimal tunings:  
* CTE: ~18/17 = 100.000, ~7/4 = 961.255 (~64/63 = 38.745)
* WE: ~18/17 = 99.9545{{c}}, ~7/4 = 961.4829{{c}} (~64/63 = 38.0624{{c}})
* POTE: ~18/17 = 100.000, ~7/4 = 961.920 (~64/63 = 38.080)
* 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 }}
{{Optimal ET sequence|legend=0| 12, 24, 36, 60c }}


Badness (Smith): 0.020939
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
 
{{Mapping| 24 38 56 48 83 108 98 102 | 0 0 0 1 0 -1 0 0 }}
 
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 (Sintel): 2.383


== Duodecim ==
== 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 297: Line 319:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~16/15 = 1\12, ~11/8 = 551.318 (~33/32 = 48.682)
* [[WE]]: ~16/15 = 99.6643{{c}}, ~11/8 = 563.1257{{c}} (~55/54 = 34.8599{{c}})
: [[error map]]: {{val| 0.000 -1.955 +13.686 +31.174 0.000 }}
: [[error map]]: {{val| -4.029 -8.334 +4.285 +19.759 -0.279 }}
* [[POTE]]: ~16/15 = 1\12, ~11/8 = 565.023 (~55/54 = 34.977)
* [[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 +13.705 }}
: error map: {{val| 0.000 -1.955 +13.686 +31.174 +10.908 }}


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


[[Badness]] (Smith): 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 316: Line 338:


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


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~36/35 = 50.000, ~5/4 = 384.226 (~81/80 = 15.774)
* [[WE]]: ~36/35 = 50.0337{{c}}, ~5/4 = 384.2919{{c}} (~81/80 = 15.9775{{c}})
: [[error map]]: {{val| 0.000 -1.955 -2.088 -3.052 }}
: [[error map]]: {{val| +0.808 -0.675 -0.406 -0.592 }}
* [[POTE]]: ~36/35 = 50.000, ~5/4 = 384.033 (~81/80 = 15.967)
* [[CWE]]: ~36/35 = 50.0000{{c}}, ~5/4 = 384.0719{{c}} (~81/80 = 15.9281{{c}})
: error map: {{val| 0.000 -1.955 -2.280 -2.859 }}
: 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]] (Smith): 0.116091
[[Badness]] (Sintel): 2.94


=== 11-limit ===
=== 11-limit ===
Line 337: Line 357:


Optimal tunings:  
Optimal tunings:  
* CTE: ~36/35 = 50.000, ~5/4 = 384.226 (~121/120 = 15.774)
* WE: ~36/35 = 50.0301{{c}}, ~5/4 = 384.2848{{c}} (~121/120 = 15.9559{{c}})
* POTE: ~36/35 = 50.000, ~5/4 = 384.054 (~121/120 = 15.946)
* CWE: ~36/35 = 50.0000{{c}}, ~5/4 = 384.0825{{c}} (~121/120 = 15.9175{{c}})


{{Multival|legend=1| 0 24 -24 0 38 -38 0 -123 -83 83 }}
{{Optimal ET sequence|legend=0| 24, 48, 72, 312bd, 384bcdd, 456bcdde, 528bcdde }}


{{Optimal ET sequence|legend=1| 24, 48, 72, 312bd, 384bcdd, 456bcdde, 528bcdde }}
Badness (Sintel): 1.20
 
Badness (Smith): 0.036248


=== 13-limit ===
=== 13-limit ===
Line 354: Line 372:


Optimal tunings:  
Optimal tunings:  
* CTE: ~36/35 = 50.000, ~5/4 = 385.420 (~121/120 = 14.580)
* WE: ~36/35 = 50.0358{{c}}, ~5/4 = 384.9267{{c}} (~121/120 = 15.3594{{c}})
* POTE: ~36/35 = 50.000, ~5/4 = 384.652 (~121/120 = 15.348)
* CWE: ~36/35 = 50.0000{{c}}, ~5/4 = 384.7662{{c}} (~121/120 = 15.2338{{c}})


{{Multival|legend=1| 0 24 -24 0 24 38 -38 0 38 -123 -83 -33 83 156 83 }}
{{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 (Smith): 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 373: Line 391:


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


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~49/48 = 33.333, ~5/4 = 386.314 (~81/80 = 13.686)
* [[WE]]: ~49/48 = 33.3519{{c}}, ~5/4 = 384.9781{{c}} (~81/80 = 15.2442{{c}})
: [[error map]]: {{val| 0.000 -1.955 0.000 -2.159 }}
: [[error map]]: {{val| +0.667 -0.899 -0.002 -0.288 }}
* [[POTE]]: ~49/48 = 33.333, ~5/4 = 384.764 (~81/80 = 15.236)
* [[CWE]]: ~49/48 = 33.3333{{c}}, ~5/4 = 385.1512{{c}} (~81/80 = 14.8488{{c}})
: error map: {{val| 0.000 -1.955 -1.549 -2.159 }}
: 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]] (Smith): 0.108016
[[Badness]] (Sintel): 2.73


=== 11-limit ===
=== 11-limit ===
Line 394: Line 410:


Optimal tunings:  
Optimal tunings:  
* CTE: ~49/48 = 33.333, ~5/4 = 385.797 (~81/80 = 14.203)
* WE: ~49/48 = 33.3504{{c}}, ~5/4 = 384.3474{{c}} (~81/80 = 15.8576{{c}})
* POTE: ~49/48 = 33.333, ~5/4 = 385.150 (~81/80 = 14.850)
* 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 (Smith): 0.043088
Badness (Sintel): 1.42


== Omicronbeta ==
== Omicronbeta ==
Line 409: Line 425:


: 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:  
[[Optimal tuning]]s:  
* [[CTE]]: ~100/99 = 16.667, ~13/8 = 840.528 (~325/324 = 7.194)
* [[WE]]: ~100/99 = 16.6768{{c}}, ~13/8 = 838.3259{{c}} (~364/363 = 4.4838{{c}})
: [[error map]]: {{val| 0.000 -1.955 -2.980 -2.159 -1.318 0.000 }}
: [[error map]]: {{val| +0.733 -0.795 -1.281 -0.104 +1.216 -0.004 }}
* [[POTE]]: ~100/99 = 16.667, ~13/8 = 837.814 (~364/363 = 4.481)
* [[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 -2.713 }}
: error map: {{val| 0.000 -1.955 -2.980 -2.159 -1.318 -2.262 }}


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


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


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