Compton family: Difference between revisions

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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.
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]]. 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 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 ===
Line 54: Line 60:


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 ====
Line 103: Line 105:


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 ====
Line 250: Line 248:


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 263:


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 278:


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


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


[[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 316:


: 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 335:


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 350:


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 369:


: 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 388:


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 403:


: 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:Temperament families]]

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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