Compton family: Difference between revisions

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Added link to narrowed compton since I realised that, because it tempers the pythagorean comma, it belongs in this family (but I don't want to outright move it here because it is too long and would clutter this page too much)
m →Gamelstearn: Add link to Stearnsma
 
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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 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.


== Compton ==
== Compton ==
3-limit as in 12edo; intervals of 5 are off by one generator. In the 7-limit (sometimes called waage), intervals of 7 are off by two generators. In the 11-limit, intervals of 11 are off by 3 generators. Thinking of 72edo might make this more concrete.
{{Main| 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.  
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 temperament]]s, it is the {{nowrap| 12 & 72 }} temperament, and [[72edo]], [[84edo]] or [[240edo]] make for good tunings.  


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
Line 14: Line 15:
: 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:
* [[CTE]]: ~256/243 = 100.000, ~5/4 = 386.314 (~81/80 = 13.686)
: [[error map]]: {{val| 0.000 -1.955 0.000 }}
* [[POTE]]: ~256/243 = 100.000, ~5/4 = 384.884 (~81/80 = 15.116)
: error map: {{val| 0.000 -1.955 -1.431 }}


{{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]] (Smith): 0.094494


== 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 }}
 
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]].  


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 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.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 33: Line 40:
{{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:
* [[CTE]]: ~200/189 = 100.000, ~5/4 = 384.922 (~126/125 = 15.078)
: [[error map]]: {{val| 0.000 -1.955 -1.392 -1.017 }}
* [[POTE]]: ~200/189 = 100.000, ~5/4 = 383.775 (~126/125 = 16.225)
: error map: {{val| 0.000 -1.955 -2.538 -1.275 }}


{{Optimal ET sequence|legend=1| 12, 48d, 60, 72, 228, 300c, 372bc, 444bc }}
{{Optimal ET sequence|legend=1| 12, 48d, 60, 72, 228, 300c, 372bc, 444bc }}


[[Badness]]: 0.035686
[[Badness]] (Smith): 0.035686


=== 11-limit ===
=== 11-limit ===
Line 46: Line 57:
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:
* CTE: ~35/33 = 100.000, ~5/4 = 384.324 (~100/99 = 15.676)
* POTE: ~35/33 = 100.000, ~5/4 = 383.266 (~100/99 = 16.734)


{{Optimal ET sequence|legend=1| 12, 48dee, 60e, 72 }}
{{Optimal ET sequence|legend=0| 12, 48dee, 60e, 72 }}


Badness: 0.022235
Badness (Smith): 0.022235


==== 13-limit ====
==== 13-limit ====
Line 59: Line 72:
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:
* CTE: ~35/33 = 100.000, ~5/4 = 384.685 (~105/104 = 15.315)
* POTE: ~35/33 = 100.000, ~5/4 = 383.963 (~105/104 = 16.037)


{{Optimal ET sequence|legend=1| 12f, 48defff, 60eff, 72, 228f }}
{{Optimal ET sequence|legend=0| 12f, 48deefff, 60eff, 72, 228f }}


Badness: 0.021852
Badness (Smith): 0.021852


===== 17-limit =====
===== 17-limit =====
Line 72: Line 87:
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:
* CTE: ~18/17 = 100.000, ~5/4 = 384.685 (~105/104 = 15.315)
* POTE: ~18/17 = 100.000, ~5/4 = 383.750 (~105/104 = 16.250)


{{Optimal ET sequence|legend=1| 12f, 60eff, 72 }}
{{Optimal ET sequence|legend=0| 12f, 60eff, 72 }}


Badness: 0.017131
Badness (Smith): 0.017131


==== Comptone ====
==== Comptone ====
Line 85: Line 102:
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:
* CTE: ~35/33 = 100.000, ~5/4 = 383.552 (~100/99 = 16.448)
* POTE: ~35/33 = 100.000, ~5/4 = 382.612 (~100/99 = 17.388)


{{Optimal ET sequence|legend=1| 12, 60e, 72, 204cdef, 276cdeff }}
{{Optimal ET sequence|legend=0| 12, 60e, 72, 204cdef, 276cdeff }}


Badness: 0.025144
Badness (Smith): 0.025144


===== 17-limit =====
===== 17-limit =====
Line 98: Line 117:
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:
* CTE: ~18/17 = 100.000, ~5/4 = 383.552 (~100/99 = 16.448)
* POTE: ~18/17 = 100.000, ~5/4 = 382.597 (~100/99 = 17.403)


{{Optimal ET sequence|legend=1| 12, 60e, 72, 204cdefg, 276cdeffgg }}
{{Optimal ET sequence|legend=0| 12, 60e, 72, 204cdefg, 276cdeffgg }}


Badness: 0.016361
Badness (Smith): 0.016361
 
== Narrowed compton ==
''See [[Substitute harmonic#Narrowed compton]].''


== 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 118: Line 136:
: mapping generators: ~16/15, ~7
: mapping generators: ~16/15, ~7


[[Optimal tuning]] ([[POTE]]): ~16/15 = 1\12, ~7/4 = 973.210 (~64/63 = 26.790)
[[Optimal tuning]]s:
* [[CTE]]: ~16/15 = 100.000, ~7/4 = 968.826 (~64/63 = 31.174)
: [[error map]]: {{val| 0.000 -1.955 +13.686 0.000 }}
* [[POTE]]: ~16/15 = 100.000, ~7/4 = 973.210 (~64/63 = 26.790)
: error map: {{val| 0.000 -1.955 +13.686 +4.384 }}


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


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


=== 11-limit ===
=== 11-limit ===
Line 131: Line 153:
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:
* CTE: ~16/15 = 100.000, ~7/4 = 973.779 (~64/63 = 26.221)
* POTE: ~16/15 = 100.000, ~7/4 = 977.277 (~64/63 = 22.723)


{{Optimal ET sequence|legend=1| 12, 36e, 48c, 108ccd }}
{{Optimal ET sequence|legend=0| 12, 36e, 48c }}


Badness: 0.058213
Badness (Smith): 0.058213


=== Catlat ===
=== Catlat ===
Line 144: Line 168:
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:
* CTE: ~16/15 = 100.000, ~7/4 = 972.823 (~64/63 = 27.177)
* POTE: ~16/15 = 100.000, ~7/4 = 972.136 (~64/63 = 27.864)


{{Optimal ET sequence|legend=1| 36, 48c, 84c }}
{{Optimal ET sequence|legend=0| 12e, 36, 48c, 84c }}


Badness: 0.081909
Badness (Smith): 0.081909


=== Catnip ===
=== Catnip ===
Line 157: Line 183:
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:
* CTE: ~16/15 = 100.000, ~7/4 = 961.874 (~64/63 = 38.126)
* POTE: ~16/15 = 100.000, ~7/4 = 967.224 (~64/63 = 32.776)


{{Optimal ET sequence|legend=1| 12, 24, 36, 72ce }}
{{Optimal ET sequence|legend=0| 12, 24, 36, 72ce }}


Badness: 0.034478
Badness (Smith): 0.034478


==== 13-limit ====
==== 13-limit ====
Line 170: Line 198:
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:
* CTE: ~16/15 = 100.000, ~7/4 = 956.375 (~40/39 = 43.625)
* POTE: ~16/15 = 100.000, ~7/4 = 962.778 (~40/39 = 37.232)


{{Optimal ET sequence|legend=1| 12f, 24, 36f, 60cf }}
{{Optimal ET sequence|legend=0| 12f, 24, 36f }}


Badness: 0.028363
Badness (Smith): 0.028363


===== 17-limit =====
===== 17-limit =====
Line 183: Line 213:
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:
* CTE: ~18/17 = 100.000, ~7/4 = 956.375 (~40/39 = 43.625)
* POTE: ~18/17 = 100.000, ~7/4 = 960.223 (~40/39 = 39.777)


{{Optimal ET sequence|legend=1| 12f, 24, 36f, 60cf }}
{{Optimal ET sequence|legend=0| 12f, 24, 36f }}


Badness: 0.023246
Badness (Smith): 0.023246


===== 19-limit =====
===== 19-limit =====
Line 196: Line 228:
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:
* CTE: ~18/17 = 100.000, ~7/4 = 956.375 (~40/39 = 43.625)
* POTE: ~18/17 = 100.000, ~7/4 = 959.835 (~40/39 = 40.165)


{{Optimal ET sequence|legend=1| 12f, 24, 36f, 60cf }}
{{Optimal ET sequence|legend=0| 12f, 24, 36f }}


Badness: 0.018985
Badness (Smith): 0.018985


==== Duodecic ====
==== Duodecic ====
Line 209: Line 243:
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:
* CTE: ~16/15 = 100.000, ~7/4 = 961.255 (~64/63 = 38.745)
* POTE: ~16/15 = 100.000, ~7/4 = 962.312 (~64/63 = 37.688)


{{Optimal ET sequence|legend=1| 12, 24, 36, 60c }}
{{Optimal ET sequence|legend=0| 12, 24, 36 }}


Badness: 0.038307
Badness (Smith): 0.038307


===== 17-limit =====
===== 17-limit =====
Line 222: Line 258:
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:
* CTE: ~18/17 = 100.000, ~7/4 = 961.255 (~64/63 = 38.745)
* POTE: ~18/17 = 100.000, ~7/4 = 961.903 (~64/63 = 38.097)


{{Optimal ET sequence|legend=1| 12, 24, 36, 60c }}
{{Optimal ET sequence|legend=0| 12, 24, 36, 60c }}


Badness: 0.027487
Badness (Smith): 0.027487


===== 19-limit =====
===== 19-limit =====
Line 235: Line 273:
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:
* CTE: ~18/17 = 100.000, ~7/4 = 961.255 (~64/63 = 38.745)
* POTE: ~18/17 = 100.000, ~7/4 = 961.920 (~64/63 = 38.080)


{{Optimal ET sequence|legend=1| 12, 24, 36, 60c }}
{{Optimal ET sequence|legend=0| 12, 24, 36, 60c }}


Badness: 0.020939
Badness (Smith): 0.020939


== Duodecim ==
== Duodecim ==
{{See also| Jubilismic clan #Duodecim }}
[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


Line 250: Line 288:
{{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:
* [[CTE]]: ~16/15 = 1\12, ~11/8 = 551.318 (~33/32 = 48.682)
: [[error map]]: {{val| 0.000 -1.955 +13.686 +31.174 0.000 }}
* [[POTE]]: ~16/15 = 1\12, ~11/8 = 565.023 (~55/54 = 34.977)
: error map: {{val| 0.000 -1.955 +13.686 +31.174 +13.705 }}


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


[[Badness]]: 0.030536
[[Badness]] (Smith): 0.030536


== 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). The name ''hours'' was named 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 266: Line 308:


{{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  
[[Optimal tuning]]s:
* [[CTE]]: ~36/35 = 50.000, ~5/4 = 384.226 (~81/80 = 15.774)
: [[error map]]: {{val| 0.000 -1.955 -2.088 -3.052 }}
* [[POTE]]: ~36/35 = 50.000, ~5/4 = 384.033 (~81/80 = 15.967)
: error map: {{val| 0.000 -1.955 -2.280 -2.859 }}


{{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]] (Smith): 0.116091


=== 11-limit ===
=== 11-limit ===
Line 284: Line 328:
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
Optimal tunings:
* CTE: ~36/35 = 50.000, ~5/4 = 384.226 (~121/120 = 15.774)
* POTE: ~36/35 = 50.000, ~5/4 = 384.054 (~121/120 = 15.946)


{{Optimal ET sequence|legend=1| 24, 48, 72, 312bd, 384bcdd, 456bcdde, 528bcdde }}
{{Optimal ET sequence|legend=1| 24, 48, 72, 312bd, 384bcdd, 456bcdde, 528bcdde }}


Badness: 0.036248
Badness (Smith): 0.036248


=== 13-limit ===
=== 13-limit ===
Line 297: Line 343:
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
Optimal tunings:
* CTE: ~36/35 = 50.000, ~5/4 = 385.420 (~121/120 = 14.580)
* POTE: ~36/35 = 50.000, ~5/4 = 384.652 (~121/120 = 15.348)


{{Optimal ET sequence|legend=1| 24, 48f, 72, 168df, 240dff }}
{{Optimal ET sequence|legend=1| 24, 48f, 72, 168df, 240dff }}


Badness: 0.026931
Badness (Smith): 0.026931


== Decades ==
== Gamelstearn ==
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).  
The gamelstearn temperament has a period of 1/36 octave and tempers out the [[gamelisma]] (1029/1024) and the [[stearnsma]] (118098/117649).  
 
It used to be named "decades".


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 314: Line 364:
: mapping generators: ~49/48, ~5
: mapping generators: ~49/48, ~5


{{Multival|legend=1| 0 36 0 57 0 -101 }}
[[Optimal tuning]]s:
 
* [[CTE]]: ~49/48 = 33.333, ~5/4 = 386.314 (~81/80 = 13.686)
[[Optimal tuning]] ([[POTE]]): ~49/48 = 1\36, ~5/4 = 384.764
: [[error map]]: {{val| 0.000 -1.955 0.000 -2.159 }}
* [[POTE]]: ~49/48 = 33.333, ~5/4 = 384.764 (~81/80 = 15.236)
: error map: {{val| 0.000 -1.955 -1.549 -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]] (Smith): 0.108016


=== 11-limit ===
=== 11-limit ===
Line 329: Line 381:
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
Optimal tunings:
* CTE: ~49/48 = 33.333, ~5/4 = 385.797 (~81/80 = 14.203)
* POTE: ~49/48 = 33.333, ~5/4 = 385.150 (~81/80 = 14.850)


{{Optimal ET sequence|legend=1| 36, 72, 396bd, 468bcd, 540bcd, 612bccdd, 684bbccdd, 756bbccdd }}
{{Optimal ET sequence|legend=1| 36, 72, 396bd, 468bcd, 540bcd, 612bccdd, 684bbccdd, 756bbccdd }}


Badness: 0.043088
Badness (Smith): 0.043088


== 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|legend=1| 72 114 167 202 249 266 | 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]]): ~100/99 = 1\72, ~13/8 = 837.814
[[Optimal tuning]]s:
* [[CTE]]: ~100/99 = 16.667, ~13/8 = 840.528 (~325/324 = 7.194)
: [[error map]]: {{val| 0.000 -1.955 -2.980 -2.159 -1.318 0.000 }}
* [[POTE]]: ~100/99 = 16.667, ~13/8 = 837.814 (~364/363 = 4.481)
: error map: {{val| 0.000 -1.955 -2.980 -2.159 -1.318 -2.713 }}


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


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


[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Compton family| ]] <!-- main article -->
[[Category:Compton family| ]] <!-- main article -->
[[Category:Compton| ]] <!-- key article -->
[[Category:Compton| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]

Latest revision as of 11:30, 2 July 2025

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 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.

Compton

Main article: 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.

Subgroup: 2.3.5

Comma list: 531441/524288

Mapping: [⟨12 19 0], ⟨0 0 1]]

mapping generators: ~256/243, ~5

Optimal tunings:

  • CTE: ~256/243 = 100.000, ~5/4 = 386.314 (~81/80 = 13.686)
error map: ⟨0.000 -1.955 0.000]
  • POTE: ~256/243 = 100.000, ~5/4 = 384.884 (~81/80 = 15.116)
error map: ⟨0.000 -1.955 -1.431]

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

Badness (Smith): 0.094494

Septimal compton

Main article: Compton

Septimal compton is also known as waage. In terms of the normal list, compton adds 413343/409600 ([-14 10 -2 1⟩) to the Pythagorean comma; however, it can also be characterized by saying it adds 225/224.

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.

Subgroup: 2.3.5.7

Comma list: 225/224, 250047/250000

Mapping: [⟨12 19 0 -22], ⟨0 0 1 2]]

Optimal tunings:

  • CTE: ~200/189 = 100.000, ~5/4 = 384.922 (~126/125 = 15.078)
error map: ⟨0.000 -1.955 -1.392 -1.017]
  • POTE: ~200/189 = 100.000, ~5/4 = 383.775 (~126/125 = 16.225)
error map: ⟨0.000 -1.955 -2.538 -1.275]

Optimal ET sequence: 12, 48d, 60, 72, 228, 300c, 372bc, 444bc

Badness (Smith): 0.035686

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:

  • CTE: ~35/33 = 100.000, ~5/4 = 384.324 (~100/99 = 15.676)
  • POTE: ~35/33 = 100.000, ~5/4 = 383.266 (~100/99 = 16.734)

Optimal ET sequence: 12, 48dee, 60e, 72

Badness (Smith): 0.022235

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:

  • CTE: ~35/33 = 100.000, ~5/4 = 384.685 (~105/104 = 15.315)
  • POTE: ~35/33 = 100.000, ~5/4 = 383.963 (~105/104 = 16.037)

Optimal ET sequence: 12f, 48deefff, 60eff, 72, 228f

Badness (Smith): 0.021852

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:

  • CTE: ~18/17 = 100.000, ~5/4 = 384.685 (~105/104 = 15.315)
  • POTE: ~18/17 = 100.000, ~5/4 = 383.750 (~105/104 = 16.250)

Optimal ET sequence: 12f, 60eff, 72

Badness (Smith): 0.017131

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:

  • CTE: ~35/33 = 100.000, ~5/4 = 383.552 (~100/99 = 16.448)
  • POTE: ~35/33 = 100.000, ~5/4 = 382.612 (~100/99 = 17.388)

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

Badness (Smith): 0.025144

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:

  • CTE: ~18/17 = 100.000, ~5/4 = 383.552 (~100/99 = 16.448)
  • POTE: ~18/17 = 100.000, ~5/4 = 382.597 (~100/99 = 17.403)

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

Badness (Smith): 0.016361

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:

  • CTE: ~16/15 = 100.000, ~7/4 = 968.826 (~64/63 = 31.174)
error map: ⟨0.000 -1.955 +13.686 0.000]
  • POTE: ~16/15 = 100.000, ~7/4 = 973.210 (~64/63 = 26.790)
error map: ⟨0.000 -1.955 +13.686 +4.384]

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

Badness (Smith): 0.050297

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:

  • CTE: ~16/15 = 100.000, ~7/4 = 973.779 (~64/63 = 26.221)
  • POTE: ~16/15 = 100.000, ~7/4 = 977.277 (~64/63 = 22.723)

Optimal ET sequence: 12, 36e, 48c

Badness (Smith): 0.058213

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:

  • CTE: ~16/15 = 100.000, ~7/4 = 972.823 (~64/63 = 27.177)
  • POTE: ~16/15 = 100.000, ~7/4 = 972.136 (~64/63 = 27.864)

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

Badness (Smith): 0.081909

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:

  • CTE: ~16/15 = 100.000, ~7/4 = 961.874 (~64/63 = 38.126)
  • POTE: ~16/15 = 100.000, ~7/4 = 967.224 (~64/63 = 32.776)

Optimal ET sequence: 12, 24, 36, 72ce

Badness (Smith): 0.034478

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:

  • CTE: ~16/15 = 100.000, ~7/4 = 956.375 (~40/39 = 43.625)
  • POTE: ~16/15 = 100.000, ~7/4 = 962.778 (~40/39 = 37.232)

Optimal ET sequence: 12f, 24, 36f

Badness (Smith): 0.028363

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:

  • CTE: ~18/17 = 100.000, ~7/4 = 956.375 (~40/39 = 43.625)
  • POTE: ~18/17 = 100.000, ~7/4 = 960.223 (~40/39 = 39.777)

Optimal ET sequence: 12f, 24, 36f

Badness (Smith): 0.023246

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:

  • CTE: ~18/17 = 100.000, ~7/4 = 956.375 (~40/39 = 43.625)
  • POTE: ~18/17 = 100.000, ~7/4 = 959.835 (~40/39 = 40.165)

Optimal ET sequence: 12f, 24, 36f

Badness (Smith): 0.018985

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:

  • CTE: ~16/15 = 100.000, ~7/4 = 961.255 (~64/63 = 38.745)
  • POTE: ~16/15 = 100.000, ~7/4 = 962.312 (~64/63 = 37.688)

Optimal ET sequence: 12, 24, 36

Badness (Smith): 0.038307

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:

  • CTE: ~18/17 = 100.000, ~7/4 = 961.255 (~64/63 = 38.745)
  • POTE: ~18/17 = 100.000, ~7/4 = 961.903 (~64/63 = 38.097)

Optimal ET sequence: 12, 24, 36, 60c

Badness (Smith): 0.027487

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:

  • CTE: ~18/17 = 100.000, ~7/4 = 961.255 (~64/63 = 38.745)
  • POTE: ~18/17 = 100.000, ~7/4 = 961.920 (~64/63 = 38.080)

Optimal ET sequence: 12, 24, 36, 60c

Badness (Smith): 0.020939

Duodecim

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:

  • CTE: ~16/15 = 1\12, ~11/8 = 551.318 (~33/32 = 48.682)
error map: ⟨0.000 -1.955 +13.686 +31.174 0.000]
  • POTE: ~16/15 = 1\12, ~11/8 = 565.023 (~55/54 = 34.977)
error map: ⟨0.000 -1.955 +13.686 +31.174 +13.705]

Optimal ET sequence: 12, 24d, 36d

Badness (Smith): 0.030536

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 named 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:

  • CTE: ~36/35 = 50.000, ~5/4 = 384.226 (~81/80 = 15.774)
error map: ⟨0.000 -1.955 -2.088 -3.052]
  • POTE: ~36/35 = 50.000, ~5/4 = 384.033 (~81/80 = 15.967)
error map: ⟨0.000 -1.955 -2.280 -2.859]

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

Badness (Smith): 0.116091

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:

  • CTE: ~36/35 = 50.000, ~5/4 = 384.226 (~121/120 = 15.774)
  • POTE: ~36/35 = 50.000, ~5/4 = 384.054 (~121/120 = 15.946)

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

Badness (Smith): 0.036248

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:

  • CTE: ~36/35 = 50.000, ~5/4 = 385.420 (~121/120 = 14.580)
  • POTE: ~36/35 = 50.000, ~5/4 = 384.652 (~121/120 = 15.348)

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

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).

It used to be named "decades".

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:

  • CTE: ~49/48 = 33.333, ~5/4 = 386.314 (~81/80 = 13.686)
error map: ⟨0.000 -1.955 0.000 -2.159]
  • POTE: ~49/48 = 33.333, ~5/4 = 384.764 (~81/80 = 15.236)
error map: ⟨0.000 -1.955 -1.549 -2.159]

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

Badness (Smith): 0.108016

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:

  • CTE: ~49/48 = 33.333, ~5/4 = 385.797 (~81/80 = 14.203)
  • POTE: ~49/48 = 33.333, ~5/4 = 385.150 (~81/80 = 14.850)

Optimal ET sequence: 36, 72, 396bd, 468bcd, 540bcd, 612bccdd, 684bbccdd, 756bbccdd

Badness (Smith): 0.043088

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:

  • CTE: ~100/99 = 16.667, ~13/8 = 840.528 (~325/324 = 7.194)
error map: ⟨0.000 -1.955 -2.980 -2.159 -1.318 0.000]
  • POTE: ~100/99 = 16.667, ~13/8 = 837.814 (~364/363 = 4.481)
error map: ⟨0.000 -1.955 -2.980 -2.159 -1.318 -2.713]

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

Badness (Smith): 0.029956

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