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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]]
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