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The '''Compton 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|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.
The '''Compton 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 cents flat, the tuning of the larger primes is not so constrained, and the point of these temperaments is to improve on it.


== Compton ==
== Compton ==
Subgroup: 2.3.5
[[Subgroup]]: 2.3.5


[[Comma]]: 531441/524288
[[Comma]]: 531441/524288
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[[Mapping]]: [{{val| 12 19 0 }}, {{val| 0 0 1 }}
[[Mapping]]: [{{val| 12 19 0 }}, {{val| 0 0 1 }}


[[POTE generator]]: ~5/4 = 384.884 or ~81/80 = 15.116
Mapping generators: ~256/243, ~5


{{Val list|legend=1| 12, 72, 84, 156, 240, 396b }}
[[Optimal tuning]] ([[POTE]]): ~5/4 = 384.884 (~81/80 = 15.116)
 
{{Val list|legend=1| 12, 48, 60, 72, 84, 156, 240, 396b, 636bbc }}


[[Badness]]: 0.094494
[[Badness]]: 0.094494
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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.


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 225/224, 250047/250000
[[Comma list]]: 225/224, 250047/250000
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[[Mapping]]: [{{val| 12 19 0 -22 }}, {{val| 0 0 1 2 }}]
[[Mapping]]: [{{val| 12 19 0 -22 }}, {{val| 0 0 1 2 }}]


[[POTE generator]]: ~5/4 = 383.7752
[[Optimal tuning]] ([[POTE]]): ~5/4 = 383.7752 (~126/125 = 16.2248)


{{Val list|legend=1| 12, 60, 72, 228, 300c, 372bc, 444bc }}
{{Val list|legend=1| 12, 48d, 60, 72, 228, 300c, 372bc, 444bc }}


[[Badness]]: 0.035686
[[Badness]]: 0.035686
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Mapping: [{{val|12 19 0 -22 -42 }}, {{val| 0 0 1 2 3 }}]
Mapping: [{{val|12 19 0 -22 -42 }}, {{val| 0 0 1 2 3 }}]


POTE generator: ~5/4 = 383.2660
Optimal tuning (POTE): ~5/4 = 383.2660 (~100/99 = 16.7340)


Optimal GPV sequence: {{Val list| 12, 60e, 72 }}
Optimal GPV sequence: {{Val list| 12, 48dee, 60e, 72 }}


Badness: 0.022235
Badness: 0.022235
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Mapping: [{{val| 12 19 0 -22 -42 -67 }}, {{val| 0 0 1 2 3 4 }}]
Mapping: [{{val| 12 19 0 -22 -42 -67 }}, {{val| 0 0 1 2 3 4 }}]


POTE generator: ~5/4 = 383.9628
Optimal tuning (POTE): ~5/4 = 383.9628 (~105/104 = 16.0372)


Optimal GPV sequence: {{Val list| 12f, 72, 84, 156, 228f, 300cf }}
Optimal GPV sequence: {{Val list| 12f, 48defff, 60eff, 72, 228f }}


Badness: 0.021852
Badness: 0.021852
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Mapping: [{{val| 12 19 0 -22 -42 -67 49 }}, {{val| 0 0 1 2 3 4 0 }}]
Mapping: [{{val| 12 19 0 -22 -42 -67 49 }}, {{val| 0 0 1 2 3 4 0 }}]


POTE generator: ~5/4 = 383.7500
Optimal tuning (POTE): ~5/4 = 383.7500 (~105/104 = 16.2500)


Optimal GPV sequence: {{Val list| 12f, 72, 84, 156g, 228fg }}
Optimal GPV sequence: {{Val list| 12f, 60eff, 72 }}


Badness: 0.017131
Badness: 0.017131
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Mapping: [{{val| 12 19 0 -22 -42 100 }}, {{val| 0 0 1 2 3 -2 }}]
Mapping: [{{val| 12 19 0 -22 -42 100 }}, {{val| 0 0 1 2 3 -2 }}]


POTE generator: ~5/4 = 382.6116
Optimal tuning (POTE): ~5/4 = 382.6116 (~100/99 = 17.3884)


Optimal GPV sequence: {{Val list| 12, 60e, 72, 204cdef, 276cdef }}
Optimal GPV sequence: {{Val list| 12, 60e, 72, 204cdef, 276cdeff }}


Badness: 0.025144
Badness: 0.025144
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Mapping: [{{val| 12 19 0 -22 -42 100 49 }}, {{val| 0 0 1 2 3 -2 0 }}]
Mapping: [{{val| 12 19 0 -22 -42 100 49 }}, {{val| 0 0 1 2 3 -2 0 }}]


POTE generator: ~5/4 = 382.5968
Optimal tuning (POTE): ~5/4 = 382.5968 (~100/99 = 17.4032)


Optimal GPV sequence: {{Val list| 12, 60e, 72, 132deg, 204cdefg }}
Optimal GPV sequence: {{Val list| 12, 60e, 72, 204cdefg, 276cdeffgg }}


Badness: 0.016361
Badness: 0.016361


== 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|12EDO]]. Catler can also be characterized as the 12&24 temperament. [[36edo|36EDO]] or [[48edo|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 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


[[Comma list]]: 81/80, 128/125
[[Comma list]]: 81/80, 128/125
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[[Mapping]]: [{{val| 12 19 28 0 }}, {{val| 0 0 0 1 }}]
[[Mapping]]: [{{val| 12 19 28 0 }}, {{val| 0 0 0 1 }}]


[[POTE generator]]: ~64/63 = 26.790
Mapping generators: ~16/15, ~7
 
[[Optimal tuning]] ([[POTE]]): ~64/63 = 26.790


{{Val list|legend=1| 12, 24, 36, 48c }}
{{Val list|legend=1| 12, 24, 36, 48c }}
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Comma list: 81/80, 99/98, 128/125
Comma list: 81/80, 99/98, 128/125


POTE generator: ~64/63 = 22.723
Mapping: [{{val| 12 19 28 0 -26 }}, {{val| 0 0 0 1 2 }}]


Mapping: [{{val| 12 19 28 0 -26 }}, {{val| 0 0 0 1 2 }}]
Optimal tuning (POTE): ~64/63 = 22.723


Optimal GPV sequence: {{Val list| 12, 36e, 48c, 108ccd }}
Optimal GPV sequence: {{Val list| 12, 36e, 48c, 108ccd }}
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Comma list: 81/80, 128/125, 540/539
Comma list: 81/80, 128/125, 540/539


POTE generator: ~64/63 = 27.864
Mapping: [{{val| 12 19 28 0 109 }}, {{val| 0 0 0 1 -2 }}]


Mapping: [{{val| 12 19 28 0 109 }}, {{val| 0 0 0 1 -2 }}]
Optimal tuning (POTE): ~64/63 = 27.864


Optimal GPV sequence: {{Val list| 36, 48c, 84c }}
Optimal GPV sequence: {{Val list| 36, 48c, 84c }}
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Comma list: 56/55, 81/80, 128/125
Comma list: 56/55, 81/80, 128/125


POTE generator: ~36/35 = 32.776
Mapping: [{{val| 12 19 28 0 8 }}, {{val| 0 0 0 1 1 }}]


Mapping: [{{val| 12 19 28 0 8 }}, {{val| 0 0 0 1 1 }}]
Optimal tuning (POTE): ~36/35 = 32.776


Optimal GPV sequence: {{Val list| 12, 24, 36, 72ce }}
Optimal GPV sequence: {{Val list| 12, 24, 36, 72ce }}
Line 157: Line 161:
Comma list: 56/55, 66/65, 81/80, 105/104
Comma list: 56/55, 66/65, 81/80, 105/104


POTE generator: ~36/35 = 37.232
Mapping: [{{val| 12 19 28 0 8 11 }}, {{val| 0 0 0 1 1 1 }}]


Mapping: [{{val| 12 19 28 0 8 11 }}, {{val| 0 0 0 1 1 1 }}]
Optimal tuning (POTE): ~36/35 = 37.232


Optimal GPV sequence: {{Val list| 12f, 24, 36f, 60cf }}
Optimal GPV sequence: {{Val list| 12f, 24, 36f, 60cf }}
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Comma list: 51/50, 56/55, 66/65, 81/80, 105/104
Comma list: 51/50, 56/55, 66/65, 81/80, 105/104


POTE generator: ~36/35 = 39.777
Mapping: [{{val| 12 19 28 0 8 11 49 }}, {{val| 0 0 0 1 1 1 0 }}]


Mapping: [{{val| 12 19 28 0 8 11 49 }}, {{val| 0 0 0 1 1 1 0 }}]
Optimal tuning (POTE): ~36/35 = 39.777


Optimal GPV sequence: {{Val list| 12f, 24, 36f, 60cf }}
Optimal GPV sequence: {{Val list| 12f, 24, 36f, 60cf }}
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Comma list: 51/50, 56/55, 66/65, 76/75, 81/80, 96/95
Comma list: 51/50, 56/55, 66/65, 76/75, 81/80, 96/95


POTE generator: ~36/35 = 40.165
Mapping: [{{val| 12 19 28 0 8 11 49 51 }}, {{val| 0 0 0 1 1 1 0 0 }}]


Mapping: [{{val| 12 19 28 0 8 11 49 51 }}, {{val| 0 0 0 1 1 1 0 0 }}]
Optimal tuning (POTE): ~36/35 = 40.165


Optimal GPV sequence: {{Val list| 12f, 24, 36f, 60cf }}
Optimal GPV sequence: {{Val list| 12f, 24, 36f, 60cf }}
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Comma list: 56/55, 81/80, 91/90, 128/125
Comma list: 56/55, 81/80, 91/90, 128/125


POTE generator: ~36/35 = 37.688
Mapping: [{{val| 12 19 28 0 8 78 }}, {{val| 0 0 0 1 1 -1 }}]


Mapping: [{{val| 12 19 28 0 8 78 }}, {{val| 0 0 0 1 1 -1 }}]
Optimal tuning (POTE): ~36/35 = 37.688


Optimal GPV sequence: {{Val list| 12, 24, 36, 60c }}
Optimal GPV sequence: {{Val list| 12, 24, 36, 60c }}
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===== 17-limit =====
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17
Comma list: 51/50, 56/55, 81/80, 91/90, 128/125
Comma list: 51/50, 56/55, 81/80, 91/90, 128/125


POTE generator: ~36/35 = 38.097
Mapping: [{{val| 12 19 28 0 8 78 49 }}, {{val| 0 0 0 1 1 -1 0 }}]


Mapping: [{{val| 12 19 28 0 8 78 49 }}, {{val| 0 0 0 1 1 -1 0 }}]
Optimal tuning (POTE): ~36/35 = 38.097


Optimal GPV sequence: {{Val list| 12, 24, 36, 60c }}
Optimal GPV sequence: {{Val list| 12, 24, 36, 60c }}
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Comma list: 51/50, 56/55, 76/75, 81/80, 91/90, 96/95
Comma list: 51/50, 56/55, 76/75, 81/80, 91/90, 96/95


POTE generator: ~36/35 = 38.080
Mapping: [{{val| 12 19 28 0 8 78 49 51 }}, {{val| 0 0 0 1 1 -1 0 0 }}]


Mapping: [{{val| 12 19 28 0 8 78 49 51 }}, {{val| 0 0 0 1 1 -1 0 0 }}]
Optimal tuning (POTE): ~36/35 = 38.080


Optimal GPV sequence: {{Val list| 12, 24, 36, 60c }}
Optimal GPV sequence: {{Val list| 12, 24, 36, 60c }}
Line 230: Line 235:


== Duodecim ==
== Duodecim ==
{{see also| Jubilismic clan #Duodecim }}
{{See also| Jubilismic clan #Duodecim }}


Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 36/35, 50/49, 64/63
[[Comma list]]: 36/35, 50/49, 64/63
Line 238: Line 243:
[[Mapping]]: [{{val| 12 19 28 34 0 }}, {{val| 0 0 0 0 1 }}]
[[Mapping]]: [{{val| 12 19 28 34 0 }}, {{val| 0 0 0 0 1 }}]


[[POTE generator]]: ~45/44 = 34.977
Mapping generators: ~16/15, ~11
 
[[Optimal tuning]] ([[POTE]]): ~45/44 = 34.977


{{Val list|legend=1| 12, 24d }}
{{Val list|legend=1| 12, 24d }}
Line 255: Line 262:
{{Multival|legend=1| 0 24 -24 38 -38 -123 }}
{{Multival|legend=1| 0 24 -24 38 -38 -123 }}


[[POTE generator]]: ~5/4 = 384.033  
Mapping generators: ~36/35, ~5
 
[[Optimal tuning]] ([[POTE]]): ~5/4 = 384.033  


{{Val list|legend=1| 24, 48, 72, 312bd, 384bcdd, 456bcdd, 528bcdd, 600bccdd }}
{{Val list|legend=1| 24, 48, 72, 312bd, 384bcdd, 456bcdd, 528bcdd, 600bccdd }}
Line 268: Line 277:
Mapping: [{{val| 24 38 0 123 83 }}, {{val| 0 0 1 -1 0 }}]
Mapping: [{{val| 24 38 0 123 83 }}, {{val| 0 0 1 -1 0 }}]


POTE generator: ~5/4 = 384.054
Optimal tuning (POTE): ~5/4 = 384.054


Optimal GPV sequence: {{Val list| 24, 48, 72, 312bd, 384bcdd, 456bcdde, 528bcdde }}
Optimal GPV sequence: {{Val list| 24, 48, 72, 312bd, 384bcdd, 456bcdde, 528bcdde }}
Line 281: Line 290:
Mapping: [{{val| 24 38 0 123 83 33 }}, {{val| 0 0 1 -1 0 1 }}]
Mapping: [{{val| 24 38 0 123 83 33 }}, {{val| 0 0 1 -1 0 1 }}]


POTE generator: ~5/4 = 384.652
Optimal tuning (POTE): ~5/4 = 384.652


Optimal GPV sequence: {{Val list| 24, 48f, 72, 168df, 240dff }}
Optimal GPV sequence: {{Val list| 24, 48f, 72, 168df, 240dff }}
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[[Mapping]]: [{{val| 36 57 0 101 }}, {{val| 0 0 1 0 }}]
[[Mapping]]: [{{val| 36 57 0 101 }}, {{val| 0 0 1 0 }}]
Mapping generators: ~49/48, ~5


{{Multival|legend=1| 0 36 0 57 0 -101 }}
{{Multival|legend=1| 0 36 0 57 0 -101 }}


[[POTE generator]]: ~5/4 = 384.764
[[Optimal tuning]] ([[POTE]]): ~5/4 = 384.764


{{Val list|legend=1| 36, 72, 252, 324bd, 396bd }}
{{Val list|legend=1| 36, 72, 252, 324bd, 396bd }}
Line 311: Line 322:
Mapping: [{{val| 36 57 0 101 41 }}, {{val| 0 0 1 0 1 }}]
Mapping: [{{val| 36 57 0 101 41 }}, {{val| 0 0 1 0 1 }}]


POTE generator: ~5/4 = 384.150
Optimal tuning (POTE): ~5/4 = 384.150


Optimal GPV sequence: {{Val list| 36, 72, 396bd, 468bcd, 540bcd, 612bccdd, 684bbccdd, 756bbccdd }}
Optimal GPV sequence: {{Val list| 36, 72, 396bd, 468bcd, 540bcd, 612bccdd, 684bbccdd, 756bbccdd }}
Line 324: Line 335:
[[Mapping]]: [{{val| 72 114 167 202 249 266 }}, {{val| 0 0 0 0 0 1 }}]
[[Mapping]]: [{{val| 72 114 167 202 249 266 }}, {{val| 0 0 0 0 0 1 }}]


[[POTE generator]]: ~13/8 = 837.814
Mapping generators: ~100/99, ~13
 
[[Optimal tuning]] ([[POTE]]): ~13/8 = 837.814


{{Val list|legend=1| 72, 144, 216c, 288cdf, 504bcdef }}
{{Val list|legend=1| 72, 144, 216c, 288cdf, 504bcdef }}
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[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Compton]]
[[Category:Compton family| ]] <!-- main article -->
[[Category:Compton family| ]] <!-- main article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
[[Category:Compton]]
Retrieved from "https://en.xen.wiki/w/Compton_family"