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Compton family: Difference between revisions

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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 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''', 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.


== 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. 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 temperaments, it is the 12 & 72 temperament, and [[72edo]], [[84edo]] or [[240edo]] make for good tunings.  


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
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[[Comma list]]: 531441/524288
[[Comma list]]: 531441/524288


[[Mapping]]: [{{val| 12 19 0 }}, {{val| 0 0 1 }}
{{Mapping|legend=1| 12 19 0 | 0 0 1 }}


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]] ([[POTE]]): ~256/243 = 1\12, ~5/4 = 384.884 (~81/80 = 15.116)
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[[Comma list]]: 225/224, 250047/250000
[[Comma list]]: 225/224, 250047/250000


[[Mapping]]: [{{val| 12 19 0 -22 }}, {{val| 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]] ([[POTE]]): ~256/243 = 1\12, ~5/4 = 383.7752 (~126/125 = 16.2248)
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Comma list: 225/224, 441/440, 4375/4356
Comma list: 225/224, 441/440, 4375/4356


Mapping: [{{val|12 19 0 -22 -42 }}, {{val| 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 tuning (POTE): ~256/243 = 1\12, ~5/4 = 383.2660 (~100/99 = 16.7340)
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Comma list: 225/224, 351/350, 364/363, 441/440
Comma list: 225/224, 351/350, 364/363, 441/440


Mapping: [{{val| 12 19 0 -22 -42 -67 }}, {{val| 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 tuning (POTE): ~256/243 = 1\12, ~5/4 = 383.9628 (~105/104 = 16.0372)
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Comma list: 221/220, 225/224, 289/288, 351/350, 441/440
Comma list: 221/220, 225/224, 289/288, 351/350, 441/440


Mapping: [{{val| 12 19 0 -22 -42 -67 49 }}, {{val| 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 tuning (POTE): ~18/17 = 1\12, ~5/4 = 383.7500 (~105/104 = 16.2500)
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Comma list: 225/224, 325/324, 441/440, 1001/1000
Comma list: 225/224, 325/324, 441/440, 1001/1000


Mapping: [{{val| 12 19 0 -22 -42 100 }}, {{val| 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 tuning (POTE): ~256/243 = 1\12, ~5/4 = 382.6116 (~100/99 = 17.3884)
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Comma list: 225/224, 273/272, 289/288, 325/324, 441/440
Comma list: 225/224, 273/272, 289/288, 325/324, 441/440


Mapping: [{{val| 12 19 0 -22 -42 100 49 }}, {{val| 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 tuning (POTE): ~18/17 = 1\12, ~5/4 = 382.5968 (~100/99 = 17.4032)
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[[Comma list]]: 81/80, 128/125
[[Comma list]]: 81/80, 128/125


[[Mapping]]: [{{val| 12 19 28 0 }}, {{val| 0 0 0 1 }}]
{{Mapping|legend=1| 12 19 28 0 | 0 0 0 1 }}


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]] ([[POTE]]): ~16/15 = 1\12, ~7/4 = 973.210 (~64/63 = 26.790)
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Comma list: 81/80, 99/98, 128/125
Comma list: 81/80, 99/98, 128/125


Mapping: [{{val| 12 19 28 0 -26 }}, {{val| 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 tuning (POTE): ~16/15 = 1\12, ~7/4 = 977.277 (~64/63 = 22.723)
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Comma list: 81/80, 128/125, 540/539
Comma list: 81/80, 128/125, 540/539


Mapping: [{{val| 12 19 28 0 109 }}, {{val| 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 tuning (POTE): ~16/15 = 1\12, ~7/4 = 972.136 (~64/63 = 27.864)
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Comma list: 56/55, 81/80, 128/125
Comma list: 56/55, 81/80, 128/125


Mapping: [{{val| 12 19 28 0 8 }}, {{val| 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 tuning (POTE): ~16/15 = 1\12, ~7/4 = 967.224 (~64/63 = 32.776)
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Comma list: 56/55, 66/65, 81/80, 105/104
Comma list: 56/55, 66/65, 81/80, 105/104


Mapping: [{{val| 12 19 28 0 8 11 }}, {{val| 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 tuning (POTE): ~16/15 = 1\12, ~7/4 = 962.778 (~40/39 = 37.232)
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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


Mapping: [{{val| 12 19 28 0 8 11 49 }}, {{val| 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 tuning (POTE): ~18/17 = 1\12, ~7/4 = 960.223 (~40/39 = 39.777)
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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


Mapping: [{{val| 12 19 28 0 8 11 49 51 }}, {{val| 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 tuning (POTE): ~18/17 = 1\12, ~7/4 = 959.835 (~40/39 = 40.165)
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Comma list: 56/55, 81/80, 91/90, 128/125
Comma list: 56/55, 81/80, 91/90, 128/125


Mapping: [{{val| 12 19 28 0 8 78 }}, {{val| 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 tuning (POTE): ~16/15 = 1\12, ~7/4 = 962.312 (~64/63 = 37.688)
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Comma list: 51/50, 56/55, 81/80, 91/90, 128/125
Comma list: 51/50, 56/55, 81/80, 91/90, 128/125


Mapping: [{{val| 12 19 28 0 8 78 49 }}, {{val| 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 tuning (POTE): ~18/17 = 1\12, ~7/4 = 961.903 (~64/63 = 38.097)
Line 228: Line 228:
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


Mapping: [{{val| 12 19 28 0 8 78 49 51 }}, {{val| 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 tuning (POTE): ~18/17 = 1\12, ~7/4 = 961.920 (~64/63 = 38.080)
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[[Comma list]]: 36/35, 50/49, 64/63
[[Comma list]]: 36/35, 50/49, 64/63


[[Mapping]]: [{{val| 12 19 28 34 0 }}, {{val| 0 0 0 0 1 }}]
{{Mapping|legend=1| 12 19 28 34 0 | 0 0 0 0 1 }}


Mapping generators: ~16/15, ~11
: mapping generators: ~16/15, ~11


[[Optimal tuning]] ([[POTE]]): ~16/15 = 1\12, ~11/8 = 565.023 (~55/54 = 34.977)
[[Optimal tuning]] ([[POTE]]): ~16/15 = 1\12, ~11/8 = 565.023 (~55/54 = 34.977)
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[[Comma list]]: 19683/19600, 33075/32768
[[Comma list]]: 19683/19600, 33075/32768


[[Mapping]]: [{{val| 24 38 0 123 }}, {{val| 0 0 1 -1 }}]
{{Mapping|legend=1| 24 38 0 123 | 0 0 1 -1 }}


{{Multival|legend=1| 0 24 -24 38 -38 -123 }}
{{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]] ([[POTE]]): ~36/35 = 1\24, ~5/4 = 384.033  
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Comma list: 243/242, 385/384, 9801/9800
Comma list: 243/242, 385/384, 9801/9800


Mapping: [{{val| 24 38 0 123 83 }}, {{val| 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 tuning (POTE): ~36/35 = 1\24, ~5/4 = 384.054
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Comma list: 243/242, 351/350, 364/363, 385/384
Comma list: 243/242, 351/350, 364/363, 385/384


Mapping: [{{val| 24 38 0 123 83 33 }}, {{val| 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 tuning (POTE): ~36/35 = 1\24, ~5/4 = 384.652
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[[Comma list]]: 1029/1024, 118098/117649
[[Comma list]]: 1029/1024, 118098/117649


[[Mapping]]: [{{val| 36 57 0 101 }}, {{val| 0 0 1 0 }}]
{{Mapping|legend=1| 36 57 0 101 | 0 0 1 0 }}


Mapping generators: ~49/48, ~5
: mapping generators: ~49/48, ~5


{{Multival|legend=1| 0 36 0 57 0 -101 }}
{{Multival|legend=1| 0 36 0 57 0 -101 }}
Line 322: Line 322:
Comma list: 540/539, 1029/1024, 4000/3993
Comma list: 540/539, 1029/1024, 4000/3993


Mapping: [{{val| 36 57 0 101 41 }}, {{val| 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 tuning (POTE): ~49/48 = 1\36, ~5/4 = 384.150
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[[Comma list]]: 225/224, 243/242, 441/440, 4375/4356
[[Comma list]]: 225/224, 243/242, 441/440, 4375/4356


[[Mapping]]: [{{val| 72 114 167 202 249 266 }}, {{val| 0 0 0 0 0 1 }}]
{{Mapping|legend=1| 72 114 167 202 249 266 | 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]] ([[POTE]]): ~100/99 = 1\72, ~13/8 = 837.814
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[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Compton family| ]] <!-- main article -->
[[Category:Compton family| ]] <!-- main article -->
[[Category:Compton| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
[[Category:Compton]]
Retrieved from "https://en.xen.wiki/w/Compton_family"