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Compton family: Difference between revisions

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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]]. 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 ==
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Mapping generators: ~256/243, ~5
Mapping generators: ~256/243, ~5


[[Optimal tuning]] ([[POTE]]): ~5/4 = 384.884 (~81/80 = 15.116)
[[Optimal tuning]] ([[POTE]]): ~256/243 = 1\12, ~5/4 = 384.884 (~81/80 = 15.116)


{{Val list|legend=1| 12, 48, 60, 72, 84, 156, 240, 396b, 636bbc }}
{{Val list|legend=1| 12, 48, 60, 72, 84, 156, 240, 396b, 636bbc }}
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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 }}]


[[Optimal tuning]] ([[POTE]]): ~5/4 = 383.7752 (~126/125 = 16.2248)
[[Optimal tuning]] ([[POTE]]): ~256/243 = 1\12, ~5/4 = 383.7752 (~126/125 = 16.2248)


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


Optimal tuning (POTE): ~5/4 = 383.2660 (~100/99 = 16.7340)
Optimal tuning (POTE): ~256/243 = 1\12, ~5/4 = 383.2660 (~100/99 = 16.7340)


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


Optimal tuning (POTE): ~5/4 = 383.9628 (~105/104 = 16.0372)
Optimal tuning (POTE): ~256/243 = 1\12, ~5/4 = 383.9628 (~105/104 = 16.0372)


Optimal GPV sequence: {{Val list| 12f, 48defff, 60eff, 72, 228f }}
Optimal GPV sequence: {{Val list| 12f, 48defff, 60eff, 72, 228f }}
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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 }}]


Optimal tuning (POTE): ~5/4 = 383.7500 (~105/104 = 16.2500)
Optimal tuning (POTE): ~18/17 = 1\12, ~5/4 = 383.7500 (~105/104 = 16.2500)


Optimal GPV sequence: {{Val list| 12f, 60eff, 72 }}
Optimal GPV sequence: {{Val list| 12f, 60eff, 72 }}
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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 }}]


Optimal tuning (POTE): ~5/4 = 382.6116 (~100/99 = 17.3884)
Optimal tuning (POTE): ~256/243 = 1\12, ~5/4 = 382.6116 (~100/99 = 17.3884)


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


Optimal tuning (POTE): ~5/4 = 382.5968 (~100/99 = 17.4032)
Optimal tuning (POTE): ~18/17 = 1\12, ~5/4 = 382.5968 (~100/99 = 17.4032)


Optimal GPV sequence: {{Val list| 12, 60e, 72, 204cdefg, 276cdeffgg }}
Optimal GPV sequence: {{Val list| 12, 60e, 72, 204cdefg, 276cdeffgg }}
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Mapping generators: ~16/15, ~7
Mapping generators: ~16/15, ~7


[[Optimal tuning]] ([[POTE]]): ~64/63 = 26.790
[[Optimal tuning]] ([[POTE]]): ~16/15 = 1\12, ~64/63 = 26.790


{{Val list|legend=1| 12, 24, 36, 48c }}
{{Val list|legend=1| 12, 24, 36, 48c }}
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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 tuning (POTE): ~16/15 = 1\12, ~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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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 tuning (POTE): ~16/15 = 1\12, ~64/63 = 27.864


Optimal GPV sequence: {{Val list| 36, 48c, 84c }}
Optimal GPV sequence: {{Val list| 36, 48c, 84c }}
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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 tuning (POTE): ~16/15 = 1\12, ~36/35 = 32.776


Optimal GPV sequence: {{Val list| 12, 24, 36, 72ce }}
Optimal GPV sequence: {{Val list| 12, 24, 36, 72ce }}
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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 tuning (POTE): ~16/15 = 1\12, ~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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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 tuning (POTE): ~18/17 = 1\12, ~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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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 tuning (POTE): ~18/17 = 1\12, ~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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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 tuning (POTE): ~16/15 = 1\12, ~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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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 tuning (POTE): ~18/17 = 1\12, ~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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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 tuning (POTE): ~18/17 = 1\12, ~36/35 = 38.080


Optimal GPV sequence: {{Val list| 12, 24, 36, 60c }}
Optimal GPV sequence: {{Val list| 12, 24, 36, 60c }}
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Mapping generators: ~16/15, ~11
Mapping generators: ~16/15, ~11


[[Optimal tuning]] ([[POTE]]): ~45/44 = 34.977
[[Optimal tuning]] ([[POTE]]): ~16/15 = 1\12, ~45/44 = 34.977


{{Val list|legend=1| 12, 24d }}
{{Val list|legend=1| 12, 24d }}
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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 so named for the following reasons – the period is 1/24 octave, and there are 24 hours per a day.


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 19683/19600, 33075/32768
[[Comma list]]: 19683/19600, 33075/32768
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Mapping generators: ~36/35, ~5
Mapping generators: ~36/35, ~5


[[Optimal tuning]] ([[POTE]]): ~5/4 = 384.033  
[[Optimal tuning]] ([[POTE]]): ~36/35 = 1\24, ~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 }}
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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 }}]


Optimal tuning (POTE): ~5/4 = 384.054
Optimal tuning (POTE): ~36/35 = 1\24, ~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 }}
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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 }}]


Optimal tuning (POTE): ~5/4 = 384.652
Optimal tuning (POTE): ~36/35 = 1\24, ~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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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 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


[[Comma list]]: 1029/1024, 118098/117649
[[Comma list]]: 1029/1024, 118098/117649
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{{Multival|legend=1| 0 36 0 57 0 -101 }}
{{Multival|legend=1| 0 36 0 57 0 -101 }}


[[Optimal tuning]] ([[POTE]]): ~5/4 = 384.764
[[Optimal tuning]] ([[POTE]]): ~49/48 = 1\36, ~5/4 = 384.764


{{Val list|legend=1| 36, 72, 252, 324bd, 396bd }}
{{Val list|legend=1| 36, 72, 252, 324bd, 396bd }}
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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 }}]


Optimal tuning (POTE): ~5/4 = 384.150
Optimal tuning (POTE): ~49/48 = 1\36, ~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 }}
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== 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, 441/440, 4375/4356
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Mapping generators: ~100/99, ~13
Mapping generators: ~100/99, ~13


[[Optimal tuning]] ([[POTE]]): ~13/8 = 837.814
[[Optimal tuning]] ([[POTE]]): ~100/99 = 1\72, ~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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