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Compton family: Difference between revisions

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== Compton ==
== Compton ==
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]]. Compton, however, does not need to be used as a 7-limit temperament (also called as ''waage''); in the 5-limit it becomes the rank two 5-limit temperament tempering out the Pythagorean comma. In terms of equal temperaments, it is the 12&72 temperament, and [[72edo|72EDO]], [[84edo|84EDO]] or [[240edo|240EDO]] make for good tunings. Possible generators are 21/20, 10/9, the secor, 6/5, 5/4, 7/5 and most importantly, 81/80.
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.
Subgroup: 2.3.5
Subgroup: 2.3.5


Line 20: Line 14:
[[Badness]]: 0.094494
[[Badness]]: 0.094494


=== 7-limit (Waage) ===
== Septimal compton ==
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]]. Compton, however, does not need to be used as a 7-limit temperament (also called as ''waage''); in the 5-limit it becomes the rank two 5-limit temperament tempering out the Pythagorean comma. In terms of equal temperaments, it is the 12&72 temperament, and [[72edo|72EDO]], [[84edo|84EDO]] or [[240edo|240EDO]] make for good tunings. Possible generators are 21/20, 10/9, the secor, 6/5, 5/4, 7/5 and most importantly, 81/80.
 
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.
 
Subgroup: 2.3.5.7
Subgroup: 2.3.5.7


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POTE generator: ~5/4 = 383.2660
POTE generator: ~5/4 = 383.2660


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


Badness: 0.022235
Badness: 0.022235


=== 13-limit ===
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


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POTE generator: ~5/4 = 383.9628
POTE generator: ~5/4 = 383.9628


Vals: {{Val list| 12f, 72, 84, 156, 228f, 300cf }}
Optimal GPV sequence: {{Val list| 12f, 72, 84, 156, 228f, 300cf }}


Badness: 0.021852
Badness: 0.021852


==== 17-limit ====
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


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POTE generator: ~5/4 = 383.7500
POTE generator: ~5/4 = 383.7500


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


Badness: 0.017131
Badness: 0.017131


=== Comptone ===
==== Comptone ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


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POTE generator: ~5/4 = 382.6116
POTE generator: ~5/4 = 382.6116


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


Badness: 0.025144
Badness: 0.025144


==== 17-limit ====
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


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POTE generator: ~5/4 = 382.5968
POTE generator: ~5/4 = 382.5968


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


Badness: 0.016361
Badness: 0.016361
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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 }}]


Vals: {{Val list| 12, 36e, 48c, 108ccd }}
Optimal GPV sequence: {{Val list| 12, 36e, 48c, 108ccd }}


Badness: 0.058213
Badness: 0.058213
Line 135: Line 135:
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 }}]


Vals: {{Val list| 36, 48c, 84c }}
Optimal GPV sequence: {{Val list| 36, 48c, 84c }}


Badness: 0.081909
Badness: 0.081909
Line 148: Line 148:
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 }}]


Vals: {{Val list| 12, 24, 36, 72ce }}
Optimal GPV sequence: {{Val list| 12, 24, 36, 72ce }}


Badness: 0.034478
Badness: 0.034478
Line 161: Line 161:
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 }}]


Vals: {{Val list| 12f, 24, 36f, 60cf }}
Optimal GPV sequence: {{Val list| 12f, 24, 36f, 60cf }}


Badness: 0.028363
Badness: 0.028363
Line 174: Line 174:
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 }}]


Vals: {{Val list| 12f, 24, 36f, 60cf }}
Optimal GPV sequence: {{Val list| 12f, 24, 36f, 60cf }}


Badness: 0.023246
Badness: 0.023246
Line 187: Line 187:
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 }}]


Vals: {{Val list| 12f, 24, 36f, 60cf }}
Optimal GPV sequence: {{Val list| 12f, 24, 36f, 60cf }}


Badness: 0.018985
Badness: 0.018985
Line 200: Line 200:
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 }}]


Vals: {{Val list| 12, 24, 36, 60c }}
Optimal GPV sequence: {{Val list| 12, 24, 36, 60c }}


Badness: 0.038307
Badness: 0.038307
Line 212: Line 212:
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 }}]


Vals: {{Val list| 12, 24, 36, 60c }}
Optimal GPV sequence: {{Val list| 12, 24, 36, 60c }}


Badness: 0.027487
Badness: 0.027487
Line 225: Line 225:
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 }}]


Vals: {{Val list| 12, 24, 36, 60c }}
Optimal GPV sequence: {{Val list| 12, 24, 36, 60c }}


Badness: 0.020939
Badness: 0.020939
Line 270: Line 270:
POTE generator: ~5/4 = 384.054
POTE generator: ~5/4 = 384.054


Vals: {{Val list| 24, 48, 72, 312bd, 384bcdd, 456bcdde, 528bcdde }}
Optimal GPV sequence: {{Val list| 24, 48, 72, 312bd, 384bcdd, 456bcdde, 528bcdde }}


Badness: 0.036248
Badness: 0.036248
Line 283: Line 283:
POTE generator: ~5/4 = 384.652
POTE generator: ~5/4 = 384.652


Vals: {{Val list| 24, 48f, 72, 168df, 240dff }}
Optimal GPV sequence: {{Val list| 24, 48f, 72, 168df, 240dff }}


Badness: 0.026931
Badness: 0.026931
Line 313: Line 313:
POTE generator: ~5/4 = 384.150
POTE generator: ~5/4 = 384.150


Vals: {{Val list| 36, 72, 396bd, 468bcd, 540bcd, 612bccdd, 684bbccdd, 756bbccdd }}
Optimal GPV sequence: {{Val list| 36, 72, 396bd, 468bcd, 540bcd, 612bccdd, 684bbccdd, 756bbccdd }}


Badness: 0.043088
Badness: 0.043088
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