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| The '''Pythagorean 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 12et, two cents flat, the tuning of the larger primes is not so constrained, and the point of these temperaments is to improve on it.
| | #REDIRECT [[Pythagorean]] |
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| = Pythagorean =
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| : ''Not to be confused with [[Pythagorean tuning]].''
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| Comma list: 531441/524288
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| [[POTE generator]]: ~5/4 = 384.884 or ~81/80 = 15.116
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| Mapping: [{{val| 12 19 0 }}, {{val| 0 0 1 }}
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| {{Val list|legend=1| 12, 72, 84, 156, 240, 396b }}
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| = Compton =
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| 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; 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]], [[84edo]] or [[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.
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| In either the 5 or 7-limit, [[240edo]] is an excellent tuning, with 81/80 coming in at 15 cents exactly. 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.
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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.
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| Comma list: 225/224, 250047/250000
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| [[POTE generator]]: ~5/4 = 383.775 or ~81/80 = 16.225
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| Mapping: [{{val| 12 19 0 -22 }}, {{val| 0 0 1 2 }}]
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| {{Val list|legend=1| 12, 60, 72, 228, 300c, 372bc, 444bc }}
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| == 11-limit ==
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| Comma list: 225/224, 441/440, 4375/4356
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| [[POTE generator]]: ~5/4 = 383.266 or ~81/80 = 16.734
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| Mapping: [{{val|12 19 0 -22 -42 }}, {{val| 0 0 1 2 3 }}]
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| {{Val list|legend=1| 12, 60e, 72 }}
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| === 13-limit ===
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| Comma list: 225/224, 441/440, 351/350, 364/363
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| POTE generator: ~5/4 = 383.963 or ~81/80 = 16.037
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| Mapping: [{{val| 12 19 0 -22 -42 -67 }}, {{val| 0 0 1 2 3 4 }}]
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| {{Val list|legend=1| 72, 228f, 300cf }}
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| Badness: 0.0219
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| === Comptone ===
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| Comma list: 225/224, 441/440, 325/324, 1001/1000
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| POTE generator: ~5/4 = 382.612 or ~81/80 = 17.388
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| Mapping: [{{val| 12 19 0 -22 -42 100 }}, {{val| 0 0 1 2 3 -2 }}]
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| {{Val list|legend=1| 12, 60e, 72, 204cdef, 276cdef }}
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| Badness: 0.0251
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| = Catler =
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| 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, 6/5, 9/7, 7/5, and most importantly, 64/63.
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| Comma list: 81/80, 128/125
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| [[POTE generator]]: ~64/63 = 26.790
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| Mapping: [{{val| 12 19 28 0 }}, {{val| 0 0 0 1 }}]
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| {{Val list|legend=1| 12, 36, 48, 132, 180 }}
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| == 11-limit ==
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| Comma list: 81/80, 99/98, 128/125
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| POTE generator: ~64/63 = 22.723
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| Mapping: [{{val| 12 19 28 0 -26 }}, {{val| 0 0 0 1 2 }}]
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| {{Val list|legend=1| 12, 48c, 108cd }}
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| Badness: 0.0582
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| == Catlat ==
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| Comma list: 81/80, 128/125, 540/539
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| POTE generator: ~64/63 = 27.864
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| Mapping: [{{val| 12 19 28 0 109 }}, {{val| 0 0 0 1 -2 }}]
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| {{Val list|legend=1| 36, 48c, 84c }}
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| Badness: 0.0819
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| == Catcall ==
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| Comma list: 56/55, 81/80, 128/125
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| POTE generator: ~36/35 = 32.776
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| Mapping: [{{val| 12 19 28 0 8 }}, {{val| 0 0 0 1 1 }}]
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| {{Val list|legend=1| 12, 24, 36, 72ce }}
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| Badness: 0.0345
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| === 13-limit ===
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| Comma list: 56/55, 66/65, 81/80, 105/104
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| POTE generator: ~36/35 = 37.232
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| Mapping: [{{val| 12 19 28 0 8 11 }}, {{val| 0 0 0 1 1 1 }}]
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| {{Val list|legend=1| 12f, 24, 36f, 60cf }}
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| Badness: 0.0284
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| === Duodecic ===
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| Comma list: 56/55, 81/80, 91/90, 128/125
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| POTE generator: ~36/35 = 37.688
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| Mapping: [{{val| 12 19 28 0 8 78 }}, {{val| 0 0 0 1 1 -1 }}]
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| {{Val list|legend=1| 12, 24, 36, 60c }}
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| Badness: 0.0383
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| ==== 17-limit ====
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| Comma list: 51/50, 56/55, 81/80, 91/90, 128/125
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| POTE generator: ~36/35 = 38.097
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| Mapping: [{{val| 12 19 28 0 8 78 49 }}, {{val| 0 0 0 1 1 -1 0 }}]
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| {{Val list|legend=1| 12, 24, 36, 60c }}
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| Badness: 0.0275
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| ==== 19-limit ====
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| Comma list: 51/50, 56/55, 76/75, 81/80, 91/90, 96/95
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| POTE generator: ~36/35 = 38.080
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| Mapping: [{{val| 12 19 28 0 8 78 49 51 }}, {{val| 0 0 0 1 1 -1 0 0 }}]
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| {{Val list|legend=1| 12, 24, 36, 60c }}
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| Badness: 0.0209
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| = Duodecim =
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| Comma list: 36/35, 50/49, 64/63
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| POTE generator: ~45/44 = 34.977
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| Mapping: [{{val| 12 19 28 34 0 }}, {{val| 0 0 0 0 1 }}]
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| {{Val list|legend=1| 12, 24d }}
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| = Omicronbeta =
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| Comma list: 225/224, 243/242, 441/440, 4375/4356
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| POTE generator: ~13/8 = 837.814
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| Mapping: [{{val| 72 114 167 202 249 266 }}, {{val| 0 0 0 0 0 1 }}]
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| {{Val list|legend=1| 72, 144, 216c, 288cdf, 504bcdef }}
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| Badness: 0.0300
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| = Hours =
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| Comma list: 19683/19600, 33075/32768
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| POTE generator: ~225/224 = 2.100
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| Mapping: [{{val| 24 38 0 123 83 }}, {{val| 0 0 1 -1 0 }}]
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| {{Multival|legend=1| 0 24 -24 38 -38 -123 }}
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| {{Val list|legend=1| 24, 48, 72, 312bd, 384bcd, 456bcd, 528bcd, 600bcd }}
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| Badness: 0.1161
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| == 11-limit ==
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| Comma list: 243/242, 385/384, 9801/9800
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| POTE generator: ~225/224 = 2.161
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| Mapping: [{{val| 24 38 0 123 83 }}, {{val| 0 0 1 -1 0 }}]
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| {{Val list|legend=1| 24, 48, 72, 312bd, 384bcd, 456bcde, 528bcde }}
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| Badness: 0.0362
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| == 13-limit ==
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| Comma list: 243/242, 351/350, 364/363, 385/384
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| POTE generator: ~225/224 = 3.955
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| Mapping: [{{val| 24 38 0 123 83 33 }}, {{val| 0 0 1 -1 0 1 }}]
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| {{Val list|legend=1| 24, 48f, 72, 168df, 240df }}
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| Badness: 0.0269
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| [[Category:Theory]]
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| [[Category:Temperament family]]
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| [[Category:Pythagorean]]
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| [[Category:Rank 2]]
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