Pythagorean family: Difference between revisions
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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. | 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. | ||
= Pythagorean = | = Pythagorean = | ||
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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; 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. | 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. | ||
In | 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. | ||
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. | ||
Comma list: 225/224, 250047/250000 | Comma list: 225/224, 250047/250000 | ||
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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 }}] | ||
{{Val list|legend=1| | {{Val list|legend=1| 72, 228f, 300cf }} | ||
Badness: 0.0219 | Badness: 0.0219 | ||
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= 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]]. 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. | 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. | ||
Comma list: 81/80, 128/125 | Comma list: 81/80, 128/125 | ||
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Badness: 0.0300 | Badness: 0.0300 | ||
=Hours= | = Hours = | ||
Comma list: 19683/19600, 33075/32768 | |||
POTE generator: ~225/224 = 2.100 | POTE generator: ~225/224 = 2.100 | ||
Mapping: [{{val| 24 38 0 123 83 }}, {{val| 0 0 1 -1 0 }}] | |||
{{Multival|legend=1| 0 24 -24 38 -38 -123 }} | |||
{{Val list|legend=1| 24, 48, 72, 312bd, 384bcd, 456bcd, 528bcd, 600bcd }} | |||
Badness: 0.1161 | Badness: 0.1161 | ||
==11-limit== | == 11-limit == | ||
Comma list: 243/242, 385/384, 9801/9800 | |||
POTE generator: ~225/224 = 2.161 | POTE generator: ~225/224 = 2.161 | ||
Mapping: [{{val| 24 38 0 123 83 }}, {{val| 0 0 1 -1 0 }}] | |||
{{Val list|legend=1| 24, 48, 72, 312bd, 384bcd, 456bcde, 528bcde }} | |||
Badness: 0.0362 | Badness: 0.0362 | ||
==13-limit== | == 13-limit == | ||
Comma list: 243/242, 351/350, 364/363, 385/384 | |||
POTE generator: ~225/224 = 3.955 | POTE generator: ~225/224 = 3.955 | ||
Mapping: [{{val| 24 38 0 123 83 33 }}, {{val| 0 0 1 -1 0 1 }}] | |||
{{Val list|legend=1| 24, 48f, 72, 168df, 240df }} | |||
Badness: 0.0269 | Badness: 0.0269 | ||
Revision as of 08:52, 11 March 2021
The Pythagorean family tempers out the Pythagorean comma, 531441/524288 = [-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.
Pythagorean
Comma list: 531441/524288
POTE generator: ~5/4 = 384.884 or ~81/80 = 15.116
Mapping: [⟨12 19 0], ⟨0 0 1]
Compton
In terms of the normal list, compton adds 413343/409600 = [-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.
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.
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.
Comma list: 225/224, 250047/250000
POTE generator: ~5/4 = 383.775 or ~81/80 = 16.225
Mapping: [⟨12 19 0 -22], ⟨0 0 1 2]]
11-limit
Comma list: 225/224, 441/440, 4375/4356
POTE generator: ~5/4 = 383.266 or ~81/80 = 16.734
Mapping: [⟨12 19 0 -22 -42], ⟨0 0 1 2 3]]
13-limit
Comma list: 225/224, 441/440, 351/350, 364/363
POTE generator: ~5/4 = 383.963 or ~81/80 = 16.037
Mapping: [⟨12 19 0 -22 -42 -67], ⟨0 0 1 2 3 4]]
Badness: 0.0219
Comptone
Comma list: 225/224, 441/440, 325/324, 1001/1000
POTE generator: ~5/4 = 382.612 or ~81/80 = 17.388
Mapping: [⟨12 19 0 -22 -42 100], ⟨0 0 1 2 3 -2]]
Badness: 0.0251
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. 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.
Comma list: 81/80, 128/125
POTE generator: ~64/63 = 26.790
Mapping: [⟨12 19 28 0], ⟨0 0 0 1]]
11-limit
Comma list: 81/80, 99/98, 128/125
POTE generator: ~64/63 = 22.723
Mapping: [⟨12 19 28 0 -26], ⟨0 0 0 1 2]]
Badness: 0.0582
Catlat
Comma list: 81/80, 128/125, 540/539
POTE generator: ~64/63 = 27.864
Mapping: [⟨12 19 28 0 109], ⟨0 0 0 1 -2]]
Badness: 0.0819
Catcall
Comma list: 56/55, 81/80, 128/125
POTE generator: ~36/35 = 32.776
Mapping: [⟨12 19 28 0 8], ⟨0 0 0 1 1]]
Badness: 0.0345
13-limit
Comma list: 56/55, 66/65, 81/80, 105/104
POTE generator: ~36/35 = 37.232
Mapping: [⟨12 19 28 0 8 11], ⟨0 0 0 1 1 1]]
Badness: 0.0284
Duodecic
Comma list: 56/55, 81/80, 91/90, 128/125
POTE generator: ~36/35 = 37.688
Mapping: [⟨12 19 28 0 8 78], ⟨0 0 0 1 1 -1]]
Badness: 0.0383
17-limit
Comma list: 51/50, 56/55, 81/80, 91/90, 128/125
POTE generator: ~36/35 = 38.097
Mapping: [⟨12 19 28 0 8 78 49], ⟨0 0 0 1 1 -1 0]]
Badness: 0.0275
19-limit
Comma list: 51/50, 56/55, 76/75, 81/80, 91/90, 96/95
POTE generator: ~36/35 = 38.080
Mapping: [⟨12 19 28 0 8 78 49 51], ⟨0 0 0 1 1 -1 0 0]]
Badness: 0.0209
Duodecim
Comma list: 36/35, 50/49, 64/63
POTE generator: ~45/44 = 34.977
Mapping: [⟨12 19 28 34 0], ⟨0 0 0 0 1]]
Omicronbeta
Comma list: 225/224, 243/242, 441/440, 4375/4356
POTE generator: ~13/8 = 837.814
Mapping: [⟨72 114 167 202 249 266], ⟨0 0 0 0 0 1]]
Badness: 0.0300
Hours
Comma list: 19683/19600, 33075/32768
POTE generator: ~225/224 = 2.100
Mapping: [⟨24 38 0 123 83], ⟨0 0 1 -1 0]]
Wedgie: ⟨⟨ 0 24 -24 38 -38 -123 ]]
Badness: 0.1161
11-limit
Comma list: 243/242, 385/384, 9801/9800
POTE generator: ~225/224 = 2.161
Mapping: [⟨24 38 0 123 83], ⟨0 0 1 -1 0]]
Badness: 0.0362
13-limit
Comma list: 243/242, 351/350, 364/363, 385/384
POTE generator: ~225/224 = 3.955
Mapping: [⟨24 38 0 123 83 33], ⟨0 0 1 -1 0 1]]
Badness: 0.0269