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 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 '''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 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. | ||
= 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. | ||
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{{Val list|legend=1| 12, 72, 84, 156, 240, 396b }} | {{Val list|legend=1| 12, 72, 84, 156, 240, 396b }} | ||
== 7-limit (aka Waage) == | === 7-limit (aka Waage) === | ||
Comma list: 225/224, 250047/250000 | Comma list: 225/224, 250047/250000 | ||
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Badness: 0.035686 | Badness: 0.035686 | ||
== 11-limit == | === 11-limit === | ||
Comma list: 225/224, 441/440, 4375/4356 | Comma list: 225/224, 441/440, 4375/4356 | ||
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Badness: 0.022235 | Badness: 0.022235 | ||
=== 13-limit === | ==== 13-limit ==== | ||
Comma list: 225/224, 351/350, 364/363, 441/440 | Comma list: 225/224, 351/350, 364/363, 441/440 | ||
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Badness: 0.021852 | Badness: 0.021852 | ||
==== 17-limit ==== | ===== 17-limit ===== | ||
Comma list: 221/220, 225/224, 289/288, 351/350, 441/440 | Comma list: 221/220, 225/224, 289/288, 351/350, 441/440 | ||
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Badness: 0.017131 | Badness: 0.017131 | ||
=== Comptone === | ==== Comptone ==== | ||
Comma list: 225/224, 325/324, 441/440, 1001/1000 | Comma list: 225/224, 325/324, 441/440, 1001/1000 | ||
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Badness: 0.025144 | Badness: 0.025144 | ||
==== 17-limit ==== | ===== 17-limit ===== | ||
Comma list: 225/224, 273/272, 289/288, 325/324, 441/440 | Comma list: 225/224, 273/272, 289/288, 325/324, 441/440 | ||
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Badness: 0.016361 | Badness: 0.016361 | ||
= 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. | ||
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{{Val list|legend=1| 12, 36, 48, 132, 180 }} | {{Val list|legend=1| 12, 36, 48, 132, 180 }} | ||
== 11-limit == | === 11-limit === | ||
Comma list: 81/80, 99/98, 128/125 | Comma list: 81/80, 99/98, 128/125 | ||
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Badness: 0.0582 | Badness: 0.0582 | ||
== Catlat == | === Catlat === | ||
Comma list: 81/80, 128/125, 540/539 | Comma list: 81/80, 128/125, 540/539 | ||
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Badness: 0.0819 | Badness: 0.0819 | ||
== Catcall == | === Catcall === | ||
Comma list: 56/55, 81/80, 128/125 | Comma list: 56/55, 81/80, 128/125 | ||
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Badness: 0.0345 | Badness: 0.0345 | ||
=== 13-limit === | ==== 13-limit ==== | ||
Comma list: 56/55, 66/65, 81/80, 105/104 | Comma list: 56/55, 66/65, 81/80, 105/104 | ||
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Badness: 0.0284 | Badness: 0.0284 | ||
=== Duodecic === | ==== Duodecic ==== | ||
Comma list: 56/55, 81/80, 91/90, 128/125 | Comma list: 56/55, 81/80, 91/90, 128/125 | ||
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Badness: 0.0383 | Badness: 0.0383 | ||
==== 17-limit ==== | ===== 17-limit ===== | ||
Comma list: 51/50, 56/55, 81/80, 91/90, 128/125 | Comma list: 51/50, 56/55, 81/80, 91/90, 128/125 | ||
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Badness: 0.0275 | Badness: 0.0275 | ||
==== 19-limit ==== | ===== 19-limit ===== | ||
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 | ||
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Badness: 0.0209 | Badness: 0.0209 | ||
= Duodecim = | == Duodecim == | ||
Comma list: 36/35, 50/49, 64/63 | Comma list: 36/35, 50/49, 64/63 | ||
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{{Val list|legend=1| 12, 24d }} | {{Val list|legend=1| 12, 24d }} | ||
= Omicronbeta = | == Omicronbeta == | ||
Comma list: 225/224, 243/242, 441/440, 4375/4356 | Comma list: 225/224, 243/242, 441/440, 4375/4356 | ||
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Badness: 0.0300 | Badness: 0.0300 | ||
= Hours = | == Hours == | ||
Comma list: 19683/19600, 33075/32768 | Comma list: 19683/19600, 33075/32768 | ||
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Badness: 0.1161 | Badness: 0.1161 | ||
== 11-limit == | === 11-limit === | ||
Comma list: 243/242, 385/384, 9801/9800 | Comma list: 243/242, 385/384, 9801/9800 | ||
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Badness: 0.0362 | Badness: 0.0362 | ||
== 13-limit == | === 13-limit === | ||
Comma list: 243/242, 351/350, 364/363, 385/384 | Comma list: 243/242, 351/350, 364/363, 385/384 | ||