Pythagorean family: Difference between revisions

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__FORCETOC__
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 = |-19 12>, and hence the fifths form a closed 12-note circle of fifths, identical to [[12edo|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 =


[[POTE_tuning|POTE generator]]: ~5/4 = 384.884 (15.116)
Comma list: 531441/524288


Map: [<12 19 0|, <0 0 1|]
[[POTE generator]]: ~5/4 = 384.884 or ~81/80 = 15.116


EDOs: [[12edo|12]], [[72edo|72]], [[84edo|84]], 156, 240, 396b
Mapping: [{{val| 12 19 0 }}, {{val| 0 0 1 }}


=Compton temperament=
{{Val list|legend=1| 12, 72, 84, 156, 240, 396b }}
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.  
 
= 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 the 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 the 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.
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.


Commas: 225/224, 250047/250000
Comma list: 225/224, 250047/250000


[[POTE_tuning|POTE generator]]: ~5/4 = 383.775 (16.225)
[[POTE generator]]: ~5/4 = 383.775 or ~81/80 = 16.225


Map: [<12 19 0 -22|, <0 0 1 2|]
Mapping: [{{val| 12 19 0 -22 }}, {{val| 0 0 1 2 }}]


EDOs: 12, [[60edo|60]], 72, 228, 300c, 372bc, 444bc
{{Val list|legend=1| 12, 60, 72, 228, 300c, 372bc, 444bc }}


==11-limit==
== 11-limit ==
Commas: 225/224, 441/440, 4375/4356
Comma list: 225/224, 441/440, 4375/4356


[[POTE_tuning|POTE generator]]: ~5/4 = 383.266 (16.734)
[[POTE generator]]: ~5/4 = 383.266 or ~81/80 = 16.734


Map: [<12 19 0 -22 -42|, <0 0 1 2 3|]
Mapping: [{{val|12 19 0 -22 -42 }}, {{val| 0 0 1 2 3 }}]


EDOs: 12, 60e, 72
{{Val list|legend=1| 12, 60e, 72 }}


==13-limit==
=== 13-limit ===
Commas: 225/224, 441/440, 351/350, 364/363
Comma list: 225/224, 441/440, 351/350, 364/363


POTE generator: ~5/4 = 383.963 (16.037)
POTE generator: ~5/4 = 383.963 or ~81/80 = 16.037


Map: [<12 19 0 -22 -42 -67|, <0 0 1 2 3 4|]
Mapping: [{{val| 12 19 0 -22 -42 -67 }}, {{val| 0 0 1 2 3 4 }}]


EDOs: 72, 228f, 300cf
{{Val list|legend=1|  72, 228f, 300cf }}


Badness: 0.0219
Badness: 0.0219


==Comptone==
=== Comptone ===
Commas: 225/224, 441/440, 325/324, 1001/1000
Comma list: 225/224, 441/440, 325/324, 1001/1000


POTE generator: ~5/4 = 382.612 (17.388)
POTE generator: ~5/4 = 382.612 or ~81/80 = 17.388


Map: [<12 19 0 -22 -42 100|, <0 0 1 2 3 -2|]
Mapping: [{{val| 12 19 0 -22 -42 100 }}, {{val| 0 0 1 2 3 -2 }}]


EDOs: 12, 60e, 72, 204cdef, 276cdef
{{Val list|legend=1| 12, 60e, 72, 204cdef, 276cdef }}


Badness: 0.0251
Badness: 0.0251


=Catler temperament=
= 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.   


Commas: 81/80, 128/125
Comma list: 81/80, 128/125


[[POTE_tuning|POTE generator]]: 26.790
[[POTE generator]]: ~64/63 = 26.790


Map: [<12 19 28 0|, <0 0 0 1|]
Mapping: [{{val| 12 19 28 0 }}, {{val| 0 0 0 1 }}]


EDOs: 12, [[36edo|36]], [[48edo|48]], 132, 180
{{Val list|legend=1| 12, 36, 48, 132, 180 }}


==11-limit==
==11-limit==

Revision as of 08:25, 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]

Template:Val list

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 the 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]]

Template:Val list

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]]

Template:Val list

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]]

Template:Val list

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]]

Template:Val list

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]]

Template:Val list

11-limit

Commas: 81/80, 99/98, 128/125

POTE generator: ~36/35 = 22.723

Map: [<12 19 28 0 -26|, <0 0 0 1 2|]

EDOs: 12, 48c, 108cd

Badness: 0.0582

Catlat

Commas: 81/80, 128/125, 540/539

POTE generator: ~36/35 = 27.864

Map: [<12 19 28 0 109|, <0 0 0 1 -2|]

EDOs: 36, 48c, 84c

Badness: 0.0819

Catcall

Commas: 56/55, 81/80, 128/125

POTE generator: ~36/35 = 32.776

Map: [<12 19 28 0 8|, <0 0 0 1 1|]

EDOs: 12, 24, 36, 72ce

Badness: 0.0345

13-limit

Commas: 56/55, 66/65, 81/80, 105/104

POTE generator: ~36/35 = 37.232

Map: [<12 19 28 0 8 11|, <0 0 0 1 1 1|]

EDOs: 12f, 24, 36f, 60cf

Badness: 0.0284

Duodecic

Commas: 56/55, 81/80, 91/90, 128/125

POTE generator: ~36/35 = 37.688

Map: [<12 19 28 0 8 78|, <0 0 0 1 1 -1|]

EDOs: 12, 24, 36, 60c

Badness: 0.0383

17-limit

Commas: 51/50, 56/55, 81/80, 91/90, 128/125

POTE generator: ~36/35 = 38.097

Map: [<12 19 28 0 8 78 49|, <0 0 0 1 1 -1 0|]

EDOs: 12, 24, 36, 60c

Badness: 0.0275

19-limit

Commas: 51/50, 56/55, 76/75, 81/80, 91/90, 96/95

POTE generator: ~36/35 = 38.080

Map: [<12 19 28 0 8 78 49 51|, <0 0 0 1 1 -1 0 0|]

EDOs: 12, 24, 36, 60c

Badness: 0.0209

Duodecim

Commas: 36/35, 50/49, 64/63

POTE generator: ~45/44 = 34.977

Map: [<12 19 28 34 0|, <0 0 0 0 1|]

EDOs: 12, 24d

Omicronbeta temperament

Commas: 225/224, 243/242, 441/440, 4375/4356

Generator: ~13/8 = 837.814

Map: [<72 114 167 202 249 266|, <0 0 0 0 0 1|]

EDOs: 72, 144, 216c, 288cdf, 504bcdef

Badness: 0.0300

Hours

Commas: 19683/19600, 33075/32768

POTE generator: ~225/224 = 2.100

Map: [<24 38 0 123 83|, <0 0 1 -1 0|]

Wedgie: <0 24 -24 38 -38 -123|

EDOs: 24, 48, 72, 312bd, 384bcd, 456bcd, 528bcd, 600bcd

Badness: 0.1161

11-limit

Commas: 243/242, 385/384, 9801/9800

POTE generator: ~225/224 = 2.161

Map: [<24 38 0 123 83|, <0 0 1 -1 0|]

EDOs: 24, 48, 72, 312bd, 384bcd, 456bcde, 528bcde

Badness: 0.0362

13-limit

Commas: 243/242, 351/350, 364/363, 385/384

POTE generator: ~225/224 = 3.955

Map: [<24 38 0 123 83 33|, <0 0 1 -1 0 1|]

EDOs: 24, 48f, 72, 168df, 240df

Badness: 0.0269

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