Compton family: Difference between revisions

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 ==

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 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: 531441/524288

[[POTE generator]]: ~5/4 = 384.884 or ~81/80 = 15.116

Mapping: [{{val| 12 19 0 }}, {{val| 0 0 1 }}

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

=== 7-limit (aka Waage)  ===

Comma list: 225/224, 250047/250000

[[POTE generator]]: ~5/4 = 383.7752

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

{{Val list|legend=1| 12, 60, 72, 228, 300c, 372bc, 444bc }}

Badness: 0.035686

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

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

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

Badness: 0.022235

POTE generator: ~5/4 = 383.9628

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

Badness: 0.021852

POTE generator: ~5/4 = 383.7500

Mapping: [{{val| 12 19 0 -22 -42 -67 49 }}, {{val| 0 0 1 2 3 4 0 }}]

Badness: 0.017131

POTE generator: ~5/4 = 382.6116

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

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

Badness: 0.025144

POTE generator: ~5/4 = 382.5968

Mapping: [{{val| 12 19 0 -22 -42 100 49 }}, {{val| 0 0 1 2 3 -2 0 }}]

{{Val list|legend=1| 12, 60e, 72, 132deg, 204cdefg }}

Badness: 0.016361

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

{{Val list|legend=1| 12, 36, 48, 132, 180 }}

POTE generator: ~64/63 = 22.723

Mapping: [{{val| 12 19 28 0 -26 }}, {{val| 0 0 0 1 2 }}]

{{Val list|legend=1| 12, 48c, 108cd }}

Badness: 0.0582

POTE generator: ~64/63 = 27.864

Mapping: [{{val| 12 19 28 0 109 }}, {{val| 0 0 0 1 -2 }}]

{{Val list|legend=1| 36, 48c, 84c }}

Badness: 0.0819

POTE generator: ~36/35 = 32.776

Mapping: [{{val| 12 19 28 0 8 }}, {{val| 0 0 0 1 1 }}]

{{Val list|legend=1| 12, 24, 36, 72ce }}

Badness: 0.0345

POTE generator: ~36/35 = 37.232

Mapping: [{{val| 12 19 28 0 8 11 }}, {{val| 0 0 0 1 1 1 }}]

{{Val list|legend=1| 12f, 24, 36f, 60cf }}

Badness: 0.0284

POTE generator: ~36/35 = 37.688

Mapping: [{{val| 12 19 28 0 8 78 }}, {{val| 0 0 0 1 1 -1 }}]

{{Val list|legend=1| 12, 24, 36, 60c }}

Badness: 0.0383

POTE generator: ~36/35 = 38.097

Mapping: [{{val| 12 19 28 0 8 78 49 }}, {{val| 0 0 0 1 1 -1 0 }}]

{{Val list|legend=1| 12, 24, 36, 60c }}

Badness: 0.0275

POTE generator: ~36/35 = 38.080

Mapping: [{{val| 12 19 28 0 8 78 49 51 }}, {{val| 0 0 0 1 1 -1 0 0 }}]

{{Val list|legend=1| 12, 24, 36, 60c }}

Badness: 0.0209

== Duodecim  ==

Comma list: 36/35, 50/49, 64/63

POTE generator: ~45/44 = 34.977

Mapping: [{{val| 12 19 28 34 0 }}, {{val| 0 0 0 0 1 }}]

{{Val list|legend=1| 12, 24d }}

Comma list: 225/224, 243/242, 441/440, 4375/4356

POTE generator: ~13/8 = 837.814

Mapping: [{{val| 72 114 167 202 249 266 }}, {{val| 0 0 0 0 0 1 }}]

{{Val list|legend=1| 72, 144, 216c, 288cdf, 504bcdef }}

Badness: 0.0300

Comma list: 19683/19600, 33075/32768

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

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

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

[[Category:Theory]]

[[Category:Temperament family]]

[[Category:Compton]]

[[Category:Rank 2]]