Pythagorean family: Difference between revisions

Line 1:

: ''Not to be confused with [[Pythagorean tuning]].''

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~~ =

= Compton =

~~: ''Not~~ to be ~~confused with~~ [[Pythagorean tuning]].''

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

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~~= 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.~~

== 7-limit (aka Waage) ==

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Badness: 0.022235

== 13-limit ==

=== 13-limit ===

Comma list: 225/224, 351/350, 364/363, 441/440

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Badness: 0.021852

== 17-limit ==

==== 17-limit ====

Comma list: 221/220, 225/224, 289/288, 351/350, 441/440

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Badness: 0.017131

== Comptone ==

=== Comptone ===

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

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Badness: 0.025144

=== 17-limit ===

==== 17-limit ====

Comma list: 225/224, 273/272, 289/288, 325/324, 441/440

@@ Line 1: / Line 1: @@
+: ''Not to be confused with [[Pythagorean tuning]].''
 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 =
+= Compton =
-: ''Not to be confused with [[Pythagorean tuning]].''
+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&amp;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
@@ Line 11: / Line 18: @@
 {{Val list|legend=1| 12, 72, 84, 156, 240, 396b }}
-= 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&amp;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.
 == 7-limit (aka Waage) ==
@@ Line 41: / Line 41: @@
 Badness: 0.022235
-== 13-limit ==
+=== 13-limit ===
 Comma list: 225/224, 351/350, 364/363, 441/440
@@ Line 52: / Line 52: @@
 Badness: 0.021852
-== 17-limit ==
+==== 17-limit ====
 Comma list: 221/220, 225/224, 289/288, 351/350, 441/440
@@ Line 63: / Line 63: @@
 Badness: 0.017131
-== Comptone ==
+=== Comptone ===
 Comma list: 225/224, 325/324, 441/440, 1001/1000
@@ Line 74: / Line 74: @@
 Badness: 0.025144
-=== 17-limit ===
+==== 17-limit ====
 Comma list: 225/224, 273/272, 289/288, 325/324, 441/440