Pythagorean family: Difference between revisions

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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.

[[POTE_tuning|POTE generator]]: 15.116

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

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

EDOs: [[12edo|12]], [[72edo|72]], [[84edo|84]], 156, 240, ~~396~~

EDOs: [[12edo|12]], [[72edo|72]], [[84edo|84]], 156, 240, 396b

=Compton temperament=

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|~~72edo]], [[~~84edo|~~84edo]] or [[~~240edo|~~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 = |-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|~~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.

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|~~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

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 The Pythagorean family tempers out the Pythagorean comma, 531441/524288 = |-19 12&gt;, 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.
-[[POTE_tuning|POTE generator]]: 15.116
+[[POTE_tuning|POTE generator]]: ~5/4 = 384.884 (15.116)
 Map: [&lt;12 19 0|, &lt;0 0 1|]
-EDOs: [[12edo|12]], [[72edo|72]], [[84edo|84]], 156, 240, 396
+EDOs: [[12edo|12]], [[72edo|72]], [[84edo|84]], 156, 240, 396b
 =Compton temperament=
-In terms of the normal list, compton adds 413343/409600 = |-14 10 -2 1&gt; 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|72edo]], [[84edo|84edo]] or [[240edo|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 = |-14 10 -2 1&gt; 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 the either the 5 or 7-limit, [[240edo|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.
-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|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