96edo: Difference between revisions

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== Theory ==

As a [[5-limit]] system, 96edo can be characterized by the fact that it tempers out both the [[Pythagorean comma]], 531441/524288, [[Würschmidt's comma]], 393216/390625, the [[unicorn comma]], 1594323/1562500, and the kwazy comma, {{monzo| -53 10 16 }}. It therefore has the same familiar 700-cent fifth as [[12edo]], and has a best major third of 387.5 cents, a bit over a cent sharp. There is therefore nothing to complain of with its representation of the 5-limit and it can be recommended as an approach to the [[würschmidt family]] of temperaments. It also tempers out the unicorn comma, and serves a way of tuning temperaments in the [[unicorn family]].

As a [[5-limit]] system, 96edo can be characterized by the fact that it tempers out both the [[Pythagorean comma]], 531441/524288, [[Würschmidt's comma]], 393216/390625, the [[unicorn comma]], 1594323/1562500, and the kwazy comma, {{monzo| -53 10 16 }}. It therefore has the same familiar 700-cent fifth as [[12edo]], and has a best major third of 387.5 cents, a bit over a cent sharp. There is therefore nothing to complain of with its representation of the 5-limit and it can be recommended as an approach to the [[würschmidt family]] of temperaments. It also tempers out the unicorn comma, and serves a way of tuning temperaments in the [[unicorn family]]. It supports [[Substitute harmonic#Sitcom|sitcom]] temperament.

In the [[7-limit]], 96 has two possible mappings for [[7/4]], a sharp one of 975 cents from the [[patent val]], and a flat one of 962.5 cents from 96d. Using the sharp mapping, 96 tempers out [[225/224]] and [[support]]s 7-limit [[würschmidt]] temperament, and using the flat mapping it tempers out [[126/125]] and supports [[worschmidt]] temperament. We can also dispense with 7 altogether, and use it as a no-sevens system, where it tempers out [[243/242]] in the 11-limit and [[676/675]] in the 13-limit. If we include 7, then the sharp mapping tempers out [[99/98]] and [[176/175]] in the 11-limit, and [[169/168]] in the 13-limit, and this provides the optimal patent val for the [[Marvel temperaments #Interpental|interpental temperament]]. With the flat 7 it tempers out [[385/384]] in the 11-limit and [[196/195]] and [[364/363]] in the 13-limit, and serves for the various temperaments of the unicorn family.

@@ Line 10: / Line 10: @@
 == Theory ==
-As a [[5-limit]] system, 96edo can be characterized by the fact that it tempers out both the [[Pythagorean comma]], 531441/524288, [[Würschmidt's comma]], 393216/390625, the [[unicorn comma]], 1594323/1562500, and the kwazy comma, {{monzo| -53 10 16 }}. It therefore has the same familiar 700-cent fifth as [[12edo]], and has a best major third of 387.5 cents, a bit over a cent sharp. There is therefore nothing to complain of with its representation of the 5-limit and it can be recommended as an approach to the [[würschmidt family]] of temperaments. It also tempers out the unicorn comma, and serves a way of tuning temperaments in the [[unicorn family]].
+As a [[5-limit]] system, 96edo can be characterized by the fact that it tempers out both the [[Pythagorean comma]], 531441/524288, [[Würschmidt's comma]], 393216/390625, the [[unicorn comma]], 1594323/1562500, and the kwazy comma, {{monzo| -53 10 16 }}. It therefore has the same familiar 700-cent fifth as [[12edo]], and has a best major third of 387.5 cents, a bit over a cent sharp. There is therefore nothing to complain of with its representation of the 5-limit and it can be recommended as an approach to the [[würschmidt family]] of temperaments. It also tempers out the unicorn comma, and serves a way of tuning temperaments in the [[unicorn family]]. It supports [[Substitute harmonic#Sitcom|sitcom]] temperament.
 In the [[7-limit]], 96 has two possible mappings for [[7/4]], a sharp one of 975 cents from the [[patent val]], and a flat one of 962.5 cents from 96d. Using the sharp mapping, 96 tempers out [[225/224]] and [[support]]s 7-limit [[würschmidt]] temperament, and using the flat mapping it tempers out [[126/125]] and supports [[worschmidt]] temperament. We can also dispense with 7 altogether, and use it as a no-sevens system, where it tempers out [[243/242]] in the 11-limit and [[676/675]] in the 13-limit. If we include 7, then the sharp mapping tempers out [[99/98]] and [[176/175]] in the 11-limit, and [[169/168]] in the 13-limit, and this provides the optimal patent val for the [[Marvel temperaments #Interpental|interpental temperament]]. With the flat 7 it tempers out [[385/384]] in the 11-limit and [[196/195]] and [[364/363]] in the 13-limit, and serves for the various temperaments of the unicorn family.