99edo: Difference between revisions

Line 2:

== Theory ==

99edo is a very strong 7-limit (and 9 odd limit) temperament, but extending it to the 11-limit requires choosing which mapping one wants to use, as both are nearly equally far off the mark. It [[tempering_out|tempers out]] 393216/390625 ([[würschmidt comma]]) and 1600000/1594323 ([[amity comma]]) in the [[5-limit]]; 2401/2400 ([[2401/2400|breedsma]]), 3136/3125 (hemimean comma), and 4375/4374 ([[4375/4374|ragisma]]) in the [[7-limit]], supporting [[hemififths]], [[amity]], [[parakleismic]], [[hemiwürschmidt]] and [[ennealimmal]] temperaments, and is pretty well a perfect tuning for [[hendecatonic]] temperament. It has a sound defined by the slight sharpness (1.075, 1.565, 0.871 cents) of its 3, 5, and 7.

99edo is a very strong 7-limit (and 9 odd limit) temperament, but extending it to the 11-limit requires choosing which mapping one wants to use, as both are nearly equally far off the mark. It [[tempering_out|tempers out]] 393216/390625 ([[würschmidt comma]]) and 1600000/1594323 ([[amity comma]]) in the [[5-limit]]; 2401/2400 ([[2401/2400|breedsma]]), 3136/3125 ([[hemimean comma]]), and 4375/4374 ([[4375/4374|ragisma]]) in the [[7-limit]], supporting [[hemififths]], [[amity]], [[parakleismic]], [[hemiwürschmidt]] and [[ennealimmal]] temperaments, and is pretty well a perfect tuning for [[hendecatonic]] temperament. It has a sound defined by the slight sharpness (1.075, 1.565, 0.871 cents) of its 3, 5, and 7.

Using the [[patent val]], 99edo is the [[optimal patent val]] for the rank four temperament tempering out [[121/120]]; zeus, the rank three temperament tempering out 121/120 and [[176/175]]; [[hemiwür]], one of the rank two 11-limit extensions of hemiwürschmidt; and [[hitchcock]] (11-limit amity), the rank two temperament which also tempers out [[2200/2187]]. Using the {{val|99 157 230 278 343}} (99e) val, it tempers out [[896/891]], [[243/242]], [[441/440]] and [[540/539]], and is an excellent tuning for the 11-limit version of hemififths temperament. Hence 99 equal divisions, in spite of the fact that it tunes 11 relatively badly, is an important 11-limit tuning in more than one way.

@@ Line 2: / Line 2: @@
 == Theory ==
-edo is a very strong 7-limit (and 9 odd limit) temperament, but extending it to the 11-limit requires choosing which mapping one wants to use, as both are nearly equally far off the mark. It [[tempering_out|tempers out]] 393216/390625 ([[würschmidt comma]]) and 1600000/1594323 ([[amity comma]]) in the [[5-limit]]; 2401/2400 ([[2401/2400|breedsma]]), 3136/3125 (hemimean comma), and 4375/4374 ([[4375/4374|ragisma]]) in the [[7-limit]], supporting [[hemififths]], [[amity]], [[parakleismic]], [[hemiwürschmidt]] and [[ennealimmal]] temperaments, and is pretty well a perfect tuning for [[hendecatonic]] temperament. It has a sound defined by the slight sharpness (1.075, 1.565, 0.871 cents) of its 3, 5, and 7.
+edo is a very strong 7-limit (and 9 odd limit) temperament, but extending it to the 11-limit requires choosing which mapping one wants to use, as both are nearly equally far off the mark. It [[tempering_out|tempers out]] 393216/390625 ([[würschmidt comma]]) and 1600000/1594323 ([[amity comma]]) in the [[5-limit]]; 2401/2400 ([[2401/2400|breedsma]]), 3136/3125 ([[hemimean comma]]), and 4375/4374 ([[4375/4374|ragisma]]) in the [[7-limit]], supporting [[hemififths]], [[amity]], [[parakleismic]], [[hemiwürschmidt]] and [[ennealimmal]] temperaments, and is pretty well a perfect tuning for [[hendecatonic]] temperament. It has a sound defined by the slight sharpness (1.075, 1.565, 0.871 cents) of its 3, 5, and 7.
 Using the [[patent val]], 99edo is the [[optimal patent val]] for the rank four temperament tempering out [[121/120]]; zeus, the rank three temperament tempering out 121/120 and [[176/175]]; [[hemiwür]], one of the rank two 11-limit extensions of hemiwürschmidt; and [[hitchcock]] (11-limit amity), the rank two temperament which also tempers out [[2200/2187]]. Using the {{val|99 157 230 278 343}} (99e) val, it tempers out [[896/891]], [[243/242]], [[441/440]] and [[540/539]], and is an excellent tuning for the 11-limit version of hemififths temperament. Hence 99 equal divisions, in spite of the fact that it tunes 11 relatively badly, is an important 11-limit tuning in more than one way.