99edo: Difference between revisions

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'''99edo''' is the [[EDO|equal division of the octave]] into 99 parts of 12.1212 [[cent]]s each. It 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 of 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 the [[EDO|equal division of the octave]] into 99 parts of 12.1212 [[cent]]s each. It 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|patent val]], 99EDO is the [[Optimal_patent_val|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 <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 [[Breedsmic_temperaments|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.

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	'''99edo''' is the [[EDO\|equal division of the octave]] into 99 parts of 12.1212 [[cent]]s each. It 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 of 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 the [[EDO\|equal division of the octave]] into 99 parts of 12.1212 [[cent]]s each. It 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\|patent val]], 99EDO is the [[Optimal_patent_val\|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 <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 [[Breedsmic_temperaments\|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.		Using the [[Patent_val\|patent val]], 99EDO is the [[Optimal_patent_val\|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 <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 [[Breedsmic_temperaments\|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.