99edo: Difference between revisions
→Temperaments: +ennealimnic/ennealim/ennealiminal |
No edit summary |
||
| Line 4: | Line 4: | ||
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. | 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]]; [[Würschmidt family #Hemiwur|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. | ||
== Intervals == | == Intervals == | ||
| Line 114: | Line 114: | ||
==Scales== | ==Scales== | ||
*[[ | *[[Tutone6]] | ||
*[[ | *[[Tutone7]] | ||
*[[ | *[[Tutone13]] | ||
*[[ | *[[Zeus7tri]] | ||
*[[ | *[[Zeus8tri]] | ||
Since | Since 99edo has a step of 12.1212 cents, it also allows one to use its MOS scales as circulating temperaments. | ||
{| class="wikitable" | {| class="wikitable" | ||
|+Circulating temperaments in 99edo | |+Circulating temperaments in 99edo | ||
| Line 374: | Line 374: | ||
==See also== | ==See also== | ||
*[[157edt]] | *[[157edt]] - relative EDT | ||
*[[58edf]] | *[[58edf]] - relative EDF | ||
*[[87edo]], [[94edo]], [[111edo]] | *[[87edo]], [[94edo]], [[111edo]] - similarly sized edos all with consistency in higher harmonics. | ||
*[[198edo]], the half-sized edo to reconcile the mappings of 11 and 13. | *[[198edo]], the half-sized edo to reconcile the mappings of 11 and 13. | ||
*[[105edo]], a similarly sized edo that supports meantone, septimal meantone, undecimal meantone and grosstone | *[[105edo]], a similarly sized edo that supports meantone, septimal meantone, undecimal meantone and grosstone | ||