99edo: Difference between revisions
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'''99edo''' is the [[EDO|equal division of the octave]] into 99 parts of 12.1212 [[cent | '''99edo''' is the [[EDO|equal division of the octave]] into 99 parts of 12.1212 [[cent]]s each. | ||
== Theory == | == Theory == | ||
99edo is a very strong 7-limit (and 9 odd limit) | 99edo is a very strong 7-limit (and [[9-odd-limit]]) tuning. 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 | Extending it to the 11-limit requires choosing which mapping one wants to use, as both are nearly equally far off the mark. Using the [[patent val]], 99edo is the [[optimal patent val]] for the rank-4 temperament tempering out [[121/120]]; zeus, the rank-3 temperament tempering out 121/120 and [[176/175]]; [[hemiwür]], one of the rank-2 11-limit extensions of hemiwürschmidt; and [[hitchcock]] (an 11-limit amity extension), the rank-2 temperament which also tempers out [[2200/2187]]. Using the {{val| 99 157 230 278 343 }} (99e) val, it tempers out [[243/242]], [[441/440]], [[540/539]] and [[896/891]], 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. | ||
The same can be said of the mapping for 13, with its patent val tempering out [[169/168]], [[351/350]] and [[352/351]], and the 99ef val tempering out [[144/143]], [[196/195]], 352/351 and [[364/363]]. | |||
Skipping 11 and 13, it is a very strong system in the 2.3.5.7.17.19.23.29 subgroup. | |||
{{Primes in edo|99|columns=11}} | {{Primes in edo|99|columns=11}} | ||
== Intervals == | |||
See [[Table of 99edo intervals]]. | |||
== Temperaments == | == Temperaments == | ||
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|} | |} | ||
==Scales== | == Scales == | ||
*[[Tutone6]] | * [[Tutone6]] | ||
*[[Tutone7]] | * [[Tutone7]] | ||
*[[Tutone13]] | * [[Tutone13]] | ||
*[[Zeus7tri]] | * [[Zeus7tri]] | ||
*[[Zeus8tri]] | * [[Zeus8tri]] | ||
Since 99edo has a step of 12.1212 cents, it also allows one to use its MOS scales as circulating temperaments. | Since 99edo has a step of 12.1212 cents, it also allows one to use its MOS scales as circulating temperaments{{clarify}}. | ||
{| class="wikitable" | {| class="wikitable" | ||
|+Circulating temperaments in 99edo | |+Circulating temperaments in 99edo | ||
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|20L 59s | |20L 59s | ||
|} | |} | ||
==Music== | == Music == | ||
*[http://www.archive.org/details/NonagintaEtNovem Nonaginta et Novem] ''[http://clones.soonlabel.com/public/micro/gene_ward_smith/mine/Nonaginta%20et%20Novem.mp3 play]'' by [[Gene Ward Smith]] | * [http://www.archive.org/details/NonagintaEtNovem Nonaginta et Novem] ''[http://clones.soonlabel.com/public/micro/gene_ward_smith/mine/Nonaginta%20et%20Novem.mp3 play]'' by [[Gene Ward Smith]] | ||
*[http://micro.soonlabel.com/gene_ward_smith/transformers/benny.mp3 Benny] Smith-Palestrina in [[zeus7tri]] | * [http://micro.soonlabel.com/gene_ward_smith/transformers/benny.mp3 Benny] Smith-Palestrina in [[zeus7tri]] | ||
==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 | ||
[[Category:Theory]] | [[Category:Theory]] | ||