99edo: Difference between revisions

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The '''99 equal divisions of the octave''' ('''99edo'''), or the '''99(-tone) equal temperament''' ('''99tet''', '''99et''') when viewed from a [[regular temperament]] perspective, is the [[EDO|equal division of the octave]] into 99 parts of 12.~~1212~~ [[cent]]s each.

{{Infobox ET

| Prime factorization = 3<sup>2</sup> × 11

| Step size = 12.121¢

| Fifth = 58\99 (703.303¢)

| Semitones = 10:7 (121¢ : 85¢)

| Consistency = 9

}}

The '''99 equal divisions of the octave''' ('''99edo'''), or the '''99(-tone) equal temperament''' ('''99tet''', '''99et''') when viewed from a [[regular temperament]] perspective, is the [[EDO|equal division of the octave]] into 99 parts of about 12.1 [[cent]]s each.

== Theory ==

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.

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.

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]].

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|20L 59s

|}

== 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]]

@@ Line 1: / Line 1: @@
-The '''99 equal divisions of the octave''' ('''99edo'''), or the '''99(-tone) equal temperament''' ('''99tet''', '''99et''') when viewed from a [[regular temperament]] perspective, is the [[EDO|equal division of the octave]] into 99 parts of 12.1212 [[cent]]s each.
+{{Infobox ET
+| Prime factorization = 3<sup>2</sup> × 11
+| Step size = 12.121¢
+| Fifth = 58\99 (703.303¢)
+| Semitones = 10:7 (121¢ : 85¢)
+| Consistency = 9
+}}
+The '''99 equal divisions of the octave''' ('''99edo'''), or the '''99(-tone) equal temperament''' ('''99tet''', '''99et''') when viewed from a [[regular temperament]] perspective, is the [[EDO|equal division of the octave]] into 99 parts of about 12.1 [[cent]]s each.
 == Theory ==
 edo 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.
-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.
+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]].
@@ Line 416: / Line 423: @@
 |20L 59s
 |}
 == 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]]