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 about 12.1 [[cent]]s each, a size~~ close to [[126/125]], the starling comma.

{{ED intro}} The step size of this system is close to [[126/125]], the starling comma.

== 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]], [[support]]ing [[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]]) 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]], [[support]]ing [[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 (1.1, 1.6, 0.9 cents) of its [[3/1|3]], [[5/1|5]], and [[7/1|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]].

The same can be said of the mapping for [[13/1|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.

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=== Prime harmonics ===

=== Octave stretch ===

99edo's approximations of harmonics 3, 5, and 7 can all be improved if slightly [[stretched and compressed tuning|compressing the octave]] is acceptable, using tunings such as [[157edt]] or [[256ed6]]. 157edt is especially performant if the 13-limit of the 99ef val is intended, but the 7-limit part is overcompressed, for which the milder 256ed6 is a better choice. If the 13-limit patent val is intended, then little to no compression, or even stretch, might be serviceable.

=== Subsets and supersets ===

Since 99 factors into {{~~factorization~~|99}}, 99edo has subset edos {{EDOs| 3, 9, 11, and 33 }}.

Since 99 factors into {{nowrap| 3<sup>2</sup> × 11 }}, 99edo has subset edos {{EDOs| 3, 9, 11, and 33 }}.

== Intervals ==

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|-

| 2.3

| {{~~monzo~~| 157 -99 }}

| {{Monzo| 157 -99 }}

| {{~~mapping~~| 99 157 }}

| {{Mapping| 99 157 }}

| −0.339

| 0.339

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| 2.3.5

| 393216/390625, 1600000/1594323

| {{~~mapping~~| 99 157 230 }}

| {{Mapping| 99 157 230 }}

| −0.451

| 0.319

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| 2.3.5.7

| 2401/2400, 3136/3125, 4375/4374

| {{~~mapping~~| 99 157 230 278 }}

| {{Mapping| 99 157 230 278 }}

| −0.416

| 0.283

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| 2.3.5.7.11

| 243/242, 441/440, 896/891, 3136/3125

| {{~~mapping~~| 99 157 230 278 343 }} (99e)

| {{Mapping| 99 157 230 278 343 }} (99e)

| −0.694

| 0.612

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| 2.3.5.7.11

| 121/120, 176/175, 1375/1372, 2200/2187

| {{~~mapping~~| 99 157 230 278 342 }} (99)

| {{Mapping| 99 157 230 278 342 }} (99)

| +0.006

| 0.881

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== See also ==

* [[58edf]] – relative [[edf]]

* [[157edt]] – relative [[edt]]

* [[58edf]] – relative [[edf]]

* [[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.

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-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, a size close to [[126/125]], the starling comma.
+{{ED intro}} The step size of this system is close to [[126/125]], the starling comma.
 == 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]], [[support]]ing [[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]]) 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]], [[support]]ing [[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 (1.1, 1.6, 0.9 cents) of its [[3/1|3]], [[5/1|5]], and [[7/1|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]].
+The same can be said of the mapping for [[13/1|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.
@@ Line 13: / Line 13: @@
 === Prime harmonics ===
 {{Harmonics in equal|99}}
+=== Octave stretch ===
+edo's approximations of harmonics 3, 5, and 7 can all be improved if slightly [[stretched and compressed tuning|compressing the octave]] is acceptable, using tunings such as [[157edt]] or [[256ed6]]. 157edt is especially performant if the 13-limit of the 99ef val is intended, but the 7-limit part is overcompressed, for which the milder 256ed6 is a better choice. If the 13-limit patent val is intended, then little to no compression, or even stretch, might be serviceable.
 === Subsets and supersets ===
-Since 99 factors into {{factorization|99}}, 99edo has subset edos {{EDOs| 3, 9, 11, and 33 }}.
+Since 99 factors into {{nowrap| 3<sup>2</sup> × 11 }}, 99edo has subset edos {{EDOs| 3, 9, 11, and 33 }}.
 == Intervals ==
@@ Line 48: / Line 51: @@
 |-
 | 2.3
-| {{monzo| 157 -99 }}
+| {{Monzo| 157 -99 }}
-| {{mapping| 99 157 }}
+| {{Mapping| 99 157 }}
 | −0.339
 | 0.339
@@ Line 56: / Line 59: @@
 | 2.3.5
 | 393216/390625, 1600000/1594323
-| {{mapping| 99 157 230 }}
+| {{Mapping| 99 157 230 }}
 | −0.451
 | 0.319
@@ Line 63: / Line 66: @@
 | 2.3.5.7
 | 2401/2400, 3136/3125, 4375/4374
-| {{mapping| 99 157 230 278 }}
+| {{Mapping| 99 157 230 278 }}
 | −0.416
 | 0.283
@@ Line 70: / Line 73: @@
 | 2.3.5.7.11
 | 243/242, 441/440, 896/891, 3136/3125
-| {{mapping| 99 157 230 278 343 }} (99e)
+| {{Mapping| 99 157 230 278 343 }} (99e)
 | −0.694
 | 0.612
@@ Line 77: / Line 80: @@
 | 2.3.5.7.11
 | 121/120, 176/175, 1375/1372, 2200/2187
-| {{mapping| 99 157 230 278 342 }} (99)
+| {{Mapping| 99 157 230 278 342 }} (99)
 | +0.006
 | 0.881
@@ Line 217: / Line 220: @@
 == See also ==
+* [[58edf]] – relative [[edf]]
 * [[157edt]] – relative [[edt]]
-* [[58edf]] – relative [[edf]]
 * [[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.