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{{Infobox ET}}
{{Infobox ET}}
{{Wikipedia|96 equal temperament}}
{{Wikipedia|96 equal temperament}}
{{EDO intro|96}}
{{ED intro}}


== Theory ==
== Theory ==
As a [[5-limit]] system, 96edo can be characterized by the fact that it tempers out both the [[Pythagorean comma]], 531441/524288, [[Würschmidt's comma]], 393216/390625, the [[unicorn comma]], 1594323/1562500, and the kwazy comma, {{monzo| -53 10 16 }}. It therefore has the same familiar 700-cent fifth as [[12edo]], and has a best major third of 387.5 cents, a bit over a cent sharp. There is therefore nothing to complain of with its representation of the 5-limit and it can be recommended as an approach to the [[würschmidt family]] of temperaments. It also tempers out the unicorn comma, and serves a way of tuning temperaments in the [[unicorn family]]. It supports [[Substitute harmonic#Sitcom|sitcom]] temperament.
As a [[5-limit]] system, 96edo can be characterized by the fact that it tempers out both the [[Pythagorean comma]], 531441/524288, [[Würschmidt's comma]], 393216/390625, the [[unicorn comma]], 1594323/1562500, and the [[kwazy comma]], {{monzo| -53 10 16 }}. It therefore has the same familiar 700{{c}} fifth as [[12edo]], and has a best major third of 387.5{{c}}, a bit over a cent sharp. There is therefore nothing to complain of with its representation of the 5-limit and it can be recommended as an approach to the [[würschmidt family]] of temperaments. It also tempers out the unicorn comma, and serves a way of tuning temperaments in the [[unicorn family]]. It supports [[Substitute harmonic#Sitcom|sitcom]] temperament.


In the [[7-limit]], 96 has two possible mappings for [[7/4]], a sharp one of 975 cents from the [[patent val]], and a flat one of 962.5 cents from 96d. Using the sharp mapping, 96 tempers out [[225/224]] and [[support]]s 7-limit [[würschmidt]] temperament, and using the flat mapping it tempers out [[126/125]] and supports [[worschmidt]] temperament. We can also dispense with 7 altogether, and use it as a no-sevens system, where it tempers out [[243/242]] in the 11-limit and [[676/675]] in the 13-limit. If we include 7, then the sharp mapping tempers out [[99/98]] and [[176/175]] in the 11-limit, and [[169/168]] in the 13-limit, and this provides the optimal patent val for the [[Marvel temperaments #Interpental|interpental temperament]]. With the flat 7 it tempers out [[385/384]] in the 11-limit and [[196/195]] and [[364/363]] in the 13-limit, and serves for the various temperaments of the unicorn family.
One notable benefit of 96edo's representation of the 5-limit is that its dramatic narrowing of [[81/80]] allows for a less dissonant [[~]][[40/27]] wolf fifth. This allows for the potential of a 12-note subset of 96edo being seen as a [[well temperament]], and as part of an equal temperament, this scale could be rotated around during the run-time of a piece of music.
 
In the [[7-limit]], 96 has two possible mappings for [[7/4]], a sharp one of 975{{c}} from the [[patent val]], and a flat one of 962.5{{c}} from 96d. Using the sharp mapping, 96 tempers out [[225/224]] and [[support]]s 7-limit [[würschmidt]] temperament, and using the flat mapping it tempers out [[126/125]] and supports [[worschmidt]] temperament. We can also dispense with 7 altogether, and use it as a no-sevens system, where it tempers out [[243/242]] in the 11-limit and [[676/675]] in the 13-limit. If we include 7, then the sharp mapping tempers out [[99/98]] and [[176/175]] in the 11-limit, and [[169/168]] in the 13-limit, and this provides the optimal patent val for the [[Marvel temperaments #Interpental|interpental temperament]]. With the flat 7 it tempers out [[385/384]] in the 11-limit and [[196/195]] and [[364/363]] in the 13-limit, and serves for the various temperaments of the unicorn family.


=== Prime harmonics ===
=== Prime harmonics ===
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=== Subsets and supersets ===
=== Subsets and supersets ===
Since 96 factors into {{factorization|96}}, 96edo has subset edos {{EDOs| 2, 3, 4, 6, 8, 12, 16, 24, 32, and 48 }}.
Since 96 factors into {{factorization|96}}, 96edo has subset edos {{EDOs| 2, 3, 4, 6, 8, 12, 16, 24, 32, and 48 }}. [[192edo]], which doubles it, corrects the 7th harmonic to near-just.


== Intervals ==
== Intervals ==
{{Interval table}}
{{Interval table}}
== Notation ==
=== Ups and downs notation ===
96edo can be notated using [[ups and downs notation]] using [[Helmholtz–Ellis]] accidentals:
{{Sharpness-sharp8|96}}
== Approximation to JI ==
{{Q-odd-limit intervals}}


== Regular temperament properties ==
== Regular temperament properties ==
 
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-1 center-2"
|-
! Periods<br>per 8ve
! Generator
! Names
|-
| 1
| 5\96
| [[Unicorn]]
|-
| 1
| 29\96
| [[Interpental]] / [[Submajor]]
|-
| 1
| 31\96
| [[Würschmidt]]
|-
|-
| 2
! rowspan="2" | [[Subgroup]]
| 13\96
! rowspan="2" | [[Comma list|Comma List]]
| [[Kwazy]] / [[Bisupermajor]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
|-
| 2
! [[TE error|Absolute]] (¢)
| 25\96
! [[TE simple badness|Relative]] (%)
| [[Vines]]
|-
|-
| 12
| 2.3.5
| 1\96
| 393216/390625, 531441/524288
| [[Compton]] / [[Substitute harmonic#Sitcom|Sitcom]]
| {{mapping| 96 152 223 }}
| +0.240
| 0.732
| 5.86
|-
|-
| 24
| 2.3.5.11
| 1\96
| 243/242, 5632/5625, 131769/131072
| [[Hours]]
| {{mapping| 96 152 223 332 }}
| +0.276
| 0.637
| 5.10
|}
|}


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|-
! Periods<br>per 8ve
! Periods<br>per 8ve
! Generator<br>(Reduced)
! Generator*
! Cents<br>(Reduced)
! Cents*
! Associated<br>Ratio
! Associated<br>ratio*
! Temperaments
! Temperament
|-
| rowspan="2" | 1
| rowspan="2" | 19\84
| rowspan="2" | 271.43
| rowspan="2" | 7/6
| [[Orwell]] (84e)
|-
|-
| 1
| 1
| 25\84
| 5\96
| 357.14
| 62.5
| 768/625
| 28/27
| [[Dodifo]]
| [[Unicorn]] / [[camahueto]] (96d)
|-
|-
| 1
| 1
| 27\84
| 21\96
| 385.71
| 262.5
| 5/4
| 64/55
| [[Mutt]]
| [[Subgroup temperaments #Spog|Spog]]
|-
|-
| 1
| 1
| 31\84
| 29\96
| 442.86
| 362.5
| 125/81
| 16/13
| [[Sensei]]
| [[Buzzardsmic clan #Demibuzzard|Demibuzzard]] / [[interpental]] (96)
|-
|-
| 1
| 1
| 41\84
| 31\96
| 585.71
| 387.5
| 7/5
| 5/4
| [[Merman]]
| [[Würschmidt]] (96) / [[worschmidt]] (96d)
|-
|-
| 2
| 2
| 5\84
| 5\96
| 71.43
| 62.5
| 25/24
| 28/27
| [[Narayana]]
| [[Monocerus]] (96d)
|-
|-
| 2
| 2
| 11\84
| 13\96
| 157.14
| 162.5
| 35/32
| 11/10
| [[Bison]]
| [[Gwazy]] (96) / [[bisupermajor]] (96d)
|-
|-
| 2
| 2
| 13\84
| 25\96
| 185.71
| 312.5
| 10/9
| 6/5
| [[Secant]]
| [[Vines]] (96d)
|-
| 3
| 11\84
| 157.14
| 35/32
| [[Nessafof]]
|-
| 7
| 5\84
| 500.00<br>(14.29)
| 4/3<br>(81/80)
| [[Absurdity]]
|-
|-
| 12
| 12
| 27\84<br>(1\84)
| 31\96<br>(1\96)
| 385.71<br>(14.29)
| 387.5<br>(12.5)
| 5/4<br>(126/125)
| 5/4<br>(126/125)
| [[Compton]]
| [[Compton]] (7-limit, 96)
|-
|-
|21
| 24
|41\84<br>(1\84)
| 31\96<br>(1\96)
|585.71<br>(14.29)
| 387.5<br>(12.5)
|91875/65536<br>(126/125)
| 5/4<br>(245/243)
|[[Akjayland]]
| [[Hours]] (96d)
|-
| 28
| 49\84<br>(1\84)
| 500.00<br>(14.29)
| 4/3<br>(105/104)
| [[Oquatonic]]
|}
|}
== Scales ==
* [[5- to 10-tone scales in 96edo]]


== History ==
== History ==
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* [http://www.tonysalinas.com/ Martin Salinas, J.A.] [[Autumn|'Autumn' conic bellophone &amp; mixed quintet.mp3]] / [[Conic_Bellophone_in_96edo|Pictures of the 96edo conic bellophone]]
* [http://www.tonysalinas.com/ Martin Salinas, J.A.] [[Autumn|'Autumn' conic bellophone &amp; mixed quintet.mp3]] / [[Conic_Bellophone_in_96edo|Pictures of the 96edo conic bellophone]]
* [[Wikipedia: Georg Friedrich Haas|Haas, Georg Friedrich]], "flow and friction"
* [[Wikipedia: Georg Friedrich Haas|Haas, Georg Friedrich]], "flow and friction"
== Instruments ==
The piano manufacturer [https://www.sauter-pianos.de/en Sauter] produces an acoustic piano tuned in 96edo:
[https://www.sauter-pianos.de/en/microtone Sauter 1/16 Tone piano]
; Lumatone
* [[Lumatone mapping for 96edo]]


== Music ==
== Music ==
=== 20th century ===


; [[Julián Carrillo]]
; [[Julián Carrillo]]
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Carrillo/Cromometrofon%eda%20%231.mp3 ''Cromometrofonía #1'']{{dead link}}
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Carrillo/Cromometrofon%eda%20%231.mp3 ''Cromometrofonía #1'']{{dead link}}
* [https://www.youtube.com/watch?v=tDHMnlQri3g ''Preludio a Colon''] (1925)
=== 21st century ===
; Various composers (Alain Bancquart, Ernst Helmuth Flammer, Werner Grimmel, Marin Imholz, Marc Kilchenmann, Bernfried E. G. Pröve, Franck Christoph Yeznikian, Sylvaine Billier, Dominik Blum, Martine Joste)
* [https://www.discogs.com/sell/release/17561818 The Carillo 1/16-Tone Piano.] (2003)
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/8xqjUkuZFaI ''microtonal improvisation in 96edo''] (2025)


; [[Shahiin Mohajeri]]
; [[Shahiin Mohajeri]]
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* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Mohajeri/Endless%20life.mp3 ''Endless life'']{{dead link}}
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Mohajeri/Endless%20life.mp3 ''Endless life'']{{dead link}}
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Mohajeri/Heroic%20elegy.mp3 ''Heroic elegy'']{{dead link}}
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Mohajeri/Heroic%20elegy.mp3 ''Heroic elegy'']{{dead link}}
; [[Juhani Nuorvala]]
* [https://www.youtube.com/watch?v=aAHkjOvplVg ''Kellot (Bells)''] (2025) – in Porcupine[7], 96edo tuning


; [[Tony Salinas]]
; [[Tony Salinas]]
* [[:File:01_-_Autumn_1.mp3|''Autumn'' for conic bellophone and mixed quintet]]  
* [[:File:01_-_Autumn_1.mp3|''Autumn'' for conic bellophone and mixed quintet]] (2007)
 
; [[Randy Wells]]
; [[Randy Wells]]
* [https://www.youtube.com/watch?v=XuS8VxDnJm4 ''Butterfly Skin''] (2021)
* [https://www.youtube.com/watch?v=XuS8VxDnJm4 ''Butterfly Skin''] (2021)
* [https://www.youtube.com/watch?v=HfZtsiGUmOA ''Dream of a memory, memory of a dream''] (2021)
* [https://www.youtube.com/watch?v=HfZtsiGUmOA ''Dream of a memory, memory of a dream''] (2021)
* [https://www.youtube.com/watch?v=QAnOwQY55bE ''The Persistence of Memory''] (2021)
* [https://www.youtube.com/watch?v=QAnOwQY55bE ''The Persistence of Memory''] (2021)
; [[Anne Veinberg]]
* [https://www.youtube.com/watch?v=IEnlgXFAfd4 ''Short impro on the 96 tone Carrillo piano''] (2014)


== See also ==
== See also ==
Line 200: Line 206:


== External links ==
== External links ==
* [http://tonalsoft.com/enc/number/3mu.aspx Tonalsoft Encyclopedia | ''3mu / triamu'']
* [http://tonalsoft.com/enc/number/3mu.aspx 3mu / triamu] on [[Tonalsoft Encyclopedia]]


[[Category:Listen]]
[[Category:Listen]]
Retrieved from "https://en.xen.wiki/w/96edo"