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__FORCETOC__
{{Interwiki
<span style="display: block; text-align: right;">[[de:96edo|Deutsch]]</span>
| de = 96-EDO
| en = 96edo
| es =
| ja =
}}
{{Infobox ET}}
{{Wikipedia|96 equal temperament}}
{{ED intro}}


=Video=
== Theory ==
<youtube>3O3H01c2SjE</youtube>
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.


-----
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.


=Properties=
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.
The 96 equal division divides the octave into 96 equal parts of exactly 12.5 cents each. As a [[5-limit|5-limit]] system, it 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, |-53 10 16&gt;. It therefore has the same familiar 700 cent fifth as [[12edo|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|Würschmidt family]] of temperaments. It also tempers out the unicorn comma, and serves a way of tuning temperaments in the [[Unicorn_family|unicorn family]].


In the [[7-limit|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 supports 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 [[Marvel_temperaments#Submajor-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 ===
{{Harmonics in equal|96}}


=History=
=== As a tuning standard ===
96 equal divisions of the octave was first used by the Mexican composer and theorist [[Julián_Carrillo|Julián Carrillo]]. It has subsequently been used by a number of other composers.
A step of 96edo is known as a '''triamu''' (third MIDI-resolution unit, 3mu, 2<sup>3</sup> = 8 equal divisions of the 12edo semitone). The internal data structure of the 3mu requires one byte, with the first two bits reserved as flags, one to indicate the byte's status as data, and one to indicate the sign (+ or &minus;) showing the direction of the pitch-bend up or down, and three other bits which are not used.  


=Links=
=== 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 }}. [[192edo]], which doubles it, corrects the 7th harmonic to near-just.


==Carrillo==
== Intervals ==
See [[Julián_Carrillo|Julián Carrillo]] .
{{Interval table}}


==Other composers==
== 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 ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 393216/390625, 531441/524288
| {{mapping| 96 152 223 }}
| +0.240
| 0.732
| 5.86
|-
| 2.3.5.11
| 243/242, 5632/5625, 131769/131072
| {{mapping| 96 152 223 332 }}
| +0.276
| 0.637
| 5.10
|}
 
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|-
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>ratio*
! Temperament
|-
| 1
| 5\96
| 62.5
| 28/27
| [[Unicorn]] / [[camahueto]] (96d)
|-
| 1
| 21\96
| 262.5
| 64/55
| [[Subgroup temperaments #Spog|Spog]]
|-
| 1
| 29\96
| 362.5
| 16/13
| [[Buzzardsmic clan #Demibuzzard|Demibuzzard]] / [[interpental]] (96)
|-
| 1
| 31\96
| 387.5
| 5/4
| [[Würschmidt]] (96) / [[worschmidt]] (96d)
|-
| 2
| 5\96
| 62.5
| 28/27
| [[Monocerus]] (96d)
|-
| 2
| 13\96
| 162.5
| 11/10
| [[Gwazy]] (96) / [[bisupermajor]] (96d)
|-
| 2
| 25\96
| 312.5
| 6/5
| [[Vines]] (96d)
|-
| 12
| 31\96<br>(1\96)
| 387.5<br>(12.5)
| 5/4<br>(126/125)
| [[Compton]] (7-limit, 96)
|-
| 24
| 31\96<br>(1\96)
| 387.5<br>(12.5)
| 5/4<br>(245/243)
| [[Hours]] (96d)
|}
 
== Scales ==
* [[5- to 10-tone scales in 96edo]]
 
== History ==
96 equal divisions of the octave was first used by the Mexican composer and theorist [[Julián Carrillo]]. It has subsequently been used by a number of other composers.
 
=== Carrillo ===
{{Main| Julián Carrillo }}
 
=== Other composers ===
Works for the [http://www.sauter-pianos.de/english/pianos/microtone.html Sauter's 1/16tone microtone piano] by the composers [http://presence.or.at/einklang/Archiv/archiv_Komponisten_E.htm#flammer Ernest Helmuth Flammer], Marc Kilchenmann, Bernfried E. G. Pröve, [http://www.musinfo.ch/index.php?content=maske_werke&pers_id=150&name=Imholz&vorname=Martin&setLanguage=en Martin Imholz], Franck Cristoph Yeznikian, [http://www.schauhoer-verlag.de/cms/file_download/13/Vita_WMG_lang.pdf Werner Grimmel], and [http://www.moderecords.com/catalog/120_121bancquart.html Alain Bancquart], are recompilated on this CD: [http://www.dominikblum.ch/carillo_d.shtml 'The Carrillo tone piano'] .
Works for the [http://www.sauter-pianos.de/english/pianos/microtone.html Sauter's 1/16tone microtone piano] by the composers [http://presence.or.at/einklang/Archiv/archiv_Komponisten_E.htm#flammer Ernest Helmuth Flammer], Marc Kilchenmann, Bernfried E. G. Pröve, [http://www.musinfo.ch/index.php?content=maske_werke&pers_id=150&name=Imholz&vorname=Martin&setLanguage=en Martin Imholz], Franck Cristoph Yeznikian, [http://www.schauhoer-verlag.de/cms/file_download/13/Vita_WMG_lang.pdf Werner Grimmel], and [http://www.moderecords.com/catalog/120_121bancquart.html Alain Bancquart], are recompilated on this CD: [http://www.dominikblum.ch/carillo_d.shtml 'The Carrillo tone piano'] .


[http://www.96edo.com/ Mohajeri, Shaahin]
* [http://www.96edo.com/ Mohajeri, Shaahin]
* [http://mac-texier.ircam.fr/textes/c00001420/ Marie, Jean-Etienne]
* [[Wikipedia: Pascale Criton|Criton, Pascale]]
* [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"
 
== 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 ==
 
=== 20th century ===
 
; [[Julián Carrillo]]
* [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 ===


[http://mac-texier.ircam.fr/textes/c00001420/ Marie, Jean-Etienne ]
; 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)


[http://en.wikipedia.org/wiki/Pascale_Criton Criton, Pascale]
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/8xqjUkuZFaI ''microtonal improvisation in 96edo''] (2025)


[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]]
; [[Shahiin Mohajeri]]
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Mohajeri/4gahforbrass-880821.mp3 ''4gah for brass'']{{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://en.wikipedia.org/wiki/Georg_Friedrich_Haas Haas, Georg Friedrich], "flow and friction"
; [[Juhani Nuorvala]]
* [https://www.youtube.com/watch?v=aAHkjOvplVg ''Kellot (Bells)''] (2025) – in Porcupine[7], 96edo tuning


=Music=
; [[Tony Salinas]]
[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Mohajeri/4gahforbrass-880821.mp3 4gah for brass] by [[Shahiin_Mohajeri|Shahiin Mohajeri]]
* [[:File:01_-_Autumn_1.mp3|''Autumn'' for conic bellophone and mixed quintet]] (2007)
; [[Randy Wells]]
* [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=QAnOwQY55bE ''The Persistence of Memory''] (2021)


[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Mohajeri/Endless%20life.mp3 Endless life] by Shahiin Mohajeri
; [[Anne Veinberg]]
* [https://www.youtube.com/watch?v=IEnlgXFAfd4 ''Short impro on the 96 tone Carrillo piano''] (2014)


[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Mohajeri/Heroic%20elegy.mp3 Heroic elegy] by Shahiin Mohajeri
== See also ==
* [[Equal-step tuning|Equal multiplications]] of MIDI-resolution units
** [[24edo]] (1mu tuning)
** [[48edo]] (2mu tuning)
** [[192edo]] (4mu tuning)
** [[384edo]] (5mu tuning)
** [[768edo]] (6mu tuning)
** [[1536edo]] (7mu tuning)
** [[3072edo]] (8mu tuning)
** [[6144edo]] (9mu tuning)
** [[12288edo]] (10mu tuning)
** [[24576edo]] (11mu tuning)
** [[49152edo]] (12mu tuning)
** [[98304edo]] (13mu tuning)
** [[196608edo]] (14mu tuning)


[[:File:01_-_Autumn_1.mp3|Autumn for conic bellophone and mixed quintet]] by Tony Salinas
== External links ==
* [http://tonalsoft.com/enc/number/3mu.aspx 3mu / triamu] on [[Tonalsoft Encyclopedia]]


[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Carrillo/Cromometrofon%eda%20%231.mp3 Cromometrofonía #1] by [[Julián_Carrillo|Julián Carrillo]]      [[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:interpental]]
[[Category:Unicorn]]
[[Category:listen]]
[[Category:Würschmidt]]
[[Category:unicorn]]
[[Category:Interpental]]
[[Category:wuerschmidt]]
[[Category:wurschmidt]]
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