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{{Interwiki
{{Interwiki
| de = 96edo
| de = 96-EDO
| en = 96edo
| en = 96edo
| es =  
| es =  
| ja =  
| ja =  
}}
}}
{{Infobox ET
{{Infobox ET}}
| Prime factorization = 2<sup>5</sup> × 3
{{Wikipedia|96 equal temperament}}
| Step size = 12.5000¢
{{ED intro}}
| Fifth = 56\96 (700.0¢) (→ [[12edo|7\12]])
| Semitones = 8:8 (100.0¢ : 100.0¢)
| Consistency = 5
}}
The '''96 equal divisions of the octave''' ('''96edo'''), or the '''96-tone equal temperament''' ('''96tet'''), '''96 equal temperament''' ('''96et''') when viewed from a [[regular temperament]] perspective, divides the octave into 96 equal parts of exactly 12.5 [[cent]]s each.


== 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]].
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 ===
{{Harmonics in equal|96}}
{{Harmonics in equal|96}}
=== As a tuning standard ===
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.
=== 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.
== Intervals ==
{{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-4 center-5 center-6"
! rowspan="2" | Subgroup
|-
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! colspan="2" | Tuning error
! colspan="2" | Tuning Error
|-
|-
! [[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
Line 34: Line 50:
|-
|-
| 2.3.5
| 2.3.5
| 3293216/390625, 531441/524288
| 393216/390625, 531441/524288
| [{{val| 96 152 223 }}]
| {{mapping| 96 152 223 }}
| +0.240
| +0.240
| 0.732
| 0.732
Line 42: Line 58:
| 2.3.5.11
| 2.3.5.11
| 243/242, 5632/5625, 131769/131072
| 243/242, 5632/5625, 131769/131072
| [{{val| 96 152 223 332 }}]
| {{mapping| 96 152 223 332 }}
| -0.550
| +0.276
| 0.637
| 0.637
| 5.10
| 5.10
|}
|}


== Scales ==
=== Rank-2 temperaments ===
Since 96edo has a step of 12.5 cents, it also allows one to use its MOS scales as circulating temperaments. It is the first 12''n''-edo which does this and the first edo which allows one to use an MOS scale with a step 20 degrees or larger as a circulating temperament{{clarify}}.
{| class="wikitable center-all left-5"
{| class="wikitable center-all"
|-
|+Circulating temperaments in 96edo
! Periods<br>per 8ve
! Tones
! Generator*
! Pattern
! Cents*
! L:s
! Associated<br>ratio*
! Temperament
|-
|-
| 5
| 1
| [[1L 4s]]
| 5\96
| 20:19
| 62.5
| 28/27
| [[Unicorn]] / [[camahueto]] (96d)
|-
|-
| 6
| 1
| [[6edo]]
| 21\96
| equal
| 262.5
| 64/55
| [[Subgroup temperaments #Spog|Spog]]
|-
|-
| 7
| 1
| [[5L 2s]]
| 29\96
| 14:13
| 362.5
| 16/13
| [[Buzzardsmic clan #Demibuzzard|Demibuzzard]] / [[interpental]] (96)
|-
|-
| 8
| 1
| [[8edo]]
| 31\96
| equal
| 387.5
| 5/4
| [[Würschmidt]] (96) / [[worschmidt]] (96d)
|-
|-
| 9
| 2
| [[6L 3s]]
| 5\96
| 11:10
| 62.5
| 28/27
| [[Monocerus]] (96d)
|-
|-
| 10
| 2
| [[6L 4s]]
| 13\96
| 10:9
| 162.5
| 11/10
| [[Gwazy]] (96) / [[bisupermajor]] (96d)
|-
|-
| 11
| 2
| [[8L 3s]]
| 25\96
| 9:8
| 312.5
| 6/5
| [[Vines]] (96d)
|-
|-
| 12
| 12
| [[12edo]]
| 31\96<br>(1\96)
| equal
| 387.5<br>(12.5)
|-
| 5/4<br>(126/125)
| 13
| [[Compton]] (7-limit, 96)
| [[5L 8s]]
| 8:7
|-
| 14
| [[12L 2s]]
| rowspan="2" |7:6
|-
| 15
| [[6L 9s]]
|-
| 16
| [[16edo]]
| equal
|-
| 17
| [[11L 6s]]
| rowspan="3" |6:5
|-
| 18
| 6L 12s
|-
| 19
| 1L 18s
|-
| 20
| 16L 4s
| rowspan="4" |5:4
|-
| 21
| 12L 9s
|-
| 22
| 8L 14s
|-
| 23
| 4L 19s
|-
|-
| 24
| 24
| [[24edo]]
| 31\96<br>(1\96)
| equal
| 387.5<br>(12.5)
|-
| 5/4<br>(245/243)
| 25
| [[Hours]] (96d)
| 21L 4s
| rowspan="7" | 4:3
|-
| 26
| 18L 8s
|-
| 27
| 15L 12s
|-
| 28
| 12L 16s
|-
| 29
| 9L 20s
|-
| 30
| 6L 24s
|-
| 31
| 3L 28s
|-
| 32
| [[32edo]]
| equal
|-
| 33
| 30L 3s
| rowspan="15" | 3:2
|-
| 34
| 28L 6s
|-
| 35
| 26L 9s
|-
| 36
| 24L 12s
|-
| 37
| 22L 15s
|-
| 38
| 20L 18s
|-
| 39
| 18L 21s
|-
| 40
| 16L 24s
|-
| 41
| 14L 27s
|-
| 42
| 12L 30s
|-
| 43
| 10L 33s
|-
| 44
| 8L 36s
|-
| 45
| 6L 39s
|-
| 46
| 4L 42s
|-
| 47
| 2L 45s
|-
| 48
| [[48edo]]
| equal
|-
| 49
| 47L 2s
| rowspan="28" |2:1
|-
| 50
| 46L 4s
|-
| 51
| 45L 6s
|-
| 52
| 44L 8s
|-
| 53
| 43L 10s
|-
| 54
| 42L 12s
|-
| 55
| 41L 14s
|-
| 56
| 40L 16s
|-
| 57
| 39L 18s
|-
| 58
| 38L 20s
|-
| 59
| 37L 22s
|-
| 60
| 36L 24s
|-
| 61
| 35L 26s
|-
| 62
| 34L 28s
|-
| 63
| 33L 30s
|-
| 64
| 32L 32s
|-
| 65
| 31L 34s
|-
| 66
| 30L 36s
|-
| 67
| 29L 38s
|-
| 68
| 28L 40s
|-
| 69
| 27L 42s
|-
| 70
| 26L 44s
|-
| 71
| 25L 46s
|-
| 72
| 24L 48s
|-
| 73
| 23L 50s
|-
| 74
| 22L 52s
|-
| 75
| 21L 54s
|-
| 76
| 20L 56s
|}
|}
== Scales ==
* [[5- to 10-tone scales in 96edo]]


== History ==
== History ==
Line 306: Line 145:
* [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 ==
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Mohajeri/4gahforbrass-880821.mp3 4gah for brass] by [[Shahiin Mohajeri]]
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Mohajeri/Endless%20life.mp3 Endless life] by Shahiin Mohajeri
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Mohajeri/Heroic%20elegy.mp3 Heroic elegy] by Shahiin Mohajeri
* [[:File:01_-_Autumn_1.mp3|Autumn for conic bellophone and mixed quintet]] by Tony Salinas
* [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Carrillo/Cromometrofon%eda%20%231.mp3 Cromometrofonía #1] by [[Julián Carrillo]]   


== Video ==
=== 20th century ===
<youtube>3O3H01c2SjE</youtube>{{dead link}}
 
; [[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 ===
 
; 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]]
* [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}}
 
; [[Juhani Nuorvala]]
* [https://www.youtube.com/watch?v=aAHkjOvplVg ''Kellot (Bells)''] (2025) – in Porcupine[7], 96edo tuning
 
; [[Tony Salinas]]
* [[: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)
 
; [[Anne Veinberg]]
* [https://www.youtube.com/watch?v=IEnlgXFAfd4 ''Short impro on the 96 tone Carrillo piano''] (2014)
 
== 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)
 
== External links ==
* [http://tonalsoft.com/enc/number/3mu.aspx 3mu / triamu] on [[Tonalsoft Encyclopedia]]


[[Category:Equal divisions of the octave]]
[[Category:Listen]]
[[Category:Listen]]
[[Category:Unicorn]]
[[Category:Unicorn]]
[[Category:Würschmidt]]
[[Category:Würschmidt]]
[[Category:Interpental]]
[[Category:Interpental]]
Retrieved from "https://en.xen.wiki/w/96edo"