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<span style="display: block; text-align: right;">[[de:96edo|Deutsch]]</span>
<span style="display: block; text-align: right;">[[de:96edo|Deutsch]]</span>


=Video=
The '''96 equal divisions of the octave''' ('''96edo''') divides the octave into 96 equal parts of exactly 12.5 [[cent]]s each.
<youtube>3O3H01c2SjE</youtube>
 
-----


=Properties=
== Theory ==
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]].
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.
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.


 
== Scales ==
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 12n-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.
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"
{| class="wikitable"
|+Circulating temperaments in 96edo
|+Circulating temperaments in 96edo
!Tones
! Tones
!Pattern
! Pattern
!L:s
! L:s
|-
|-
|5
| 5
|[[1L 4s]]
| [[1L 4s]]
|20:19
| 20:19
|-
|-
|6
| 6
|[[6edo]]
| [[6edo]]
|equal
| equal
|-
|-
|7
| 7
|[[5L 2s]]
| [[5L 2s]]
|14:13
| 14:13
|-
|-
|8
| 8
|[[8edo]]
| [[8edo]]
|equal
| equal
|-
|-
|9
| 9
|[[6L 3s]]
| [[6L 3s]]
|11:10
| 11:10
|-
|-
|10
| 10
|[[6L 4s]]
| [[6L 4s]]
|10:9
| 10:9
|-
|-
|11
| 11
|[[8L 3s]]
| [[8L 3s]]
|9:8
| 9:8
|-
|-
|12
| 12
|[[12edo]]
| [[12edo]]
|equal
| equal
|-
|-
| 13
| 13
|[[5L 8s]]
| [[5L 8s]]
|8:7
| 8:7
|-
|-
|14
| 14
|[[12L 2s]]
| [[12L 2s]]
| rowspan="2" |7:6
| rowspan="2" |7:6
|-
|-
|15
| 15
|[[6L 9s]]
| [[6L 9s]]
|-
|-
|16
| 16
|[[16edo]]
| [[16edo]]
|equal
| equal
|-
|-
|17
| 17
|[[11L 6s]]
| [[11L 6s]]
| rowspan="3" |6:5
| rowspan="3" |6:5
|-
|-
|18
| 18
|6L 12s
| 6L 12s
|-
|-
| 19
| 19
|1L 18s
| 1L 18s
|-
|-
|20
| 20
|16L 4s
| 16L 4s
| rowspan="4" |5:4
| rowspan="4" |5:4
|-
|-
|21
| 21
|12L 9s
| 12L 9s
|-
|-
|22
| 22
|8L 14s
| 8L 14s
|-
|-
|23
| 23
|4L 19s
| 4L 19s
|-
|-
|24
| 24
|[[24edo]]
| [[24edo]]
|equal
| equal
|-
|-
|25
| 25
|21L 4s
| 21L 4s
| rowspan="7" |4:3
| rowspan="7" | 4:3
|-
|-
|26
| 26
|18L 8s
| 18L 8s
|-
|-
|27
| 27
|15L 12s
| 15L 12s
|-
|-
|28
| 28
|12L 16s
| 12L 16s
|-
|-
|29
| 29
|9L 20s
| 9L 20s
|-
|-
|30
| 30
|6L 24s
| 6L 24s
|-
|-
|31
| 31
|3L 28s
| 3L 28s
|-
|-
|32
| 32
|[[32edo]]
| [[32edo]]
|equal
| equal
|-
|-
|33
| 33
|30L 3s
| 30L 3s
| rowspan="15" |3:2
| rowspan="15" | 3:2
|-
|-
| 34
| 34
|28L 6s
| 28L 6s
|-
|-
|35
| 35
|26L 9s
| 26L 9s
|-
|-
|36
| 36
|24L 12s
| 24L 12s
|-
|-
|37
| 37
|22L 15s
| 22L 15s
|-
|-
| 38
| 38
|20L 18s
| 20L 18s
|-
|-
|39
| 39
|18L 21s
| 18L 21s
|-
|-
|40
| 40
|16L 24s
| 16L 24s
|-
|-
|41
| 41
|14L 27s
| 14L 27s
|-
|-
|42
| 42
|12L 30s
| 12L 30s
|-
|-
|43
| 43
|10L 33s
| 10L 33s
|-
|-
|44
| 44
|8L 36s
| 8L 36s
|-
|-
|45
| 45
|6L 39s
| 6L 39s
|-
|-
|46
| 46
|4L 42s
| 4L 42s
|-
|-
|47
| 47
|2L 45s
| 2L 45s
|-
|-
|48
| 48
|[[48edo]]
| [[48edo]]
|equal
| equal
|-
|-
|49
| 49
|47L 2s
| 47L 2s
| rowspan="28" |2:1
| rowspan="28" |2:1
|-
|-
|50
| 50
|46L 4s
| 46L 4s
|-
|-
|51
| 51
|45L 6s
| 45L 6s
|-
|-
|52
| 52
|44L 8s
| 44L 8s
|-
|-
|53
| 53
|43L 10s
| 43L 10s
|-
|-
|54
| 54
|42L 12s
| 42L 12s
|-
|-
|55
| 55
|41L 14s
| 41L 14s
|-
|-
|56
| 56
|40L 16s
| 40L 16s
|-
|-
|57
| 57
|39L 18s
| 39L 18s
|-
|-
|58
| 58
|38L 20s
| 38L 20s
|-
|-
|59
| 59
|37L 22s
| 37L 22s
|-
|-
|60
| 60
|36L 24s
| 36L 24s
|-
|-
|61
| 61
|35L 26s
| 35L 26s
|-
|-
|62
| 62
|34L 28s
| 34L 28s
|-
|-
|63
| 63
|33L 30s
| 33L 30s
|-
|-
|64
| 64
|32L 32s
| 32L 32s
|-
|-
|65
| 65
|31L 34s
| 31L 34s
|-
|-
|66
| 66
|30L 36s
| 30L 36s
|-
|-
|67
| 67
|29L 38s
| 29L 38s
|-
|-
|68
| 68
|28L 40s
| 28L 40s
|-
|-
|69
| 69
|27L 42s
| 27L 42s
|-
|-
|70
| 70
|26L 44s
| 26L 44s
|-
|-
|71
| 71
|25L 46s
| 25L 46s
|-
|-
|72
| 72
|24L 48s
| 24L 48s
|-
|-
|73
| 73
|23L 50s
| 23L 50s
|-
|-
|74
| 74
|22L 52s
| 22L 52s
|-
|-
|75
| 75
|21L 54s
| 21L 54s
|-
|-
|76
| 76
|20L 56s
| 20L 56s
|}
|}
=History=
 
== 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.
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.


=Links=
=== Carrillo ===
{{Main| Julián Carrillo }}


==Carrillo==
=== Other composers ===
See [[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]
* [http://en.wikipedia.org/wiki/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]]
* [http://en.wikipedia.org/wiki/Georg_Friedrich_Haas Haas, Georg Friedrich], "flow and friction"


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


[http://en.wikipedia.org/wiki/Pascale_Criton Criton, Pascale]
== Video ==
 
<youtube>3O3H01c2SjE</youtube>
[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://en.wikipedia.org/wiki/Georg_Friedrich_Haas Haas, Georg Friedrich], "flow and friction"
 
=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]]   


[[Category:Equal divisions of the octave]]
[[Category:Equal divisions of the octave]]

Revision as of 17:06, 5 December 2021

The 96 equal divisions of the octave (96edo) divides the octave into 96 equal parts of exactly 12.5 cents each.

Theory

As a 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>. 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.

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 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 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.

Scales

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 12n-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[clarification needed].

Circulating temperaments in 96edo
Tones Pattern L:s
5 1L 4s 20:19
6 6edo equal
7 5L 2s 14:13
8 8edo equal
9 6L 3s 11:10
10 6L 4s 10:9
11 8L 3s 9:8
12 12edo equal
13 5L 8s 8:7
14 12L 2s 7:6
15 6L 9s
16 16edo equal
17 11L 6s 6:5
18 6L 12s
19 1L 18s
20 16L 4s 5:4
21 12L 9s
22 8L 14s
23 4L 19s
24 24edo equal
25 21L 4s 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 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 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

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 article: Julián Carrillo

Other composers

Works for the Sauter's 1/16tone microtone piano by the composers Ernest Helmuth Flammer, Marc Kilchenmann, Bernfried E. G. Pröve, Martin Imholz, Franck Cristoph Yeznikian, Werner Grimmel, and Alain Bancquart, are recompilated on this CD: 'The Carrillo tone piano' .

Music

Video

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