96edo: Difference between revisions

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

{{Interwiki

| de = 96edo

| en = 96edo

| es =

| ja =

}}

The '''96 equal divisions of the octave''' ('''96edo''') divides the octave into 96 equal parts of exactly 12.5 [[cent]]s each.

== Theory ==

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~~>~~. 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]] 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]].

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]], 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 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 ===

== 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 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 center-all"

|+Circulating temperaments in 96edo

! Tones

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* [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]

* [[Wikipedia: Pascale Criton|Criton, Pascale]]

* [http://www.tonysalinas.com/ Martin Salinas, J.A.] [[Autumn|'Autumn' conic bellophone & 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"

* [[Wikipedia: Georg Friedrich Haas|Haas, Georg Friedrich]], "flow and friction"

== Music ==

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== Video ==

<youtube>3O3H01c2SjE</youtube>{{dead link}}

[[Category:Equal divisions of the octave]]

~~[[Category:Interpental]]~~

[[Category:Listen]]

[[Category:Unicorn]]

[[Category:Würschmidt]]

[[Category:Interpental]]

~~{{todo|rework}}~~

@@ Line 1: / Line 1: @@
-<span style="display: block; text-align: right;">[[de:96edo|Deutsch]]</span>
+{{Interwiki
+| de = 96edo
+| en = 96edo
+| es =
+| ja =
+}}
 The '''96 equal divisions of the octave''' ('''96edo''') divides the octave into 96 equal parts of exactly 12.5 [[cent]]s each.
 == Theory ==
-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]] 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]].
-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]], 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 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 ===
+{{Primes in edo|96}}
 == 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 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 center-all"
 |+Circulating temperaments in 96edo
 ! Tones
@@ Line 263: / Line 270: @@
 * [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]
+* [[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]]
-* [http://en.wikipedia.org/wiki/Georg_Friedrich_Haas Haas, Georg Friedrich], "flow and friction"
+* [[Wikipedia: Georg Friedrich Haas|Haas, Georg Friedrich]], "flow and friction"
 == Music ==
@@ Line 275: / Line 282: @@
 == Video ==
-<youtube>3O3H01c2SjE</youtube>
+<youtube>3O3H01c2SjE</youtube>{{dead link}}
 [[Category:Equal divisions of the octave]]
-[[Category:Interpental]]
 [[Category:Listen]]
 [[Category:Unicorn]]
 [[Category:Würschmidt]]
+[[Category:Interpental]]
-{{todo|rework}}

96edo: Difference between revisions

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