192edo: Difference between revisions

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==Theory==

192edo is closely related to [[96edo]], but the patent ~~vals~~ differ on the mapping for 7. Using the patent val, it tempers out the [[pythagorean comma]], 531441/524288 in the 3-limit; the [[~~Würschmidt family|Würschmidt~~ comma]], 393216/390625 and the unicorn comma, 1594323/1562500 in the 5-limit; 2401/2400, 3136/3125, and 1594323/1568000 in the 7-limit; 243/242, 441/440, 5632/5625, and 131769/131072 in the 11-limit; 676/675, 1287/1280, 1573/1568, and 2704/2695 in the 13-limit. Using the alternative 192f val, 351/350, 1001/1000, 1575/1573, 3584/3575, and 5445/5408 are tempered out in the 13-limit.

== Theory ==

=== ~~Significance in digital audio~~ ===

192edo is closely related to [[96edo]], but the [[patent val]]s differ on the mapping for [[harmonic]] [[7/1|7]]. Using the patent val, it [[tempering out|tempers out]] the [[pythagorean comma]], 531441/524288 in the 3-limit; the [[würschmidt comma]], 393216/390625 and the [[unicorn comma]], 1594323/1562500 in the 5-limit; [[2401/2400]], [[3136/3125]], and 1594323/1568000 in the 7-limit; [[243/242]], [[441/440]], [[5632/5625]], and [[131769/131072]] in the 11-limit; [[676/675]], [[1287/1280]], [[1573/1568]], and 2704/2695 in the 13-limit. Using the alternative 192f val, [[351/350]], [[1001/1000]], [[1575/1573]], 3584/3575, and 5445/5408 are tempered out in the 13-limit.

~~Its adjacent~~ step is known as '''~~Tetramu~~''' (fourth MIDI-resolution unit, 4mu, 2<sup>4</sup> = 16 equal divisions of the [[12edo]] semitone).

=== As a tuning standard ===

A step of 192edo is known as a '''tetramu''' (fourth MIDI-resolution unit, 4mu, 2<sup>4</sup> = 16 equal divisions of the [[12edo]] semitone). The internal data structure of the 4mu 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 −) showing the direction of the pitch-bend up or down, and two other bits which are not used.

=== Odd harmonics ===

==See also==

*[[Equal-step tuning|Equal multiplications]] of MIDI-resolution units

=== Subsets and supersets ===

**[[24edo]] (1mu tuning)

Since 192 factors into {{factorization|192}}, 192edo has subset edos {{EDOs| 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, and 96 }}.

**[[48edo]] (2mu tuning)

**[[96edo]] (3mu tuning)

== See also ==

**[[384edo]] (5mu tuning)

* [[Equal-step tuning|Equal multiplications]] of MIDI-resolution units

**[[768edo]] (6mu tuning)

** [[24edo]] (1mu tuning)

**[[1536edo]] (7mu tuning)

** [[48edo]] (2mu tuning)

**[[3072edo]] (8mu tuning)

** [[96edo]] (3mu tuning)

**[[6144edo]] (9mu tuning)

** [[384edo]] (5mu tuning)

**[[12288edo]] (10mu tuning)

** [[768edo]] (6mu tuning)

**[[24576edo]] (11mu tuning)

** [[1536edo]] (7mu tuning)

**[[49152edo]] (12mu tuning)

** [[3072edo]] (8mu tuning)

**[[98304edo]] (13mu tuning)

** [[6144edo]] (9mu tuning)

**[[196608edo]] (14mu tuning)

** [[12288edo]] (10mu tuning)

** [[24576edo]] (11mu tuning)

** [[49152edo]] (12mu tuning)

** [[98304edo]] (13mu tuning)

** [[196608edo]] (14mu tuning)

[~~[Category~~:~~Equal divisions of the octave~~|~~###]~~] ~~~~

== External links ==

* [http://tonalsoft.com/enc/number/4mu.aspx Tonalsoft Encyclopedia | ''4mu / tetramu'']

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
 {{EDO intro|192}}
-==Theory==
-edo is closely related to [[96edo]], but the patent vals differ on the mapping for 7. Using the patent val, it tempers out the [[pythagorean comma]], 531441/524288 in the 3-limit; the [[Würschmidt family|Würschmidt comma]], 393216/390625 and the unicorn comma, 1594323/1562500 in the 5-limit; 2401/2400, 3136/3125, and 1594323/1568000 in the 7-limit; 243/242, 441/440, 5632/5625, and 131769/131072 in the 11-limit; 676/675, 1287/1280, 1573/1568, and 2704/2695 in the 13-limit. Using the alternative 192f val, 351/350, 1001/1000, 1575/1573, 3584/3575, and 5445/5408 are tempered out in the 13-limit.
+== Theory ==
-=== Significance in digital audio ===
+edo is closely related to [[96edo]], but the [[patent val]]s differ on the mapping for [[harmonic]] [[7/1|7]]. Using the patent val, it [[tempering out|tempers out]] the [[pythagorean comma]], 531441/524288 in the 3-limit; the [[würschmidt comma]], 393216/390625 and the [[unicorn comma]], 1594323/1562500 in the 5-limit; [[2401/2400]], [[3136/3125]], and 1594323/1568000 in the 7-limit; [[243/242]], [[441/440]], [[5632/5625]], and [[131769/131072]] in the 11-limit; [[676/675]], [[1287/1280]], [[1573/1568]], and 2704/2695 in the 13-limit. Using the alternative 192f val, [[351/350]], [[1001/1000]], [[1575/1573]], 3584/3575, and 5445/5408 are tempered out in the 13-limit.
-Its adjacent step is known as '''Tetramu''' (fourth MIDI-resolution unit, 4mu, 2<sup>4</sup> = 16 equal divisions of the [[12edo]] semitone).
+=== As a tuning standard ===
+A step of 192edo is known as a '''tetramu''' (fourth MIDI-resolution unit, 4mu, 2<sup>4</sup> = 16 equal divisions of the [[12edo]] semitone). The internal data structure of the 4mu 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 two other bits which are not used.
 === Odd harmonics ===
-{{harmonics in equal|192}}
+{{Harmonics in equal|192}}
-==See also==
-*[[Equal-step tuning|Equal multiplications]] of MIDI-resolution units
+=== Subsets and supersets ===
-**[[24edo]] (1mu tuning)
+Since 192 factors into {{factorization|192}}, 192edo has subset edos {{EDOs| 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, and 96 }}.
-**[[48edo]] (2mu tuning)
-**[[96edo]] (3mu tuning)
+== See also ==
-**[[384edo]] (5mu tuning)
+* [[Equal-step tuning|Equal multiplications]] of MIDI-resolution units
-**[[768edo]] (6mu tuning)
+** [[24edo]] (1mu tuning)
-**[[1536edo]] (7mu tuning)
+** [[48edo]] (2mu tuning)
-**[[3072edo]] (8mu tuning)
+** [[96edo]] (3mu tuning)
-**[[6144edo]] (9mu tuning)
+** [[384edo]] (5mu tuning)
-**[[12288edo]] (10mu tuning)
+** [[768edo]] (6mu tuning)
-**[[24576edo]] (11mu tuning)
+** [[1536edo]] (7mu tuning)
-**[[49152edo]] (12mu tuning)
+** [[3072edo]] (8mu tuning)
-**[[98304edo]] (13mu tuning)
+** [[6144edo]] (9mu tuning)
-**[[196608edo]] (14mu tuning)
+** [[12288edo]] (10mu tuning)
+** [[24576edo]] (11mu tuning)
+** [[49152edo]] (12mu tuning)
+** [[98304edo]] (13mu tuning)
+** [[196608edo]] (14mu tuning)
-[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
+== External links ==
+* [http://tonalsoft.com/enc/number/4mu.aspx Tonalsoft Encyclopedia | ''4mu / tetramu'']