192edo: Difference between revisions

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'''192edo''' is the [[EDO|equal division of the octave]] into 192 parts of 6.25 [[cent]]s each exactly. Its adjacent step is known as '''Tetramu''' (fourth MIDI-resolution unit, 4mu, 2<sup>4</sup> = 16 equal divisions of the [[12edo]] semitone). It 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.

==~~See also~~==

== Theory ==

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

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.

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

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

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

[[~~Category:Equal divisions~~ of the ~~octave~~|~~###~~]] <!-~~- 3-digit~~ number ~~-->~~

=== 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 ===

=== Subsets and supersets ===

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

== See also ==

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

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

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

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

* [[Intercom scales]] (the scales listed are tuned to 192edo)

== External links ==

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

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-'''192edo''' is the [[EDO|equal division of the octave]] into 192 parts of 6.25 [[cent]]s each exactly. Its adjacent step is known as '''Tetramu''' (fourth MIDI-resolution unit, 4mu, 2<sup>4</sup> = 16 equal divisions of the [[12edo]] semitone). It 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.
+{{Infobox ET}}
+{{ED intro}}
-==See also==
+== Theory ==
-*[[Equal-step tuning|Equal multiplications]] of MIDI-resolution units
+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.
-**[[24edo]] (1mu tuning)
-**[[48edo]] (2mu tuning)
-**[[96edo]] (3mu 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)
-[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
+=== 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}}
+=== Subsets and supersets ===
+Since 192 factors into {{factorization|192}}, 192edo has subset edos {{EDOs| 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, and 96 }}.
+== See also ==
+* [[Equal-step tuning|Equal multiplications]] of MIDI-resolution units
+** [[24edo]] (1mu tuning)
+** [[48edo]] (2mu tuning)
+** [[96edo]] (3mu 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)
+* [[Intercom scales]] (the scales listed are tuned to 192edo)
+== External links ==
+* [http://tonalsoft.com/enc/number/4mu.aspx 4mu / tetramu] on [[Tonalsoft Encyclopedia]]