224edo: Difference between revisions
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The '''224 equal divisions of the octave''' (''' | {{Infobox ET | ||
| Prime factorization = 2<sup>5</sup> × 7 | |||
| Step size = 5.35714¢ | |||
| Fifth = 131\224 (701.79¢) | |||
| Semitones = 21:17 (112.50¢ : 91.07¢) | |||
| Consistency = 15 | |||
}} | |||
The '''224 equal divisions of the octave''' ('''224edo'''), or the '''224(-tone) equal temperament''' ('''224tet''', '''224et''') when viewed from a [[regular temperament]] perspective, is the [[EDO|equal division of the octave]] into 224 parts of about 5.36 [[cent]]s each. | |||
== Theory == | == Theory == | ||
224edo is a very strong [[13-limit]] system, tempering out [[32805/32768]] in the [[5-limit]]; [[4375/4374]], 16875/16807 and 65625/65536 in the [[7-limit]]; [[540/539]], 1375/1372, [[4000/3993]] and notably, the [[quartisma]] in the [[11-limit]]; and [[625/624]], [[729/728]], [[1575/1573]] and [[2200/2197]] in the [[13-limit]], leading to an abundance of precisely-tuned essentially tempered chords. It defines the [[optimal patent val]] for the [[octoid]] in the 7-, 11- and 13-limit, and for [[mirkwai]], the 7-limit [[planar temperament]] tempering out 16875/16807. It also provides an excellent tuning for [[indra]] and [[shibi]] temperaments. It is the twelfth [[The Riemann zeta function and tuning #Zeta EDO lists|zeta integral edo]]. | 224edo is a very strong [[13-limit]] system, tempering out [[32805/32768]] in the [[5-limit]]; [[4375/4374]], 16875/16807 and 65625/65536 in the [[7-limit]]; [[540/539]], 1375/1372, [[4000/3993]] and notably, the [[quartisma]] in the [[11-limit]]; and [[625/624]], [[729/728]], [[1575/1573]] and [[2200/2197]] in the [[13-limit]], leading to an abundance of precisely-tuned essentially tempered chords. It defines the [[optimal patent val]] for the [[octoid]] in the 7-, 11- and 13-limit, and for [[mirkwai]], the 7-limit [[planar temperament]] tempering out 16875/16807. It also provides an excellent tuning for [[indra]] and [[shibi]] temperaments. It is the twelfth [[The Riemann zeta function and tuning #Zeta EDO lists|zeta integral edo]]. | ||
224 = 32 × 7, and has divisors 2, 4, 8, 16, 32, 7, 14, 28, 56, and 112. | 224 = 32 × 7, and has divisors {{EDOs| 2, 4, 8, 16, 32, 7, 14, 28, 56, and 112 }}. | ||
=== Prime harmonics === | === Prime harmonics === | ||
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== Music == | == Music == | ||
[http://www.archive.org/details/Dreyfus Dreyfus] [http://www.archive.org/download/Dreyfus/Genewardsmith-Dreyfus.mp3 play] by [[Gene Ward Smith]] | * [http://www.archive.org/details/Dreyfus Dreyfus] [http://www.archive.org/download/Dreyfus/Genewardsmith-Dreyfus.mp3 play] by [[Gene Ward Smith]] | ||
[[Category:Theory]] | |||
[[Category:Equal divisions of the octave]] | [[Category:Equal divisions of the octave]] | ||
[[Category:Indra]] | [[Category:Indra]] | ||
[[Category:Mirkwai]] | [[Category:Mirkwai]] | ||
[[Category:Octoid]] | [[Category:Octoid]] | ||
[[Category:Quartismic]] | [[Category:Quartismic]] | ||
[[Category:Shibi]] | [[Category:Shibi]] | ||
[[Category: | [[Category:Listen]] | ||
[[Category:Zeta]] | |||