224edo: Difference between revisions

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~~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 ==

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 chord]]s, including [[swetismic chords]], [[squbemic chords]], and [[petrmic triad]] in the 13-odd-limit, in addition to [[nicolic chords]] in the 15-odd-limit. 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 {{EDOs| 2, 4, 8, 16, 32, 7, 14, 28, 56, and 112 }}.~~

=== Prime harmonics ===

=== Subsets and supersets ===

Since 224 = 32 × 7, 224edo has subset edos {{EDOs| 2, 4, 8, 16, 32, 7, 14, 28, 56, and 112 }}.

== Regular temperament properties ==

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! rowspan="2" | [[Comma list|Comma List]]

! rowspan="2" | [[Mapping]]

! rowspan="2" | Optimal<br>8ve ~~stretch~~ (¢)

! rowspan="2" | Optimal<br>8ve Stretch (¢)

! colspan="2" | Tuning ~~error~~

! colspan="2" | Tuning Error

|-

! [[TE error|Absolute]] (¢)

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== Music ==

* [https://www.archive.org/details/Dreyfus Dreyfus] [https://www.archive.org/download/Dreyfus/Genewardsmith-Dreyfus.mp3 play] by [~~[Gene Ward Smith]~~]

; [[Gene Ward Smith]]

* [https://www.archive.org/details/Dreyfus ''Dreyfus''] [https://www.archive.org/download/Dreyfus/Genewardsmith-Dreyfus.mp3 play] – octoid[72]

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

[[Category:Indra]]

[[Category:Mirkwai]]

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[[Category:Shibi]]

[[Category:Listen]]

~~[[Category:Zeta]]~~

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-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.
+{{EDO intro|224}}
 == Theory ==
 edo 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 chord]]s, including [[swetismic chords]], [[squbemic chords]], and [[petrmic triad]] in the 13-odd-limit, in addition to [[nicolic chords]] in the 15-odd-limit. 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]].
-= 32 × 7, and has divisors {{EDOs| 2, 4, 8, 16, 32, 7, 14, 28, 56, and 112 }}.
 === Prime harmonics ===
 {{Harmonics in equal|224|columns=11}}
+=== Subsets and supersets ===
+Since 224 = 32 × 7, 224edo has subset edos {{EDOs| 2, 4, 8, 16, 32, 7, 14, 28, 56, and 112 }}.
 == Regular temperament properties ==
@@ Line 15: / Line 16: @@
 ! rowspan="2" | [[Comma list|Comma List]]
 ! rowspan="2" | [[Mapping]]
-! rowspan="2" | Optimal<br>8ve stretch (¢)
+! rowspan="2" | Optimal<br>8ve Stretch (¢)
-! colspan="2" | Tuning error
+! colspan="2" | Tuning Error
 |-
 ! [[TE error|Absolute]] (¢)
@@ Line 212: / Line 213: @@
 == Music ==
-* [https://www.archive.org/details/Dreyfus Dreyfus] [https://www.archive.org/download/Dreyfus/Genewardsmith-Dreyfus.mp3 play] by [[Gene Ward Smith]]
+; [[Gene Ward Smith]]
+* [https://www.archive.org/details/Dreyfus ''Dreyfus''] [https://www.archive.org/download/Dreyfus/Genewardsmith-Dreyfus.mp3 play] – octoid[72]
-[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
 [[Category:Indra]]
 [[Category:Mirkwai]]
@@ Line 221: / Line 222: @@
 [[Category:Shibi]]
 [[Category:Listen]]
-[[Category:Zeta]]