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{{Infobox ET}}
{{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.
{{ED intro}}


== 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 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]].  
224edo is a very strong [[13-limit]] system. It is the twelfth [[the Riemann zeta function and tuning #Zeta EDO lists|zeta integral edo]] and is the second-smallest edo after [[87edo|87]] to approximate all of the first 16 harmonics of the harmonic series with [[minimal consistent EDOs|no greater than 25%]] relative error.  


224 = 32 × 7, and has divisors {{EDOs| 2, 4, 8, 16, 32, 7, 14, 28, 56, and 112 }}.
As an equal temperament, 224et [[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 chords]] 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. [[217edo]], only a bit smaller, has a worse 13-limit, but it achieves a much higher [[consistency limit]], almost [[31-odd-limit|31-odd]].
 
224edo tempers the [[syntonic comma]] to 1/56th of the octave (4 steps) and as a corollary supports the [[barium]] temperament. As a consequence of this, the 224bb val (flattening the fifth by one step) is a tuning for [[meantone]] and is very close (0.15 cents) to the [[quarter-comma meantone]] fifth. The generator however reduces to [[112edo]], being 65\112; that said, the use of both types of fifth enables creation of a closed circle of 24 notes per octave, generated as 16 patent fifths plus 8 bb fifths (as in [[quadrant]] temperament)<ref>[http://www.youtube.com/@Xen-p6p @Xen-p6p] (2026), YouTube post on [https://www.youtube.com/watch?v=Hmjx4wvLG7Q Uccellini - «Aria Sopra La Bergamasca» (1642), arranged for Organ, tuned into Adaptive Just Intonation] rendered by [[Claudi Meneghin]] (2024).</ref>, although a different distribution than the quarter-octave distribution specified by quadrant might be desired for a well-tempered 24 note tuning system.


=== Prime harmonics ===
=== Prime harmonics ===
{{Harmonics in equal|224|columns=11}}
{{Harmonics in equal|224|columns=11}}
{{Harmonics in equal|224|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 224edo (continued)}}
=== Subsets and supersets ===
Since 224 factors into primes as {{nowrap| 2<sup>5</sup> × 7 }}, 224edo has subset edos {{EDOs| 2, 4, 8, 16, 32, 7, 14, 28, 56, and 112 }}.
== Intervals ==
{{Todo|create page|comment=Table of 224edo intervals}}
{{Interval table}}
== Notation ==
=== Sagittal ===
224edo can be written in Sagittal using ''almost'' the entire Athenian extension, by virtue of its apotome being equal to 21 edosteps, which is the maximum equal division of the apotome (eda) supported by Athenian. It is identical to [[217edo]]'s Sagittal notation, but it uses the 55C for the +6/-6 alteration instead of 11/7C.<ref>[https://sagittal.org/sagittal.pdf Sagittal – A Microtonal Notation System] by [[George Secor|George D. Secor]] and [[Dave Keenan|David C. Keenan]]</ref>
{| class="wikitable"
|+ Sagittal notation
! colspan="2" |Steps
! 0
! 1
! 2
! 3
! 4
! 5
! 6
! 7
! 8
! 9
! 10
! 11
! 12
! 13
! 14
! 15
! 16
! 17
! 18
! 19
! 20
! 21
|-
! rowspan="2" |Symbol
! Evo
| rowspan="2" | <big>{{sagittal||//|}}</big>
| rowspan="2" | <big>{{sagittal||(}}</big>
| rowspan="2" | <big>{{sagittal|)|(}}</big>
| rowspan="2" | <big>{{sagittal|~|(}}</big>
| rowspan="2" | <big>{{sagittal|/|}}</big>
| rowspan="2" | <big>{{sagittal||)}}</big>
| rowspan="2" | <big>{{sagittal||\}}</big>
| rowspan="2" | <big>{{sagittal|(|(}}</big>
| rowspan="2" | <big>{{sagittal|//|}}</big>
| rowspan="2" | <big>{{sagittal|/|)}}</big>
| rowspan="2" | <big>{{sagittal|/|\}}</big>
| <small>{{sagittal|#}}{{sagittal|\!/}}</small>
| <small>{{sagittal|#}}{{sagittal|\!)}}</small>
| <small>{{sagittal|#}}{{sagittal|\\!}}</small>
| <small>{{sagittal|#}}{{sagittal|(!(}}</small>
| <small>{{sagittal|#}}{{sagittal|!/}}</small>
| <small>{{sagittal|#}}{{sagittal|!)}}</small>
| <small>{{sagittal|#}}{{sagittal|\!}}</small>
| <small>{{sagittal|#}}{{sagittal|~!(}}</small>
| <small>{{sagittal|#}}{{sagittal|)!(}}</small>
| <small>{{sagittal|#}}{{sagittal|!(}}</small>
| <small>{{sagittal|#}}</small>
|-
! Revo
| <big>{{sagittal|(|)}}</big>
| <big>{{sagittal|(|\}}</big>
| <big>{{sagittal|)||(}}</big>
| <big>{{sagittal|~||(}}</big>
| <big>{{sagittal|/||}}</big>
| <big>{{sagittal|||)}}</big>
| <big>{{sagittal|||\}}</big>
| <big>{{sagittal|(||(}}</big>
| <big>{{sagittal|//||}}</big>
| <big>{{sagittal|/||)}}</big>
| <big>{{sagittal|/||\}}</big>
|}
Because it uses the entire Athenian system (except for {{sagittal|(|}} {{sagittal|(!}} {{sagittal|)||~}} {{sagittal|)!!~}} since it tempers [[1240029/1239040]]), it allows no accidental enharmonic respellings
=== Ups-and-downs notation ===
The 4-up (quup) alteration maps to the pythagorean/syntonic comma.
{| class="wikitable" style="text-align:center;"
|+ Ups-and-downs notation
! rowspan="6" | 224edosteps
! 0
! 1
! 2
! 3
! 4
! 5
! 6
! 7
! 8
! 9
! 10
|-
| rowspan="2" | h
| ^
| ^^
| ^^^
| v>
| >
| ^>
| ^^>
| ^^^>
| v>>
| >>
|-
| <<<<#
| ^<<<<#
| vvv<<<#
| vv<<<#
| v<<<#
| <<<#
| ^<<<#
| vvv<<#
| vv<<#
| v<<#
|-
! 11
! 12
! 13
! 14
! 15
! 16
! 17
! 18
! 19
! 20
! 21
|-
| ^>>
| ^^>>
| ^^^>>
| v>>>
| >>>
| ^>>>
| ^^>>>
| ^^^>>>
| v>>>>
| >>>>
| rowspan="2" |#
|-
| <<#
| ^<<#
| vvv<#
| vv<#
| v<#
| <#
| ^<#
| vvv#
| vv#
| v#
|}
== Approximation to JI ==
=== Interval mappings ===
{{Q-odd-limit intervals}}


== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! rowspan="2" | Optimal<br>8ve stretch (¢)
Line 22: Line 184:
|-
|-
| 2.3
| 2.3
| {{monzo| -355 224 }}
| {{Monzo| -355 224 }}
| [{{val| 224 355 }}]
| {{Mapping| 224 355 }}
| +0.053
| +0.053
| 0.0534
| 0.0534
Line 30: Line 192:
| 2.3.5
| 2.3.5
| 32805/32768, {{monzo| -5 -32 24 }}
| 32805/32768, {{monzo| -5 -32 24 }}
| [{{val| 224 355 520 }}]
| {{Mapping| 224 355 520 }}
| +0.122
| +0.122
| 0.1059
| 0.1059
Line 37: Line 199:
| 2.3.5.7
| 2.3.5.7
| 4375/4374, 16875/16807, 32805/32768
| 4375/4374, 16875/16807, 32805/32768
| [{{val| 224 355 520 629 }}]
| {{Mapping| 224 355 520 629 }}
| +0.018
| +0.018
| 0.2009
| 0.2009
Line 44: Line 206:
| 2.3.5.7.11
| 2.3.5.7.11
| 540/539, 1375/1372, 4000/3993, 32805/32768
| 540/539, 1375/1372, 4000/3993, 32805/32768
| [{{val| 224 355 520 629 775 }}]
| {{Mapping| 224 355 520 629 775 }}
| -0.012
| −0.012
| 0.1899
| 0.1899
| 3.54
| 3.54
Line 51: Line 213:
| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 540/539, 625/624, 729/728, 1375/1372, 2200/2197
| 540/539, 625/624, 729/728, 1375/1372, 2200/2197
| [{{val| 224 355 520 629 775 829 }}]
| {{Mapping| 224 355 520 629 775 829 }}
| -0.035
| −0.035
| 0.1805
| 0.1805
| 3.37
| 3.37
|-
| 2.3.5.7.11.13.17
| 375/374, 540/539, 625/624, 715/714, 729/728, 2200/2197
| {{Mapping| 224 355 520 629 775 829 916 }}
| −0.106
| 0.2420
| 4.52
|}
|}
* 224et has a lower relative error than any previous equal temperaments in the 13-limit, being the first to beat [[72edo|72]]. The next equal temperament that does better in terms of either absolute or relative error is [[270edo|270]].
* It is also notable in the 11- and 17-limit, with lower absolute errors than any previous equal temperaments. In the 11-limit it is the first to beat [[152edo|152]] and is superseded by [[239edo|239]]. In the 17-limit it is the first to beat [[217edo|217]] and is superseded by 270.


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods<br>per 8ve
! Periods<br>per 8ve
! Generator<br>(Reduced)
! Generator*
! Cents<br>(Reduced)
! Cents*
! Associated<br>Ratio
! Associated<br>ratio*
! Temperaments
! Temperament
|-
|-
| 1
| 1
Line 76: Line 248:
| 316.07
| 316.07
| 6/5
| 6/5
| [[Counterkleismic]]
| [[Counterkleismic]] / counterlytic
|-
|-
| 1
| 1
Line 100: Line 272:
| 498.21
| 498.21
| 4/3
| 4/3
| [[Helmholtz]] / [[pontiac]] / [[ponta]]
| [[Pontiac]] / [[ponta]]
|-
|-
| 1
| 1
Line 129: Line 301:
| 33\224
| 33\224
| 176.79
| 176.79
| 448/405, 195/176
| 195/176
| [[Quatracot]]
| [[Quatracot]]
|-
|-
Line 193: Line 365:
|-
|-
| 32
| 32
| 174\224<br>(6\224)
| 50\224<br>(1\224)
| 932.14<br>(32.14)
| 267.86<br>(5.36)
| 245/143<br>(???)
| 245/143<br>(???)
| [[Germanium]]
| [[Germanium]]
|-
| 32
| 93\224<br>(2\224)
| 498.21<br>(10.71)
| 4/3<br>(???)
| [[Bezique]]
|-
|-
| 56
| 56
Line 204: Line 382:
| [[Barium]]
| [[Barium]]
|}
|}
<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct


== Music ==
== Music ==
* [https://www.archive.org/details/Dreyfus Dreyfus] [https://www.archive.org/download/Dreyfus/Genewardsmith-Dreyfus.mp3 play] by [[Gene Ward Smith]]
; [[Mercury Amalgam]]
* [https://www.youtube.com/watch?v=iFi1zKsRBfY ''Kindness Is A Weakness''] (2023) – Octant[24], Hemigamera[26], Oquatonic[56], Bezique[64] in 224edo tuning
 
; [[Gene Ward Smith]]
* ''Dreyfus'' (archived 2010) – [https://soundcloud.com/genewardsmith/genewardsmith-dreyfus SoundCloud] | [https://www.archive.org/details/Dreyfus details] | [https://www.archive.org/download/Dreyfus/Genewardsmith-Dreyfus.mp3 play] – Octoid[72] in 224edo tuning
 
== References ==


[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
[[Category:Listen]]
[[Category:Canopic]]
[[Category:Indra]]
[[Category:Indra]]
[[Category:Mirkwai]]
[[Category:Shibi]]
[[Category:Octoid]]
[[Category:Octoid]]
[[Category:Quartismic]]
[[Category:Shibi]]
[[Category:Listen]]
[[Category:Zeta]]
Retrieved from "https://en.xen.wiki/w/224edo"