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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.
{{EDO intro|224}}


== 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, 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 ===
=== Prime harmonics ===
{{Harmonics in equal|224|columns=11}}
{{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 ==
== Regular temperament properties ==
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! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! 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]] (¢)
! [[TE error|Absolute]] (¢)
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== Music ==
== 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:Indra]]
[[Category:Mirkwai]]
[[Category:Mirkwai]]
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[[Category:Shibi]]
[[Category:Shibi]]
[[Category:Listen]]
[[Category:Listen]]
[[Category:Zeta]]

Revision as of 13:10, 10 February 2023

← 223edo 224edo 225edo →
Prime factorization 25 × 7
Step size 5.35714 ¢ 
Fifth 131\224 (701.786 ¢)
Semitones (A1:m2) 21:17 (112.5 ¢ : 91.07 ¢)
Consistency limit 15
Distinct consistency limit 15

Template:EDO intro

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, 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 zeta integral edo.

Prime harmonics

Approximation of prime harmonics in 224edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 -0.17 -0.60 +0.82 +0.47 +0.54 +2.19 +2.49 -1.49 -1.01 +1.39
Relative (%) +0.0 -3.2 -11.2 +15.2 +8.7 +10.2 +40.8 +46.4 -27.8 -18.8 +26.0
Steps
(reduced) 		224
(0) 		355
(131) 		520
(72) 		629
(181) 		775
(103) 		829
(157) 		916
(20) 		952
(56) 		1013
(117) 		1088
(192) 		1110
(214)

Subsets and supersets

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

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢) 		Tuning Error
Absolute (¢) Relative (%)
2.3 [-355 224⟩ [⟨224 355]] +0.053 0.0534 1.00
2.3.5 32805/32768, [-5 -32 24⟩ [⟨224 355 520]] +0.122 0.1059 1.98
2.3.5.7 4375/4374, 16875/16807, 32805/32768 [⟨224 355 520 629]] +0.018 0.2009 3.75
2.3.5.7.11 540/539, 1375/1372, 4000/3993, 32805/32768 [⟨224 355 520 629 775]] -0.012 0.1899 3.54
2.3.5.7.11.13 540/539, 625/624, 729/728, 1375/1372, 2200/2197 [⟨224 355 520 629 775 829]] -0.035 0.1805 3.37

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator
(Reduced) 		Cents
(Reduced) 		Associated
Ratio 		Temperaments
1 43\224 230.36 8/7 Gamera
1 59\224 316.07 6/5 Counterkleismic
1 65\224 348.21 11/9 Eris
1 71\224 380.36 56/45 Quanharuk
1 87\224 466.07 55/42 Hemiseptisix
1 93\224 498.21 4/3 Helmholtz / pontiac / ponta
1 103\224 551.79 11/8 Emkay
1 111\224 594.64 55/39 Gaster
2 93\224
(19\224) 		498.21
(101.79) 		4/3
(35/33) 		Bipont
2 31\224 166.07 11/10 Pogo
2 33\224 176.79 448/405, 195/176 Quatracot
2 39\224 208.93 44/39 Abigail
2 43\224 230.36 8/7 Hemigamera
4 71\224
(15\224) 		380.36
(80.36) 		81/65
(22/21) 		Quasithird
4 93\224
(19\224) 		498.21
(101.79) 		4/3
(35/33) 		Quadrant
7 97\224
(1\224) 		519.64
(5.36) 		27/20
(325/324) 		Brahmagupta
7 93\224
(3\224) 		498.21
(16.07) 		4/3
(99/98) 		Septant
8 93\224
(9\224) 		498.21
(48.21) 		4/3
(36/35) 		Octant
8 109\224
(3\224) 		583.93
(16.07) 		7/5
(100/99) 		Octoid
14 93\224
(3\224) 		498.21
(16.07) 		4/3
(105/104) 		Silicon
28 93\224
(3\224) 		498.21
(16.07) 		4/3
(126/125) 		Oquatonic
32 50\224
(1\224) 		267.86
(5.36) 		245/143
(???) 		Germanium
32 93\224
(2\224) 		498.21
(10.71) 		4/3
(???) 		Bezique
56 93\224
(3\224) 		498.21
(16.07) 		4/3
(126/125) 		Barium

Music

Gene Ward Smith
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