224edo: Difference between revisions

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{{Infobox ET
{{Infobox ET}}
| Prime factorization = 2<sup>5</sup> × 7
{{ED intro}}
| 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. 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 ===
{{Primes in edo|224}}
{{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]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
Line 28: 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 36: 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 43: 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 50: 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 57: 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 octave
|-
! Generator<br>(reduced)
! Periods<br>per 8ve
! Cents<br>(reduced)
! Generator*
! Associated<br>ratio
! Cents*
! Temperaments
! Associated<br>ratio*
! Temperament
|-
|-
| 1
| 1
Line 82: Line 248:
| 316.07
| 316.07
| 6/5
| 6/5
| [[Counterkleismic]]
| [[Counterkleismic]] / counterlytic
|-
|-
| 1
| 1
Line 106: Line 272:
| 498.21
| 498.21
| 4/3
| 4/3
| [[Helmholtz]] / [[pontiac]] / [[ponta]]
| [[Pontiac]] / [[ponta]]
|-
|-
| 1
| 1
Line 113: Line 279:
| 11/8
| 11/8
| [[Emkay]]
| [[Emkay]]
|-
| 1
| 111\224
| 594.64
| 55/39
| [[Gaster temperament|Gaster]]
|-
|-
| 2
| 2
Line 129: Line 301:
| 33\224
| 33\224
| 176.79
| 176.79
| 448/405, 195/176
| 195/176
| [[Quatracot]]
| [[Quatracot]]
|-
|-
Line 145: Line 317:
|-
|-
| 4
| 4
| 15\224
| 71\224<br>(15\224)
| 80.36
| 380.36<br>(80.36)
| 22/21
| 81/65<br>(22/21)
| [[Quasithird]]
| [[Quasithird]]
|-
|-
| 4
| 4
| 37\224<br>(19\224)
| 93\224<br>(19\224)
| 198.21<br>(101.79)
| 498.21<br>(101.79)
| 28/25<br>(35/33)
| 4/3<br>(35/33)
| [[Quadrant]]
| [[Quadrant]]
|-
|-
Line 159: Line 331:
| 97\224<br>(1\224)
| 97\224<br>(1\224)
| 519.64<br>(5.36)
| 519.64<br>(5.36)
| 27/20<br>&nbsp;
| 27/20<br>(325/324)
| [[Brahmagupta]]
| [[Brahmagupta]]
|-
|-
Line 167: Line 339:
| 4/3<br>(99/98)
| 4/3<br>(99/98)
| [[Septant]]
| [[Septant]]
|-
| 8
| 3\224
| 16.07
| 100/99
| [[Octoid]]
|-
|-
| 8
| 8
Line 179: Line 345:
| 4/3<br>(36/35)
| 4/3<br>(36/35)
| [[Octant]]
| [[Octant]]
|-
| 8
| 109\224<br>(3\224)
| 583.93<br>(16.07)
| 7/5<br>(100/99)
| [[Octoid]]
|-
| 14
| 93\224<br>(3\224)
| 498.21<br>(16.07)
| 4/3<br>(105/104)
| [[Silicon]]
|-
|-
| 28
| 28
| 3\224
| 93\224<br>(3\224)
| 16.07
| 498.21<br>(16.07)
| 126/125
| 4/3<br>(126/125)
| [[Oquatonic]]
| [[Oquatonic]]
|-
| 32
| 50\224<br>(1\224)
| 267.86<br>(5.36)
| 245/143<br>(???)
| [[Germanium]]
|-
| 32
| 93\224<br>(2\224)
| 498.21<br>(10.71)
| 4/3<br>(???)
| [[Bezique]]
|-
| 56
| 93\224<br>(3\224)
| 498.21<br>(16.07)
| 4/3<br>(126/125)
| [[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 ==
* [http://www.archive.org/details/Dreyfus Dreyfus] [http://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


[[Category:Theory]]
; [[Gene Ward Smith]]
[[Category:Equal divisions of the octave]]
* ''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:Listen]]
[[Category:Canopic]]
[[Category:Indra]]
[[Category:Indra]]
[[Category:Mirkwai]]
[[Category:Shibi]]
[[Category:Octoid]]
[[Category:Octoid]]
[[Category:Quartismic]]
[[Category:Shibi]]
[[Category:Listen]]
[[Category:Zeta]]

Latest revision as of 10:28, 6 June 2026

← 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

224 equal divisions of the octave (abbreviated 224edo or 224ed2), also called 224-tone equal temperament (224tet) or 224 equal temperament (224et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 224 equal parts of about 5.36 ¢ each. Each step represents a frequency ratio of 21/224, or the 224th root of 2.

Theory

224edo is a very strong 13-limit system. It is the twelfth zeta integral edo and is the second-smallest edo after 87 to approximate all of the first 16 harmonics of the harmonic series with no greater than 25% relative error.

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 chords, 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.

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)[1], 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

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)
Approximation of prime harmonics in 224edo (continued)
Harmonic 37 41 43 47 53 59 61 67 71 73 79
Error Absolute (¢) +0.44 -0.49 -2.59 -1.22 -0.29 +1.54 -2.60 +1.05 +2.45 +2.57 -0.25
Relative (%) +8.2 -9.2 -48.3 -22.8 -5.4 +28.8 -48.5 +19.6 +45.7 +47.9 -4.7
Steps
(reduced) 		1167
(47) 		1200
(80) 		1215
(95) 		1244
(124) 		1283
(163) 		1318
(198) 		1328
(208) 		1359
(15) 		1378
(34) 		1387
(43) 		1412
(68)

Subsets and supersets

Since 224 factors into primes as 25 × 7, 224edo has subset edos 2, 4, 8, 16, 32, 7, 14, 28, 56, and 112.

Intervals

Steps Cents Approximate ratios Ups and downs notation
0 0 1/1 D
1 5.36 ^D, ^5E♭♭
2 10.71 ^^D, ^6E♭♭
3 16.07 ^3D, ^7E♭♭
4 21.43 ^4D, ^8E♭♭
5 26.79 63/62, 64/63, 65/64, 66/65 ^5D, ^9E♭♭
6 32.14 55/54, 56/55 ^6D, ^10E♭♭
7 37.5 46/45, 47/46 ^7D, v10E♭
8 42.86 41/40 ^8D, v9E♭
9 48.21 36/35, 37/36 ^9D, v8E♭
10 53.57 33/32, 65/63 ^10D, v7E♭
11 58.93 30/29 v10D♯, v6E♭
12 64.29 v9D♯, v5E♭
13 69.64 51/49 v8D♯, v4E♭
14 75 47/45 v7D♯, v3E♭
15 80.36 22/21 v6D♯, vvE♭
16 85.71 41/39 v5D♯, vE♭
17 91.07 39/37, 58/55 v4D♯, E♭
18 96.43 37/35, 55/52 v3D♯, ^E♭
19 101.79 35/33 vvD♯, ^^E♭
20 107.14 50/47 vD♯, ^3E♭
21 112.5 16/15 D♯, ^4E♭
22 117.86 ^D♯, ^5E♭
23 123.21 29/27, 44/41 ^^D♯, ^6E♭
24 128.57 14/13 ^3D♯, ^7E♭
25 133.93 27/25 ^4D♯, ^8E♭
26 139.29 13/12 ^5D♯, ^9E♭
27 144.64 25/23, 62/57 ^6D♯, ^10E♭
28 150 12/11 ^7D♯, v10E
29 155.36 35/32 ^8D♯, v9E
30 160.71 34/31, 45/41 ^9D♯, v8E
31 166.07 ^10D♯, v7E
32 171.43 v10D𝄪, v6E
33 176.79 31/28, 41/37, 72/65 v9D𝄪, v5E
34 182.14 10/9 v8D𝄪, v4E
35 187.5 39/35 v7D𝄪, v3E
36 192.86 19/17 v6D𝄪, vvE
37 198.21 37/33, 65/58 v5D𝄪, vE
38 203.57 9/8 E
39 208.93 44/39 ^E, ^5F♭
40 214.29 ^^E, ^6F♭
41 219.64 42/37 ^3E, ^7F♭
42 225 41/36, 74/65 ^4E, ^8F♭
43 230.36 8/7 ^5E, ^9F♭
44 235.71 47/41, 55/48, 63/55 ^6E, ^10F♭
45 241.07 23/20, 54/47 ^7E, v10F
46 246.43 ^8E, v9F
47 251.79 37/32 ^9E, v8F
48 257.14 29/25, 65/56 ^10E, v7F
49 262.5 57/49, 64/55 v10E♯, v6F
50 267.86 v9E♯, v5F
51 273.21 41/35, 48/41 v8E♯, v4F
52 278.57 47/40, 74/63 v7E♯, v3F
53 283.93 33/28 v6E♯, vvF
54 289.29 13/11 v5E♯, vF
55 294.64 32/27 F
56 300 44/37, 69/58 ^F, ^5G♭♭
57 305.36 31/26, 37/31, 68/57 ^^F, ^6G♭♭
58 310.71 ^3F, ^7G♭♭
59 316.07 6/5 ^4F, ^8G♭♭
60 321.43 65/54 ^5F, ^9G♭♭
61 326.79 29/24 ^6F, ^10G♭♭
62 332.14 40/33, 63/52 ^7F, v10G♭
63 337.5 62/51 ^8F, v9G♭
64 342.86 39/32, 50/41 ^9F, v8G♭
65 348.21 11/9 ^10F, v7G♭
66 353.57 v10F♯, v6G♭
67 358.93 16/13 v9F♯, v5G♭
68 364.29 58/47 v8F♯, v4G♭
69 369.64 26/21 v7F♯, v3G♭
70 375 36/29, 41/33 v6F♯, vvG♭
71 380.36 v5F♯, vG♭
72 385.71 5/4 v4F♯, G♭
73 391.07 v3F♯, ^G♭
74 396.43 44/35 vvF♯, ^^G♭
75 401.79 29/23 vF♯, ^3G♭
76 407.14 62/49 F♯, ^4G♭
77 412.5 33/26 ^F♯, ^5G♭
78 417.86 14/11 ^^F♯, ^6G♭
79 423.21 60/47 ^3F♯, ^7G♭
80 428.57 41/32 ^4F♯, ^8G♭
81 433.93 ^5F♯, ^9G♭
82 439.29 49/38, 58/45 ^6F♯, ^10G♭
83 444.64 75/58 ^7F♯, v10G
84 450 35/27, 48/37 ^8F♯, v9G
85 455.36 ^9F♯, v8G
86 460.71 30/23, 47/36 ^10F♯, v7G
87 466.07 55/42, 72/55 v10F𝄪, v6G
88 471.43 21/16 v9F𝄪, v5G
89 476.79 54/41 v8F𝄪, v4G
90 482.14 37/28 v7F𝄪, v3G
91 487.5 v6F𝄪, vvG
92 492.86 v5F𝄪, vG
93 498.21 4/3 G
94 503.57 ^G, ^5A♭♭
95 508.93 51/38, 55/41 ^^G, ^6A♭♭
96 514.29 35/26, 74/55 ^3G, ^7A♭♭
97 519.64 27/20 ^4G, ^8A♭♭
98 525 42/31, 65/48 ^5G, ^9A♭♭
99 530.36 ^6G, ^10A♭♭
100 535.71 ^7G, v10A♭
101 541.07 41/30 ^8G, v9A♭
102 546.43 37/27, 48/35 ^9G, v8A♭
103 551.79 11/8 ^10G, v7A♭
104 557.14 40/29, 69/50 v10G♯, v6A♭
105 562.5 18/13 v9G♯, v5A♭
106 567.86 25/18, 68/49 v8G♯, v4A♭
107 573.21 39/28 v7G♯, v3A♭
108 578.57 v6G♯, vvA♭
109 583.93 v5G♯, vA♭
110 589.29 45/32, 52/37 v4G♯, A♭
111 594.64 31/22, 55/39 v3G♯, ^A♭
112 600 41/29, 58/41 vvG♯, ^^A♭
113 605.36 44/31 vG♯, ^3A♭
114 610.71 37/26, 64/45 G♯, ^4A♭
115 616.07 ^G♯, ^5A♭
116 621.43 63/44 ^^G♯, ^6A♭
117 626.79 56/39 ^3G♯, ^7A♭
118 632.14 36/25, 49/34 ^4G♯, ^8A♭
119 637.5 13/9 ^5G♯, ^9A♭
120 642.86 29/20 ^6G♯, ^10A♭
121 648.21 16/11 ^7G♯, v10A
122 653.57 35/24, 54/37 ^8G♯, v9A
123 658.93 60/41 ^9G♯, v8A
124 664.29 69/47 ^10G♯, v7A
125 669.64 v10G𝄪, v6A
126 675 31/21, 65/44 v9G𝄪, v5A
127 680.36 40/27 v8G𝄪, v4A
128 685.71 52/35, 55/37 v7G𝄪, v3A
129 691.07 76/51 v6G𝄪, vvA
130 696.43 v5G𝄪, vA
131 701.79 3/2 A
132 707.14 ^A, ^5B♭♭
133 712.5 ^^A, ^6B♭♭
134 717.86 56/37 ^3A, ^7B♭♭
135 723.21 41/27 ^4A, ^8B♭♭
136 728.57 32/21 ^5A, ^9B♭♭
137 733.93 55/36 ^6A, ^10B♭♭
138 739.29 23/15, 72/47 ^7A, v10B♭
139 744.64 ^8A, v9B♭
140 750 37/24, 54/35 ^9A, v8B♭
141 755.36 65/42 ^10A, v7B♭
142 760.71 45/29, 76/49 v10A♯, v6B♭
143 766.07 v9A♯, v5B♭
144 771.43 64/41 v8A♯, v4B♭
145 776.79 47/30 v7A♯, v3B♭
146 782.14 11/7 v6A♯, vvB♭
147 787.5 52/33 v5A♯, vB♭
148 792.86 49/31 v4A♯, B♭
149 798.21 46/29, 65/41 v3A♯, ^B♭
150 803.57 35/22 vvA♯, ^^B♭
151 808.93 75/47 vA♯, ^3B♭
152 814.29 8/5 A♯, ^4B♭
153 819.64 69/43 ^A♯, ^5B♭
154 825 29/18, 66/41 ^^A♯, ^6B♭
155 830.36 21/13 ^3A♯, ^7B♭
156 835.71 47/29 ^4A♯, ^8B♭
157 841.07 13/8 ^5A♯, ^9B♭
158 846.43 75/46 ^6A♯, ^10B♭
159 851.79 18/11 ^7A♯, v10B
160 857.14 41/25, 64/39 ^8A♯, v9B
161 862.5 51/31 ^9A♯, v8B
162 867.86 33/20 ^10A♯, v7B
163 873.21 48/29 v10A𝄪, v6B
164 878.57 v9A𝄪, v5B
165 883.93 5/3 v8A𝄪, v4B
166 889.29 v7A𝄪, v3B
167 894.64 52/31, 57/34, 62/37 v6A𝄪, vvB
168 900 37/22 v5A𝄪, vB
169 905.36 27/16 B
170 910.71 22/13 ^B, ^5C♭
171 916.07 56/33 ^^B, ^6C♭
172 921.43 63/37 ^3B, ^7C♭
173 926.79 41/24, 70/41 ^4B, ^8C♭
174 932.14 ^5B, ^9C♭
175 937.5 55/32 ^6B, ^10C♭
176 942.86 50/29 ^7B, v10C
177 948.21 64/37 ^8B, v9C
178 953.57 ^9B, v8C
179 958.93 40/23, 47/27 ^10B, v7C
180 964.29 v10B♯, v6C
181 969.64 7/4 v9B♯, v5C
182 975 65/37, 72/41 v8B♯, v4C
183 980.36 37/21 v7B♯, v3C
184 985.71 v6B♯, vvC
185 991.07 39/22 v5B♯, vC
186 996.43 16/9 C
187 1001.79 66/37 ^C, ^5D♭♭
188 1007.14 34/19 ^^C, ^6D♭♭
189 1012.5 70/39 ^3C, ^7D♭♭
190 1017.86 9/5 ^4C, ^8D♭♭
191 1023.21 56/31, 65/36, 74/41 ^5C, ^9D♭♭
192 1028.57 ^6C, ^10D♭♭
193 1033.93 ^7C, v10D♭
194 1039.29 31/17 ^8C, v9D♭
195 1044.64 64/35, 75/41 ^9C, v8D♭
196 1050 11/6 ^10C, v7D♭
197 1055.36 46/25, 57/31 v10C♯, v6D♭
198 1060.71 24/13 v9C♯, v5D♭
199 1066.07 50/27 v8C♯, v4D♭
200 1071.43 13/7 v7C♯, v3D♭
201 1076.79 41/22, 54/29 v6C♯, vvD♭
202 1082.14 v5C♯, vD♭
203 1087.5 15/8 v4C♯, D♭
204 1092.86 47/25 v3C♯, ^D♭
205 1098.21 66/35 vvC♯, ^^D♭
206 1103.57 70/37 vC♯, ^3D♭
207 1108.93 55/29, 74/39 C♯, ^4D♭
208 1114.29 ^C♯, ^5D♭
209 1119.64 21/11 ^^C♯, ^6D♭
210 1125 ^3C♯, ^7D♭
211 1130.36 ^4C♯, ^8D♭
212 1135.71 ^5C♯, ^9D♭
213 1141.07 29/15 ^6C♯, ^10D♭
214 1146.43 64/33 ^7C♯, v10D
215 1151.79 35/18, 72/37 ^8C♯, v9D
216 1157.14 ^9C♯, v8D
217 1162.5 45/23 ^10C♯, v7D
218 1167.86 55/28 v10C𝄪, v6D
219 1173.21 63/32, 65/33 v9C𝄪, v5D
220 1178.57 v8C𝄪, v4D
221 1183.93 v7C𝄪, v3D
222 1189.29 v6C𝄪, vvD
223 1194.64 v5C𝄪, vD
224 1200 2/1 D

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.[2]

Sagittal notation
Steps 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
Symbol Evo
Revo

Because it uses the entire Athenian system (except for 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.

Ups-and-downs notation
224edosteps 0 1 2 3 4 5 6 7 8 9 10
h ^ ^^ ^^^ v> > ^> ^^> ^^^> v>> >>
<<<<# ^<<<<# vvv<<<# vv<<<# v<<<# <<<# ^<<<# vvv<<# vv<<# v<<#
11 12 13 14 15 16 17 18 19 20 21
^>> ^^>> ^^^>> v>>> >>> ^>>> ^^>>> ^^^>>> v>>>> >>>> #
<<# ^<<# vvv<# vv<# v<# <# ^<# vvv# vv# v#

Approximation to JI

Interval mappings

The following table shows how 15-odd-limit intervals are represented in 224edo. Prime harmonics are in bold.

As 224edo is consistent in the 15-odd-limit, the mappings by direct approximation and through the patent val are identical.

15-odd-limit intervals in 224edo
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
13/11, 22/13 0.076 1.4
3/2, 4/3 0.169 3.2
9/5, 10/9 0.261 4.9
13/7, 14/13 0.273 5.1
9/8, 16/9 0.339 6.3
11/7, 14/11 0.349 6.5
5/3, 6/5 0.430 8.0
11/8, 16/11 0.468 8.7
13/8, 16/13 0.544 10.2
5/4, 8/5 0.599 11.2
11/6, 12/11 0.637 11.9
13/12, 24/13 0.713 13.3
15/8, 16/15 0.769 14.3
11/9, 18/11 0.806 15.1
7/4, 8/7 0.817 15.2
13/9, 18/13 0.882 16.5
7/6, 12/7 0.986 18.4
11/10, 20/11 1.067 19.9
13/10, 20/13 1.143 21.3
9/7, 14/9 1.156 21.6
15/11, 22/15 1.236 23.1
15/13, 26/15 1.312 24.5
7/5, 10/7 1.416 26.4
15/14, 28/15 1.586 29.6

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
2.3.5.7.11.13.17 375/374, 540/539, 625/624, 715/714, 729/728, 2200/2197 [⟨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 72. The next equal temperament that does better in terms of either absolute or relative error is 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 152 and is superseded by 239. In the 17-limit it is the first to beat 217 and is superseded by 270.

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperament
1 43\224 230.36 8/7 Gamera
1 59\224 316.07 6/5 Counterkleismic / counterlytic
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 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\224 176.79 195/176 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

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

Music

Mercury Amalgam
Gene Ward Smith

References

  1. ↑ @Xen-p6p (2026), YouTube post on Uccellini - «Aria Sopra La Bergamasca» (1642), arranged for Organ, tuned into Adaptive Just Intonation rendered by Claudi Meneghin (2024).
  2. ↑ Sagittal – A Microtonal Notation System by George D. Secor and David C. Keenan
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