224edo: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older editNewer edit →
VisualWikitext
ArrowHead294 (talk | contribs)
15,213 edits
mNo edit summary
Line 11: Line 11:


=== Subsets and supersets ===
=== Subsets and supersets ===
Since 224 = 32 × 7, 224edo has subset edos {{EDOs| 2, 4, 8, 16, 32, 7, 14, 28, 56, and 112 }}.
Since {{nowrap|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 ==
{| class="wikitable center-4 center-5 center-6"
{{comma basis begin}}
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
|-
| 2.3
| 2.3
Line 48: Line 40:
| 540/539, 1375/1372, 4000/3993, 32805/32768
| 540/539, 1375/1372, 4000/3993, 32805/32768
| {{mapping| 224 355 520 629 775 }}
| {{mapping| 224 355 520 629 775 }}
| -0.012
| &minus;0.012
| 0.1899
| 0.1899
| 3.54
| 3.54
Line 55: Line 47:
| 540/539, 625/624, 729/728, 1375/1372, 2200/2197
| 540/539, 625/624, 729/728, 1375/1372, 2200/2197
| {{mapping| 224 355 520 629 775 829 }}
| {{mapping| 224 355 520 629 775 829 }}
| -0.035
| &minus;0.035
| 0.1805
| 0.1805
| 3.37
| 3.37
Line 62: Line 54:
| 375/374, 540/539, 625/624, 715/714, 729/728, 2200/2197
| 375/374, 540/539, 625/624, 715/714, 729/728, 2200/2197
| {{mapping| 224 355 520 629 775 829 916 }}
| {{mapping| 224 355 520 629 775 829 916 }}
| -0.106
| &minus;0.106
| 0.2420
| 0.2420
| 4.52
| 4.52
|}
{{comma basis end}}
* 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]].  
* 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 In the 11-limit it is the first to beat [[152edo|152]] and is superseded by [[239edo|239]]. In the 17-limit it is the first to beat [[217edo|217]] and is superseded by 270.  
* 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"
{{rank-2 begin}}
|+Table of rank-2 temperaments by generator
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>Ratio*
! Temperaments
|-
|-
| 1
| 1
Line 221: Line 207:
| 4/3<br>(126/125)
| 4/3<br>(126/125)
| [[Barium]]
| [[Barium]]
|}
{{rank-2 end}}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
{{orf}}


== Music ==
== Music ==
; [[Gene Ward Smith]]
; [[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
* ''Dreyfus'' (archived 2010) &ndash; [https://soundcloud.com/genewardsmith/genewardsmith-dreyfus SoundCloud] | [https://www.archive.org/details/Dreyfus details] | [https://www.archive.org/download/Dreyfus/Genewardsmith-Dreyfus.mp3 play] &ndash; octoid[72] in 224edo tuning


; [[Mercury Amalgam]]
; [[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
* [https://www.youtube.com/watch?v=iFi1zKsRBfY ''Kindness Is A Weakness''] (2023) &ndash; octant[24], hemigamera[26], oquatonic[56], bezique[64] in 224edo tuning


[[Category:Indra]]
[[Category:Indra]]
[[Category:Listen]]
[[Category:Mirkwai]]
[[Category:Mirkwai]]
[[Category:Octoid]]
[[Category:Octoid]]
[[Category:Quartismic]]
[[Category:Quartismic]]
[[Category:Shibi]]
[[Category:Shibi]]
[[Category:Listen]]

Revision as of 05:53, 16 November 2024

← 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 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. It is the twelfth zeta integral edo.

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.

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

Template:Comma basis begin |- | 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 Template:Comma basis end

  • 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

Template:Rank-2 begin |- | 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 Template:Rank-2 end Template:Orf

Music

Gene Ward Smith
Mercury Amalgam
Retrieved from "https://en.xen.wiki/index.php?title=224edo&oldid=165975"