224edo: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older editNewer edit →
VisualWikitext
Eliora (talk | contribs)
autopatrolled, emailconfirmed
3,381 edits
→Theory: finalize the writing
Cleanup; linking; clarify the title row of the rank-2 temp table
Line 3: Line 3:


== 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 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 [[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 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 [[The Riemann zeta function and tuning #Zeta EDO lists|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.
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.
Line 26: Line 26:
| 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 33: Line 33:
| 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 40: Line 40:
| 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 47: Line 47:
| 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
Line 54: Line 54:
| 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
Line 64: Line 64:
|+Table of rank-2 temperaments by generator
|+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
! Temperaments
|-
|-
Line 213: Line 213:
| [[Barium]]
| [[Barium]]
|}
|}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct


== Music ==
== Music ==
; [[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]
* ''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


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


[[Category:Indra]]
[[Category:Indra]]

Revision as of 09:08, 24 October 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 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

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* Cents* Associated
Ratio* 		Temperaments
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 it is distinct

Music

Gene Ward Smith
Mercury Amalgam
Retrieved from "https://en.xen.wiki/index.php?title=224edo&oldid=126369"