224edo: Difference between revisions
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== 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. | ||
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| 2.3 | | 2.3 | ||
| {{monzo| -355 224 }} | | {{monzo| -355 224 }} | ||
| | | {{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 }} | ||
| | | {{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 | ||
| | | {{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 | ||
| | | {{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 | ||
| | | {{mapping| 224 355 520 629 775 829 }} | ||
| -0.035 | | -0.035 | ||
| 0.1805 | | 0.1805 | ||
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|+Table of rank-2 temperaments by generator | |+Table of rank-2 temperaments by generator | ||
! Periods<br>per 8ve | ! Periods<br>per 8ve | ||
! Generator | ! Generator* | ||
! Cents | ! Cents* | ||
! Associated<br>Ratio | ! Associated<br>Ratio* | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
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| [[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'' (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''] | * [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]] | ||