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212edo: Difference between revisions

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→Theory: full 23-limit isn't too much of a stretch
m Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct"
 
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It [[tempering out|tempers out]] the same commas ([[15625/15552]], [[32805/32768]], [[amity comma|1600000/1594323]], etc.) as 53edo in the [[5-limit]]. In the [[7-limit]], it tempers out 2401/2400 ([[breedsma]]), 390625/388962 ([[dimcomp comma]]), and 4802000/4782969 ([[canousma]]). In the [[11-limit]], [[385/384]], [[1375/1372]], [[6250/6237]], [[9801/9800]], and [[14641/14580]]; in the [[13-limit]], [[325/324]], [[625/624]], [[676/675]], [[1001/1000]], [[1716/1715]], [[2080/2079]], and [[10648/10647]].  
It [[tempering out|tempers out]] the same commas ([[15625/15552]], [[32805/32768]], [[amity comma|1600000/1594323]], etc.) as 53edo in the [[5-limit]]. In the [[7-limit]], it tempers out 2401/2400 ([[breedsma]]), 390625/388962 ([[dimcomp comma]]), and 4802000/4782969 ([[canousma]]). In the [[11-limit]], [[385/384]], [[1375/1372]], [[6250/6237]], [[9801/9800]], and [[14641/14580]]; in the [[13-limit]], [[325/324]], [[625/624]], [[676/675]], [[1001/1000]], [[1716/1715]], [[2080/2079]], and [[10648/10647]].  


It is the [[optimal patent val]] for 7- and 13-limit [[quadritikleismic]] temperament, the 7-limit [[Kleismic rank three family #Rank-3 kleismic|rank-3 kleismic]] temperament, and the 13-limit rank-3 [[agni]] temperament. It enables [[marveltwin chords]], [[keenanismic chords]], [[sinbadmic chords]], and [[lambeth chords]] in the 13-odd-limit in addition to [[island chords]] in the 15-odd-limit.  
It is the [[optimal patent val]] for 7- and 13-limit [[quadritikleismic]] temperament, the 7-limit [[rank-3 kleismic]] temperament, and the 13-limit rank-3 [[agni]] temperament. It enables [[marveltwin chords]], [[keenanismic chords]], [[sinbadmic chords]], and [[lambeth chords]] in the 13-odd-limit in addition to [[island chords]] in the 15-odd-limit.  


To the 13-limit we may add the [[prime harmonic|prime]] [[23/1|23]] without introducing too much extra error, tempering out [[484/483]] and [[507/506]]. The 212gh val shows some potential if the full [[23-limit]] is desired, where it notably tempers out [[289/288]] and [[361/360]]. Also, using 212bb val (where fifth is flattened by single step, approximately 1/4 comma) gives a tuning almost identical to the POTE tuning for 5-limit meantone. This is related to the fact that 212edo splits steps of 53edo, which are mapped to a syntonic comma, in four.
To the 13-limit we may add the [[prime harmonic|prime]] [[23/1|23]] without introducing too much extra error, tempering out [[484/483]] and [[507/506]]. The 212gh val shows some potential if the full [[23-limit]] is desired, where due to the flatness of harmonics 17 and 19, it tempers out the square superparticulars for both of them, being [[289/288]] and [[361/360]] respectively. Also, using 212bb val (where fifth is flattened by single step, approximately 1/4 comma) gives a tuning almost identical to the POTE tuning for 5-limit meantone. This is related to the fact that 212edo splits steps of 53edo, which are mapped to a syntonic comma, in four.


=== Prime harmonics ===
=== Prime harmonics ===
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! rowspan="2" | [[Comma list]]
! rowspan="2" | [[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
|-
|-
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| 2.3.5.7
| 2.3.5.7
| 2401/2400, 15625/15552, 32805/32768
| 2401/2400, 15625/15552, 32805/32768
| {{mapping| 212 336 492 595 }}
| {{Mapping| 212 336 492 595 }}
| +0.243
| +0.243
| 0.244
| 0.244
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| 2.3.5.7.11
| 2.3.5.7.11
| 385/384, 1375/1372, 6250/6237, 14641/14580
| 385/384, 1375/1372, 6250/6237, 14641/14580
| {{mapping| 212 336 492 595 733 }}
| {{Mapping| 212 336 492 595 733 }}
| +0.325
| +0.325
| 0.273
| 0.273
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| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 325/324, 385/384, 625/624, 1375/1372, 10648/10647
| 325/324, 385/384, 625/624, 1375/1372, 10648/10647
| {{mapping| 212 336 492 595 733 784 }}
| {{Mapping| 212 336 492 595 733 784 }}
| +0.396
| +0.396
| 0.296
| 0.296
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| 2.3.5.7.11.13.17
| 2.3.5.7.11.13.17
| 289/288, 325/324, 385/384, 442/441, 625/624, 10648/10647
| 289/288, 325/324, 385/384, 442/441, 625/624, 10648/10647
| {{mapping| 212 336 492 595 733 784 866 }} (212g)
| {{Mapping| 212 336 492 595 733 784 866 }} (212g)
| +0.447
| +0.447
| 0.301
| 0.301
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| 2.3.5.7.11.13.17.19
| 2.3.5.7.11.13.17.19
| 289/288, 325/324, 361/360, 385/384, 442/441, 513/512, 625/624
| 289/288, 325/324, 361/360, 385/384, 442/441, 513/512, 625/624
| {{mapping| 212 336 492 595 733 784 866 900 }} (212gh)
| {{Mapping| 212 336 492 595 733 784 866 900 }} (212gh)
| +0.485
| +0.485
| 0.299
| 0.299
| 5.27
| 5.27
|-
| 2.3.5.7.11.13.17.19.23
| 289/288, 323/322, 325/324, 361/360, 385/384, 442/441, 484/483, 507/506
| {{Mapping| 212 336 492 595 733 784 866 900 959 }} (212gh)
| +0.430
| 0.321
| 5.67
|}
|}
* 212et (212gh val) has a lower absolute error in the 19-limit than any previous equal temperaments, past [[193edo|193]] and followed by [[217edo|217]].
* 212et (212gh val) has a lower absolute error in the 19-limit than any previous equal temperaments, past [[193edo|193]] and followed by [[217edo|217]].
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|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | 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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|-
|-
| 2
| 2
| 97\212<br />(9\212)
| 97\212<br>(9\212)
| 549.06<br />(50.94)
| 549.06<br>(50.94)
| 11/8<br />(36/35)
| 11/8<br>(36/35)
| [[Kleischismic]]
| [[Kleischismic]]
|-
|-
| 4
| 4
| 56\212<br />(3\212)
| 56\212<br>(3\212)
| 316.98<br />(16.98)
| 316.98<br>(16.98)
| 6/5<br />(126/125)
| 6/5<br>(126/125)
| [[Quadritikleismic]]
| [[Quadritikleismic]]
|-
|-
| 4
| 4
| 88\212<br />(18\212)
| 88\212<br>(18\212)
| 498.11<br />(101.89)
| 498.11<br>(101.89)
| 4/3<br />(35/33)
| 4/3<br>(35/33)
| [[Quadrant]]
| [[Quadrant]]
|-
|-
| 53
| 53
| 41\212<br />(1\212)
| 41\212<br>(1\212)
| 232.08<br />(5.66)
| 232.08<br>(5.66)
| 8/7<br />(225/224)
| 8/7<br>(225/224)
| [[Schismerc]] / [[cartography]]
| [[Schismerc]] / [[cartography]]
|}
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
<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 ==
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