212edo: Difference between revisions
→Regular temperament properties: + 23-limit |
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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 [[ | 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 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. | ||
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! rowspan="2" | [[Comma list]] | ! rowspan="2" | [[Comma list]] | ||
! rowspan="2" | [[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" | Optimal<br | ! rowspan="2" | Optimal<br>8ve stretch (¢) | ||
! colspan="2" | Tuning error | ! colspan="2" | Tuning error | ||
|- | |- | ||
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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 | ! Periods<br>per 8ve | ||
! Generator* | ! Generator* | ||
! Cents* | ! Cents* | ||
! Associated<br | ! Associated<br>ratio* | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
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|- | |- | ||
| 2 | | 2 | ||
| 97\212<br | | 97\212<br>(9\212) | ||
| 549.06<br | | 549.06<br>(50.94) | ||
| 11/8<br | | 11/8<br>(36/35) | ||
| [[Kleischismic]] | | [[Kleischismic]] | ||
|- | |- | ||
| 4 | | 4 | ||
| 56\212<br | | 56\212<br>(3\212) | ||
| 316.98<br | | 316.98<br>(16.98) | ||
| 6/5<br | | 6/5<br>(126/125) | ||
| [[Quadritikleismic]] | | [[Quadritikleismic]] | ||
|- | |- | ||
| 4 | | 4 | ||
| 88\212<br | | 88\212<br>(18\212) | ||
| 498.11<br | | 498.11<br>(101.89) | ||
| 4/3<br | | 4/3<br>(35/33) | ||
| [[Quadrant]] | | [[Quadrant]] | ||
|- | |- | ||
| 53 | | 53 | ||
| 41\212<br | | 41\212<br>(1\212) | ||
| 232.08<br | | 232.08<br>(5.66) | ||
| 8/7<br | | 8/7<br>(225/224) | ||
| [[Schismerc]] / [[cartography]] | | [[Schismerc]] / [[cartography]] | ||
|} | |} | ||
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[ | <nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[normal forms|minimal form]] in parentheses if distinct | ||
== Music == | == Music == | ||