212edo: Difference between revisions
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== Regular temperament properties == | == Regular temperament properties == | ||
{ | {{comma basis begin}} | ||
|- | |- | ||
| 2.3.5.7 | | 2.3.5.7 | ||
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| 0.299 | | 0.299 | ||
| 5.27 | | 5.27 | ||
{{comma basis end}} | |||
* 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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Note: temperaments supported by 53et are not included. | Note: temperaments supported by 53et are not included. | ||
{ | {{rank-2 begin}} | ||
|- | |- | ||
| 1 | | 1 | ||
| Line 122: | Line 107: | ||
|- | |- | ||
| 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]] | ||
{{rank-2 end}} | |||
{{orf}} | |||
== Music == | == Music == | ||
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[[Category:53edo]] | [[Category:53edo]] | ||
[[Category:Agni]] | |||
[[Category:Kleismic]] | [[Category:Kleismic]] | ||
[[Category: | [[Category:Listen]] | ||
[[Category:Quadritikleismic]] | [[Category:Quadritikleismic]] | ||
Revision as of 05:38, 16 November 2024
| ← 211edo | 212edo | 213edo → |
Theory
212edo is distinctly consistent in the 15-odd-limit with harmonics of 3 through 13 all tuned flat. 212 = 4 × 53, and it shares the 3rd, 5th, and 13th harmonics with 53edo, but the mapping differs for 7 and 11.
The equal temperament tempers out the same commas (15625/15552, 32805/32768, 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 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.
The 212gh val shows some potential beyond 15-odd-limit. 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
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | -0.07 | -1.41 | -0.90 | -2.26 | -2.79 | +2.59 | +2.49 | +0.03 | +0.61 | -1.64 |
| Relative (%) | +0.0 | -1.2 | -24.9 | -15.9 | -40.0 | -49.3 | +45.8 | +43.9 | +0.5 | +10.8 | -29.0 | |
| Steps (reduced) |
212 (0) |
336 (124) |
492 (68) |
595 (171) |
733 (97) |
784 (148) |
867 (19) |
901 (53) |
959 (111) |
1030 (182) |
1050 (202) | |
Subsets and supersets
Since 212 factors into 22 × 53, 212edo has subset edos 2, 4, 53, and 106. As such, it can be used to tune the period-53 cartography temperament and the period-106 boiler temperment.
A step of 212edo is exactly 50 türk sents.
Regular temperament properties
Template:Comma basis begin |- | 2.3.5.7 | 2401/2400, 15625/15552, 32805/32768 | [⟨212 336 492 595]] | +0.243 | 0.244 | 4.30 |- | 2.3.5.7.11 | 385/384, 1375/1372, 6250/6237, 14641/14580 | [⟨212 336 492 595 733]] | +0.325 | 0.273 | 4.82 |- | 2.3.5.7.11.13 | 325/324, 385/384, 625/624, 1375/1372, 10648/10647 | [⟨212 336 492 595 733 784]] | +0.396 | 0.296 | 5.23 |- | 2.3.5.7.11.13.17 | 289/288, 325/324, 385/384, 442/441, 625/624, 10648/10647 | [⟨212 336 492 595 733 784 866]] (212g) | +0.447 | 0.301 | 5.32 |- | 2.3.5.7.11.13.17.19 | 289/288, 325/324, 361/360, 385/384, 442/441, 513/512, 625/624 | [⟨212 336 492 595 733 784 866 900]] (212gh) | +0.485 | 0.299 | 5.27 Template:Comma basis end
- 212et (212gh val) has a lower absolute error in the 19-limit than any previous equal temperaments, past 193 and followed by 217.
Rank-2 temperaments
Note: temperaments supported by 53et are not included.
Template:Rank-2 begin
|-
| 1
| 15\212
| 84.91
| 21/20
| Amicable / amorous / pseudoamical
|-
| 1
| 31\212
| 175.47
| 448/405
| Sesquiquartififths
|-
| 1
| 41\212
| 232.08
| 8/7
| Quadrawell
|-
| 1
| 67\212
| 379.25
| 56/45
| Marthirds
|-
| 2
| 11\212
| 62.26
| 28/27
| Eagle
|-
| 2
| 15\212
| 84.91
| 21/20
| Floral
|-
| 2
| 31\212
| 175.47
| 448/405
| Bisesqui
|-
| 2
| 97\212
(9\212)
| 549.06
(50.94)
| 11/8
(36/35)
| Kleischismic
|-
| 4
| 56\212
(3\212)
| 316.98
(16.98)
| 6/5
(126/125)
| Quadritikleismic
|-
| 4
| 88\212
(18\212)
| 498.11
(101.89)
| 4/3
(35/33)
| Quadrant
|-
| 53
| 41\212
(1\212)
| 232.08
(5.66)
| 8/7
(225/224)
| Schismerc / cartography
Template:Rank-2 end
Template:Orf