212edo: Difference between revisions
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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 [[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. | ||
The 212gh val shows some potential | 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. | ||
=== Prime harmonics === | === Prime harmonics === | ||
| Line 18: | Line 18: | ||
=== Subsets and supersets === | === Subsets and supersets === | ||
Since 212 factors into {{ | Since 212 factors into primes as {{nowrap| 2<sup>2</sup> × 53 }}, 212edo has subset edos {{EDOs| 2, 4, 53, and 106 }}. As such, it can be used to tune the 53rd-octave [[cartography]] temperament and the 106th-octave [[boiler]] temperment. | ||
A step of 212edo is exactly 50 [[türk sent]]s. | A step of 212edo is exactly 50 [[türk sent]]s. | ||
| Line 36: | Line 36: | ||
| 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 }} | ||
| +0.243 | | +0.243 | ||
| 0.244 | | 0.244 | ||
| Line 43: | Line 43: | ||
| 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 }} | ||
| +0.325 | | +0.325 | ||
| 0.273 | | 0.273 | ||
| Line 50: | Line 50: | ||
| 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 }} | ||
| +0.396 | | +0.396 | ||
| 0.296 | | 0.296 | ||
| Line 57: | Line 57: | ||
| 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) | ||
| +0.447 | | +0.447 | ||
| 0.301 | | 0.301 | ||
| Line 64: | Line 64: | ||
| 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) | ||
| +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]]. | ||
Latest revision as of 20:25, 12 August 2025
| ← 211edo | 212edo | 213edo → |
212 equal divisions of the octave (abbreviated 212edo or 212ed2), also called 212-tone equal temperament (212tet) or 212 equal temperament (212et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 212 equal parts of about 5.66 ¢ each. Each step represents a frequency ratio of 21/212, or the 212th root of 2.
Theory
212edo is distinctly consistent in the 15-odd-limit with harmonics of 3 through 13 all tuned flat. It shares the 3rd, 5th, and 13th harmonics with 53edo, but the mapping differs for 7 and 11.
It 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.
To the 13-limit we may add the prime 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.
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) | |
Octave stretch
212edo can benefit from slightly stretching the octave, using tunings such as 124edf or 336edt. This improves the approximated harmonics 5, 7, 11, 13 and brings the flat mappings of 17 and 19 closer; only the 23 becomes less accurate as it is tuned sharp already.
Subsets and supersets
Since 212 factors into primes as 22 × 53, 212edo has subset edos 2, 4, 53, and 106. As such, it can be used to tune the 53rd-octave cartography temperament and the 106th-octave boiler temperment.
A step of 212edo is exactly 50 türk sents.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 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 |
| 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 | [⟨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 193 and followed by 217.
Rank-2 temperaments
Note: temperaments supported by 53et are not included.
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 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 |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct