293edo
| ← 292edo | 293edo | 294edo → |
293 equal divisions of the octave (abbreviated 293edo or 293ed2), also called 293-tone equal temperament (293tet) or 293 equal temperament (293et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 293 equal parts of about 4.1 ¢ each. Each step represents a frequency ratio of 21/293, or the 293rd root of 2.
Theory
293edo is only consistent to the 5-odd-limit and it does not approximate prime harmonics well all the way into the 41st, with none approximated within 20% relative error, and all primes besides 13 have over 30% error. The first harmonic that it approximates well is the 43rd, which is 10% flat compared to the just intonated interval.
Nonetheless, a number of mappings can be considered.
Using the patent val, ⟨293 464 680 823], 293edo tempers out the parakleisma and [-40 15 7⟩ in the 5-limit and the marvel comma in the 7-limit.
The 293bb val, with ⟨293 463 680 823], is a tuning close to the POTE tuning for the meantone temperament.
293edo nonetheless has good approximations to 6/5, 11/7, 17/11, 19/17, 24/23, 25/17, 25/19, and respectively their octave inversions. 21/16, which is a composite octave-reduced harmonic, is also well represented.
In the 17-limit, although inconsistent, 293edo in the patent val is a tuning for the Symmetry454 temperament which is constructed from a calendar layout by the same name. See the dedicated page.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -1.61 | -1.33 | +1.82 | +0.87 | +1.58 | -0.94 | +1.15 | +1.53 | +1.46 | +0.21 | -1.65 |
| Relative (%) | -39.4 | -32.5 | +44.5 | +21.2 | +38.7 | -22.9 | +28.1 | +37.3 | +35.7 | +5.1 | -40.4 | |
| Steps (reduced) |
464 (171) |
680 (94) |
823 (237) |
929 (50) |
1014 (135) |
1084 (205) |
1145 (266) |
1198 (26) |
1245 (73) |
1287 (115) |
1325 (153) | |
Subsets and supersets
293edo is the 62nd prime edo.
Regular temperament properties
Commas
293edo tempers out the 2.43 [1590 293⟩ comma in the patent val, equating a stack of 293 43rd harmonics with 1590 octaves.
Using the patent val, it tempers out 225/224, 2500000/2470629, and 344373768/341796875 in the 7-limit; 6250/6237, 8019/8000, 14700/14641, and 16896/16807 in the 11-limit; 351/350, 625/624, 1625/1617, and 13122/13013 in the 13-limit; 715/714, 850/847, 1089/1088, 1377/1375, 2058/2057, and 2880/2873 in the 17-limit.
Using the 293b val, it tempers out 16875/16807, 20000/19683, and 65625/65536 in the 7-limit; 896/891, 6875/6804, 9375/9317, and 12005/11979 in the 11-limit; 352/351, 364/363, 1716/1715, and 8125/8019 in the 13-limit.
Using the 293bcf val, it tempers out 2401/2400, 179200/177147, and 1959552/1953125 in the 7-limit; 896/891, 2200/2187, 26411/26244, and 43923/43750 in the 11-limit; 847/845, 1001/1000, 1716/1715, 2197/2187, and 6656/6615 in the 13-limit.
Using the 293d val, it tempers out 1029/1024, 19683/19600, and 48828125/48771072 in the 7-limit; 540/539, 2835/2816, 4375/4356, and 1835008/1830125 in the 11-limit; 364/363, 625/624, 2205/2197, and 4459/4455 in the 13-limit; 273/272, 833/832, 1089/1088, 1377/1375, 2295/2288, and 2500/2499 in the 17-limit.
Using the 293deg val, it tempers out 385/384, 441/440, 24057/24010, and 234375/234256 in the 11-limit; 625/624, 847/845, 1001/1000, and 1575/1573 in the 13-limit; 561/560, 1225/1224, 1275/1274, and 2025/2023 in the 17-limit.
Using the well-approximated intervals, 6/5, 11/7, 17/11, 19/17, 24/23, 25/17, 25/19 and 21/16, 293edo tempers out 2376/2375, 304175/304128, 2599200/2598977 .
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated Ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 11\293 | 45.06 | 36/35 | Quartonic (293bcd) |
| 1 | 62\293 | 253.92 | 52/45 | Symmetry454 |
| 1 | 118\293 | 483.28 | 320/243 | Hemiseven (293de) |
| 1 | 143\293 | 585.66 | 7/5 | Merman (293ef) |
| 1 | 170\293 | 696.25 | 3/2 | Meantone (293bb) |
Scales
The 33L 19s maximally even scale of 293edo is a leap year pattern of a proposed calendar. It employs 62\293 as a generator, described as "accumulator" by the creator of the calendar himself. Likewise, a 71-note cycle with 260\293 generator can be constructed by analogy.
The corresponding rank two temperament is therefore called Symmetry454.
Music
- Whiplash (2022) – using the Symmetry454[52] scale.