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 does not approximate prime harmonics well all the way into the 41st, with none approximated within 30% error. The first harmonic that it approximates well is the 43rd, which is 10% flat compared to the just intonated interval. As such, it is only consistent to the 5-odd-limit.
Using the patent val, the equal temperament tempers out the parakleisma and [-40 15 7⟩. It also tempers out the marvel comma in the 7-limit.
The 293bb val, with 170\293 fifth, is a good tuning for the meantone temperament.
When it comes to the intervals that are not octave-reduced prime harmonics, some which are well-approximated are 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. These numbers are related to poor approximation of prime harmonics by cancelling out of the errors. For example, 19th and 17th harmonics have +36 and +37 error respectively, which together cancels out to 1.
One step of 293edo is at the edge of human pitch perception of 3.5 cents. When combined with low harmonicity, this opens 293edo to a wide range of interpretations. For example, 293edo also can be interpreted as a dual-interval tuning, with two notes instead of one assigned to a particular interval.
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 | -32 | +45 | +21 | +39 | -23 | +28 | +37 | +36 | +5 | -40 | |
| 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.