293edo
293EDO is the equal division of the octave into 293 parts of 4.0956 cents each.
293EDO is the 62nd prime EDO.
Theory
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -1.61 | -1.33 | +1.82 | +0.87 | +1.58 | -0.94 | +1.15 | +1.53 | +1.46 | +0.21 |
| Relative (%) | -39.4 | -32.5 | +44.5 | +21.2 | +38.7 | -22.9 | +28.1 | +37.3 | +35.7 | +5.1 | |
| Steps (reduced) |
464 (171) |
680 (94) |
823 (237) |
929 (50) |
1014 (135) |
1084 (205) |
1145 (266) |
1198 (26) |
1245 (73) |
1287 (115) | |
293 edo does not approximate prime harmonics well all the way into the 41st, unless 30-relative cent errors are considered "well", in which case it equally represents all of them. The first harmonic that it approximates within 1 standard deviation of one step is 43rd, which is 10 cents flat compared to the just intonated interval.
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 harmoincs 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.
Likewise, 293edo also can be interpreted as a dual-interval tuning, with two notes instead of one assigned to a particular interval.
| Degree | Name | Cents | Approximate ratios |
|---|---|---|---|
| 0 | Unison, prime | 0.0000 | 1/1 exact |
| 1 | Limit-tone | 4.0596 | 423/422 |
| 5 | Minor leap week interval | 20.4778 | 85/84 |
| 6 | Major leap week interval | 24.5734 | 71/70 |
| 11 | Bundle of 2 | 45.0512 | |
| 17 | Bundle of 3 | 69.6246 | |
| 18 | Vicesimotertial quarter-tone | 73.7201 | 24/23 |
| 45 | Minor subcycle | 184.3003 | |
| 47 | Undevicesimal meantone | 192.4915 | 19/17 |
| 56 | Minor septimal second | 229.3515 | 8/7, 214/187 |
| 57 | Major septimal second | 233.4471 | 8/7, 266/233 |
| 62 | Leap week accumulator | 253.9249 | 755/652 |
| 77 | Minor third | 315.3584 | 6/5 |
| 79 | Major subcycle | 323.5495 | |
| 115 | 21st harmonic | 470.9898 | 21/16 |
| 116 | 25 over 19 | 475.0853 | 25/19 |
| 125 | 43rd harmonic | 511.9454 | 43/32 |
| 130 | Vengeance superfourth | 532.4232 | 34/25 |
| 163 | Vengeance subfifth | 667.5768 | 25/17 |
| 168 | 43rd subharmonic | 688.0546 | 64/43 |
| 171 | Perfect fifth | 700.3413 | 3/2 |
| 172 | "Major" fifth | 704.4369 | 347/231 |
| 191 | Undecimal minor sixth | 782.2526 | 11/7 |
| 216 | Major sixth | 884.6416 | 5/3 |
| 236 | Minor harmonic seventh | 966.5529 | 7/4, 187/107 |
| 237 | Major harmonic seventh | 970.6485 | 7/4, 233/133 |
| 260 | Leap day accumulator | 1064.8464 | 468/253 |
| 293 | Perfect octave | 1200.0000 | 2/1 exact |
Tempered commas
293edo tempers out the [1590 012 293⟩ comma in the patent val, equating a stack of 293 43rd harmonics with 1590 octaves.
293edo tempers out 1224440064/1220703125 (parakleisma) and 1121008359375/1099511627776 in the 5-limit. 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.
Scales
33L 19s maximally even scale of 293edo has a real life application - it is a leap year pattern of a proposed calendar. Using MOS, 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.
- LeapWeek[52]
- LeapDay[71]
Music
- Whiplash by Cinnamon Mavka - using the LeapWeek[52] scale.