53edf
| ← 52edf | 53edf | 54edf → |
53 equal divisions of the octave (abbreviated 53edo or 53ed2), also called 53-tone equal temperament (53tet) or 53 equal temperament (53et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 53 equal parts of about 22.6 ¢ each. Each step represents a frequency ratio of 21/53, or the 53rd root of 2.
Theory
53edf corresponds to 90.6041edo, similar to every fifth step of 453edo. It is related to the regular temperament which tempers out [-44 44 53 -53⟩ in the 7-limit, which is supported by 90-, 91-, 181-, 453-, 544-, 634-, 725-, 997-, 1087-, and 1178edo.
Harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +5.24 | +5.24 | -4.98 | -4.74 | -5.81 | -3.64 | -4.51 | +1.59 | +1.94 | -2.03 | +1.72 |
| Relative (%) | +39.6 | +39.6 | -37.6 | -35.8 | -43.9 | -27.5 | -34.1 | +12.0 | +14.7 | -15.3 | +13.0 | |
| Steps (reduced) |
91 (38) |
144 (38) |
210 (51) |
254 (42) |
313 (48) |
335 (17) |
370 (52) |
385 (14) |
410 (39) |
440 (16) |
449 (25) | |
| Harmonic | 37 | 41 | 43 | 47 | 53 | 59 | 61 | 67 | 71 | 73 | 79 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.03 | -5.51 | +4.74 | -3.56 | +0.36 | +0.11 | -4.62 | +5.13 | -2.55 | +2.34 | -1.97 |
| Relative (%) | +0.2 | -41.6 | +35.8 | -26.9 | +2.7 | +0.8 | -34.9 | +38.7 | -19.2 | +17.7 | -14.8 | |
| Steps (reduced) |
472 (48) |
485 (8) |
492 (15) |
503 (26) |
519 (42) |
533 (3) |
537 (7) |
550 (20) |
557 (27) |
561 (31) |
571 (41) | |
Related temperament
7-limit 453&544&634
Comma: |-44 44 53 -53>
POTE generators: ~5/4 = 386.2004, ~3796875/3764768 = 13.2434
Mapping: [<1 1 0 0|, <0 53 0 44|, <0 0 1 1|]
EDOs: 90, 91, 181, 453, 544, 634, 725, 997, 1087, 1178
7-limit 453&1178
Commas: 2460375/2458624, |6 -1 38 -33>
POTE generator: ~3796875/3764768 = 13.2432
Mapping: [<1 1 -1 -1|, <0 53 301 345|]