653edo
| ← 652edo | 653edo | 654edo → |
653 equal divisions of the octave (abbreviated 653edo or 653ed2), also called 653-tone equal temperament (653tet) or 653 equal temperament (653et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 653 equal parts of about 1.84 ¢ each. Each step represents a frequency ratio of 21/653, or the 653rd root of 2.
Theory
653edo is distinctly consistent to the 21-odd-limit, tempering out [39 -29 3⟩ (tricot comma) and [-20 -24 25⟩ (counterhanson comma) in the 5-limit; 2401/2400, 65625/65536, and [7 -27 13 2⟩ in the 7-limit; 3025/3024, 41503/41472, 496125/495616, and 1953125/1948617 in the 11-limit; 2080/2079, 4459/4455, 6656/6655, 10985/10976, and 170625/170368 in the 13-limit; 1225/1224, 2058/2057, 2431/2430, 2500/2499, 4914/4913, and 11271/11264 in the 17-limit; 1445/1444, 1521/1520, 1540/1539, 1729/1728, 3136/3135, 4200/4199, and 4394/4389 in the 19-limit.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | +0.036 | -0.403 | -0.373 | -0.016 | -0.711 | -0.208 | +0.190 | +0.210 | -0.481 | -0.166 |
| Relative (%) | +0.0 | +1.9 | -21.9 | -20.3 | -0.9 | -38.7 | -11.3 | +10.3 | +11.4 | -26.2 | -9.0 | |
| Steps (reduced) |
653 (0) |
1035 (382) |
1516 (210) |
1833 (527) |
2259 (300) |
2416 (457) |
2669 (57) |
2774 (162) |
2954 (342) |
3172 (560) |
3235 (623) | |
| Harmonic | 37 | 41 | 43 | 47 | 53 | 59 | 61 | 67 | 71 | 73 | |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.417 | -0.885 | -0.645 | -0.269 | -0.610 | -0.672 | +0.420 | -0.287 | +0.395 | +0.082 |
| Relative (%) | +22.7 | -48.1 | -35.1 | -14.7 | -33.2 | -36.6 | +22.9 | -15.6 | +21.5 | +4.5 | |
| Steps (reduced) |
3402 (137) |
3498 (233) |
3543 (278) |
3627 (362) |
3740 (475) |
3841 (576) |
3873 (608) |
3961 (43) |
4016 (98) |
4042 (124) | |
Subsets and supersets
653edo is the 119th prime edo.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [1035 -653⟩ | [⟨653 1035]] | −0.0113 | 0.0113 | 0.61 |
| 2.3.5 | [39 -29 3⟩, [-20 -24 25⟩ | [⟨653 1035 1516]] | +0.0503 | 0.0875 | 4.76 |
| 2.3.5.7 | 2401/2400, 65625/65536, [7 -27 13 2⟩ | [⟨653 1035 1516 1833]] | +0.0709 | 0.0838 | 4.56 |
| 2.3.5.7.11 | 2401/2400, 3025/3024, 65625/65536, 1953125/1948617 | [⟨653 1035 1516 1833 2259]] | +0.0576 | 0.0795 | 4.33 |
| 2.3.5.7.11.13 | 2080/2079, 2401/2400, 3025/3024, 10985/10976, 65625/65536 | [⟨653 1035 1516 1833 2259 2416]] | +0.0801 | 0.0882 | 4.80 |
| 2.3.5.7.11.13.17 | 1225/1224, 2058/2057, 2080/2079, 2401/2400, 4914/4913, 10985/10976 | [⟨653 1035 1516 1833 2259 2416 2669]] | +0.0759 | 0.0823 | 4.48 |
| 2.3.5.7.11.13.17.19 | 1225/1224, 1445/1444, 1521/1520, 1540/1539, 2058/2057, 2080/2079, 2401/2400 | [⟨653 1035 1516 1833 2259 2416 2669 2774]] | +0.0608 | 0.0867 | 4.72 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 21\653 | 38.59 | 45/44 | Hemitert |
| 1 | 42\653 | 77.18 | 256/245 | Tertiaseptal |
| 1 | 172/653 | 316.08 | 6/5 | Counterhanson |
| 1 | 308/653 | 566.00 | 81920/59049 | Tricot |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct