653edo
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, and the 23-odd-limit if not for the 23/13 and its octave complement barely missing the mark. Although the 25 is flat enough to create more inconsistencies, the 29 and 31 blend well with the lower primes, together making it a fairly strong 31-limit system.
As an equal temperament, it tempers out [39 -29 3⟩ (alphatricot 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; 875/874, 1105/1104 among others in the 23-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 | 79 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.417 | -0.885 | -0.645 | -0.269 | -0.610 | -0.672 | +0.420 | -0.287 | +0.395 | +0.082 | -0.678 |
| Relative (%) | +22.7 | -48.1 | -35.1 | -14.7 | -33.2 | -36.6 | +22.9 | -15.6 | +21.5 | +4.5 | -36.9 | |
| Steps (reduced) |
3402 (137) |
3498 (233) |
3543 (278) |
3627 (362) |
3740 (475) |
3841 (576) |
3873 (608) |
3961 (43) |
4016 (98) |
4042 (124) |
4116 (198) | |
Subsets and supersets
653edo is the 119th prime edo. As such, it does not contain any nontrivial subset 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 |
| 2.3.5.7.11.13.17.19.23 | 875/874, 1105/1104, 1225/1224, 1445/1444, 1521/1520, 1540/1539, 2058/2057, 2080/2079 | [⟨653 1035 1516 1833 2259 2416 2669 2774 2954]] | +0.0489 | 0.0884 | 4.81 |
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 | Alphatricot |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct