202edo
202 equal divisions of the octave (abbreviated 202edo or 202ed2), also called 202-tone equal temperament (202tet) or 202 equal temperament (202et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 202 equal parts of about 5.94 ¢ each. Each step represents a frequency ratio of 21/202, or the 202nd root of 2.
Theory
202edo is consistent to the 9-odd-limit with a flat tendency in harmonics 3, 5, and 7. It also has a decent harmonic 11, though it is sharp unlike the previous harmonics, with 11/9 barely exceeding 50% relative error. Despite this, it is most notable in the 11-limit, providing the optimal patent val for many temperaments tempering out 243/242.
Using the patent val, 202et tempers out 2401/2400, 19683/19600 and 65625/65536 in the 7-limit, and 243/242, 441/440, 4000/3993 in the 11-limit. It also notably tempers out the quartisma, equating a stack of five 33/32 quartertones with 7/6. It is the optimal patent val for the 11-limit rank-2 temperaments harry and tertiaseptal, the rank-3 temperament jove tempering out 243/242 and 441/440, which also tempers out 540/539, and the rank-4 rastmic temperament, which tempers out 243/242.
It extends less well to the 13-limit, with harmonic 13 being about halfway between its steps. Nonetheless, the patent val tempers out 351/350, 364/363, 676/675, 729/728, and 2080/2079, supporting jovial and jovis, as well as 13-limit harry. Primes 17 and 23 are quite sharp, but prime 19 is accurate. 202edo can thus be considered a 2.3.5.7.11.13.19-subgroup temperament with a mostly flat tendency, with the exception of prime 11. The intervals 11/9, 13/11, and their octave complements are the only inconsistencies in the no-17 21-odd-limit, and the no-11 no-17 21-odd limit is completely consistent, though one may also want to exclude prime 13 given its inaccuracy, giving us the 2.3.5.7.19 subgroup.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | -0.96 | -0.18 | -0.51 | +1.16 | -2.90 | +1.98 | -0.48 | +1.43 | -1.85 | +1.50 |
| Relative (%) | +0.0 | -16.2 | -2.9 | -8.6 | +19.5 | -48.9 | +33.3 | -8.1 | +24.0 | -31.2 | +25.2 | |
| Steps (reduced) |
202 (0) |
320 (118) |
469 (65) |
567 (163) |
699 (93) |
747 (141) |
826 (18) |
858 (50) |
914 (106) |
981 (173) |
1001 (193) | |
Subsets and supersets
Since 202 factors into 2 × 101, 202edo contains 2edo and 101edo as subset edos.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3.5 | [-13 17 -6⟩, [23 6 -14⟩ | [⟨202 320 469]] | +0.2280 | 0.2710 | 4.56 |
| 2.3.5.7 | 2401/2400, 19683/19600, 65625/65536 | [⟨202 320 469 567]] | +0.2164 | 0.2356 | 3.97 |
| 2.3.5.7.11 | 243/242, 441/440, 4000/3993, 65625/65536 | [⟨202 320 469 567 699]] | +0.1061 | 0.3049 | 5.13 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 13\202 | 77.23 | 256/245 | Tertiaseptal |
| 1 | 51\202 | 302.97 | 25/21 | Quinmite |
| 1 | 85\202 | 504.95 | 104976/78125 | Countermeantone |
| 1 | 87\202 | 516.83 | 27/20 | Larry |
| 2 | 12\202 | 71.29 | 25/24 | Narayana |
| 2 | 87\202 (14\202) |
516.83 (83.17) |
27/20 (21/20) |
Harry |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct
Scales
Music
- Home Planet Nostalgia – in Oktone scale