320edo
| ← 319edo | 320edo | 321edo → |
320 equal divisions of the octave (abbreviated 320edo or 320ed2), also called 320-tone equal temperament (320tet) or 320 equal temperament (320et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 320 equal parts of exactly 3.75 ¢ each. Each step represents a frequency ratio of 21/320, or the 320th root of 2.
Theory
320edo is consistent in the 19-odd-limit and a fairly good tuning for the 19-limit. It has a flat tendency for most prime harmonics from 3 to 19, with the sole exception of 17.
The equal temperament tempers out 65625/65536 (horwell comma) and 420175/419904 (wizma) in the 7-limit and 441/440, 8019/8000 and 9801/9800 in the 11-limit, and so supports the varuna temperament, the rank-3 temperament tempering out 441/440, 8019/8000 and 9801/9800, for which it provides the optimal patent val. It also provides the optimal patent val for the rank-4 werckismic temperament tempering out 441/440. It tempers out 729/728, 1001/1000, 1575/1573, 4225/4224 and 6656/6655 in the 13-limit, leading to further temperaments for which it provides the optimal patent val, such as tempering out 441/440 with 729/728, 1001/1000 or both, or with 8019/8000, leading to an extension of varuna.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | -0.71 | -0.06 | -1.33 | -0.07 | -0.53 | +0.04 | -1.26 | +1.73 | +1.67 | -1.29 |
| Relative (%) | +0.0 | -18.8 | -1.7 | -35.4 | -1.8 | -14.1 | +1.2 | -33.7 | +46.0 | +44.6 | -34.3 | |
| Steps (reduced) |
320 (0) |
507 (187) |
743 (103) |
898 (258) |
1107 (147) |
1184 (224) |
1308 (28) |
1359 (79) |
1448 (168) |
1555 (275) |
1585 (305) | |
Subsets and supersets
Since 320 factors into 26 × 5, 320edo has subset edos 2, 4, 5, 10, 16, 20, 32, 40, 64, 80, and 160.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-507 320⟩ | [⟨320 507]] | +0.2224 | 0.2224 | 5.93 |
| 2.3.5 | [23 6 -14⟩, [-28 25 -5⟩ | [⟨320 507 743]] | +0.1574 | 0.2036 | 5.43 |
| 2.3.5.7 | 65625/65536, 235298/234375, 321489/320000 | [⟨320 507 743 898]] | +0.2361 | 0.2229 | 5.94 |
| 2.3.5.7.11 | 441/440, 8019/8000, 41503/41472, 65625/65536 | [⟨320 507 743 898 1107]] | +0.1928 | 0.2173 | 5.80 |
| 2.3.5.7.11.13 | 441/440, 729/728, 1001/1000, 4225/4224, 6656/6655 | [⟨320 507 743 898 1107 1184]] | +0.1845 | 0.1993 | 5.31 |
| 2.3.5.7.11.13.17 | 441/440, 729/728, 833/832, 1001/1000, 1089/1088, 4225/4224 | [⟨320 507 743 898 1107 1184 1308]] | +0.1565 | 0.1968 | 5.25 |
| 2.3.5.7.11.13.17.19 | 441/440, 513/512, 729/728, 833/832, 969/968, 1001/1000, 1521/1520 | [⟨320 507 743 898 1107 1184 1308 1359]] | +0.1741 | 0.1899 | 5.06 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 7\320 | 26.25 | [-2 13 -8⟩ | Sfourth (5-limit) |
| 1 | 131\320 | 491.25 | 3645/2744 | Fifthplus |
| 1 | 157\320 | 588.75 | 45/32 | Untriton (5-limit) |
| 1 | 93\320 | 348.75 | 6144/3757 | Hectosaros leap week |
| 2 | 19\320 | 71.25 | 25/24 | Narayana |
| 5 | 133\320 (5\320) |
498.75 (18.75) |
4/3 (81/80) |
Pental |
| 8 | 133\320 (9\320) |
566.25 (33.75) |
104/75 (55/54) |
Octowerck |
| 10 | 19\320 (13\320) |
71.25 (48.75) |
25/24 (36/35) |
Decavish |
| 10 | 133\320 (5\320) |
498.75 (18.75) |
4/3 (81/80) |
Decal |
| 20 | 151\320 (7\320) |
566.25 (26.25) |
165/119 (?) |
Soviet ferris wheel |
| 32 | 133\320 (3\320) |
498.75 (11.25) |
4/3 (?) |
Bezique |
| 80 | 99\320 (3\320) |
371.25 (11.25) |
2275/1836 (?) |
Mercury |
| 80 | 133\320 (1\320) |
498.75 (3.75) |
4/3 (245/243) |
Octogintic |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct