320edo
The 320 equal divisions of the octave (320edo), or the 320(-tone) equal temperament (320tet, 320et) when viewed from a regular temperament perspective, divides the octave into 320 equal parts of precisely 3.75 cents each.
Theory
320et tempers out 65625/65536 (horwell) 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 a rank-3 temperament.
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) | |
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 Octave |
Generator (Reduced) |
Cents (Reduced) |
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 (no-5, no-11) |
| 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 |
| 80 | 133\320 (1\320) |
498.75 (3.75) |
4/3 (245/243) |
Octogintic |