301edo
| ← 300edo | 301edo | 302edo → |
301 equal divisions of the octave (abbreviated 301edo or 301ed2), also called 301-tone equal temperament (301tet) or 301 equal temperament (301et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 301 equal parts of about 3.99 ¢ each. Each step represents a frequency ratio of 21/301, or the 301st root of 2.
Theory
301edo is a strong 7-limit system, and distinctly consistent through the 17-odd-limit. The equal temperament tempers out 32805/32768 in the 5-limit, 2401/2400 in the 7-limit, 3025/3024, 5632/5625, 8019/8000 in the 11-limit, 729/728, 847/845, 1001/1000, 1716/1715, 2200/2197 in the 13-limit, and 561/560, 833/832, 1089/1088, 1156/1155, 1275/1274 and 1701/1700 in the 17-limit. Since it tempers out both 32805/32768 and 2401/2400, it supports the sesquiquartififths temperament.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | -0.29 | +0.40 | -0.06 | -1.15 | +0.67 | -1.30 | +1.49 | +1.63 | -1.01 | -0.85 |
| Relative (%) | +0.0 | -7.4 | +10.0 | -1.4 | -28.9 | +16.8 | -32.6 | +37.4 | +40.8 | -25.2 | -21.3 | |
| Steps (reduced) |
301 (0) |
477 (176) |
699 (97) |
845 (243) |
1041 (138) |
1114 (211) |
1230 (26) |
1279 (75) |
1362 (158) |
1462 (258) |
1491 (287) | |
Subsets and supersets
Since 301 factors into 7 × 43, 301edo has 7edo and 43edo as its subsets. This is related to the proposal of the deaf French mathematician and acoustician Joseph Sauveur to divide the octave in 43 parts called merides, and those into seven more parts called heptamerides. Back in the days of slide rules and log tables, this made sense since by multiplying the log base ten of the interval in question by 1000, one came close to how many heptamerides it constituted.
301edo also tempers out [168 -43 -43⟩ and 5250987/5242880, so it supports the meridic temperament.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-477 301⟩ | [⟨301 477]] | +0.0927 | 0.0927 | 2.33 |
| 2.3.5 | 32805/32768, [3 45 -32⟩ | [⟨301 477 699]] | +0.0048 | 0.1456 | 3.65 |
| 2.3.5.7 | 2401/2400, 32805/32768, 1959552/1953125 | [⟨301 477 699 845]] | +0.0085 | 0.1262 | 3.17 |
| 2.3.5.7.11 | 2401/2400, 3025/3024, 5632/5625, 8019/8000 | [⟨301 477 699 845 1041]] | +0.0734 | 0.1720 | 4.31 |
| 2.3.5.7.11.13 | 729/728, 847/845, 1001/1000, 1716/1715, 3025/3024 | [⟨301 477 699 845 1041 1114]] | +0.0310 | 0.1834 | 4.60 |
| 2.3.5.7.11.13.17 | 561/560, 729/728, 833/832, 847/845, 1001/1000, 1089/1088 | [⟨301 477 699 845 1041 1114 1230]] | +0.0721 | 0.1973 | 4.95 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 25\301 | 99.67 | 18/17 | Quintaschis |
| 1 | 44\301 | 175.42 | 448/405 | Sesquiquartififths / sesquart (301e) |
| 1 | 68\301 | 271.10 | 90/77 | Quasiorwell |
| 1 | 76\301 | 302.99 | 25/21 | Quinmite |
| 1 | 125\301 | 498.34 | 4/3 | Helmholtz |
| 7 | 125\301 (4\301) |
498.34 (15.95) |
4/3 (245/243) |
Septant |
| 43 | 125\301 (1\301) |
498.34 (3.99) |
4/3 (540/539) |
Meridic |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct