359edo
| ← 358edo | 359edo | 360edo → |
(semiconvergent)
359 equal divisions of the octave (abbreviated 359edo or 359ed2), also called 359-tone equal temperament (359tet) or 359 equal temperament (359et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 359 equal parts of about 3.34 ¢ each. Each step represents a frequency ratio of 21/359, or the 359th root of 2.
Theory
359edo contains a very close approximation of the pure 3/2 fifth of 701.955 cents, with the 210\359 step of 701.94986 cents. In the 5-limit it tempers out the würschmidt comma and the counterschisma; in the 7-limit 2401/2400 and 3136/3125, supporting hemiwürschmidt; in the 11-limit, 8019/8000, providing the optimal patent val for 11-limit hera.
359edo supports a type of exaggerated Hornbostel mode, with an approximation of the blown fifth that he described of the pan flutes of some regions of South America [citation needed ]; the Pythagorean fifth (701.955¢) minus the Pythagorean comma (23.46¢) = 678.495¢; in 359edo this is the step 203\359 of 678.55153¢.
Pythagorean diatonic scale: 61 61 27 61 61 61 27
Exaggerated Hornbostel superdiatonic scale: 47 47 47 15 47 47 47 47 15 (fails in the position of Phi and the square root of Pi [+1\359 step of each one] [clarification needed ]).
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | -0.01 | +1.43 | +0.53 | +0.21 | -1.53 | -1.33 | -0.02 | +0.14 | -0.05 | +1.48 |
| Relative (%) | +0.0 | -0.2 | +42.8 | +16.0 | +6.4 | -45.8 | -39.9 | -0.6 | +4.1 | -1.5 | +44.4 | |
| Steps (reduced) |
359 (0) |
569 (210) |
834 (116) |
1008 (290) |
1242 (165) |
1328 (251) |
1467 (31) |
1525 (89) |
1624 (188) |
1744 (308) |
1779 (343) | |
Subsets and supersets
359edo is the 72nd prime edo. 718edo, which doubles it, provides a good correction to the harmonics 5, 13, 17, and 31.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-569 359⟩ | [⟨359 569]] | +0.0016 | 0.0016 | 0.05 |
| 2.3.5 | 393216/390625, [-69 45 -1⟩ | [⟨359 569 834]] | −0.2042 | 0.2910 | 8.71 |
| 2.3.5.7 | 2401/2400, 3136/3125, [-18 24 -5 -3⟩ | [⟨359 569 834 1008]] | −0.2007 | 0.2521 | 7.54 |
| 2.3.5.7.11 | 2401/2400, 3136/3125, 8019/8000, 42592/42525 | [⟨359 569 834 1008 1242]] | −0.1729 | 0.2322 | 6.95 |
| 2.3.5.7.11.13 | 729/728, 847/845, 1001/1000, 1716/1715, 3136/3125 | [⟨359 569 834 1008 1242 1328]] (359f) | −0.2257 | 0.2426 | 7.26 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 58\359 | 193.87 | 28/25 | Hemiwürschmidt |
| 1 | 116\359 | 387.74 | 5/4 | Würschmidt (5-limit) |
| 1 | 149\359 | 498.05 | 4/3 | Counterschismic |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct