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. Due to the fifth being reached at the extremely divisible number of 210 steps, 359edo turns out to be important as an accurate supporting edo of various temperaments that divide the fifth into multiple parts.
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 678.495 ¢ Pythagorean diminished sixth; in 359edo this is reached using 203 steps, or 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 distinct