360edo
| ← 359edo | 360edo | 361edo → |
360 equal divisions of the octave (abbreviated 360edo or 360ed2), also called 360-tone equal temperament (360tet) or 360 equal temperament (360et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 360 equal parts of about 3.33 ¢ each. Each step represents a frequency ratio of 21/360, or the 360th root of 2.
Theory
360edo is consistent to the 7-odd-limit, but harmonic 3 is about halfway between its steps. It can also be used with 2.5.9.13 subgroup.
In the 5-limit, the patent val supports the misty temperament, and in the 7-limit 360edo supports the trimisty (name proposed by Eliora) 63 & 99 temperament with the comma basis {10976/10935, 2097152/2083725}, which is similar to the misty temperament but has a period of 1/9- rather than 1/3-octave.
360edo provides the optimal patent val in the 11-limit, and otherwise a good tuning in the 13-limit for degrees, the 140 & 220 temperament with period 1\20. Aside from that, it provides the optimal patent val for the 41 & 360 temperament with comma basis {10976/10935, 16384000000/16209796869}, on which it has lower badness than any other 7-limit temperament for which 360edo gives the optimal patent val. It also supports 12 & 360 with the comma basis {390625/388962, 67108864/66430125}.
Aside from the patent val, there is a number of mappings to be considered. The 360d val, ⟨360 571 836 1010], tempers out 3136/3125, 5120/5103, and extends the misty temperament in to the 7-limit. It is also a tuning for the 12th-octave magnesium temperament.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +1.38 | +0.35 | +1.17 | -0.58 | -1.32 | -0.53 | -1.60 | -1.62 | -0.85 | -0.78 | -1.61 |
| Relative (%) | +41.3 | +10.6 | +35.2 | -17.3 | -39.5 | -15.8 | -48.1 | -48.7 | -25.4 | -23.4 | -48.2 | |
| Steps (reduced) |
571 (211) |
836 (116) |
1011 (291) |
1141 (61) |
1245 (165) |
1332 (252) |
1406 (326) |
1471 (31) |
1529 (89) |
1581 (141) |
1628 (188) | |
Subsets and supersets
360 is the 13th highly composite edo, with many proper divisors: 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180. One step of 360edo is known as the Dröbisch angle, an interval size measure first proposed by Moritz Dröbisch in the 19th century at first merely by the name "angle".
Table of intervals
Eliora proposes notating 360edo with calendar dates, Jan 1 being the tonic, Jan 2 being the next step, etc, and each month having even 30 days. The notation is convenient because 1 month in this scenario is equal to 1 semitone, and corresponds to 12edo.
Any other notation system involving the number 360 can also be used.
See: Table of 360edo intervals
Regular temperament properties
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 119\360 | 396.67 | 44/35 | Squarschmidt |
| 2 | 53\360 | 176.67 | 448/405 | Quatracot |
| 3 | 149\360 (29\360) |
703.33 (303.33) |
4/3 (135/128) |
Misty |
| 4 | 23\360 | 76.67 | 4302592/4100625 | Reenactment |
| 9 | 149\360 (29\360) |
703.33 (36.67) |
4/3 (135/128) |
Trimisty |
| 12 | 73\360 (13\360) |
243.333 (43.333) |
3145728/2734375 (?) |
Magnesium (360d) |
| 20 | 149\360 (5\360) |
703.33 (43.33) |
4/3 (126/125) |
Degrees |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct
Music
- Idyllic Tribe (2022)
Application as a logarithmic scale outside of music
360edo is used in the eyeborg, which maps its scale degrees onto color hues, thus converting color into sound waves. The device was originally intended to help colorblind individuals.