360edo
| ← 359edo | 360edo | 361edo → |
Theory
360edo is consistent to the 7-odd-limit, but harmonic 3 is about halfway between its steps. Its 5-limit 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. In addition, 360edo 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. 360edo tempers out the linus comma, meaning 15/14 corresponds to 1/10 of the octave, 36 steps.
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.
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.
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 |
| 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.