360edo
| ← 359edo | 360edo | 361edo → |
One step of 360edo is known as the Dröbisch angle.
Theory
360edo is consistent in the 7-limit. Its 5-limit patent val supports 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 1/10th 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 the degrees temperament, the 80&140 temperament with period 20. Eliora proposes a 7-limit reenactment temperament for 360edo, defined as 188 & 360 and named after the YouTubers cs188 and radicalfaith360. It has a comma basis 2097152/2083725 and [0, -19, -10, 19⟩.
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.
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, such as calling steps degrees, deriving them from Moritz Dröbisch's proposal of calling the step an angle.
| Degree | Name | Calendar notation (if unison is Jan 1) | Ratio |
|---|---|---|---|
| 0 | Prime, unison | January 1 | 1/1 |
| 1 | Degree, grad, schisma | January 2 | 32805/32768 |
| 30 | Dodecaphonic semitone | February 1 | 89/84 |
| 36 | Septimal diatonic semitone, decioctave | February 6 | 15/14 |
| 60 | Dodecaphonic major second | March 1 | |
| 90 | Dodecaphonic minor third | April 1 | |
| 116 | Classical major third | April 26 | |
| 120 | May 1 | ||
| 150 | June 1 | ||
| 180 | Symmetric tritone | July 1 | |
| 210 | Dodecaphonic perfect fifth | August 1 | 442/295 |
| 211 | Just perfect fifth | August 2 | 3/2 |
| 240 | September 1 | ||
| 270 | October 1 | ||
| 291 | Harmonic seventh | October 21 | |
| 300 | November 1 | ||
| 330 | December 1 | ||
| 360 | Octave | January 1 |
Regular temperament properties
Rank-2 temperaments
| Periods per 8ve |
Generator (reduced) |
Cents (reduced) |
Associated ratio |
Temperaments |
|---|---|---|---|---|
| 1 | 119\360 | 396.67 | 44/35 | Squarschmidt |
| 2 | 53\360 | 176.67 | 448/405 | Quatracot |
| 3 | 211\360 (91\360) |
703.33 (303.33) |
3/2 | Misty |
| 4 | 23\360 | 76.67 | 4302592/4100625 | Reenactment |
| 9 | 211\360 (11\360) |
703.33 (36.67) |
3/2 | Trimisty |
| 20 | 211\360 (13\360) |
703.33 (43.33) |
3/2 (45/44) |
Degrees |
Music
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.