360edo

← 359edo

360edo

361edo →

Prime factorization	23 × 32 × 5 (highly composite)
Step size	3.33333 ¢
Fifth	211\360 (703.333 ¢)
Semitones (A1:m2)	37:25 (123.3 ¢ : 83.33 ¢)
Dual sharp fifth	211\360 (703.333 ¢)
Dual flat fifth	210\360 (700 ¢) (→ 7\12)
Dual major 2nd	61\360 (203.333 ¢)
Consistency limit	7
Distinct consistency limit	7

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

Approximation of odd harmonics in 360edo

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.

Table of selected intervals

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

• Idyllic Tribe by Eliora

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.