360edo: Difference between revisions
m Moving from Category:Highly melodic to Category:Highly composite using Cat-a-lot |
→Theory: miscellaneous cleanup |
||
| Line 1: | Line 1: | ||
{{Infobox ET}} | {{Infobox ET}} | ||
'''360 equal divisions of the octave''' ('''360edo'''), or '''360-tone equal temperament''' ('''360tet'''), '''360 equal temperament''' ('''360et''') when viewed from a [[regular temperament]] perspective, is the tuning system that divides the [[octave]] into 360 [[equal]] parts of {{ExactlyOrAbout| {{#expr: 1200/360 round 16}} }}{{cent}} each, a step size known as '''the Dröbisch angle'''. | '''360 equal divisions of the octave''' ('''360edo'''), or '''360-tone equal temperament''' ('''360tet'''), '''360 equal temperament''' ('''360et''') when viewed from a [[regular temperament]] perspective, is the tuning system that divides the [[octave]] into 360 [[equal]] parts of {{ExactlyOrAbout| {{#expr: 1200/360 round 16}} }}{{cent}} each, a step size known as '''the Dröbisch angle'''. | ||
== Theory == | == Theory == | ||
{{Primes in edo|360|columns=10}} | {{Primes in edo|360|columns=10}} | ||
360 has 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. | 360 has 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. | ||
360 is the 13th [[highly composite EDO]]. | |||
Its 5-limit patent val [[support]]s [[misty]] temperament. | Its 5-limit patent val [[support]]s [[misty]] temperament. | ||
360edo is consistent in the 7-limit. In it, it 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 [[15/14 equal-step tuning|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 [[Hemimage_temperaments#Degrees|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 {{Monzo|0, -19, -10, 19}}. | |||
=== Proposed notation === | |||
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]]. | |||
=== Miscellaneous properties === | |||
In the 360b val, 360edo's fifth is the same as 12edo. Coincidentally, the difference between a just fifth and a 12edo one is known as the grad, being a variant of translation of "degree", and 1/360th of a circle is a degree. | In the 360b val, 360edo's fifth is the same as 12edo. Coincidentally, the difference between a just fifth and a 12edo one is known as the grad, being a variant of translation of "degree", and 1/360th of a circle is a degree. | ||
360edo is used in the [[wikipedia:Eyeborg|eyeborg]], which maps its scale degrees onto color hues, thus converting color into sound waves. The device was originally intended to help colorblind individuals. | |||
== Rank two temperaments by generator == | == Rank two temperaments by generator == | ||
Revision as of 20:01, 14 November 2022
| ← 359edo | 360edo | 361edo → |
360 equal divisions of the octave (360edo), or 360-tone equal temperament (360tet), 360 equal temperament (360et) when viewed from a regular temperament perspective, is the tuning system that divides the octave into 360 equal parts of about 3.33 ¢ each, a step size known as the Dröbisch angle.
Theory
Script error: No such module "primes_in_edo". 360 has 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.
360 is the 13th highly composite EDO.
Its 5-limit patent val supports misty temperament.
360edo is consistent in the 7-limit. In it, it 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⟩.
Proposed notation
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.
Miscellaneous properties
In the 360b val, 360edo's fifth is the same as 12edo. Coincidentally, the difference between a just fifth and a 12edo one is known as the grad, being a variant of translation of "degree", and 1/360th of a circle is a degree.
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.
Rank two temperaments by generator
| Periods
per octave |
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 |
Table of intervals
| Step | 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 |