360edo: Difference between revisions

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== Theory ==

360edo is [[consistent]] to the [[7-odd-limit]], but [[harmonic]] [[3/1|3]] is about halfway between its steps. ~~Its 5-limit [[patent val]] [[support]]s 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 is [[consistent]] to the [[7-odd-limit]], but [[harmonic]] [[3/1|3]] is about halfway between its steps. It can also be used with 2.5.9.13 subgroup.

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.

In the 5-limit, the [[patent val]] [[support]]s 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 ===

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| 4/3<br>(135/128)

| [[Trimisty]]

|-

|12

|73\360

(13\360)

|243.333

(43.333)

|3145728/2734375

(?)

|[[Magnesium]] (360d)

|-

| 20

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 == Theory ==
-edo is [[consistent]] to the [[7-odd-limit]], but [[harmonic]] [[3/1|3]] is about halfway between its steps. Its 5-limit [[patent val]] [[support]]s 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.
+edo is [[consistent]] to the [[7-odd-limit]], but [[harmonic]] [[3/1|3]] is about halfway between its steps. It can also be used with 2.5.9.13 subgroup.
-edo provides the optimal patent val in the 11-limit, and otherwise a good tuning in the 13-limit for [[degrees]], the 140 &amp; 220 temperament with period 1\20.
+In the 5-limit, the [[patent val]] [[support]]s 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.
+edo provides the optimal patent val in the 11-limit, and otherwise a good tuning in the 13-limit for [[degrees]], the 140 &amp; 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 ===
@@ Line 58: / Line 62: @@
 | 4/3<br>(135/128)
 | [[Trimisty]]
+|-
+|12
+|73\360
+(13\360)
+|243.333
+(43.333)
+|3145728/2734375
+(?)
+|[[Magnesium]] (360d)
 |-
 | 20