360edo: Difference between revisions

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== Theory ==

Its 23-limit patent val is <360 571 836 1011 1245 1332 1471 1529 1628|. This val tempers out the kalisma, the triaphonisma, the septendecimal bridge comma, the misty comma, hemimage, dimicomp, 2*(14/15)^10, 289/288, 352/351, 589824/588245 and 2560000000/2542277421. Its 5-limit patent val ~~supports~~ misty temperament, and the 7-limit patent val supports the 63&99 temperament with wedgie <<9 -36 9 -78 -11 122|| which tempers out misty but has a period of 1/9 rather than 1/3 octave. Two other seven limit temperaments it supports and also provides the optimal patent val for are 41&360 = <<11 76 51 95 50 -95|| and 12&360 = <<12 -48 -108 -104 -205 -116||; neither is very good though 41&360 has a TE badness lower than any alternative 7-limit temperament for which 360 gives the optimal patent val. Much better is [[Hemimage_temperaments#Degrees|degrees temperament]], the 80&140 temperament, for which 360 supplies the optimal patent val in the 11-limit and which it supports and provides an excellent tuning for in the 13-limit.

Its 23-limit patent val is <360 571 836 1011 1245 1332 1471 1529 1628|. This val tempers out the kalisma, the triaphonisma, the septendecimal bridge comma, the misty comma, hemimage, dimicomp, 2*(14/15)^10, 289/288, 352/351, 589824/588245 and 2560000000/2542277421. Its 5-limit patent val [[support]]s misty temperament, and the 7-limit patent val supports the 63&99 temperament with wedgie <<9 -36 9 -78 -11 122|| which tempers out misty but has a period of 1/9 rather than 1/3 octave. Two other seven limit temperaments it supports and also provides the optimal patent val for are 41&360 = <<11 76 51 95 50 -95|| and 12&360 = <<12 -48 -108 -104 -205 -116||; neither is very good though 41&360 has a TE badness lower than any alternative 7-limit temperament for which 360 gives the optimal patent val. Much better is [[Hemimage_temperaments#Degrees|degrees temperament]], the 80&140 temperament, for which 360 supplies the optimal patent val in the 11-limit and which it supports and provides an excellent tuning for in the 13-limit.

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.

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 == Theory ==
 {{Primes in edo|360|columns=10}}
-Its 23-limit patent val is &lt;360 571 836 1011 1245 1332 1471 1529 1628|. This val tempers out the kalisma, the triaphonisma, the septendecimal bridge comma, the misty comma, hemimage, dimicomp, 2*(14/15)^10, 289/288, 352/351, 589824/588245 and 2560000000/2542277421. Its 5-limit patent val supports misty temperament, and the 7-limit patent val supports the 63&amp;99 temperament with wedgie &lt;&lt;9 -36 9 -78 -11 122|| which tempers out misty but has a period of 1/9 rather than 1/3 octave. Two other seven limit temperaments it supports and also provides the optimal patent val for are 41&amp;360 = &lt;&lt;11 76 51 95 50 -95|| and 12&amp;360 = &lt;&lt;12 -48 -108 -104 -205 -116||; neither is very good though 41&amp;360 has a TE badness lower than any alternative 7-limit temperament for which 360 gives the optimal patent val. Much better is [[Hemimage_temperaments#Degrees|degrees temperament]], the 80&amp;140 temperament, for which 360 supplies the optimal patent val in the 11-limit and which it supports and provides an excellent tuning for in the 13-limit.
+Its 23-limit patent val is &lt;360 571 836 1011 1245 1332 1471 1529 1628|. This val tempers out the kalisma, the triaphonisma, the septendecimal bridge comma, the misty comma, hemimage, dimicomp, 2*(14/15)^10, 289/288, 352/351, 589824/588245 and 2560000000/2542277421. Its 5-limit patent val [[support]]s misty temperament, and the 7-limit patent val supports the 63&amp;99 temperament with wedgie &lt;&lt;9 -36 9 -78 -11 122|| which tempers out misty but has a period of 1/9 rather than 1/3 octave. Two other seven limit temperaments it supports and also provides the optimal patent val for are 41&amp;360 = &lt;&lt;11 76 51 95 50 -95|| and 12&amp;360 = &lt;&lt;12 -48 -108 -104 -205 -116||; neither is very good though 41&amp;360 has a TE badness lower than any alternative 7-limit temperament for which 360 gives the optimal patent val. Much better is [[Hemimage_temperaments#Degrees|degrees temperament]], the 80&amp;140 temperament, for which 360 supplies the optimal patent val in the 11-limit and which it supports and provides an excellent tuning for in the 13-limit.
 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.