83edo: Difference between revisions

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== Theory ==

The [[3/1|harmonic 3]] is 6.5 cents sharp and the [[5/1|5]] is 4 cents sharp, with [[7/1|7]], [[11/1|11]], and [[13/1|13]] more accurate but a little flat. Using the [[patent val]], it [[tempering out|tempers out]] [[15625/15552]] in the 5-limit and [[686/675]], [[4000/3969]] and [[6144/6125]] in the 7-limit, and provides the [[optimal patent val]] for the 7-limit 27 & 56 temperament with wedgie {{multival| 5 18 17 17 13 -11 }}. In the 11-limit it tempers out [[121/120]], [[176/175]] and [[385/384]], and in the 13-limit [[91/90]], [[169/168]] and [[196/195]], and it provides the optimal patent val for the 11-limit 22 & 61 temperament and the 13-limit 15 & 83 temperament.

Every odd harmonic between the 7th and the 17th is tuned flatly. As a consequence, this tuning provides a good approximation of the 7:9:11:13:15:17 [[hexad]], and especially of the 9:11:13 [[triad]].

=== Odd harmonics ===

The 3/1 is 6.5 cents sharp and the 5/1 is 4 cents sharp, with 7, 11, and 13 more accurate but a little flat. It tempers out 15625/15552 in the 5-limit and 686/675, 4000/3969 and 6144/6125 in the 7-limit, and provides the optimal patent val for the 7-limit 27&56 temperament with wedgie {{multival|5 18 17 17 13 -11}}. In the 11-limit it tempers out 121/120, 176/175 and 385/384, and in the 13-limit 91/90, 169/168 and 196/195, and it provides the optimal patent val for the 11-limit 22&61 temperament and the 13-limit 15&83 temperament. 83edo is the 23rd [[prime EDO]].

~~Every odd harmonic between the 7th~~ and ~~the 17th~~ is ~~tuned flatly. As a consequence, this tuning provides a good approximation of the 7:9:11:13:15:17 [[hexad]], and especially of~~ the ~~9:11:13~~ [[~~triad~~]].

=== Subsets and supersets ===

83edo is the 23rd [[prime edo]].

== Intervals ==

~~{{Todo| expand }}~~

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 == Theory ==
+The [[3/1|harmonic 3]] is 6.5 cents sharp and the [[5/1|5]] is 4 cents sharp, with [[7/1|7]], [[11/1|11]], and [[13/1|13]] more accurate but a little flat. Using the [[patent val]], it [[tempering out|tempers out]] [[15625/15552]] in the 5-limit and [[686/675]], [[4000/3969]] and [[6144/6125]] in the 7-limit, and provides the [[optimal patent val]] for the 7-limit 27 &amp; 56 temperament with wedgie {{multival| 5 18 17 17 13 -11 }}. In the 11-limit it tempers out [[121/120]], [[176/175]] and [[385/384]], and in the 13-limit [[91/90]], [[169/168]] and [[196/195]], and it provides the optimal patent val for the 11-limit 22 & 61 temperament and the 13-limit 15 & 83 temperament.
+Every odd harmonic between the 7th and the 17th is tuned flatly. As a consequence, this tuning provides a good approximation of the 7:9:11:13:15:17 [[hexad]], and especially of the 9:11:13 [[triad]].
+=== Odd harmonics ===
 {{Harmonics in equal|83}}
-The 3/1 is 6.5 cents sharp and the 5/1 is 4 cents sharp, with 7, 11, and 13 more accurate but a little flat. It tempers out 15625/15552 in the 5-limit and 686/675, 4000/3969 and 6144/6125 in the 7-limit, and provides the optimal patent val for the 7-limit 27&amp;56 temperament with wedgie {{multival|5 18 17 17 13 -11}}. In the 11-limit it tempers out 121/120, 176/175 and 385/384, and in the 13-limit 91/90, 169/168 and 196/195, and it provides the optimal patent val for the 11-limit 22&61 temperament and the 13-limit 15&83 temperament. 83edo is the 23rd [[prime EDO]].
-Every odd harmonic between the 7th and the 17th is tuned flatly. As a consequence, this tuning provides a good approximation of the 7:9:11:13:15:17 [[hexad]], and especially of the 9:11:13 [[triad]].
+=== Subsets and supersets ===
+edo is the 23rd [[prime edo]].
 == Intervals ==
 {{Interval table}}
-{{Todo| expand }}