83edo: Difference between revisions

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== Theory ==

The [[3/1|harmonic 3]] is 6.5{{c}} sharp and the [[5/1|5]] is 4{{c}} 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 {{nowrap|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 {{nowrap|22 & 61}} temperament and the 13-limit 15 & 83 temperament.

The [[3/1|harmonic 3]] is 6.5{{c}} sharp and the [[5/1|5]] is 4{{c}} 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. 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 {{nowrap|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]].

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 == Theory ==
-The [[3/1|harmonic 3]] is 6.5{{c}} sharp and the [[5/1|5]] is 4{{c}} 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 {{nowrap|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 {{nowrap|22 &amp; 61}} temperament and the 13-limit 15 & 83 temperament.
+The [[3/1|harmonic 3]] is 6.5{{c}} sharp and the [[5/1|5]] is 4{{c}} 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. 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 {{nowrap|22 &amp; 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]].