83edo: Difference between revisions

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-The '''83 equal temperament''' ('''83edo''') divides the octave into 83 equal parts of 14.458 [[cent]]s each. 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 &lt;&lt;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&amp;61 temperament and the 13-limit 15&amp;83 temperament. 83 is the 23rd prime number.
+{{Infobox ET}}
+{{ED intro}}
-{| class="wikitable center-all"
+== 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. 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.
-|+ Approximation of [[prime interval]]s in 83 EDO
-|-
-! colspan="2" | Prime number
-! 3
-! 5
-! 7
-! 11
-! 13
-! 17
-! 19
-! 23
-|-
-! rowspan="2" | Error
-! absolute ([[cent|¢]])
-| +6.48
-| +4.05
-| -0.15
-| -1.92
-| -1.97
-| -3.75
-| +6.10
-| -6.59
-|-
-! [[Relative error|relative]] (%)
-| +44.8
-| +28.0
-| -1.0
-| -13.3
-| -13.6
-| -25.9
-| +42.2
-| -45.6
-|-
-! colspan="2" | Degree ([[octave reduction|reduced]])
-| 132 (49)
-| 193 (27)
-| 233 (67)
-| 287 (38)
-| 307 (58)
-| 339 (7)
-| 353 (21)
-| 375 (43)
-|}
-[[Category:Equal divisions of the octave]]
+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]].
-[[Category:Prime EDO]]
-[[Category:todo:expand]]
+=== Odd harmonics ===
-[[Category:todo:explain its xenharmonic value]]
+{{Harmonics in equal|83}}
+=== Subsets and supersets ===
+edo is the 23rd [[prime edo]], following [[79edo]] and before [[89edo]].
+== Intervals ==
+{{Interval table}}
+== Instruments ==
+A [[Lumatone mapping for 83edo]] has now been demonstrated (see the Ripple and Miracle mapping for full gamut coverage).
+== Music ==
+* [https://youtube.com/shorts/vYhK-A74Kpg?si=n8bjn_AOYAPY69qx ''Microtonal improvisation in 83edo''] - [[Bryan Deister]] (Mar 2025)
+* [https://www.youtube.com/shorts/FFNFQ4H-2D4 ''83edo waltz''] (2025)