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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&56 temperament with wedgie <<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. 83 is the 23rd prime number.
{{EDO intro|83}}


{| class="wikitable center-all"
== Theory ==
|-
{{Harmonics in equal|83}}
|+ Approximation of [[prime interval]]s in 83 EDO
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]].
|-
 
! colspan="2" | Prime number
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]].
! 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|##]] <!-- 2-digit number -->
[[Category:Equal divisions of the octave|##]] <!-- 2-digit number -->
[[Category:Prime EDO]]
[[Category:Prime EDO]]


[[Category:todo:expand]]
{{todo|expand}}
[[Category:todo:explain its xenharmonic value]]
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