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→Instruments: Insert music section after this, starting with Bryan Deister's ''microtonal improvisation in 82edo'' (2025)
 
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The 82 equal temperament divides the octave into 82 equal parts of 14.634 cents each. The patent val is contorted in the 11-limit, from 82 = 2*41, and in the 13-limit the patent val tempers out 169/168 and 676/675, and in the 17-limit tempers out 273/272. It provides the optimal patent val for [[Magic_family#Soothsaying|soothsaying]] temperament and supports [[Gamelismic clan#Baladic|baladic]] temperament.
{{Infobox ET}}
{{ED intro}}
 
== Theory ==
82edo's [[patent val]] is [[contorted]] in the [[11-limit]], since {{nowrap|82 {{=}} 2 × 41}}. In the [[13-limit]] the patent val tempers out [[169/168]] and [[676/675]], and in the [[17-limit]] tempers out [[273/272]]. It provides the optimal patent val for [[soothsaying]] temperament and [[support]]s [[baladic]] temperament. The 82d val tempers out [[50/49]] and is an excellent tuning for [[astrology]] and [[byhearted]], surpassing their optimal patent vals. The alternative 82e val tempers out [[121/120]] instead.
 
=== Prime harmonics ===
{{Harmonics in equal|82}}
 
=== Subsets and supersets ===
82edo contains [[2edo]] and [[41edo]] as subsets. [[164edo]], which doubles it, is a notable tuning.
 
A step of 82edo is exactly 30 [[mina]]s.
 
== Intervals ==
{| class="wikitable right-1 right-2 left-3 left-4 left-5"
|-
! rowspan="2" | #
! rowspan="2" | Cents
! rowspan="2" | Approximate ratios*
! colspan="2" | Additional ratios
|-
! Using the 82e val
! Using the patent val
|-
| 0
| 0.000
| 1/1
| 1/1
| 1/1
|-
| 1
| 14.634
| ''65/64'', 91/90
| ''55/54''
|
|-
| 2
| 29.268
| 49/48, 50/49, ''81/80'', ''126/125''
|
| 45/44, 55/54
|-
| 3
| 43.902
| 40/39
| ''33/32'', ''45/44''
|
|-
| 4
| 58.537
| ''25/24'', 28/27, ''36/35''
|
| 33/32
|-
| 5
| 73.171
| 26/25, ''27/26''
| 22/21
|
|-
| 6
| 87.805
| 19/18, 20/19, 21/20
|
| ''22/21''
|-
| 7
| 102.439
| 17/16, 18/17
|
|
|-
| 8
| 117.073
| 15/14, 16/15
|
|
|-
| 9
| 131.707
| 14/13, 13/12
|
|
|-
| 10
| 146.341
|
|
| 12/11
|-
| 11
| 160.976
|
| 11/10, ''12/11''
|
|-
| 12
| 175.610
| 10/9, 21/19
|
| ''11/10''
|-
| 13
| 190.244
| 19/17
|
|
|-
| 14
| 204.878
| 9/8
|
|
|-
| 15
| 219.512
| 17/15
|
|
|-
| 16
| 234.146
| 8/7
|
|
|-
| 17
| 248.780
| 15/13
| 22/19
|
|-
| 18
| 263.415
| 7/6
|
| ''22/19''
|-
| 19
| 278.049
| 20/17
|
| ''13/11''
|-
| 20
| 292.683
| 19/16
| 13/11
|
|-
| 21
| 307.317
|
|
|
|-
| 22
| 321.951
| 6/5
|
|
|-
| 23
| 336.585
| 17/14
| ''11/9''
|
|-
| 24
| 351.220
|
|
| 11/9
|-
| 25
| 365.854
| 16/13, 21/17, 26/21
|
|
|-
| 26
| 380.488
| 5/4
|
|
|-
| 27
| 395.122
|
|
|
|-
| 28
| 409.756
| 19/15, 24/19
|
| ''14/11''
|-
| 29
| 424.390
|
| 14/11
|
|-
| 30
| 439.024
| 9/7
| ''22/17''
|
|-
| 31
| 453.659
| 13/10
|
| 22/17
|-
| 32
| 468.293
| 17/13, 21/16
|
|
|-
| 33
| 482.927
|
|
|
|-
| 34
| 497.561
| 4/3
|
|
|-
| 35
| 512.195
|
|
|
|-
| 36
| 526.829
| 19/14
|
| ''15/11''
|-
| 37
| 541.463
| 26/19
| ''11/8'', 15/11
|
|-
| 38
| 556.098
|
|
| 11/8
|-
| 39
| 570.732
| ''18/13''
|
|
|-
| 40
| 585.366
| 7/5
|
|
|-
| 41
| 600.000
| 17/12, 24/17
|
|
|-
| …
| …
|
|
|
|}
<nowiki />* As a no-11 19-limit temperament
 
== Notation ==
=== Ups and downs notation ===
60edo can be notated using [[ups and downs notation]] using [[Helmholtz–Ellis]] accidentals:
 
{{Sharpness-sharp8}}
 
== Approximation to JI ==
=== Zeta peak index ===
{{ZPI
| zpi = 448
| steps = 81.9541455954050
| step size = 14.6423343356444
| tempered height = 6.653983
| pure height = 5.154524
| integral = 0.941321
| gap = 14.718732
| octave = 1200.67141552284
| consistent = 8
| distinct = 8
}}
 
== Instruments ==
* [[Lumatone mapping for 82edo]]
 
== Music ==
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/p9xUY8EU7Zg ''microtonal improvisation in 82edo''] (2025)
Retrieved from "https://en.xen.wiki/w/82edo"