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82edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
'''82edo''', or 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 [[soothsaying]] temperament and [[support]]s [[baladic]] temperament.
{{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}}
{{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 ==
== Intervals ==
[[Category:Equal divisions of the octave|##]]
{| class="wikitable right-1 right-2 left-3 left-4 left-5"
{| class="wikitable right-1 right-2 left-3 left-4 left-5"
|+
!Degree
!Cents
!21-odd-limit
no-11 ratios
!Additional ratios
with 11s (82e val)
!Additional ratios
with 11s (patent val)
|-
|-
|0
! rowspan="2" | #
|0.000
! rowspan="2" | Cents
|1/1
! rowspan="2" | Approximate ratios*
|1/1
! colspan="2" | Additional ratios
|1/1
|-
! Using the 82e val
! Using the patent val
|-
|-
|1
| 0
|14.634
| 0.000
|
| 1/1
|
| 1/1
|
| 1/1
|-
| 1
| 14.634
| ''65/64'', 91/90
| ''55/54''
|  
|-
|-
|2
| 2
|29.268
| 29.268
|
| 49/48, 50/49, ''81/80'', ''126/125''
|
|  
|
| 45/44, 55/54
|-
|-
|3
| 3
|43.902
| 43.902
|
| 40/39
|
| ''33/32'', ''45/44''
|
|  
|-
|-
|4
| 4
|58.537
| 58.537
|
| ''25/24'', 28/27, ''36/35''
|
|  
|
| 33/32
|-
|-
|5
| 5
|73.171
| 73.171
|
| 26/25, ''27/26''
|22/21
| 22/21
|
|  
|-
|-
|6
| 6
|87.805
| 87.805
|21/20, 20/19, 19/18
| 19/18, 20/19, 21/20
|
|  
|22/21
| ''22/21''
|-
|-
|7
| 7
|102.439
| 102.439
|18/17, 17/16
| 17/16, 18/17
|
|  
|
|  
|-
|-
|8
| 8
|117.073
| 117.073
|16/15, 15/14
| 15/14, 16/15
|
|  
|
|  
|-
|-
|9
| 9
|131.707
| 131.707
|14/13, 13/12
| 14/13, 13/12
|
|  
|
|  
|-
|-
|10
| 10
|146.341
| 146.341
|
|  
|
|  
|12/11
| 12/11
|-
|-
|11
| 11
|160.976
| 160.976
|
|  
|12/11, 11/10
| 11/10, ''12/11''
|
|  
|-
|-
|12
| 12
|175.610
| 175.610
|10/9, 21/19
| 10/9, 21/19
|
|  
|11/10
| ''11/10''
|-
|-
|13
| 13
|190.244
| 190.244
|19/17
| 19/17
|
|  
|
|  
|-
|-
|14
| 14
|204.878
| 204.878
|9/8
| 9/8
|
|  
|
|  
|-
|-
|15
| 15
|219.512
| 219.512
|17/15
| 17/15
|
|  
|
|  
|-
|-
|16
| 16
|234.146
| 234.146
|8/7
| 8/7
|
|  
|
|  
|-
|-
|17
| 17
|248.780
| 248.780
|15/13
| 15/13
|22/19
| 22/19
|
|  
|-
|-
|18
| 18
|263.415
| 263.415
|7/6
| 7/6
|
|  
|22/19
| ''22/19''
|-
|-
|19
| 19
|278.049
| 278.049
|20/17
| 20/17
|
|  
|13/11
| ''13/11''
|-
|-
| 20
| 20
|292.683
| 292.683
|19/16
| 19/16
| 13/11
| 13/11
|
|
|-
|-
|21
| 21
|307.317
| 307.317
|
|  
|
|  
|
|  
|-
|-
|22
| 22
|321.951
| 321.951
|6/5
| 6/5
|
|  
|
|  
|-
|-
|23
| 23
|336.585
| 336.585
|17/14
| 17/14
|11/9
| ''11/9''
|
|  
|-
|-
|24
| 24
|351.220
| 351.220
|
|  
|
|  
|11/9
| 11/9
|-
|-
|25
| 25
|365.854
| 365.854
|16/13, 21/17, 26/21
| 16/13, 21/17, 26/21
|
|  
|
|  
|-
|-
|26
| 26
|380.488
| 380.488
|5/4
| 5/4
|
|  
|
|  
|-
|-
|27
| 27
|395.122
| 395.122
|
|  
|
|  
|
|  
|-
|-
|28
| 28
|409.756
| 409.756
|24/19, 19/15
| 19/15, 24/19
|
|  
|14/11
| ''14/11''
|-
|-
|29
| 29
|424.390
| 424.390
|
|  
|14/11
| 14/11
|
|  
|-
|-
|30
| 30
|439.024
| 439.024
|9/7
| 9/7
|22/17
| ''22/17''
|
|  
|-
|-
|31
| 31
|453.659
| 453.659
|13/10
| 13/10
|
|  
|22/17
| 22/17
|-
|-
|32
| 32
|468.293
| 468.293
|17/13, 21/16
| 17/13, 21/16
|
|  
|
|  
|-
|-
|33
| 33
|482.927
| 482.927
|
|  
|
|  
|
|  
|-
|-
|34
| 34
|497.561
| 497.561
|4/3
| 4/3
|
|  
|
|  
|-
|-
|35
| 35
|512.195
| 512.195
|
|  
|
|  
|
|  
|-
|-
|36
| 36
|526.829
| 526.829
|19/14
| 19/14
|
|  
|15/11
| ''15/11''
|-
|-
|37
| 37
|541.463
| 541.463
|26/19
| 26/19
|15/11, 11/8
| ''11/8'', 15/11
|
|  
|-
|-
|38
| 38
|556.098
| 556.098
|
|  
|
|  
|11/8
| 11/8
|-
|-
|39
| 39
|570.732
| 570.732
|18/13
| ''18/13''
|
|  
|
|  
|-
|-
|40
| 40
|585.366
| 585.366
|7/5
| 7/5
|
|  
|
|  
|-
|-
|41
| 41
|600.000
| 600.000
|24/17, 17/12
| 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"