82edo: Difference between revisions

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→Instruments: Insert music section after this, starting with Bryan Deister's ''microtonal improvisation in 82edo'' (2025)
 
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'''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 supports [[baladic]] temperament.
{{Infobox ET}}
{{ED intro}}


[[Category:Edo]]
== 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)

Latest revision as of 21:30, 14 July 2025

← 81edo 82edo 83edo →
Prime factorization 2 × 41
Step size 14.6341 ¢ 
Fifth 48\82 (702.439 ¢) (→ 24\41)
Semitones (A1:m2) 8:6 (117.1 ¢ : 87.8 ¢)
Consistency limit 9
Distinct consistency limit 9

82 equal divisions of the octave (abbreviated 82edo or 82ed2), also called 82-tone equal temperament (82tet) or 82 equal temperament (82et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 82 equal parts of about 14.6 ¢ each. Each step represents a frequency ratio of 21/82, or the 82nd root of 2.

Theory

82edo's patent val is contorted in the 11-limit, since 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 supports 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

Approximation of prime harmonics in 82edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 +0.48 -5.83 -2.97 +4.78 -6.38 -2.52 -4.83 +0.99 -5.19 -3.57
Relative (%) +0.0 +3.3 -39.8 -20.3 +32.7 -43.6 -17.2 -33.0 +6.8 -35.4 -24.4
Steps
(reduced) 		82
(0) 		130
(48) 		190
(26) 		230
(66) 		284
(38) 		303
(57) 		335
(7) 		348
(20) 		371
(43) 		398
(70) 		406
(78)

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 minas.

Intervals

# Cents Approximate ratios* 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
… …

* 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:

Step offset 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
Sharp symbol  
Flat symbol

Approximation to JI

Zeta peak index

Tuning Strength Octave (cents) Integer limit
ZPI Steps
per 8ve 		Step size
(cents) 		Height Integral Gap Size Stretch Consistent Distinct
Tempered Pure
448zpi 81.954146 14.642334 6.653983 5.154524 0.941321 14.718732 1200.671416 0.671416 8 8

Instruments

Music

Bryan Deister
Retrieved from "https://en.xen.wiki/index.php?title=82edo&oldid=204529"