# 82edo

 ← 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, from 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 & 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
with 11's (82e Val)
with 11's (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