82edo: Difference between revisions

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== Theory ==
== 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 [[support]]s [[baladic]] temperament.
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 [[support]]s [[baladic]] temperament. The alternative 82e val tempers out [[121/120]] instead.  


=== Prime harmonics ===
=== Prime harmonics ===
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! #
! #
! Cents
! Cents
! 21-odd-limit<br>no-11 ratios
! Approximate Ratios*
! Additional Ratios<br>with 11's (82e Val)
! Additional Ratios<br>with 11's (82e Val)
! Additional Ratios<br>with 11's (Patent Val)
! Additional Ratios<br>with 11's (Patent Val)
Line 30: Line 30:
| 1
| 1
| 14.634
| 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
|  
|  
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| 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
|  
|  
|  
|  
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| 8
| 8
| 117.073
| 117.073
| 16/15, 15/14
| 15/14, 16/15
|  
|  
|  
|  
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| 160.976
| 160.976
|  
|  
| 12/11, 11/10
| 11/10, ''12/11''
|  
|  
|-
|-
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| 10/9, 21/19
| 10/9, 21/19
|  
|  
| 11/10
| ''11/10''
|-
|-
| 13
| 13
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| 7/6
| 7/6
|  
|  
| 22/19
| ''22/19''
|-
|-
| 19
| 19
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| 20/17
| 20/17
|  
|  
| 13/11
| ''13/11''
|-
|-
| 20
| 20
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| 336.585
| 336.585
| 17/14
| 17/14
| 11/9
| ''11/9''
|  
|  
|-
|-
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| 28
| 28
| 409.756
| 409.756
| 24/19, 19/15
| 19/15, 24/19
|  
|  
| 14/11
| ''14/11''
|-
|-
| 29
| 29
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| 439.024
| 439.024
| 9/7
| 9/7
| 22/17
| ''22/17''
|  
|  
|-
|-
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| 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
|  
|  
|-
|-
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| 39
| 39
| 570.732
| 570.732
| 18/13
| ''18/13''
|  
|  
|  
|  
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| 41
| 41
| 600.000
| 600.000
| 24/17, 17/12
| 17/12, 24/17
|  
|  
|  
|  
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|
|
|}
|}
<nowiki>*</nowiki> as a no-11 19-limit temperament

Revision as of 09:48, 21 June 2024

← 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

Template:EDO intro

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 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) 		Additional Ratios
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

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