82edo: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older editNewer edit →
VisualWikitext
Royalmilktea (talk | contribs)
autopatrolled, emailconfirmed
64 edits
→Interval table: expanded upon with approx. ratios
Cleanup
Line 1: Line 1:
{{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.
{{EDO intro|82}}


== 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.
=== Prime harmonics ===
{{Harmonics in equal|82}}
{{Harmonics in equal|82}}


== 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
! Cents
!21-odd-limit
! 21-odd-limit<br>no-11 ratios
no-11 ratios
! Additional Ratios<br>with 11's (82e Val)
!Additional ratios
! Additional Ratios<br>with 11's (Patent Val)
with 11s (82e val)
!Additional ratios
with 11s (patent val)
|-
|-
|0
| 0
|0.000
| 0.000
|1/1
| 1/1
|1/1
| 1/1
|1/1
| 1/1
|-
|-
|1
| 1
|14.634
| 14.634
|
|  
|
|  
|
|  
|-
|-
|2
| 2
|29.268
| 29.268
|
|  
|
|  
|
|  
|-
|-
|3
| 3
|43.902
| 43.902
|
|  
|
|  
|
|  
|-
|-
|4
| 4
|58.537
| 58.537
|
|  
|
|  
|
|  
|-
|-
|5
| 5
|73.171
| 73.171
|
|  
|22/21
| 22/21
|
|  
|-
|-
|6
| 6
|87.805
| 87.805
|21/20, 20/19, 19/18
| 21/20, 20/19, 19/18
|
|  
|22/21
| 22/21
|-
|-
|7
| 7
|102.439
| 102.439
|18/17, 17/16
| 18/17, 17/16
|
|  
|
|  
|-
|-
|8
| 8
|117.073
| 117.073
|16/15, 15/14
| 16/15, 15/14
|
|  
|
|  
|-
|-
|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
| 12/11, 11/10
|
|  
|-
|-
|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
| 24/19, 19/15
|
|  
|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
| 15/11, 11/8
|
|  
|-
|-
|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
| 24/17, 17/12
|
|  
|
|  
|-
|-
|...
| …
|...
| …
|
|
|
|
|
|
|}
|}

Revision as of 05:26, 10 September 2023

← 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.

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)

Intervals

# Cents 21-odd-limit
no-11 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
2 29.268
3 43.902
4 58.537
5 73.171 22/21
6 87.805 21/20, 20/19, 19/18 22/21
7 102.439 18/17, 17/16
8 117.073 16/15, 15/14
9 131.707 14/13, 13/12
10 146.341 12/11
11 160.976 12/11, 11/10
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 24/19, 19/15 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 15/11, 11/8
38 556.098 11/8
39 570.732 18/13
40 585.366 7/5
41 600.000 24/17, 17/12
… …
Retrieved from "https://en.xen.wiki/index.php?title=82edo&oldid=123814"