84ed5: Difference between revisions

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Division of the 5th harmonic '''into 84 equal parts''' (84ed5) is related to [[36edo]] but the 5/1 rather than the 2/1 being just. The octave is compressed (about 5.8656 cents) and the step size is about 33.1704 cents.{{Infobox ET|84ed5}}
{{Infobox ET}}
==Intervals==
 
{| class="wikitable"
Division of the 5th harmonic '''into 84 equal parts''' (84ed5) is related to [[36edo]] but the 5/1 rather than the 2/1 being just. The octave is compressed (about 5.8656 cents) and the step size is about 33.1704 cents.
|+
 
|-
== Intervals ==
!degree!!cents value!!corresponding
{{Interval table}}
JI intervals
 
!comments
== Harmonics ==
|-
{{Harmonics in equal
|0||0.0000||exact 1/1
| steps = 84
|
| num = 5
|-
| denom = 1
|1||33.1704||1990656/1953125
}}
|Valentine comma
{{Harmonics in equal
|-
| steps = 84
|2||66.3408||
| num = 5
|
| denom = 1
|-
| start = 12
| 3 |3
| collapsed = 1
|99.5112
}}
|256/243
 
|limma
==Music==
|-
* [https://www.youtube.com/watch?v=rTTym6vAFhs Not From This World] by Francium
| 4 |4
 
|132.6816
{{todo|expand}}
|27/25
|large limma, BP small semitone, Zarlino semitone
|-
| 5 |5
|165.8520
|
|
|-
| 6 |6
|199.0224
|
|pseudo-9/8
|-
| 7 |7
|232.1928
|
|
|-
| 8 |8
|265.3632
|
|
|-
|9
|298.5336
|
|pseudo-32/27
|-
|10
|331.7040
|
|
|-
|11
|364.8744
|100/81
|grave major third
|-
|12
|398.0448
|
|pseudo-81/64
|-
|13
|431.2152
|
|
|-
|14
|464.3856
|
|
|-
|15
|497.5560
|
|pseudo-4/3
|-
|16
|530.7264
|
|
|-
|17
|563.8968
|
|
|-
|18
|597.0672
|
|
|-
|19
|630.2376
|
|
|-
|20
|663.4080
|
|
|-
|21
|696.5784
|
|pseudo-3/2
|-
|22
|729.7488
|
|
|-
|23
|762.9192
|
|
|-
|24
|796.0896
|
|pseudo-128/81
|-
|25
|829.2600
|
|
|-
|26
|862.4304
|400/243
|grave major sixth
|-
|27
|895.6008
|
|pseudo-27/16
|-
|28
|928.7712
|
|
|-
|29
|961.9416
|
|
|-
|30
|995.1120
|
|pseudo-16/9
|-
|31
|1028.2824
|
|
|-
|32
|1061.4528
|
|
|-
|33
|1094.6232
|
|
|-
|34
|1127.7936
|
|
|-
|35
|1160.9640
|
|
|-
|36
|1194.1344
|
|pseudo-2/1
|-
|37
|1227.3049
|128/63
|septimal comma, Archytas' comma plus one octave
|-
|38
|1260.4753
|524288/253125
|Passion comma plus one octave
|-
|39
|1293.6457
|
|
|-
|40
|1326.8161
|
|
|-
|41
|1359.9865
|
|
|-
|42
|1393.1569
|
|
|-
|43
|1426.3273
|729/320
|acute major second plus one octave
|-
|44
|1459.4977
|
|
|-
|45
|1492.6681
|
|
|-
|46
|1525.8385
|
|
|-
|47
|1559.0089
|
|
|-
|48
|1592.1793
|
|
|-
|49
|1625.3497
|
|
|-
|50
|1658.5201
|
|
|-
|51
|1691.6905
|
|
|-
|52
|1724.8609
|
|
|-
|53
|1758.0313
|
|
|-
|54
|1791.2017
|
|
|-
|55
|1824.3721
|
|
|-
|56
|1857.5425
|
|
|-
|57
|1890.7129
|
|
|-
|58
|1923.8833
|243/80
|acute fifth plus one octave
|-
|59
|1957.0537
|
|
|-
|60
|1990.2241
|
|
|-
|61
|2023.3945
|
|
|-
|62
|2056.5649
|
|
|-
|63
|2089.7353
|
|
|-
|64
|2122.9057
|
|
|-
|65
|2156.0761
|
|
|-
|66
|2189.2465
|
|
|-
|67
|2222.4169
|
|
|-
|68
|2255.5873
|
|
|-
|69
|2288.7577
|
|
|-
|70
|2321.9281
|
|
|-
|71
|2355.0985
|
|
|-
|72
|2388.2689
|
|
|-
|73
|2421.4393
|81/20
|syntonic comma, Didymus comma plus two octaves
|-
|74
|2454.6097
|
|
|-
|75
|2487.7801
|
|
|-
|76
|2520.9505
|
|
|-
|77
|2554.1209
|2187/500
|Gorgo limma plus two octaves
|-
|78
|2587.2913
|
|
|-
|79
|2620.4617
|
|
|-
|80
|2653.6321
|
|
|-
|81
|2686.8025
|
|
|-
|82
|2719.9729
|
|
|-
|83
|2753.1433
|
|
|-
|84
|2786.3137
|exact 5/1
|just major third plus two octaves
|}

Latest revision as of 04:10, 22 December 2024

← 83ed5 84ed5 85ed5 →
Prime factorization 22 × 3 × 7
Step size 33.1704 ¢ 
Octave 36\84ed5 (1194.13 ¢) (→ 3\7ed5)
Twelfth 57\84ed5 (1890.71 ¢) (→ 19\28ed5)
Consistency limit 5
Distinct consistency limit 5

Division of the 5th harmonic into 84 equal parts (84ed5) is related to 36edo but the 5/1 rather than the 2/1 being just. The octave is compressed (about 5.8656 cents) and the step size is about 33.1704 cents.

Intervals

Steps Cents Approximate ratios
0 0 1/1
1 33.2
2 66.3 26/25
3 99.5
4 132.7 40/37, 41/38
5 165.9 11/10
6 199 28/25, 37/33
7 232.2
8 265.4
9 298.5
10 331.7 23/19, 40/33
11 364.9 21/17, 37/30
12 398 39/31
13 431.2
14 464.4 17/13
15 497.6 4/3
16 530.7 19/14, 34/25
17 563.9
18 597.1 41/29
19 630.2
20 663.4 22/15
21 696.6
22 729.7 29/19, 35/23
23 762.9
24 796.1
25 829.3 21/13
26 862.4 23/14, 28/17
27 895.6
28 928.8
29 961.9
30 995.1 16/9
31 1028.3 38/21
32 1061.5
33 1094.6
34 1127.8
35 1161 41/21
36 1194.1
37 1227.3
38 1260.5 29/14
39 1293.6
40 1326.8 28/13
41 1360
42 1393.2 38/17
43 1426.3
44 1459.5
45 1492.7
46 1525.8 41/17
47 1559
48 1592.2
49 1625.3
50 1658.5
51 1691.7
52 1724.9
53 1758
54 1791.2 31/11
55 1824.4
56 1857.5 38/13, 41/14
57 1890.7
58 1923.9
59 1957.1 31/10
60 1990.2 41/13
61 2023.4
62 2056.6 23/7
63 2089.7
64 2122.9
65 2156.1
66 2189.2 39/11
67 2222.4
68 2255.6
69 2288.8 15/4
70 2321.9
71 2355.1 39/10
72 2388.3
73 2421.4
74 2454.6 33/8
75 2487.8
76 2521
77 2554.1
78 2587.3
79 2620.5
80 2653.6 37/8
81 2686.8
82 2720
83 2753.1
84 2786.3 5/1

Harmonics

Approximation of harmonics in 84ed5
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -5.9 -11.2 -11.7 +0.0 +16.1 +14.6 +15.6 +10.7 -5.9 -5.0 +10.2
Relative (%) -17.7 -33.9 -35.4 +0.0 +48.4 +43.9 +47.0 +32.2 -17.7 -15.1 +30.7
Steps
(reduced) 		36
(36) 		57
(57) 		72
(72) 		84
(0) 		94
(10) 		102
(18) 		109
(25) 		115
(31) 		120
(36) 		125
(41) 		130
(46)
Approximation of harmonics in 84ed5
Harmonic 13 14 15 16 17 18 19 20 21 22 23
Error Absolute (¢) +4.3 +8.7 -11.2 +9.7 +4.3 +4.8 +10.7 -11.7 +3.3 -10.9 +11.7
Relative (%) +13.0 +26.2 -33.9 +29.3 +12.9 +14.5 +32.3 -35.4 +10.0 -32.8 +35.2
Steps
(reduced) 		134
(50) 		138
(54) 		141
(57) 		145
(61) 		148
(64) 		151
(67) 		154
(70) 		156
(72) 		159
(75) 		161
(77) 		164
(80)

Music

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