32edf: Difference between revisions

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'''32EDF''' is the [[EDF|equal division of the just perfect fifth]] into 32 parts of 21.9361 [[cent|cents]] each, corresponding to 54.7044 [[edo]] (similar to every seventh step of [[383edo]]). It is related to the regular temperament which tempers out |127 -127 32> in the 5-limit, which is supported by 164, 383, 547, 711, 875, and 1258 EDOs.
{{Infobox ET}}
{{ED intro}}
 
== Theory ==
32edf corresponds to 54.7044[[edo]], similar to every seventh step of [[383edo]]. It is related to the [[regular temperament]] which [[tempering out|tempers out]] {{monzo| 127 -127 32 }} in the [[5-limit]], which is supported by {{EDOs| 164-, 383-, 547-, 711-, 875-, and 1258edo }}.


Lookalikes: [[55edo]], [[87edt]]
Lookalikes: [[55edo]], [[87edt]]


==Intervals==
=== Harmonics ===
{| class="wikitable"
{{Harmonics in equal|32|3|2}}
{{Harmonics in equal|32|3|2|start=12|collapsed=1}}
 
== Intervals ==
{| class="wikitable mw-collapsible"
|+ Intervals of 32edf
|-
|-
! | degree
! | degree
Line 11: Line 20:
! | comments
! | comments
|-
|-
| | 0
| colspan="2"| 0
| | 0.0000
| | '''exact [[1/1]]'''
| | '''exact [[1/1]]'''
| |  
| |  
Line 28: Line 36:
| | 3
| | 3
| | 65.8083
| | 65.8083
| | [[27/26]]
| | [[27/26]], 28/27
| |  
| |  
|-
|-
Line 38: Line 46:
| | 5
| | 5
| | 109.6805
| | 109.6805
| | 49/46
| | 49/46, 16/15
| |  
| |  
|-
|-
Line 48: Line 56:
| | 7
| | 7
| | 153.5527
| | 153.5527
| | 59/54
| | 59/54, 18/11
| |  
| |  
|-
|-
Line 67: Line 75:
|-
|-
| | 11
| | 11
| | 241.2970
| | 241.297
| | (23/20)
| | (23/20)
| |  
| |  
Line 73: Line 81:
| | 12
| | 12
| | 263.2331
| | 263.2331
| |  
| |7/6
| |  
| |  
|-
|-
Line 108: Line 116:
| | 19
| | 19
| | 416.7858
| | 416.7858
| |  
| |14/11
| |  
| |  
|-
|-
| | 20
| | 20
| | 438.7219
| | 438.7219
| |  
| |9/7
| |  
| |  
|-
|-
Line 129: Line 137:
| | 504.5302
| | 504.5302
| | 87/65
| | 87/65
| |  
| |pseudo-4/3
|-
|-
| | 24
| | 24
Line 153: Line 161:
| | 28
| | 28
| | 614.2106
| | 614.2106
| |  
| |10/7
| |  
| |  
|-
|-
| | 29
| | 29
| | 636.1467
| | 636.1467
| | 39/27
| |13/9
| |  
| |  
|-
|-
Line 175: Line 183:
| | '''exact [[3/2]]'''
| | '''exact [[3/2]]'''
| | just perfect fifth
| | just perfect fifth
|-
|33
|723.8911
|243/160
|
|-
|34
|745.8372
|20/13
|
|-
|35
|766.7633
|81/52, 14/9
|
|-
|36
|790.6994
|
|
|-
|37
|811.6355
|147/92, 8/5
|
|-
|38
|833.5716
|123/76
|
|-
|39
|855.5077
|59/36, 18/11
|
|-
|40
|877.4438
|
|
|-
|41
|899.3798
|195/116
|
|-
|42
|922.3159
|63/37
|
|-
|43
|943.252
|69/40
|
|-
|44
|965.1881
|7/4
|
|-
|45
|987.1242
|
|
|-
|46
|1009.0603
|351/196
|
|-
|47
|1030.9964
|78/43
|
|-
|48
|1052.9325
|90/49, 147/80
|
|-
|49
|1076.8686
|387/208
|
|-
|50
|1096.847
|147/78
|
|-
|51
|1118.7408
|21/11
|
|-
|52
|1140.6769
|27/14
|
|-
|53
|1162.613
|45/23
|
|-
|54
|1184.5451
|111/56
|
|-
|55
|1206.4852
|261/130
|pseudo-2/1
|-
|56
|1228.4213
|61/30
|
|-
|57
|1250.3575
|243/118
|
|-
|58
|1272.2934
|171/82
|
|-
|59
|1294.2395
|207/98
|
|-
|60
|1316.1656
|15/7
|
|-
|61
|1338.1017
|13/6
|
|-
|62
|1360.0378
|351/160
|
|-
|63
|1381.9739
|20/9
|
|-
|64
|1403.91
|'''exact''' 9/4
|
|}
|}


[[Category:Edf]]
{{Todo|cleanup|expand}}
[[Category:Edonoi]]

Latest revision as of 17:08, 17 January 2025

← 31edf 32edf 33edf →
Prime factorization 25
Step size 21.9361 ¢ 
Octave 55\32edf (1206.49 ¢)
Twelfth 87\32edf (1908.44 ¢)
Consistency limit 3
Distinct consistency limit 3

32 equal divisions of the perfect fifth (abbreviated 32edf or 32ed3/2) is a nonoctave tuning system that divides the interval of 3/2 into 32 equal parts of about 21.9 ¢ each. Each step represents a frequency ratio of (3/2)1/32, or the 32nd root of 3/2.

Theory

32edf corresponds to 54.7044edo, similar to every seventh step of 383edo. It is related to the regular temperament which tempers out [127 -127 32⟩ in the 5-limit, which is supported by 164-, 383-, 547-, 711-, 875-, and 1258edo.

Lookalikes: 55edo, 87edt

Harmonics

Approximation of harmonics in 32edf
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +6.49 +6.49 -8.97 -0.43 -8.97 +9.33 -2.48 -8.97 +6.06 -5.40 -2.48
Relative (%) +29.6 +29.6 -40.9 -2.0 -40.9 +42.5 -11.3 -40.9 +27.6 -24.6 -11.3
Steps
(reduced) 		55
(23) 		87
(23) 		109
(13) 		127
(31) 		141
(13) 		154
(26) 		164
(4) 		173
(13) 		182
(22) 		189
(29) 		196
(4)
Approximation of harmonics in 32edf
Harmonic 13 14 15 16 17 18 19 20 21 22 23
Error Absolute (¢) -9.44 -6.12 +6.06 +4.00 +8.73 -2.48 -8.34 -9.40 -6.12 +1.09 -10.06
Relative (%) -43.0 -27.9 +27.6 +18.3 +39.8 -11.3 -38.0 -42.8 -27.9 +5.0 -45.9
Steps
(reduced) 		202
(10) 		208
(16) 		214
(22) 		219
(27) 		224
(0) 		228
(4) 		232
(8) 		236
(12) 		240
(16) 		244
(20) 		247
(23)

Intervals

Intervals of 32edf
degree cents value corresponding
JI intervals 		comments
0 exact 1/1
1 21.9361 81/80
2 43.8722 40/39
3 65.8083 27/26, 28/27
4 87.7444
5 109.6805 49/46, 16/15
6 131.6166 41/38
7 153.5527 59/54, 18/11
8 175.4888
9 197.4248 65/58
10 219.3609 42/37
11 241.297 (23/20)
12 263.2331 7/6
13 285.1692
14 307.1053 117/98
15 329.0414 52/43
16 350.9775 60/49, 49/40
17 372.9136 129/104
18 394.8497 49/39
19 416.7858 14/11
20 438.7219 9/7
21 460.6580 (30/23)
22 482.5941 37/28
23 504.5302 87/65 pseudo-4/3
24 526.4663 61/45
25 548.4023 81/59
26 570.3384 57/41
27 592.2745 69/49
28 614.2106 10/7
29 636.1467 13/9
30 658.0828 117/80
31 680.0189 40/27
32 701.9550 exact 3/2 just perfect fifth
33 723.8911 243/160
34 745.8372 20/13
35 766.7633 81/52, 14/9
36 790.6994
37 811.6355 147/92, 8/5
38 833.5716 123/76
39 855.5077 59/36, 18/11
40 877.4438
41 899.3798 195/116
42 922.3159 63/37
43 943.252 69/40
44 965.1881 7/4
45 987.1242
46 1009.0603 351/196
47 1030.9964 78/43
48 1052.9325 90/49, 147/80
49 1076.8686 387/208
50 1096.847 147/78
51 1118.7408 21/11
52 1140.6769 27/14
53 1162.613 45/23
54 1184.5451 111/56
55 1206.4852 261/130 pseudo-2/1
56 1228.4213 61/30
57 1250.3575 243/118
58 1272.2934 171/82
59 1294.2395 207/98
60 1316.1656 15/7
61 1338.1017 13/6
62 1360.0378 351/160
63 1381.9739 20/9
64 1403.91 exact 9/4
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