37edf: Difference between revisions

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'''37EDF''' is the [[EDF|equal division of the just perfect fifth]] into 37 parts of 18.9718 [[cent|cents]] each, corresponding to 63.2519 [[edo]] (similar to every fourth step of [[253edo]]). It is related to the regular temperament which tempers out 385/384, 12005/11979, and 820125/819896 in the 11-limit, which is supported by [[63edo]], [[190edo]], and [[253edo]] among others.
{{Infobox ET}}
{{ED intro}}


[[Category:Edf]]
== Theory ==
[[Category:Edonoi]]
37edf corresponds to 63.2519[[edo]], similar to every fourth step of [[253edo]]. It is related to the [[regular temperament]] which [[tempering out|tempers out]] 385/384, 12005/11979, and 820125/819896 in the [[11-limit]], which is supported by [[63edo]], [[190edo]], and [[253edo]] among others.
 
=== Harmonics ===
{{Harmonics in equal|37|3|2}}
{{Harmonics in equal|37|3|2|start=12|collapsed=1}}
 
== Intervals ==
{| class="wikitable mw-collapsible"
|+ Intervals of 37edf
|-
! | degree
! | cents value
! | corresponding <br>JI intervals
! | comments
|-
|  colspan="2"| 0
| | '''exact [[1/1]]'''
| |
|-
| | 1
| | 18.9718
| |
| |
|-
| | 2
| | 37.9435
| | [[45/44]]
| |
|-
| | 3
| | 56.9153
| |
| |
|-
| | 4
| | 75.887
| |25/24
| |
|-
| | 5
| | 94.8588
| |
| |
|-
| | 6
| | 113.8305
| | [[16/15]]
| |
|-
| | 7
| | 132.8023
| |
| |
|-
| | 8
| | 151.7741
| | [[12/11]]
| |
|-
| | 9
| | 170.7458
| |
| |
|-
| | 10
| | 189.7176
| |10/9
| |
|-
| | 11
| | 208.6893
| |9/8
| |
|-
| | 12
| | 227.6611
| |8/7
| |
|-
| | 13
| | 246.6328
| | [[15/13]]
| |
|-
| | 14
| | 265.6046
| | [[7/6]]
| |
|-
| | 15
| | 284.5764
| | 33/28
| |
|-
| | 16
| | 303.5481
| |25/21
| |
|-
| | 17
| | 322.5199
| |6/5
| |
|-
| | 18
| | 341.4916
| |11/9
| |
|-
| | 19
| | 360.4634
| |27/22
| |
|-
| | 20
| | 379.4351
| |5/4
| |
|-
| | 21
| | 398.4069
| | 34/27
| |
|-
| | 22
| | 417.3786
| | [[14/11]]
| |
|-
| | 23
| | 436.3504
| | [[9/7]]
| |
|-
| | 24
| | 455.3222
| | [[13/10]]
| |
|-
| | 25
| | 474.2939
| |
| |
|-
| | 26
| | 493.2657
| |4/3
| |
|-
| | 27
| | 512.2374
| |
| |
|-
| | 28
| | 531.2092
| |15/11
| |
|-
| | 29
| | 550.1809
| | [[11/8]]
| |pseudo-25/18
|-
| | 30
| | 569.1527
| |
| | real 25/18
|-
| | 31
| | 588.1245
| | [[45/32]], 7/5
| |
|-
| | 32
| | 607.0962
| |64/45, 10/7
| |
|-
| | 33
| | 626.068
| |
| |real 36/25
|-
| | 34
| | 645.0397
| |16/11
| |pseudo-36/25
|-
| | 35
| | 664.0115
| | [[22/15]]
| |
|-
| | 36
| | 682.9832
| |40/27
| |
|-
| | 37
| | 701.955
| | '''exact [[3/2]]'''
| | just perfect fifth
|-
|38
|720.9268
|
|
|-
|39
|739.8985
|135/88
|
|-
|40
|758.8703
|
|
|-
|41
|777.842
|25/16
|
|-
|42
|796.8138
|
|
|-
|43
|815.7855
|8/5
|
|-
|44
|834.7573
|
|
|-
|45
|853.7291
|18/11
|
|-
|46
|872.7008
|
|
|-
|47
|891.6726
|5/3
|
|-
|48
|910.6443
|27/16
|
|-
|49
|929.6161
|12/7
|
|-
|50
|948.5978
|45/26
|
|-
|51
|967.5596
|7/4
|
|-
|52
|986.5314
|99/56
|
|-
|53
|1005.5031
|25/14
|
|-
|54
|1024.4749
|9/5
|
|-
|55
|1043.4466
|
|
|-
|56
|1062.4184
|
|
|-
|57
|1081.3901
|15/8
|
|-
|58
|1100.3619
|17/9
|
|-
|59
|1119.3336
|21/11
|
|-
|60
|1138.3054
|27/14
|
|-
|61
|1157.2772
|39/20
|
|-
|62
|1176.2489
|
|
|-
|63
|1195.2007
|2/1
|
|-
|64
|1214.1924
|
|
|-
|65
|1233.1642
|45/22
|
|-
|66
|1252.1359
|33/16
|pseudo-25/12
|-
|67
|1271.1077
|
|real 25/12
|-
|68
|1290.0795
|135/64, 21/10
|
|-
|69
|1309.0512
|32/15, 15/7
|
|-
|70
|1328.023
|
|real 54/25
|-
|71
|1347.9947
|24/11
|pseudo-54/25
|-
|72
|1365.9668
|11/5
|
|-
|73
|1385.9382
|20/9
|
|-
|74
|1403.91
|'''exact''' 9/4
|
|}
 
==Related regular temperaments==
===7-limit 63&amp;190===
Commas: 2460375/2458624, 514714375/509607936
 
POTE generator: ~1728/1715 = 18.957
 
Mapping: [&lt;1 1 3 2|, &lt;0 37 -43 51|]
 
EDOs: 63, 190, 253
 
===11-limit 63&amp;190===
Commas: 385/384, 12005/11979, 820125/819896
 
POTE generator: ~99/98 = 18.957
 
Mapping: [&lt;1 1 3 2 3|, &lt;0 37 -43 51 29|]
 
EDOs: 63, 190, 253
 
===13-limit 63&amp;190===
Commas: 385/384, 1575/1573, 2200/2197, 4459/4455
 
POTE generator: ~99/98 = 18.959
 
Mapping: [&lt;1 1 3 2 3 4|, &lt;0 37 -43 51 29 -19|]
 
EDOs: 63, 190, 253
 
{{Todo|cleanup|expand}}

Latest revision as of 17:11, 17 January 2025

← 36edf 37edf 38edf →
Prime factorization 37 (prime)
Step size 18.9718 ¢ 
Octave 63\37edf (1195.22 ¢)
Twelfth 100\37edf (1897.18 ¢)
Consistency limit 3
Distinct consistency limit 3

37 equal divisions of the perfect fifth (abbreviated 37edf or 37ed3/2) is a nonoctave tuning system that divides the interval of 3/2 into 37 equal parts of about 19 ¢ each. Each step represents a frequency ratio of (3/2)1/37, or the 37th root of 3/2.

Theory

37edf corresponds to 63.2519edo, similar to every fourth step of 253edo. It is related to the regular temperament which tempers out 385/384, 12005/11979, and 820125/819896 in the 11-limit, which is supported by 63edo, 190edo, and 253edo among others.

Harmonics

Approximation of harmonics in 37edf
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -4.78 -4.78 +9.41 +2.53 +9.41 +8.15 +4.63 +9.41 -2.24 +3.50 +4.63
Relative (%) -25.2 -25.2 +49.6 +13.4 +49.6 +42.9 +24.4 +49.6 -11.8 +18.4 +24.4
Steps
(reduced) 		63
(26) 		100
(26) 		127
(16) 		147
(36) 		164
(16) 		178
(30) 		190
(5) 		201
(16) 		210
(25) 		219
(34) 		227
(5)
Approximation of harmonics in 37edf
Harmonic 13 14 15 16 17 18 19 20 21 22 23
Error Absolute (¢) -1.14 +3.37 -2.24 -0.15 +8.73 +4.63 +5.89 -7.02 +3.37 -1.28 -2.35
Relative (%) -6.0 +17.7 -11.8 -0.8 +46.0 +24.4 +31.0 -37.0 +17.7 -6.8 -12.4
Steps
(reduced) 		234
(12) 		241
(19) 		247
(25) 		253
(31) 		259
(0) 		264
(5) 		269
(10) 		273
(14) 		278
(19) 		282
(23) 		286
(27)

Intervals

Intervals of 37edf
degree cents value corresponding
JI intervals 		comments
0 exact 1/1
1 18.9718
2 37.9435 45/44
3 56.9153
4 75.887 25/24
5 94.8588
6 113.8305 16/15
7 132.8023
8 151.7741 12/11
9 170.7458
10 189.7176 10/9
11 208.6893 9/8
12 227.6611 8/7
13 246.6328 15/13
14 265.6046 7/6
15 284.5764 33/28
16 303.5481 25/21
17 322.5199 6/5
18 341.4916 11/9
19 360.4634 27/22
20 379.4351 5/4
21 398.4069 34/27
22 417.3786 14/11
23 436.3504 9/7
24 455.3222 13/10
25 474.2939
26 493.2657 4/3
27 512.2374
28 531.2092 15/11
29 550.1809 11/8 pseudo-25/18
30 569.1527 real 25/18
31 588.1245 45/32, 7/5
32 607.0962 64/45, 10/7
33 626.068 real 36/25
34 645.0397 16/11 pseudo-36/25
35 664.0115 22/15
36 682.9832 40/27
37 701.955 exact 3/2 just perfect fifth
38 720.9268
39 739.8985 135/88
40 758.8703
41 777.842 25/16
42 796.8138
43 815.7855 8/5
44 834.7573
45 853.7291 18/11
46 872.7008
47 891.6726 5/3
48 910.6443 27/16
49 929.6161 12/7
50 948.5978 45/26
51 967.5596 7/4
52 986.5314 99/56
53 1005.5031 25/14
54 1024.4749 9/5
55 1043.4466
56 1062.4184
57 1081.3901 15/8
58 1100.3619 17/9
59 1119.3336 21/11
60 1138.3054 27/14
61 1157.2772 39/20
62 1176.2489
63 1195.2007 2/1
64 1214.1924
65 1233.1642 45/22
66 1252.1359 33/16 pseudo-25/12
67 1271.1077 real 25/12
68 1290.0795 135/64, 21/10
69 1309.0512 32/15, 15/7
70 1328.023 real 54/25
71 1347.9947 24/11 pseudo-54/25
72 1365.9668 11/5
73 1385.9382 20/9
74 1403.91 exact 9/4

Related regular temperaments

7-limit 63&190

Commas: 2460375/2458624, 514714375/509607936

POTE generator: ~1728/1715 = 18.957

Mapping: [<1 1 3 2|, <0 37 -43 51|]

EDOs: 63, 190, 253

11-limit 63&190

Commas: 385/384, 12005/11979, 820125/819896

POTE generator: ~99/98 = 18.957

Mapping: [<1 1 3 2 3|, <0 37 -43 51 29|]

EDOs: 63, 190, 253

13-limit 63&190

Commas: 385/384, 1575/1573, 2200/2197, 4459/4455

POTE generator: ~99/98 = 18.959

Mapping: [<1 1 3 2 3 4|, <0 37 -43 51 29 -19|]

EDOs: 63, 190, 253

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