53edt: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
'''53EDT''' is the [[Edt|equal division of the third harmonic]] into 53 parts of 35.8859 [[cent|cents]] each, corresponding to 33.4393 [[edo]]. It has a generally sharp tendency, in the sense that if 3 is pure, 5, 7, 11, 13, and 17 are all sharp. It tempers out 413343/390625 and 823543/820125 in the 7-limit; [[891/875|891/875,]] 3087/3025, and 164025/161051 in the 11-limit; 637/625, 729/715, 847/845, and 1575/1573 in the 13-limit; 189/187, 429/425, 459/455, and 833/825 in the 17-limit; 171/169, 247/245, 361/357, and 855/847 in the 19-limit (no-twos subgroup).
{{ED intro}}


== Harmonics ==
== Theory ==
{{Harmonics in equal
53edt corresponds to 33.4393…[[edo]]. It has a generally sharp tendency, in the sense that if 3 is pure, 5, 7, 11, 13, and 17 are all sharp. It tempers out 413343/390625 and 823543/820125 in the 7-limit; [[891/875]], 3087/3025, and 164025/161051 in the 11-limit; 637/625, 729/715, 847/845, and 1575/1573 in the 13-limit; 189/187, 429/425, 459/455, and 833/825 in the 17-limit; 171/169, 247/245, 361/357, and 855/847 in the 19-limit (no-twos subgroup).
| steps = 53
 
| num = 3
=== Harmonics ===
| denom = 1
{{Harmonics in equal|53|3|1|columns=11}}
}}
{{Harmonics in equal|53|3|1|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 53edt (continued)}}
{{Harmonics in equal
| steps = 53
| num = 3
| denom = 1
| start = 12
| collapsed = 1
}}


== Intervals ==
== Intervals ==
{| class="wikitable"
{| class="wikitable center-1 right-2 right-3"
|-
|-
! Degree
! #
! Cents value
! Cents
! Hekts
! Hekts
! Corresponding <br>JI intervals
! Approximate ratios
! Comments
|-
|-
| colspan="3" | 0
| 0
| '''exact [[1/1]]'''
| 0.0000
|
| 0.0000
| [[1/1]]
|-
|-
| 1
| 1
Line 33: Line 26:
| 24.5283
| 24.5283
| [[50/49]], [[49/48]]
| [[50/49]], [[49/48]]
|
|-
|-
| 2
| 2
Line 39: Line 31:
| 49.0566
| 49.0566
| [[25/24]]
| [[25/24]]
|
|-
|-
| 3
| 3
Line 45: Line 36:
| 73.5849
| 73.5849
| [[17/16]], [[16/15]]
| [[17/16]], [[16/15]]
|
|-
|-
| 4
| 4
Line 51: Line 41:
| 98.1132
| 98.1132
| 38/35
| 38/35
|
|-
|-
| 5
| 5
Line 57: Line 46:
| 122.6415
| 122.6415
| 51/46, 132/119
| 51/46, 132/119
|
|-
|-
| 6
| 6
Line 63: Line 51:
| 147.1698
| 147.1698
| 17/15
| 17/15
|
|-
|-
| 7
| 7
Line 69: Line 56:
| 171.6981
| 171.6981
| 15/13
| 15/13
|
|-
|-
| 8
| 8
Line 75: Line 61:
| 196.2264
| 196.2264
| 33/28, 13/11
| 33/28, 13/11
|
|-
|-
| 9
| 9
Line 81: Line 66:
| 220.7547
| 220.7547
| 6/5
| 6/5
|
|-
|-
| 10
| 10
Line 87: Line 71:
| 245.283
| 245.283
| [[16/13]]
| [[16/13]]
|
|-
|-
| 11
| 11
Line 93: Line 76:
| 269.8113
| 269.8113
| [[5/4]], 49/39
| [[5/4]], 49/39
|
|-
|-
| 12
| 12
Line 99: Line 81:
| 594.3396
| 594.3396
| [[9/7]], 50/39
| [[9/7]], 50/39
|
|-
|-
| 13
| 13
Line 105: Line 86:
| 318.8679
| 318.8679
| 21/16
| 21/16
|
|-
|-
| 14
| 14
Line 111: Line 91:
| 343.3962
| 343.3962
| [[4/3]], 171/128
| [[4/3]], 171/128
|
|-
|-
| 15
| 15
Line 117: Line 96:
| 367.9245
| 367.9245
| [[15/11]]
| [[15/11]]
|
|-
|-
| 16
| 16
Line 123: Line 101:
| 392.4528
| 392.4528
| 39/28
| 39/28
|
|-
|-
| 17
| 17
Line 129: Line 106:
| 416.9811
| 416.9811
| [[10/7]]
| [[10/7]]
|
|-
|-
| 18
| 18
Line 135: Line 111:
| 441.5094
| 441.5094
| 35/24
| 35/24
|
|-
|-
| 19
| 19
Line 141: Line 116:
| 466.0377
| 466.0377
| 126/85, 40/27
| 126/85, 40/27
|
|-
|-
| 20
| 20
Line 147: Line 121:
| 490.566
| 490.566
|  
|  
| pseudo-3/2
|-
|-
| 21
| 21
Line 153: Line 126:
| 515.0943
| 515.0943
| [[17/11]]
| [[17/11]]
|
|-
|-
| 22
| 22
Line 159: Line 131:
| 539.6226
| 539.6226
| [[30/19]]
| [[30/19]]
|
|-
|-
| 23
| 23
Line 165: Line 136:
| 564.1509
| 564.1509
| [[13/8]]
| [[13/8]]
|
|-
|-
| 24
| 24
| 861.2626
| 861.2626
| 588.67945
| 588.67945
|
|  
|  
|-
|-
Line 177: Line 146:
| 613.20755
| 613.20755
| 42/25, 32/19
| 42/25, 32/19
|
|-
|-
| 26
| 26
Line 183: Line 151:
| 637.73585
| 637.73585
| [[12/7]]
| [[12/7]]
|
|-
|-
| 27
| 27
Line 189: Line 156:
| 662.26415
| 662.26415
| [[7/4]]
| [[7/4]]
| Off by ~0.104¢<ref group=Note name=Note01/>
|-
|-
| 28
| 28
Line 195: Line 161:
| 686.79245
| 686.79245
| 25/14, 57/32
| 25/14, 57/32
|
|-
|-
| 29
| 29
| 1040.6924
| 1040.6924
| 711.32075
| 711.32075
|
|  
|  
|-
|-
Line 207: Line 171:
| 735.8491
| 735.8491
| 24/13
| 24/13
|
|-
|-
| 31
| 31
Line 213: Line 176:
| 760.3774
| 760.3774
| [[19/10]]
| [[19/10]]
|
|-
|-
| 32
| 32
Line 219: Line 181:
| 784.9057
| 784.9057
| 33/17
| 33/17
|
|-
|-
| 33
| 33
Line 225: Line 186:
| 809.434
| 809.434
|  
|  
| pseudooctave
|-
|-
| 34
| 34
Line 231: Line 191:
| 833.9623
| 833.9623
| 85/42, 81/40
| 85/42, 81/40
|
|-
|-
| 35
| 35
Line 237: Line 196:
| 858.4906
| 858.4906
| 95/46
| 95/46
|
|-
|-
| 36
| 36
Line 243: Line 201:
| 883.0189
| 883.0189
| 21/10
| 21/10
|
|-
|-
| 37
| 37
Line 249: Line 206:
| 907.5472
| 907.5472
| [[14/13|28/13]]
| [[14/13|28/13]]
|
|-
|-
| 38
| 38
Line 255: Line 211:
| 932.0755
| 932.0755
| [[11/5]]
| [[11/5]]
|
|-
|-
| 39
| 39
Line 261: Line 216:
| 956.6038
| 956.6038
| 9/4, [[64/57|128/57]]
| 9/4, [[64/57|128/57]]
|
|-
|-
| 40
| 40
Line 267: Line 221:
| 981.1321
| 981.1321
| 16/7
| 16/7
|
|-
|-
| 41
| 41
Line 273: Line 226:
| 1005.3304
| 1005.3304
| 7/3, 117/50
| 7/3, 117/50
|
|-
|-
| 42
| 42
Line 279: Line 231:
| 1303.1887
| 1303.1887
| 12/5, 117/49
| 12/5, 117/49
|
|-
|-
| 43
| 43
Line 285: Line 236:
| 1054.717
| 1054.717
| [[39/32|39/16]]
| [[39/32|39/16]]
|
|-
|-
| 44
| 44
Line 291: Line 241:
| 1079.2453
| 1079.2453
| 5/2
| 5/2
|
|-
|-
| 45
| 45
Line 297: Line 246:
| 1130.7736
| 1130.7736
| 28/11, 33/13
| 28/11, 33/13
|
|-
|-
| 46
| 46
Line 303: Line 251:
| 1128.3019
| 1128.3019
| 13/5
| 13/5
|
|-
|-
| 47
| 47
Line 309: Line 256:
| 1152.8302
| 1152.8302
| 45/17
| 45/17
|
|-
|-
| 48
| 48
Line 315: Line 261:
| 1177.3585
| 1177.3585
| 119/44
| 119/44
|
|-
|-
| 49
| 49
Line 321: Line 266:
| 1201.8868
| 1201.8868
| 105/38
| 105/38
|
|-
|-
| 50
| 50
Line 327: Line 271:
| 1226.4151
| 1226.4151
| 48/17, 45/16
| 48/17, 45/16
|
|-
|-
| 51
| 51
Line 333: Line 276:
| 1250.9434
| 1250.9434
| [[36/25|72/25]]
| [[36/25|72/25]]
|
|-
|-
| 52
| 52
Line 339: Line 281:
| 1275.4717
| 1275.4717
| 147/50, 144/49
| 147/50, 144/49
|
|-
|-
| 53
| 53
| 1901.955
| 1901.955
| 1300
| 1300.0000
| '''exact [[3/1]]'''
| [[3/1]]
|
|}
|}


==Notes==
{{Todo|unify precision|review}}
{{reflist|group=Note|refs=
<ref name=Note01>7/4 is notably a [[generator]] of this scale</ref>
}}

Latest revision as of 13:10, 2 January 2026

← 52edt 53edt 54edt →
Prime factorization 53 (prime)
Step size 35.8859 ¢ 
Octave 33\53edt (1184.24 ¢)
Consistency limit 3
Distinct consistency limit 3

53 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 53edt or 53ed3), is a nonoctave tuning system that divides the interval of 3/1 into 53 equal parts of about 35.9 ¢ each. Each step represents a frequency ratio of 31/53, or the 53rd root of 3.

Theory

53edt corresponds to 33.4393…edo. It has a generally sharp tendency, in the sense that if 3 is pure, 5, 7, 11, 13, and 17 are all sharp. It tempers out 413343/390625 and 823543/820125 in the 7-limit; 891/875, 3087/3025, and 164025/161051 in the 11-limit; 637/625, 729/715, 847/845, and 1575/1573 in the 13-limit; 189/187, 429/425, 459/455, and 833/825 in the 17-limit; 171/169, 247/245, 361/357, and 855/847 in the 19-limit (no-twos subgroup).

Harmonics

Approximation of harmonics in 53edt
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -15.8 +0.0 +4.4 +12.8 -15.8 +4.5 -11.4 +0.0 -3.0 +11.5 +4.4
Relative (%) -43.9 +0.0 +12.1 +35.6 -43.9 +12.4 -31.8 +0.0 -8.3 +31.9 +12.1
Steps
(reduced) 		33
(33) 		53
(0) 		67
(14) 		78
(25) 		86
(33) 		94
(41) 		100
(47) 		106
(0) 		111
(5) 		116
(10) 		120
(14)
Approximation of harmonics in 53edt (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +9.3 -11.3 +12.8 +8.7 +11.4 -15.8 -1.7 +17.1 +4.5 -4.3 -9.5 -11.4
Relative (%) +26.0 -31.5 +35.6 +24.3 +31.8 -43.9 -4.8 +47.8 +12.4 -12.0 -26.5 -31.8
Steps
(reduced) 		124
(18) 		127
(21) 		131
(25) 		134
(28) 		137
(31) 		139
(33) 		142
(36) 		145
(39) 		147
(41) 		149
(43) 		151
(45) 		153
(47)

Intervals

# Cents Hekts Approximate ratios
0 0.0000 0.0000 1/1
1 35.8859 24.5283 50/49, 49/48
2 71.7719 49.0566 25/24
3 107.6578 73.5849 17/16, 16/15
4 143.5438 98.1132 38/35
5 179.4297 122.6415 51/46, 132/119
6 215.3157 147.1698 17/15
7 251.2016 171.6981 15/13
8 287.0875 196.2264 33/28, 13/11
9 322.9735 220.7547 6/5
10 358.8594 245.283 16/13
11 394.7454 269.8113 5/4, 49/39
12 430.6313 594.3396 9/7, 50/39
13 466.5173 318.8679 21/16
14 502.4032 343.3962 4/3, 171/128
15 538.2892 367.9245 15/11
16 574.1751 392.4528 39/28
17 610.061 416.9811 10/7
18 645.947 441.5094 35/24
19 681.8329 466.0377 126/85, 40/27
20 717.7189 490.566
21 753.6048 515.0943 17/11
22 789.4908 539.6226 30/19
23 825.3767 564.1509 13/8
24 861.2626 588.67945
25 897.1486 613.20755 42/25, 32/19
26 933.0345 637.73585 12/7
27 968.9205 662.26415 7/4
28 1004.8064 686.79245 25/14, 57/32
29 1040.6924 711.32075
30 1076.5783 735.8491 24/13
31 1112.4642 760.3774 19/10
32 1148.3502 784.9057 33/17
33 1184.2361 809.434
34 1220.1221 833.9623 85/42, 81/40
35 1256.008 858.4906 95/46
36 1291.894 883.0189 21/10
37 1327.7799 907.5472 28/13
38 1363.6658 932.0755 11/5
39 1399.5518 956.6038 9/4, 128/57
40 1435.4377 981.1321 16/7
41 1471.3237 1005.3304 7/3, 117/50
42 1507.2096 1303.1887 12/5, 117/49
43 1543.0956 1054.717 39/16
44 1578.9815 1079.2453 5/2
45 1614.8675 1130.7736 28/11, 33/13
46 1650.7534 1128.3019 13/5
47 1686.6393 1152.8302 45/17
48 1722.5253 1177.3585 119/44
49 1758.4112 1201.8868 105/38
50 1794.2972 1226.4151 48/17, 45/16
51 1830.1831 1250.9434 72/25
52 1866.0691 1275.4717 147/50, 144/49
53 1901.955 1300.0000 3/1
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