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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>
}}
Retrieved from "https://en.xen.wiki/w/53edt"