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'''[[EDF|Division of the just perfect fifth]] into 58 equal parts''' (58EDF) is related to [[99edo|99 edo]], but with the 3/2 rather than the 2/1 being just. The octave is about 1.8354 cents compressed and the step size is about 12.1027 cents (corresponding to 99.1517 edo). It is consistent to the [[11-odd-limit|12-integer-limit]]. In comparison, 99edo is only consistent up to the [[9-odd-limit|10-integer-limit]].
{{Infobox ET}}
{{ED intro}}


Lookalikes: [[99edo]], [[157edt]]
== Theory ==
{| class="wikitable"
58edf corresponds to 99.1517…edo. It is related to [[99edo]], but with the [[3/2|perfect fifth]] rather than the [[2/1|octave]] being just. The octave is [[stretched and compressed tuning|compressed]] by about 1.84 cents. 58edf is [[consistent]] to the [[integer limit|12-integer-limit]]. In comparison, 99edo is only consistent up to the 10-integer-limit. 58edf has a flat tendency, with [[prime harmonic]]s 2, [[3/1|3]], [[5/1|5]], [[7/1|7]], and [[11/1|11]] all tuned flat of just.  
|-
 
! |Degrees
=== Harmonics ===
! |Cents Value
{{Harmonics in equal|58|3|2|intervals=integer|columns=11}}
|Five limit
{{Harmonics in equal|58|3|2|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 58edf (continued)}}
|Seven limit
 
|Eleven limit
=== Subsets and supersets ===
|Thirteen limit
Since 58 factors into primes as {{nowrap| 2 × 29 }}, 58edf contains [[2edf]] and [[29edf]] as subset edts.
|-
 
| |1
== See also ==
| |12.1027
* [[99edo]] – relative edo
|2048/2025
* [[157edt]] – relative edt
|126/125
* [[256ed6]] – relative ed6
|99/98
|91/90
|-
| |2
| |24.2053
|81/80
|64/63
|55/54
|55/54
|-
| |3
| |36.308
|128/125
|49/48
|49/48
|49/48
|-
| |4
| |48.4107
|250/243
|36/35
|33/32
|33/32
|-
| |5
| |60.5134
|648/625
|28/27
|28/27
|26/25
|-
| |6
| |72.616
|25/24
|25/24
|22/21
|22/21
|-
| |7
| |84.7187
|256/243
|21/20
|21/20
|21/20
|-
| |8
| |96.8214
|135/128
|135/128
|81/77
|52/49
|-
| |9
| |108.92405
|16/15
|16/15
|16/15
|16/15
|-
| |10
| |121.0267
|2187/2048
|15/14
|15/14
|15/14
|-
| |11
| |133.1294
|27/25
|27/25
|27/25
|13/12
|-
| |12
| |145.2321
|625/576
|49/45
|49/45
|49/45
|-
| |13
| |157.3347
|800/729
|35/32
|11/10
|11/10
|-
| |14
| |169.4374
|1125/1024
|54/49
|54/49
|54/49
|-
| |15
| |181.54
|10/9
|10/9
|10/9
|10/9
|-
| |16
| |193.6428
|4096/3645
|28/25
|28/25
|28/25
|-
| |17
| |205.7454
|9/8
|9/8
|9/8
|9/8
|-
| |18
| |217.8481
|256/225
|245/216
|112/99
|91/80
|-
| |19
| |229.9508
|729/640
|8/7
|8/7
|8/7
|-
| |20
| |242.05345
|144/125
|144/125
|63/55
|52/45
|-
| |21
| |254.1561
|125/108
|81/70
|81/70
|15/13
|-
| |22
| |266.2587
|729/625
|7/6
|7/6
|7/6
|-
| |23
| |278.3615
|75/64
|75/64
|33/28
|33/28
|-
| |24
| |290.4641
|32/27
|32/27
|32/27
|13/11
|-
| |25
| |302.5668
|1215/1024
|25/21
|25/21
|25/21
|-
| |26
| |314.6695
|6/5
|6/5
|6/5
|6/5
|-
| |27
| |326.7722
|3125/2592
|98/81
|98/81
|91/75
|-
| |28
| |338.8748
|243/200
|128/105
|11/9
|11/9
|-
| |29
| |350.9775
|625/512
|49/40
|49/40
|49/40
|-
| |30
| |363.0802
|100/81
|100/81
|27/22
|16/13
|-
| |31
| |375.18285
|3888/3125
|56/45
|56/45
|56/45
|-
| |32
| |387.2855
|5/4
|5/4
|5/4
|5/4
|-
| |33
| |399.3882
|512/405
|63/50
|63/50
|49/39
|-
| |34
| |411.4909
|81/64
|80/63
|80/63
|33/26
|-
| |35
| |423.5935
|32/25
|32/25
|14/11
|14/11
|-
| |36
| |435.6962
|625/486
|9/7
|9/7
|9/7
|-
| |37
| |447.7989
|162/125
|35/27
|35/27
|13/10
|-
| |38
| |459.90155
|125/96
|64/49
|55/42
|55/42
|-
| |39
| |472.0042
|320/243
|21/16
|21/16
|21/16
|-
| |40
| |484.1069
|675/512
|250/189
|250/189
|65/49
|-
| |41
| |469.2096
|4/3
|4/3
|4/3
|4/3
|-
| |42
| |508.3122
|8192/6075
|75/56
|66/49
|66/49
|-
| |43
| |520.4149
|27/20
|27/20
|27/20
|27/20
|-
| |44
| |532.5176
|512/375
|49/36
|49/36
|49/36
|-
| |45
| |544.6203
|1000/729
|48/35
|11/8
|11/8
|-
| |46
| |556.7229
|864/625
|112/81
|112/81
|91/66
|-
| |47
| |568.8256
|25/18
|25/18
|25/18
|18/13
|-
| |48
| |580.9283
|1024/729
|7/5
|7/5
|7/5
|-
| |49
| |593.03095
|45/32
|45/32
|45/32
|45/32
|-
| |50
| |605.1336
|64/45
|64/45
|64/45
|64/45
|-
| |51
| |617.2362
|729/512
|10/7
|10/7
|10/7
|-
| |52
| |629.339
|36/25
|36/25
|36/25
|13/9
|-
| |53
| |641.4416
|625/432
|81/56
|81/56
|75/52
|-
| |54
| |653.5443
|729/500
|35/24
|16/11
|16/11
|-
| |55
| |665.647
|375/256
|72/49
|72/49
|72/49
|-
| |56
| |677.7497
|40/27
|40/27
|40/27
|40/27
|-
| |57
| |689.8523
|6075/4096
|112/75
|49/33
|49/33
|-
| |58
| |701.955
|3/2
|3/2
|3/2
|3/2
|-
| |59
| |714.0577
|1024/675
|189/125
|189/125
|91/60
|-
| |60
| |726.16035
|243/160
|32/21
|32/21
|32/21
|-
| |61
| |738.263
|192/125
|49/32
|49/32
|49/32
|-
| |62
| |750.3657
|125/81
|54/35
|54/35
|20/13
|-
| |63
| |762.4684
|972/625
|14/9
|14/9
|14/9
|-
| |64
| |774.571
|25/16
|25/16
|11/7
|11/7
|-
| |65
| |786.6737
|128/81
|63/40
|63/40
|52/33
|-
| |66
| |798.7764
|405/256
|100/63
|100/63
|78/49
|-
| |67
| |810.87905
|8/5
|8/5
|8/5
|8/5
|-
| |68
| |822.9817
|3125/1944
|45/28
|45/28
|45/28
|-
| |69
| |835.0844
|81/50
|81/50
|44/27
|13/8
|-
| |70
| |847.1871
|625/384
|49/30
|49/30
|49/30
|-
| |71
| |859.2897
|400/243
|105/64
|18/11
|18/11
|-
| |72
| |871.3924
|3375/2048
|81/49
|81/49
|81/49
|-
| |73
| |883.4951
|5/3
|5/3
|5/3
|5/3
|-
| |74
| |895.5978
|2048/1215
|42/25
|42/25
|42/25
|-
| |75
| |907.7004
|27/16
|27/16
|27/16
|22/13
|-
| |76
| |919.8031
|128/75
|128/75
|56/33
|56/33
|-
| |77
| |931.9058
|1250/729
|12/7
|12/7
|12/7
|-
| |78
| |944.00845
|216/125
|140/81
|140/81
|26/15
|-
| |79
| |956.1111
|125/72
|125/72
|110/63
|45/26
|-
| |80
| |968.2138
|1280/729
|7/4
|7/4
|7/4
|-
| |81
| |980.3165
|225/128
|225/128
|99/56
|99/56
|-
|82
|992.4191
|16/9
|16/9
|16/9
|16/9
|-
|83
|1004.5218
|3645/2048
|25/14
|25/14
|25/14
|-
|84
|1016.6245
|9/5
|9/5
|9/5
|9/5
|-
|85
|1028.7272
|2048/1125
|49/27
|49/27
|49/27
|-
|86
|1040.8298
|729/400
|64/35
|11/6
|11/6
|-
|87
|1052.9325
|1152/625
|90/49
|90/49
|90/49
|-
|88
|1065.0352
|50/27
|50/27
|50/27
|13/7
|-
|89
|1077.13785
|4096/2187
|28/15
|28/15
|28/15
|-
|90
|1089.2405
|15/8
|15/8
|15/8
|15/8
|-
|91
|1101.3432
|256/135
|189/100
|154/81
|49/26
|-
|92
|1113.4459
|243/128
|40/21
|40/21
|40/21
|-
|93
|1125.5485
|48/25
|48/25
|21/11
|21/11
|-
|94
|1137.6512
|625/324
|27/14
|27/14
|25/13
|-
|95
|1149.7539
|243/125
|35/18
|35/18
|35/18
|-
|96
|1161.8566
|125/64
|49/25
|49/25
|49/25
|-
|97
|1173.9592
|160/81
|63/32
|63/32
|63/32
|-
|98
|1186.0619
|2025/1024
|125/63
|125/63
|125/63
|-
|99
|1198.1646
|2/1
|2/1
|2/1
|2/1
|}
[[Category:Edf]]
[[Category:Edonoi]]

Latest revision as of 13:20, 18 April 2025

← 57edf 58edf 59edf →
Prime factorization 2 × 29
Step size 12.1027 ¢ 
Octave 99\58edf (1198.16 ¢)
Twelfth 157\58edf (1900.12 ¢)
Consistency limit 12
Distinct consistency limit 12

58 equal divisions of the perfect fifth (abbreviated 58edf or 58ed3/2) is a nonoctave tuning system that divides the interval of 3/2 into 58 equal parts of about 12.1 ¢ each. Each step represents a frequency ratio of (3/2)1/58, or the 58th root of 3/2.

Theory

58edf corresponds to 99.1517…edo. It is related to 99edo, but with the perfect fifth rather than the octave being just. The octave is compressed by about 1.84 cents. 58edf is consistent to the 12-integer-limit. In comparison, 99edo is only consistent up to the 10-integer-limit. 58edf has a flat tendency, with prime harmonics 2, 3, 5, 7, and 11 all tuned flat of just.

Harmonics

Approximation of harmonics in 58edf
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -1.84 -1.84 -3.67 -2.70 -3.67 -4.28 -5.51 -3.67 -4.53 -0.10 -5.51
Relative (%) -15.2 -15.2 -30.3 -22.3 -30.3 -35.4 -45.5 -30.3 -37.5 -0.8 -45.5
Steps
(reduced) 		99
(41) 		157
(41) 		198
(24) 		230
(56) 		256
(24) 		278
(46) 		297
(7) 		314
(24) 		329
(39) 		343
(53) 		355
(7)
Approximation of harmonics in 58edf (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +1.15 +5.98 -4.53 +4.76 -3.37 -5.51 -2.29 +5.73 +5.98 -1.94 +5.83 +4.76
Relative (%) +9.5 +49.4 -37.5 +39.3 -27.9 -45.5 -18.9 +47.4 +49.4 -16.0 +48.1 +39.3
Steps
(reduced) 		367
(19) 		378
(30) 		387
(39) 		397
(49) 		405
(57) 		413
(7) 		421
(15) 		429
(23) 		436
(30) 		442
(36) 		449
(43) 		455
(49)

Subsets and supersets

Since 58 factors into primes as 2 × 29, 58edf contains 2edf and 29edf as subset edts.

See also

Retrieved from "https://en.xen.wiki/index.php?title=58edf&oldid=192829"