58edf: Difference between revisions

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-unuseful table. The interval table of 99edo already covers both sharp and flat mappings
Line 11: Line 11:
=== Subsets and supersets ===
=== Subsets and supersets ===
Since 58 factors into primes as {{nowrap| 2 × 29 }}, 58edf contains [[2edf]] and [[29edf]] as subsets.
Since 58 factors into primes as {{nowrap| 2 × 29 }}, 58edf contains [[2edf]] and [[29edf]] as subsets.
== Intervals ==
{{todo|inline=1|complete table|text=The [[mapping]] for 11 and 13 should differ from [[99edo]].}}
{| class="wikitable"
|-
! Degrees
! Cents Value
! Five limit
! Seven limit
! Eleven limit
! Thirteen limit
|-
| |1
| |12.1027
|2048/2025
|126/125
|99/98
|91/90
|-
| |2
| |24.2053
|81/80
|64/63
| colspan="2" |55/54
|-
| |3
| |36.308
|128/125
| colspan="3" |49/48
|-
| |4
| |48.4107
|250/243
|36/35
| colspan="2" |33/32
|-
| |5
| |60.5134
|648/625
| colspan="2" |28/27
|26/25
|-
| |6
| |72.616
| colspan="2" |25/24
| colspan="2" |22/21
|-
| |7
| |84.7187
|256/243
| colspan="3" |21/20
|-
| |8
| |96.8214
| colspan="2" |135/128
|81/77
|52/49
|-
| |9
| |108.92405
| colspan="4" |16/15
|-
| |10
| |121.0267
|2187/2048
| colspan="3" |15/14
|-
| |11
| |133.1294
| colspan="3" |27/25
|13/12
|-
| |12
| |145.2321
|625/576
| colspan="3" |49/45
|-
| |13
| |157.3347
|800/729
|35/32
| colspan="2" |11/10
|-
| |14
| |169.4374
|1125/1024
| colspan="3" |54/49
|-
| |15
| |181.54
|10/9
| colspan="3" |10/9
|-
| |16
| |193.6428
|4096/3645
| colspan="3" |28/25
|-
| |17
| |205.7454
| colspan="4" |9/8
|-
| |18
| |217.8481
|256/225
|245/216
|112/99
|91/80
|-
| |19
| |229.9508
|729/640
| colspan="3" |8/7
|-
| |20
| |242.05345
| colspan="2" |144/125
|63/55
|52/45
|-
| |21
| |254.1561
|125/108
| colspan="2" |81/70
|15/13
|-
| |22
| |266.2587
|729/625
| colspan="3" |7/6
|-
| |23
| |278.3615
| colspan="2" |75/64
| colspan="2" |33/28
|-
| |24
| |290.4641
|32/27
| colspan="2" |32/27
|13/11
|-
| |25
| |302.5668
|1215/1024
| colspan="3" |25/21
|-
| |26
| |314.6695
| colspan="4" |6/5
|-
| |27
| |326.7722
|3125/2592
| colspan="2" |98/81
|91/75
|-
| |28
| |338.8748
|243/200
|128/105
| colspan="2" |11/9
|-
| |29
| |350.9775
|625/512
| colspan="3" |49/40
|-
| |30
| |363.0802
| colspan="2" |100/81
|27/22
|16/13
|-
| |31
| |375.18285
|3888/3125
| colspan="3" |56/45
|-
| |32
| |387.2855
| colspan="4" |5/4
|-
| |33
| |399.3882
|512/405
| colspan="2" |63/50
|49/39
|-
| |34
| |411.4909
|81/64
| colspan="2" |80/63
|33/26
|-
| |35
| |423.5935
| colspan="2" |32/25
| colspan="2" |14/11
|-
| |36
| |435.6962
|625/486
| colspan="3" |9/7
|-
| |37
| |447.7989
|162/125
| colspan="2" |35/27
|13/10
|-
| |38
| |459.90155
|125/96
|64/49
| colspan="2" |55/42
|-
| |39
| |472.0042
|320/243
| colspan="3" |21/16
|-
| |40
| |484.1069
|675/512
| colspan="2" |250/189
|65/49
|-
| |41
| |469.2096
| colspan="4" |4/3
|-
| |42
| |508.3122
|8192/6075
|75/56
| colspan="2" |66/49
|-
| |43
| |520.4149
| colspan="4" |27/20
|-
| |44
| |532.5176
|512/375
| colspan="3" |49/36
|-
| |45
| |544.6203
|1000/729
|48/35
| colspan="2" |11/8
|-
| |46
| |556.7229
|864/625
| colspan="2" |112/81
|91/66
|-
| |47
| |568.8256
| colspan="3" |25/18
|18/13
|-
| |48
| |580.9283
|1024/729
| colspan="3" |7/5
|-
| |49
| |593.03095
| colspan="4" |45/32
|-
| |50
| |605.1336
| colspan="4" |64/45
|-
| |51
| |617.2362
|729/512
| colspan="3" |10/7
|-
| |52
| |629.339
| colspan="3" |36/25
|13/9
|-
| |53
| |641.4416
|625/432
| colspan="2" |81/56
|75/52
|-
| |54
| |653.5443
|729/500
|35/24
| colspan="2" |16/11
|-
| |55
| |665.647
|375/256
| colspan="3" |72/49
|-
| |56
| |677.7497
| colspan="4" |40/27
|-
| |57
| |689.8523
|6075/4096
|112/75
| colspan="2" |49/33
|-
| |58
| |701.955
| colspan="4" |3/2
|-
| |59
| |714.0577
|1024/675
| colspan="2" |189/125
|91/60
|-
| |60
| |726.16035
|243/160
| colspan="3" |32/21
|-
| |61
| |738.263
|192/125
| colspan="3" |49/32
|-
| |62
| |750.3657
|125/81
| colspan="2" |54/35
|20/13
|-
| |63
| |762.4684
|972/625
| colspan="3" |14/9
|-
| |64
| |774.571
| colspan="2" |25/16
| colspan="2" |11/7
|-
| |65
| |786.6737
|128/81
| colspan="2" |63/40
|52/33
|-
| |66
| |798.7764
|405/256
| colspan="2" |100/63
|78/49
|-
| |67
| |810.87905
| colspan="4" |8/5
|-
| |68
| |822.9817
|3125/1944
| colspan="3" |45/28
|-
| |69
| |835.0844
| colspan="2" |81/50
|44/27
|13/8
|-
| |70
| |847.1871
|625/384
| colspan="3" |49/30
|-
| |71
| |859.2897
|400/243
|105/64
| colspan="2" |18/11
|-
| |72
| |871.3924
|3375/2048
| colspan="3" |81/49
|-
| |73
| |883.4951
| colspan="4" |5/3
|-
| |74
| |895.5978
|2048/1215
| colspan="3" |42/25
|-
| |75
| |907.7004
| colspan="3" |27/16
|22/13
|-
| |76
| |919.8031
| colspan="2" |128/75
|56/33
|56/33
|-
| |77
| |931.9058
|1250/729
| colspan="3" |12/7
|-
| |78
| |944.00845
|216/125
| colspan="2" |140/81
|26/15
|-
| |79
| |956.1111
| colspan="2" |125/72
|110/63
|45/26
|-
| |80
| |968.2138
|1280/729
| colspan="3" |7/4
|-
| |81
| |980.3165
|225/128
|225/128
| colspan="2" |99/56
|-
|82
|992.4191
| colspan="4" |16/9
|-
|83
|1004.5218
|3645/2048
| colspan="3" |25/14
|-
|84
|1016.6245
| colspan="4" |9/5
|-
|85
|1028.7272
|2048/1125
| colspan="3" |49/27
|-
|86
|1040.8298
|729/400
|64/35
| colspan="2" |11/6
|-
|87
|1052.9325
|1152/625
| colspan="3" |90/49
|-
|88
|1065.0352
| colspan="4" |50/27
|-
|89
|1077.13785
|4096/2187
| colspan="3" |28/15
|-
|90
|1089.2405
| colspan="4" |15/8
|-
|91
|1101.3432
|256/135
|189/100
|154/81
|49/26
|-
|92
|1113.4459
|243/128
| colspan="3" |40/21
|-
|93
|1125.5485
| colspan="4" |48/25
|-
|94
|1137.6512
|625/324
| colspan="2" |27/14
|25/13
|-
|95
|1149.7539
|243/125
| colspan="2" |35/18
|35/18
|-
|96
|1161.8566
|125/64
| colspan="2" |49/25
|49/25
|-
|97
|1173.9592
|160/81
| colspan="3" |63/32
|-
|98
|1186.0619
|2025/1024
| colspan="3" |125/63
|-
|99
|1198.1646
| colspan="4" |2/1
|-
|100
|1210.2672
|4096/2025
|252/125
|99/49
|91/45
|-
|101
|1222.3699
|81/40
|128/63
| colspan="2" |55/27
|-
|102
|1234.4726
|256/125
| colspan="3" |49/24
|-
|103
|1246.5753
|500/243
|72/35
| colspan="2" |33/16
|-
|104
|1258.6779
|1296/625
| colspan="2" |56/27
|52/25
|-
|105
|1270.7806
| colspan="2" |25/12
| colspan="2" |44/21
|-
|106
|1282.8833
|512/243
| colspan="3" |21/10
|-
|107
|1294.98595
| colspan="2" |135/64
|162/77
|104/49
|-
|108
|1307.0886
| colspan="4" |32/15
|-
|109
|1319.1913
|2187/1024
| colspan="3" |15/7
|-
|110
|1331.294
| colspan="3" |54/25
|13/6
|-
|111
|1343.3966
|625/288
| colspan="3" |98/45
|-
|112
|1355.4993
|1600/729
|35/16
| colspan="2" |11/5
|-
|113
|1367.602
|1125/512
| colspan="3" |108/49
|-
|114
|1379.7047
| colspan="4" |20/9
|-
|115
|1391.8073
|8192/3645
| colspan="3" |56/25
|-
|116
|1403.91
| colspan="4" |9/4
|}


== See also ==
== See also ==
* [[99edo]] – relative edo
* [[99edo]] – relative edo
* [[157edt]] – relative edt
* [[157edt]] – relative edt

Revision as of 14:40, 22 March 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 3/2 rather than the 2/1 being just. The octave is about 1.8354 cents compressed. 58edf is consistent to the 12-integer-limit. In comparison, 99edo is only consistent up to the 10-integer-limit.

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 subsets.

See also

Retrieved from "https://en.xen.wiki/index.php?title=58edf&oldid=187694"