38ed7/3: Difference between revisions

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== Intervals ==
{{Infobox ET}}
{| class="wikitable"
{{ED intro}}
!Degrees
! colspan="2" |Enneatonic
!ed43\36
!Pyrite<sub>v</sub>
!ed6\5~32ed1213¢
!18ed4\7
!ed35\29
!''ed29\24=r¢<sub>v</sub>''
!ed23\19
!Pyrite<sub>r¢</sub>~31ed1187¢
!ed17\14
!Golden<sub>v</sub>
!ed39\32
!ed11\9~ed7/3
!30ed1160¢
!31edo
!(21ed8/5+25ed7/4)/2~32ed1240¢
!Golden<sub>v</sub>
!ed16\13
!ed37\30
!18edf~39cet~ed21\17
!31ed1213¢~26ed9/5
!12.5πcET~ed56\45
!ed91\73
!8ed6/5~''ed5\4=r¢<sub>^</sub>''
|-
| rowspan="2" |1
| colspan="2" |G^
| rowspan="2" |37.7193
| rowspan="2" |37.8367
| rowspan="2" |37.8947
37.90625
| rowspan="2" |38.0952
| rowspan="2" |38.1126
| rowspan="2" |''38.1579''
| rowspan="2" |38.22715
| rowspan="2" |38.2808
38.2903
| rowspan="2" |38.3459
| rowspan="2" |38.4171
| rowspan="2" |38.4868
| rowspan="2" |38.5965
38.6019
| rowspan="2" |38.6667
| rowspan="2" |38.7097
| rowspan="2" |38.747
38.75(3)
| rowspan="2" |38.7855
| rowspan="2" |38.8664
| rowspan="2" |38.9474
| rowspan="2" |38.9975
39(.0093)
| rowspan="2" |39.129


39.1383
== Theory ==
| rowspan="2" |39.2699
While 38ed7/3 fails to accurately represent low [[prime interval|prime harmonics]], it provides great approximations of the [[13/1|13th]], [[17/1|17th]], [[19/1|19th]], and a multitude of higher primes, and also handles the interval of [[5/3]] well. But 38ed7/3 should, most of all, be noted for the exceptional quality of its approximation to [[11/9]], which is a mere 0.0088 cents off from just. Its natural subgroup in the [[19-limit]] is 5/3.7/3.11/9.13.17.19, but this can extend to include higher primes, especially [[29/1|29]], [[31/1|31]], and [[37/1|37]].


39.29825
38ed7/3 possesses a shimmering octave at 31 steps in, therefore making this a potential octave-compressed version of [[31edo]], one that sacrifices its notable accuracy in the [[7-limit]] (though a number of 7-limit intervals are still portrayed passably due to the common flat tendency of harmonics 2, 3, 5, and 7) in favor of a huge number of high primes.
| rowspan="2" |39.3655
| rowspan="2" |39.4552
''39.4737''
|-
|Jbv
|''Abv''
|-
|2
|Jb
|''Ab''
|75.4386
|75.6734
|75.7895
75.8125
|76.1905
|76.22505
|''76.3158''
|76.4543
|76.5615
76.58065
|76.6917
|76.83425
|769.7368
|77.193
77.2037
|77.3333
|77.4194
|77.494
77.5(061)
|77.571
|77.7328
|77.8947
|77.995
78(.0186)
|78.2581


156.5533
=== Harmonics ===
|78.5398
{{Harmonics in equal|38|7|3|columns=11}}
{{Harmonics in equal|38|7|3|columns=12|start=12|collapsed=1|title=Approximation of harmonics in 38ed7/3 (continued)}}


78.5965
== Intervals ==
|78.7311
{| class="wikitable center-1 right-2"
|78.9103
! #
''78.9474''
! Cents
|-
|-
| rowspan="2" |3
| 1
|Jb^
| 38.6
|''Ab^''
| rowspan="2" |113.1579
| rowspan="2" |113.5101
| rowspan="2" |113.6842
113.71875
| rowspan="2" |114.2857
| rowspan="2" |114.3376
| rowspan="2" |''114.4737''
| rowspan="2" |114.6814
| rowspan="2" |114.8423
114.871
| rowspan="2" |115.0376
| rowspan="2" |115.2514
| rowspan="2" |115.6053
| rowspan="2" |115.7895
115.8056
| rowspan="2" |116
| rowspan="2" |116.129
| rowspan="2" |116.241
116.25(91)
| rowspan="2" |116.35655
| rowspan="2" |116.5992
| rowspan="2" |116.8421
| rowspan="2" |116.9925
117(.0279)
| rowspan="2" |117.3871
 
117.415
| rowspan="2" |117.8097
 
117.8947
| rowspan="2" |118.0966
| rowspan="2" |118.3655
''118.42105''
|-
|-
| colspan="2" |G#v
| 2
| 77.2
|-
|-
|4
| 3
| colspan="2" |G#
| 115.8
|150.8772
|151.3468
|151.57895
151.625
|152.38095
|152.4501
|''152.6316''
|152.9086
|153.1231
153.1613
|153.3835
|153.6685
|153.9473
|154.386
154.4075
|154.6667
|154.8387
|154.9879
155(.01215)
|155.1421
|155.4656
|155.7895
|155.99
156(.03715)
|156.5161
 
156.5533
|157.0796
 
157.193
|157.46215
|157.8206
''157.8947''
|-
|-
| rowspan="2" |5
| 4
| colspan="2" |G#^
| 154.4
| rowspan="2" |188.5965
| rowspan="2" |189.1835
| rowspan="2" |189.4737
189.53125
| rowspan="2" |190.4762
| rowspan="2" |190.5626
| rowspan="2" |''190.7895''
| rowspan="2" |191.1357
| rowspan="2" |191.4039
191.4516
| rowspan="2" |191.2293
| rowspan="2" |192.0856
| rowspan="2" |192.4342
| rowspan="2" |192.98245
193.0093
| rowspan="2" |193.3333
| rowspan="2" |193.5484
| rowspan="2" |193.7348
193.7(6)5(2)
| rowspan="2" |193.9276
| rowspan="2" |194.332
| rowspan="2" |194.7368
| rowspan="2" |194.9875
195(.0464)
| rowspan="2" |195.6452
 
195.6916
| rowspan="2" |196.3495
 
196.4912
| rowspan="2" |196.827t
| rowspan="2" |197.2758
''197.3684''
|-
|-
|Jv
| 5
|''Av''
| 193.0
|-
|-
|6
| 6
|J
| 231.6
|''A''
|226.3158
|227.0202
|227.3684
227.4375
|228.5714
|228.6751
|''228.9474''
|229.3629
|229.6846
229.7419
|230.0752
|230.5028
|230.92105
|231.57895
231.6112
|232
|232.2581
|232.4818
232.5(182)
|232.7131
|233.1984
|233.6842
|233.985
234(.0557)
|234.7742
 
234.8299
|235.61945
 
235.7895
|236.1932
|236.731
''236.8421''
|-
|-
|7
| 7
|J^/Av
| 270.2
|''A^/Bv''
|264.0351
|264.85685
|265.2632
266.34375
|266.6667
|266.7877
|''267.1053''
|267.59
|267.9654
268.0323
|268.42105
|268.9199
|269.4079
|270.1754
270.2131
|270.6667
|270.9677
|271.2288
271.(271)25
|271.4986
|272.0648
|272.6316
|272.9825
273(.065)
|273.9032
 
273.9682
|274.8894
 
275.0877
|275.5588
|276.1861
''276.3158''
|-
|-
|8
| 8
|A
| 308.8
|''B''
|301.7544
|302.69355
|303.1579
303.25
|304.7619
|304.9002
|''305.2632''
|305.8172
|306.2462
306.3226
|306.7669
|307.337
|307.8947
|308.7719
308.8149
|309.3333
|309.6774
|309.9757
310(.0243)
|310.5841
|310.9312
|311.57895
|311.98
312(.0743)
|313.0323
 
313.10655
|314.1593
 
314.386
|314.9243
|315.6413
''315.7895''
|-
|-
|9
| 9
|A^/Bbv
| 347.4
|B^/Cbv
|339.4737
|340.5302
|341.0526
341.15625
|342.8571
|343.0127
|''343.42105''
|344.0443
|344.5269
344.6129
|345.1128
|345.75415
|346.3816
|347.3684
347.4168
|348
|348.3871
|348.7227
348.75
 
348.7773
|349.0697
|349.7976
|350.5263
|350.9775
351(.0836)
|352.1613
 
352.2449
|353.4292
 
353.6842
|354.2898
|355.0964
''355.2632''
|-
|-
|10
| 10
|Bb
| 386.0
|''Cb''
|377.193
|378.3669
|378.9474
379.0625
|380.9524
|381.1252
|''381.57895''
|382.2715
|382.8077
382.9032
|383.45865
|384.1713
|384.6842
|385.9649
386.0187
|386.6667
|387.0968
|387.4697
387.5(304)
|387.8552
|388.664
|389.4737
|389.975
390(.0929)
|391.2903
 
391.3832
|392.6991
 
392.9825
|393.6554
|394.5516
''394.7368''
|-
|11
|Bb^/A#v
|''Cb^/B#''v
|414.9123
|416.2036
|416.8421
416.96875
|419.0476
|419.2377
|''419.7368''
|420.4986
|421.0885
421.19355
|421.8045
|422.5884
|423.3552
|424.5614
424.6205
|425.3333
|425.80645
|426.2166
426.25
 
426.2834
|426.6407
|427.5304
|428.42105
|428.9725
429(.1022)
|430.41935
 
430.5215
|431.969
 
432.2807
|433.0209
|434.0068
''434.2105''
|-
|12
|A#
|''B#''
|452.6316
|454.0403
|454.7368
454.875
|457.1429
|457.3503
|''457.8947''
|458.7258
|459.3693
459.4839
|460.1504
|461.0055
|461.8421
|463.1579
463.2224
|464
|464.5161
|464.9636
465(.0364)
|465.4262
|466.3968
|467.3684
|467.97
468(.1115)
|469.5484
 
469.6598
|471.2389
 
471.57895
|472.38645
|473.4619
''473.6842''
|-
|13
|A#^/Bv
|''B#^/Cv''
|490.3509
|491.877
|491.6316
491.78125
|495.2381
|495.4628
|''496.0526''
|496.9529
|497.65
497.7742
|498.4962
|499.4227
|500.32895
|501.7544
501.8243
|502.6667
|503.2258
|503.7106
503.7(89)5
|504.2117
|505.2632
|506.3158
|506.9675
507(.1207)
|508.6774
 
508.7981
|510.5088
 
510.8772
|511.752
|512.9171
''513.1579''
|-
|-
|14
| 11
|B
| 424.6
|''C''
|528.0702
|529.7137
|530.5263
530.6875
|533.3333
|533.5753
|''534.2105''
|535.1801
|535.9308
536.0645
|536.8421
|537.8398
|538.8158
|540.3509
540.4261
|541.3333
|541.9355
|542.4575
542.5(425)
|542.99725
|544.12955
|545.2632
|545.965
546(.13)
|547.80645
 
547.9365
|549.7787
 
550.1754
|551.1175
|552.37225
''552.6316''
|-
|-
|15
| 12
|B^/Cv
| 463.2
|''C^/Qv''
|565.7895
|567.5504
|568.42105
568.59375
|571.4286
|571.6878
|''572.3684''
|573.4072
|574.2116
574.3548
|575.188
|576.2569
|377.3026
|578.9474
579.028
|580
|580.6452
|581.2045
581.2(94)5
|581.7828
|582.99595
|583.2105
|584.9625
585(.1393)
|586.9355
 
587.0748
|589.0486
 
589.4737
|590.4831
|591.8274
''592.1053''
|-
|-
|16
| 13
|C
| 502.7
|''Q''
|603.5088
|605.3871
|606.3158
606.5
|609.5238
|609.8004
|''610.5263''
|611.63435
|612.4923
612.6452
|612.5338
|614.674
|615.8947
|617.5439
617.6299
|618.6667
|619.3548
|619.9515
620(.0486)
|620.5683
|621.86235
|623.1579
|623.96
624(.1486)
|626.0645
 
626.2131
|628.3185
 
628.7719
|629.8486
|631.2826
''631.57895''
|-
|-
|17
| 14
|C^/Qbv
| 540.4
|''Q^/Dbv''
|641.2281
|643.2238
|644.2105
644.40625
|647.61905
|647.9129
|''648.6842''
|649.8615
|650.7731
650.9355
|651.8797
|653.0912
|654.2763
|656.14035
656.2317
|657.3333
|658.0645
|658.6984
658.75
 
658.8016
|659.3538
|660.7587
|661.1053
|662.9575
663(.1579)
|665.19355
 
665.3514
|667.5884
 
668.0702
|669.2141
|670.7377
''671.0526''
|-
|-
|18
| 15
|Qb
| 579.0
|''Db''
|678.9474
|681.0605
|682.1053
682.3125
|685.7143
|686.0254
|''686.8421''
|688.0886
|689.0538
689.2258
|690.2256
|691.5083
|692.7632
|694.7368
694.8336
|696
|696.7742
|697.4454
697.5(5465)
|698.1393
|699.5951
|701.0526
|701.955
702(.1672)
|704.3226
 
704.4897
|706.85835
 
707.3684
|708.5797
|710.1929
''710.5263''
|-
|-
|19
| 16
|Qb^/C#v
| 617.6
|''Db^/Q#v''
|716.6667
|718.8972
|720(.21875)
|723.8095
|724.1379
|''725''
|726.3158
|727.3347
727.5161
|728.5714
|729.9254
|731.25
|733.3333
733.43545
|734.6667
|735.4839
|736.(19)2(3)5
736.3077
|736.9248
|738.4615
|740
|740.9525
741(.1765)
|743.4516
 
743.6281
|746.1283
 
746.6667
|747.9452
|749.6481
''750''
|-
|-
|20
| 17
|C#
| 656.2
|''Q#''
|754.386
|756.7339
|757.8947
758.125
|761.9048
|762.25045
|''763.1579''
|764.54294
|765.6154
765.80645
|766.9173
|768.34255
|769.3684
|771.9298
772.0373
|773.3333
|774.19355
|774.9393
775(.0607)
|775.7104
|777.3279
|778.9474
|779.95
780(.1858)
|782.58065
 
782.7664
|785.3982
 
785.9649
|787.3107
|789.1032
''789.4737''
|-
|-
|21
| 18
|C#^/Qv
| 694.8
|''Q#/Dv''
|792.1053
|794.5706
|795.7895
796.03125
|800
|800.363
|''801.3158''
|802.7701
|803.8962
804.0968
|805.2632
|806.7597
|808.2237
|810.5263
810.6392
|812
|812.9032
|813.6863
813.75
 
813.8138
|814.4959
|816.1943
|817.8947
|818.9475
819(.19505)
|821.7097
 
821.9047
|824.6681
 
825.2632
|826.6763
|828.5584
''828.9474''
|-
|-
|22
| 19
|Q
| 733.4
|''D''
|829.8246
|832.4073
|833.6842
833.9375
|838.0952
|838.4755
|''839.473''7
|840.99723
|842.177
842.3871
|843.609
|845.1768
|846.1053
|849.1228
849.24105
|850.6667
|851.6129
|852.(4332)5[668]
|853.2814
|855.0607
|856.8421
|857.945
858(.2043)
|860.8387
861.043
|863.938
 
864.5614
|866.0418
|868.0135
''868.42105''
|-
|-
|23
| 20
|Q^/Dv
| 772.0
|''D^/Sv''
|867.5439
|870.24395
|871.57895
871.84375
|876.1905
|876.588
|''877.6316''
|879.2244
|880.45775
880.6774
|881.9549
|883.5939
|885.1974
|887.7193
887.8429
|889.3333
|890.3226
|891.1802
891.25
 
891.3198
|892.0669
|893.9271
|895.7895
|896.9425
897(.2136)
|899.9677
 
900.1813
|903.2079
 
903.85965
|905.40735
|907.4687
''907.8947''
|-
|-
|24
| 21
|D
| 810.6
|''S''
|905.2632
|908.0806
|909.4737
909.75
|914.2857
|914.7005
|''915.7895''
|917.45152
|918.7385
918.9677
|920.30075
|922.0111
|923.6842
|926.3158
926.4448
|928
|929.0323
|929.9272
930(.0729)
|930.8524
|932.7935
|934.7368
|935.94
936(.2229)
|939.0968
 
939.31965
|942.4778
 
943.1579
|944.7729
|946.9239
''947.3684''
|-
|-
| rowspan="2" |25
| 22
|D^
| 849.2
|''S^''
| rowspan="2" |942.9825
| rowspan="2" |945.9173
| rowspan="2" |947.3684
947.65625
| rowspan="2" |952.38095
| rowspan="2" |952.8131
| rowspan="2" |''953.9474''
| rowspan="2" |955.6787
| rowspan="2" |957.0193
957.2581
| rowspan="2" |958.6466
| rowspan="2" |960.4282
| rowspan="2" |962.17105
| rowspan="2" |964.9123
965.04665
| rowspan="2" |966.6667
| rowspan="2" |967.7419
| rowspan="2" |968.(6)7(41)5
968.8259
| rowspan="2" |969.63795
| rowspan="2" |971.6599
| rowspan="2" |973.6842
| rowspan="2" |974.9375
975(.2322)
| rowspan="2" |978.2258
 
978.458
| rowspan="2" |981.7477
 
982.4561
| rowspan="2" |984.1384
| rowspan="2" |986.379
''986.8421''
|-
|-
| colspan="2" |Ebv
| 23
| 887.8
|-
|-
|26
| 24
| colspan="2" |Eb
| 926.4
|980.70175
|983.754
|985.2632
985.5625
|990.4762
|990.9256
|''992.1053''
|993.90582
|995.3001
995.5484
|996.9925
|998.8453
|1000.6579
|1003.5088
1003.6485
|1005.3333
|1006.4516
|1007.4211
1007.5(789)
|1008.4235
|1010.5263
|1012.6316
|1013.935
1014(.2415)
|1017.3548
 
1017.5963
|1021.0176
 
1021.7544
|1023.504
|1025.8342
''1026.3158''
|-
|-
| rowspan="2" |27
| 25
| colspan="2" |Eb^
| 965.0
| rowspan="2" |1018.42105
| rowspan="2" |1021.1591
| rowspan="2" |1023.1579
1023.46875
| rowspan="2" |1028.5714
| rowspan="2" |1029.0381
| rowspan="2" |''1030.2632''
| rowspan="2" |1032.132
| rowspan="2" |1033.5808
1033.8387
| rowspan="2" |1035.33835
| rowspan="2" |1037.2624
| rowspan="2" |1039.1447
| rowspan="2" |1042.1053
1042.2504
| rowspan="2" |1044
| rowspan="2" |1045.1613
| rowspan="2" |1046.1681
1046.25
 
1046.332
| rowspan="2" |1047.0209
| rowspan="2" |1049.3927
| rowspan="2" |1051.57895
| rowspan="2" |1052.9325
1053(.2508)
| rowspan="2" |1056.4839
 
1056.7346
| rowspan="2" |1060.2875
 
1061.0526
| rowspan="2" |1062.8695
| rowspan="2" |1065.2893
''1065.7895''
|-
|-
|D#v
| 26
|''S#v''
| 1003.6
|-
|-
|28
| 27
|D#
| 1042.3
|''S#''
|1056.14035
|1059.4274
|1061.0526
1061.375
|1066.6667
|1067.1506
|''1068.42105''
|1070.3601
|1071.8616
1072.129
|1073.6842
|1075.6796
|1077.6315
|1080.70175
1080.85225
|1083.6667
|1083.871
|1084.91505
1085(.085)
|1085.9945
|1088.2591
|1090.5263
|1091.93
1092(.26)
|1095.6129
 
1095.8729
|1099.5574
 
1100.3509
|1102.235
|1104.7445
''1105.2632''
|-
|-
| rowspan="2" |29
| 28
|D#^
| 1080.9
|''S#^''
| rowspan="2" |1093.85965
| rowspan="2" |1097.7264
| rowspan="2" |1098.9474
1099.28125
| rowspan="2" |1104.7619
| rowspan="2" |1105.2632
| rowspan="2" |''1106.57895''
| rowspan="2" |1108.58726
| rowspan="2" |1110.1424
1110.4194
| rowspan="2" |1112.0301
| rowspan="2" |1114.0967
| rowspan="2" |1116.1184
| rowspan="2" |1119.29825
1119.4541
| rowspan="2" |1122.3333
| rowspan="2" |1122.5806
| rowspan="2" |1123.662
1123.75
 
1123.83805
| rowspan="2" |1124.78
| rowspan="2" |1127.1255
| rowspan="2" |1129.4737
| rowspan="2" |1130.9275
1131(.26935)
| rowspan="2" |1134.7419
 
1135.0112
| rowspan="2" |1138.8273
 
1139.6491
| rowspan="2" |1141.6006
| rowspan="2" |1144.1997
''1144.7368''
|-
|-
| colspan="2" |Ev
| 29
| 1119.5
|-
|-
|30
| 30
| colspan="2" |E
| 1158.1
|1131.57895
|1135.1008
|1136.8421
1137.1875
|1142.8571
|1143.3757
|''1144.7368''
|1146.8144
|1148.4231
1148.7097
|1150.3759
|1152.5138
|1154.0526
|1157.8947
1158.0559
|1160
|1161.2903
|1162.409
1162.5(911)
|1163.5655
|1165.9919
|1168.42105
|1169.925
1170(.2786)
|1173.871
 
1174.1496
|1178.0972
 
1178.9474
|1180.9661
|1183.6548
''1184.2105''
|-
|-
|31
| 31
| colspan="2" |E^/Fbv
| 1196.7
|1169.29825
|1172.9375
|1174.7368
1175.09625
|1180.9524
|1181.4882
|''1182.8947''
|1185.04155
|1186.7039
1187
|1188.7218
|1190.93095
|1193.0921
|1196.4912
1196.6578
|1198.6667
|1200
|1201.15595
1201.25
 
1201.3441
|1202.3511
|1204.8583
|1207.3684
|1208.9225
1209(.2879)
|1213(.2879)
|1217.36715
 
1218.2456
|1220.33165
|1223.11
''1223.6842''
|-
|-
|32
| 32
| colspan="2" |Fb
| 1235.3
|1207.0715
|1210.7742
|1212.6316
1213
|1219.0476
|1219.6007
|''1221.0526''
|1223.2687
|1224.9847
1225.2903
|1227.0677
|1229.3481
|1231.57895
|1235.0877
1235.2567
|1237.3333
|1238.7097
|1239.9029
1240(.0972)
|1241.1366
|1243.7247
|1246.3158
|1247.92
1248(.2972)
|1252.129
1252.4262
|1256.6371
 
1257.5439
|1259.6972
|1262.56515
''1263.1579''
|-
|-
|33
| 33
| colspan="2" |Fb^/E#v
| 1273.9
|1244.7368
|1248.6109
|1250.5263
1250.90625
|1257.1429
|1257.71235
|''1259.2105''
|1261.4958
|1263.2655
1263.58065
|1265.4135
|1267.7652
|1270.0658
|1273.68425
1273.8616
|1276
|1277.4194
|1278.6499
1278.75
 
1278.8502
|1279.9221
|1282.5911
|1284.2632
|1286.9175
1287(.3065)
|1291.2581
1291.5645
|1295.907
 
1296.8421
|1299.0627
|1302.0203
''1302.6316''
|-
|-
|34
| 34
| colspan="2" |E#
| 1312.5
|1282.4561
|1286.4476
|1288.42105
1288.8125
|1295.2381
|1295.8258
|''1297.3684''
|1299.723
|1301.5462
1301.871
|1303.7594
|1306.1823
|1308.5526
|1312.2807
1312.4634
|1314.6667
|1316.129
|1317.3968
1317.5
 
1317.6032
|1318.7076
|1321.4575
|1323.2105
|1325.915
1326(.3158)
|1330.3871
1330.7028
|1335.1769
 
1336.1404
|1338.4283
|1341.4755
''1342.1053''
|-
|-
|35
| 35
| colspan="2" |E#^/Fv
| 1351.1
|1320.1754
|1324.2843
|1326.3158
1326.71875
|1333.3333
|1333.9383
|''1335.5263''
|1337.9501
|1339.827
1340.1613
|1342.1053
|1344.59945
|1347.0395
|1350.8772
1531.0654
|1353.3333
|1354.8387
|1356.1438
1356.25
 
1356.3563
|1357.4931
|1360.3239
|1362.1579
|1364.9125
1365(.3251)
|1369.5161
1369.8412
|1374.4468
 
1375.4386
|1377.7938
|1380.9306
''1381.57895''
|-
|-
|36
| 36
| colspan="2" |F
| 1389.7
|1357.8947
|1362.121
|1364.2105
1364.625
|1371.4286
|1372.0508
|''1373.6842''
|1376.1773
|1378.1078
1378.4516
|1380.4511
|1383.0166
|1385.5263
|1389.4737
1389.6672
|1392
|1393.5484
|1394.8908
1395(.1093)
|1396.27865
|1399.1903
|1401.1053
|1403.91
1404(.3343)
|1408.6452
1408.9795
|1413.7167
 
1414.7368
|1417.1593
|1420.3858
''1421.0526''
|-
|-
|37
| 37
| colspan="2" |F^/Gv
| 1428.3
|1395.614
|1399.9577
|1402.1053
1402.53125
|1409.5238
|1410.1633
|''1411.8421''
|1414.4044
|1416.3885
1416.7419
|1418.797
|1421.4337
|1424.0132
|1428.0702
1428.269
|1430.6667
|1432.2581
|1433.6377
1433.75
 
1433.8623
|1435.0642
|1438.0567
|1440.0526
|1442.9075
1443(.34365)
|1447.7742
1448.1178
|1452.9866
 
1454.0351
|1456.5249
|1459.841
''1460.5263''
|-
|-
|38
| 38
| colspan="2" |G
| 1466.9
|1433.3333
|1437.79435
|1440(.4375)
|1447.61905
|1448.2759
|''1450''
|1452.6316
|1454.6693
1455.0323
|1457.1429
|1459.8508
|1462.5
|1466.6617
1466.8709
|1469.3333
|1470.6977
|1472.3847
1472.(61)5(4)
|1473.8497
|1476.9231
|1480
|1481.905
1482(.3529)
|1486.9032
1487.2561
|1492.2565
1493.3333
|1495.8904
|1499.2961
''1500''
|}
|}

Latest revision as of 18:22, 14 May 2026

← 37ed7/3 38ed7/3 39ed7/3 →
Prime factorization 2 × 19
Step size 38.6019 ¢ 
Octave 31\38ed7/3 (1196.66 ¢)
(semiconvergent)
Twelfth 49\38ed7/3 (1891.49 ¢)
Consistency limit 8
Distinct consistency limit 8

38 equal divisions of 7/3 (abbreviated 38ed7/3) is a nonoctave tuning system that divides the interval of 7/3 into 38 equal parts of about 38.6 ¢ each. Each step represents a frequency ratio of (7/3)1/38, or the 38th root of 7/3.

Theory

While 38ed7/3 fails to accurately represent low prime harmonics, it provides great approximations of the 13th, 17th, 19th, and a multitude of higher primes, and also handles the interval of 5/3 well. But 38ed7/3 should, most of all, be noted for the exceptional quality of its approximation to 11/9, which is a mere 0.0088 cents off from just. Its natural subgroup in the 19-limit is 5/3.7/3.11/9.13.17.19, but this can extend to include higher primes, especially 29, 31, and 37.

38ed7/3 possesses a shimmering octave at 31 steps in, therefore making this a potential octave-compressed version of 31edo, one that sacrifices its notable accuracy in the 7-limit (though a number of 7-limit intervals are still portrayed passably due to the common flat tendency of harmonics 2, 3, 5, and 7) in favor of a huge number of high primes.

Harmonics

Approximation of harmonics in 38ed7/3
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -3.3 -10.5 -6.7 -7.0 -13.8 -10.5 -10.0 +17.7 -10.3 +17.7 -17.1
Relative (%) -8.7 -27.1 -17.3 -18.1 -35.8 -27.1 -26.0 +45.8 -26.7 +45.8 -44.4
Steps
(reduced) 		31
(31) 		49
(11) 		62
(24) 		72
(34) 		80
(4) 		87
(11) 		93
(17) 		99
(23) 		103
(27) 		108
(32) 		111
(35)
Approximation of harmonics in 38ed7/3 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) -1.3 -13.8 -17.4 -13.4 -2.5 +14.3 -2.1 -13.7 +17.7 +14.3 +14.6 +18.1
Relative (%) -3.4 -35.8 -45.2 -34.6 -6.5 +37.1 -5.4 -35.4 +45.8 +37.2 +37.8 +46.9
Steps
(reduced) 		115
(1) 		118
(4) 		121
(7) 		124
(10) 		127
(13) 		130
(16) 		132
(18) 		134
(20) 		137
(23) 		139
(25) 		141
(27) 		143
(29)

Intervals

# Cents
1 38.6
2 77.2
3 115.8
4 154.4
5 193.0
6 231.6
7 270.2
8 308.8
9 347.4
10 386.0
11 424.6
12 463.2
13 502.7
14 540.4
15 579.0
16 617.6
17 656.2
18 694.8
19 733.4
20 772.0
21 810.6
22 849.2
23 887.8
24 926.4
25 965.0
26 1003.6
27 1042.3
28 1080.9
29 1119.5
30 1158.1
31 1196.7
32 1235.3
33 1273.9
34 1312.5
35 1351.1
36 1389.7
37 1428.3
38 1466.9
Retrieved from "https://en.xen.wiki/index.php?title=38ed7/3&oldid=230345"