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38ed7/3: Difference between revisions

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{{Infobox ET}}
{{ED intro}}
== Theory ==
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]].
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 ===
{{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)}}
== Intervals ==
== Intervals ==
{| class="wikitable"
{| class="wikitable center-1 right-2"
!Degrees
! #
! colspan="2" |Enneatonic
! Cents
!ed43\36
!Pyrite<sub>v</sub>
!ed35\29
!''ed29\24=r¢<sub>v</sub>''
!ed17\14
!ed11\9~ed7/3
!Golden
!ed16\13
!ed21\17
!''ed5\4=r¢<sub>^</sub>''
|-
|-
| rowspan="2" |1
| 1
| colspan="2" |G^
| 38.6
| rowspan="2" |37.7193
| rowspan="2" |37.8367
| rowspan="2" |38.1126
| rowspan="2" |''38.1579''
| rowspan="2" |38.3459
| rowspan="2" |38.5965
38.6019
| rowspan="2" |38.7855
| rowspan="2" |38.8664
| rowspan="2" |39.0093
| rowspan="2" |''39.4737''
|-
|-
|Jbv
| 2
|''Abv''
| 77.2
|-
|-
|2
| 3
|Jb
| 115.8
|''Ab''
|75.4386
|75.6734
|76.22505
|''76.3158''
|76.6917
|77.193
77.2037
|77.571
|77.7328
|78.0186
|''78.9474''
|-
|-
| rowspan="2" |3
| 4
|Jb^
| 154.4
|''Ab^''
| rowspan="2" |113.1579
| rowspan="2" |113.5101
| rowspan="2" |114.3376
| rowspan="2" |''114.4737''
| rowspan="2" |115.0376
| rowspan="2" |115.7895
115.8056
| rowspan="2" |116.35655
| rowspan="2" |116.5992
| rowspan="2" |117.0279
| rowspan="2" |''118.42105''
|-
|-
| colspan="2" |G#v
| 5
| 193.0
|-
|-
|4
| 6
| colspan="2" |G#
| 231.6
|150.8772
|151.3468
|152.4501
|''152.6316''
|153.3835
|154.386
154.4075
|155.1421
|155.4656
|156.03715
|''157.8947''
|-
|-
| rowspan="2" |5
| 7
| colspan="2" |G#^
| 270.2
| rowspan="2" |188.5965
| rowspan="2" |189.1835
| rowspan="2" |190.5626
| rowspan="2" |''190.7895''
| rowspan="2" |191.2293
| rowspan="2" |192.98245
193.0093
| rowspan="2" |193.9276
| rowspan="2" |194.332
| rowspan="2" |195.0464
| rowspan="2" |''197.3684''
|-
|-
|Jv
| 8
|''Av''
| 308.8
|-
|-
|6
| 9
|J
| 347.4
|''A''
|226.3158
|227.0202
|228.6751
|''228.9474''
|230.0752
|231.57895
231.6112
|232.7131
|233.1984
|234.0557
|''236.8421''
|-
|-
|7
| 10
|J^/Av
| 386.0
|''A^/Bv''
|264.0351
|264.85685
|266.7877
|''267.1053''
|268.42105
|270.1754
270.2131
|271.4986
|272.0648
|273.065
|''276.3158''
|-
|-
|8
| 11
|A
| 424.6
|''B''
|301.7544
|302.69355
|304.9002
|''305.2632''
|306.7669
|308.7719
308.8149
|310.5841
|310.9312
|312.0743
|''315.7895''
|-
|-
|9
| 12
|A^/Bbv
| 463.2
|B^/Cbv
|339.4737
|340.5302
|343.0127
|''343.42105''
|345.1128
|347.3684
347.4168
|349.0697
|349.7976
|351.0836
|''355.2632''
|-
|-
|10
| 13
|Bb
| 502.7
|''Cb''
|377.193
|378.3669
|381.1252
|''381.57895''
|383.45865
|385.9649
386.0187
|387.8552
|388.664
|390.0929
|''394.7368''
|-
|-
|11
| 14
|Bb^/A#v
| 540.4
|''Cb^/B#''v
|414.9123
|416.2036
|419.2377
|''419.7368''
|421.8045
|424.5614
424.6205
|426.6407
|427.5304
|429.1022
|''434.2105''
|-
|-
|12
| 15
|A#
| 579.0
|''B#''
|452.6316
|454.0403
|457.3503
|''457.8947''
|460.1504
|463.1579
463.2224
|465.4262
|466.3968
|468.1115
|''473.6842''
|-
|-
|13
| 16
|A#^/Bv
| 617.6
|''B#^/Cv''
|490.3509
|491.877
|495.4628
|''496.0526''
|498.4962
|501.7544
501.8243
|504.2117
|505.2632
|507.1207
|''513.1579''
|-
|-
|14
| 17
|B
| 656.2
|''C''
|528.0702
|529.7137
|533.5753
|''534.2105''
|536.8421
|540.3509
540.4261
|542.99725
|544.12955
|546.13
|''552.6316''
|-
|-
|15
| 18
|B^/Cv
| 694.8
|''C^/Qv''
|565.7895
|567.5504
|571.6878
|''572.3684''
|575.188
|578.9474
579.028
|581.7828
|582.99595
|585.1393
|''592.1053''
|-
|-
|16
| 19
|C
| 733.4
|''Q''
|603.5088
|605.3871
|609.8004
|''610.5263''
|612.5338
|617.5439
617.6299
|620.5683
|621.86235
|624.1486
|''631.57895''
|-
|-
|17
| 20
|C^/Qbv
| 772.0
|''Q^/Dbv''
|641.2281
|643.2238
|647.9129
|''648.6842''
|651.8797
|656.14035
656.2317
|659.3538
|660.7587
|663.1579
|''671.0526''
|-
|-
|18
| 21
|Qb
| 810.6
|''Db''
|678.9474
|681.0605
|686.0254
|''686.8421''
|690.2256
|694.7368
694.8336
|698.1393
|699.5951
|702.1672
|''710.5263''
|-
|-
|19
| 22
|Qb^/C#v
| 849.2
|''Db^/Q#v''
|716.6667
|718.8972
|724.1379
|''725''
|728.5714
|733.3333
733.43545
|736.9248
|738.4615
|741.1765
|''750''
|-
|-
|20
| 23
|C#
| 887.8
|''Q#''
|754.386
|756.7339
|762.25045
|''763.1579''
|766.9173
|771.9298
772.0373
|775.7104
|777.3279
|780.1858
|''789.4737''
|-
|-
|21
| 24
|C#^/Qv
| 926.4
|''Q#/Dv''
|792.1053
|794.5706
|800.363
|''801.3158''
|805.2632
|810.5263
810.6392
|814.4959
|816.1943
|819.19505
|''828.9474''
|-
|-
|22
| 25
|Q
| 965.0
|''D''
|829.8246
|832.4073
|838.4755
|''839.473''7
|843.609
|849.1228
849.24105
|853.2814
|855.0607
|858.2043
|''868.42105''
|-
|-
|23
| 26
|Q^/Dv
| 1003.6
|''D^/Sv''
|867.5439
|870.24395
|876.588
|''877.6316''
|881.9549
|887.7193
887.8429
|892.0669
|893.9271
|897.2136
|''907.8947''
|-
|-
|24
| 27
|D
| 1042.3
|''S''
|905.2632
|908.0806
|914.7005
|''915.7895''
|920.30075
|926.3158
926.4448
|930.8524
|932.7935
|936.2229
|''947.3684''
|-
|-
| rowspan="2" |25
| 28
|D^
| 1080.9
|''S^''
| rowspan="2" |942.9825
| rowspan="2" |945.9173
| rowspan="2" |952.8131
| rowspan="2" |''953.9474''
| rowspan="2" |958.6466
| rowspan="2" |964.9123
965.04665
| rowspan="2" |969.63795
| rowspan="2" |971.6599
| rowspan="2" |975.2322
| rowspan="2" |''986.8421''
|-
|-
| colspan="2" |Ebv
| 29
| 1119.5
|-
|-
|26
| 30
| colspan="2" |Eb
| 1158.1
|980.70175
|983.754
|990.9256
|''992.1053''
|996.9925
|1003.5088
1003.6485
|1008.4235
|1010.5263
|1014.2415
|''1026.3158''
|-
|-
| rowspan="2" |27
| 31
| colspan="2" |Eb^
| 1196.7
| rowspan="2" |1018.42105
| rowspan="2" |1021.1591
| rowspan="2" |1029.0381
| rowspan="2" |''1030.2632''
| rowspan="2" |1035.33835
| rowspan="2" |1042.1053
1042.2504
| rowspan="2" |1047.0209
| rowspan="2" |1049.3927
| rowspan="2" |1053.2508
| rowspan="2" |''1065.7895''
|-
|-
|D#v
| 32
|''S#v''
| 1235.3
|-
|-
|28
| 33
|D#
| 1273.9
|''S#''
|1056.14035
|1059.4274
|1067.1506
|''1068.42105''
|1073.6842
|1080.70175
1080.85225
|1085.9945
|1088.2591
|1092.26
|''1105.2632''
|-
|-
| rowspan="2" |29
| 34
|D#^
| 1312.5
|''S#^''
| rowspan="2" |1093.85965
| rowspan="2" |1097.7264
| rowspan="2" |1102.2632
| rowspan="2" |''1106.57895''
| rowspan="2" |1112.0301
| rowspan="2" |1119.29825
1119.4541
| rowspan="2" |1124.78
| rowspan="2" |1127.1255
| rowspan="2" |1131.26935
| rowspan="2" |''1144.7368''
|-
|-
| colspan="2" |Ev
| 35
| 1351.1
|-
|-
|30
| 36
| colspan="2" |E
| 1389.7
|1131.57895
|1135.1008
|1143.3757
|''1144.7368''
|1150.3759
|1157.8947
1158.0559
|1163.5655
|1165.9919
|1170.2786
|''1184.2105''
|-
|-
|31
| 37
| colspan="2" |E^/Fbv
| 1428.3
|1169.29825
|1172.9375
|1181.4882
|''1182.8947''
|1188.7218
|1196.4912
1196.6578
|1202.3511
|1204.8583
|1209.2879
|''1224.2105''
|-
|-
|32
| 38
| colspan="2" |Fb
| 1466.9
|1207.0715
|1210.7742
|1219.6007
|''1221.0526''
|1227.0677
|1235.0877
1235.2567
|1241.1366
|1243.7247
|1248.2972
|''1263.1579''
|-
|33
| colspan="2" |Fb^/E#v
|1244.7368
|1248.6109
|1257.71235
|''1259.2105''
|1265.4135
|1273.68425
1273.8616
|1279.9221
|1282.5911
|1287.3065
|''1302.6316''
|-
|34
| colspan="2" |E#
|1282.4561
|1286.4476
|1295.8258
|''1297.3684''
|1303.7594
|1312.2807
1312.4634
|1318.7076
|1321.4575
|1326.3158
|''1342.1053''
|-
|35
| colspan="2" |E#^/Fv
|1320.1754
|1324.2843
|1333.9383
|''1335.5263''
|1342.1053
|1380.8772
1531.0654
|1357.4931
|1360.3239
|1365.3251
|''1381.57895''
|-
|36
| colspan="2" |F
|1357.8947
|1326.121
|1372.0508
|''1373.6842''
|1380.4511
|1389.4737
1389.6672
|1396.27865
|1399.1903
|1404.3343
|''1421.0526''
|-
|37
| colspan="2" |F^/Gv
|1395.614
|1399.9577
|1410.1633
|''1411.8421''
|1418.797
|1428.0702
1428.269
|1435.0642
|1438.0567
|1443.34365
|''1460.5263''
|-
|38
| colspan="2" |G
|1433.3333
|1437.79435
|1448.2759
|''1450''
|1457.1429
|1466.6617
1466.8709
|1473.8497
|1476.9231
|1482.3529
|''1500''
|}
|}
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