38ed7/3: Difference between revisions

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{{Infobox ET}}
{{ED intro}}
While 38ed7/3 fails to accurately represent low primes, it provides great approximations of the 13th, 17th, 19th, and a multitude of higher prime harmonics, 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 7/3.5/3.11/9.13.17.19, 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 stretch 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.
== Intervals ==
== Intervals ==
{| class="wikitable"
{| class="wikitable"
!Degrees
!Degrees
! colspan="2" |Enneatonic
! colspan="2" |Enneatonic
!ed43\36
! colspan="2" |ed11\9~ed7/3
!Pyrite<sub>v</sub>
!ed35\29
!''ed29\24=r¢<sub>v</sub>''
!ed17\14
!Golden<sub>v</sub>
!ed11\9~ed7/3
!Golden<sub>v</sub>
!ed16\13
!ed21\17
!''ed5\4=r¢<sub>^</sub>''
|-
|-
| rowspan="2" |1
| rowspan="2" |1
| colspan="2" |G^
| colspan="2" |G^
| rowspan="2" |37.7193
| rowspan="2" |37.8367
| rowspan="2" |38.1126
| rowspan="2" |''38.1579''
| rowspan="2" |38.3459
| rowspan="2" |38.4171
| rowspan="2" |38.5965
| rowspan="2" |38.5965
38.6019
| rowspan="2" |38.6019
| rowspan="2" |38.7855
| rowspan="2" |38.8664
| rowspan="2" |39.0093
| rowspan="2" |''39.4737''
|-
|-
|Jbv
|Jbv
Line 36: Line 23:
|Jb
|Jb
|''Ab''
|''Ab''
|75.4386
|75.6734
|76.22505
|''76.3158''
|76.6917
|76.83425
|77.193
|77.193
77.2037
|77.2037
|77.571
|77.7328
|78.0186
|''78.9474''
|-
|-
| rowspan="2" |3
| rowspan="2" |3
|Jb^
|Jb^
|''Ab^''
|''Ab^''
| rowspan="2" |113.1579
| rowspan="2" |113.5101
| rowspan="2" |114.3376
| rowspan="2" |''114.4737''
| rowspan="2" |115.0376
| rowspan="2" |115.2514
| rowspan="2" |115.7895
| rowspan="2" |115.7895
115.8056
| rowspan="2" |115.8056
| rowspan="2" |116.35655
| rowspan="2" |116.5992
| rowspan="2" |117.0279
| rowspan="2" |''118.42105''
|-
|-
| colspan="2" |G#v
| colspan="2" |G#v
Line 69: Line 36:
|4
|4
| colspan="2" |G#
| colspan="2" |G#
|150.8772
|151.3468
|152.4501
|''152.6316''
|153.3835
|230.5028
|154.386
|154.386
154.4075
|154.4075
|155.1421
|155.4656
|156.03715
|''157.8947''
|-
|-
| rowspan="2" |5
| rowspan="2" |5
| colspan="2" |G#^
| colspan="2" |G#^
| rowspan="2" |188.5965
| rowspan="2" |189.1835
| rowspan="2" |190.5626
| rowspan="2" |''190.7895''
| rowspan="2" |191.2293
| rowspan="2" |192.0856
| rowspan="2" |192.98245
| rowspan="2" |192.98245
193.0093
| rowspan="2" |193.0093
| rowspan="2" |193.9276
| rowspan="2" |194.332
| rowspan="2" |195.0464
| rowspan="2" |''197.3684''
|-
|-
|Jv
|Jv
Line 103: Line 50:
|J
|J
|''A''
|''A''
|226.3158
|227.0202
|228.6751
|''228.9474''
|230.0752
|307.337
|231.57895
|231.57895
231.6112
|231.6112
|232.7131
|233.1984
|234.0557
|''236.8421''
|-
|-
|7
|7
|J^/Av
|J^/Av
|''A^/Bv''
|''A^/Bv''
|264.0351
|264.85685
|266.7877
|''267.1053''
|268.42105
|268.9199
|270.1754
|270.1754
270.2131
|270.2131
|271.4986
|272.0648
|273.065
|''276.3158''
|-
|-
|8
|8
|A
|A
|''B''
|''B''
|301.7544
|302.69355
|304.9002
|''305.2632''
|306.7669
|384.1713
|308.7719
|308.7719
308.8149
|308.8149
|310.5841
|310.9312
|312.0743
|''315.7895''
|-
|-
|9
|9
|A^/Bbv
|A^/Bbv
|B^/Cbv
|B^/Cbv
|339.4737
|340.5302
|343.0127
|''343.42105''
|345.1128
|345.75415
|347.3684
|347.3684
347.4168
|347.4168
|349.0697
|349.7976
|351.0836
|''355.2632''
|-
|-
|10
|10
|Bb
|Bb
|''Cb''
|''Cb''
|377.193
|378.3669
|381.1252
|''381.57895''
|383.45865
|384.1713
|385.9649
|385.9649
386.0187
|386.0187
|387.8552
|388.664
|390.0929
|''394.7368''
|-
|-
|11
|11
|Bb^/A#v
|Bb^/A#v
|''Cb^/B#''v
|''Cb^/B#''v
|414.9123
|416.2036
|419.2377
|''419.7368''
|421.8045
|422.5884
|424.5614
|424.5614
424.6205
|424.6205
|426.6407
|427.5304
|429.1022
|''434.2105''
|-
|-
|12
|12
|A#
|A#
|''B#''
|''B#''
|452.6316
|454.0403
|457.3503
|''457.8947''
|460.1504
|461.0055
|463.1579
|463.1579
463.2224
|463.2224
|465.4262
|466.3968
|468.1115
|''473.6842''
|-
|-
|13
|13
|A#^/Bv
|A#^/Bv
|''B#^/Cv''
|''B#^/Cv''
|490.3509
|491.877
|495.4628
|''496.0526''
|498.4962
|499.4227
|501.7544
|501.7544
501.8243
|502.6667
|504.2117
|505.2632
|507.1207
|''513.1579''
|-
|-
|14
|14
|B
|B
|''C''
|''C''
|528.0702
|529.7137
|533.5753
|''534.2105''
|536.8421
|537.8398
|540.3509
|540.3509
540.4261
|540.4261
|542.99725
|544.12955
|546.13
|''552.6316''
|-
|-
|15
|15
|B^/Cv
|B^/Cv
|''C^/Qv''
|''C^/Qv''
|565.7895
|567.5504
|571.6878
|''572.3684''
|575.188
|576.2569
|578.9474
|578.9474
579.028
|579.028
|581.7828
|582.99595
|585.1393
|''592.1053''
|-
|-
|16
|16
|C
|C
|''Q''
|''Q''
|603.5088
|605.3871
|609.8004
|''610.5263''
|612.5338
|614.674
|617.5439
|617.5439
617.6299
|617.6299
|620.5683
|621.86235
|624.1486
|''631.57895''
|-
|-
|17
|17
|C^/Qbv
|C^/Qbv
|''Q^/Dbv''
|''Q^/Dbv''
|641.2281
|643.2238
|647.9129
|''648.6842''
|651.8797
|653.0912
|656.14035
|656.14035
656.2317
|656.2317
|659.3538
|660.7587
|663.1579
|''671.0526''
|-
|-
|18
|18
|Qb
|Qb
|''Db''
|''Db''
|678.9474
|681.0605
|686.0254
|''686.8421''
|690.2256
|691.5083
|694.7368
|694.7368
694.8336
|694.8336
|698.1393
|699.5951
|702.1672
|''710.5263''
|-
|-
|19
|19
|Qb^/C#v
|Qb^/C#v
|''Db^/Q#v''
|''Db^/Q#v''
|716.6667
|733.{{Overline|3}}
|718.8972
|733.43545
|724.1379
|''725''
|728.5714
|729.9254
|733.3333
733.43545
|736.9248
|738.4615
|741.1765
|''750''
|-
|-
|20
|20
|C#
|C#
|''Q#''
|''Q#''
|754.386
|756.7339
|762.25045
|''763.1579''
|766.9173
|768.34255
|771.9298
|771.9298
772.0373
|772.0373
|775.7104
|777.3279
|780.1858
|''789.4737''
|-
|-
|21
|21
|C#^/Qv
|C#^/Qv
|''Q#/Dv''
|''Q#/Dv''
|792.1053
|794.5706
|800.363
|''801.3158''
|805.2632
|806.7597
|810.5263
|810.5263
810.6392
|810.6392
|814.4959
|816.1943
|819.19505
|''828.9474''
|-
|-
|22
|22
|Q
|Q
|''D''
|''D''
|829.8246
|832.4073
|838.4755
|''839.473''7
|843.609
|845.1768
|849.1228
|849.1228
849.24105
|849.24105
|853.2814
|855.0607
|858.2043
|''868.42105''
|-
|-
|23
|23
|Q^/Dv
|Q^/Dv
|''D^/Sv''
|''D^/Sv''
|867.5439
|870.24395
|876.588
|''877.6316''
|881.9549
|883.5939
|887.7193
|887.7193
887.8429
|887.8429
|892.0669
|893.9271
|897.2136
|''907.8947''
|-
|-
|24
|24
|D
|D
|''S''
|''S''
|905.2632
|908.0806
|914.7005
|''915.7895''
|920.30075
|922.0111
|926.3158
|926.3158
926.4448
|926.4448
|930.8524
|932.7935
|936.2229
|''947.3684''
|-
|-
| rowspan="2" |25
| rowspan="2" |25
|D^
|D^
|''S^''
|''S^''
| rowspan="2" |942.9825
| rowspan="2" |945.9173
| rowspan="2" |952.8131
| rowspan="2" |''953.9474''
| rowspan="2" |958.6466
| rowspan="2" |960.4282
| rowspan="2" |964.9123
| rowspan="2" |964.9123
965.04665
| rowspan="2" |965.04665
| rowspan="2" |969.63795
| rowspan="2" |971.6599
| rowspan="2" |975.2322
| rowspan="2" |''986.8421''
|-
|-
| colspan="2" |Ebv
| colspan="2" |Ebv
Line 424: Line 171:
|26
|26
| colspan="2" |Eb
| colspan="2" |Eb
|980.70175
|983.754
|990.9256
|''992.1053''
|996.9925
|998.8453
|1003.5088
|1003.5088
1003.6485
|1003.6485
|1008.4235
|1010.5263
|1014.2415
|''1026.3158''
|-
|-
| rowspan="2" |27
| rowspan="2" |27
| colspan="2" |Eb^
| colspan="2" |Eb^
| rowspan="2" |1018.42105
| rowspan="2" |1021.1591
| rowspan="2" |1029.0381
| rowspan="2" |''1030.2632''
| rowspan="2" |1035.33835
| rowspan="2" |1037.2624
| rowspan="2" |1042.1053
| rowspan="2" |1042.1053
1042.2504
| rowspan="2" |1042.2504
| rowspan="2" |1047.0209
| rowspan="2" |1049.3927
| rowspan="2" |1053.2508
| rowspan="2" |''1065.7895''
|-
|-
|D#v
|D#v
Line 458: Line 185:
|D#
|D#
|''S#''
|''S#''
|1056.14035
|1059.4274
|1067.1506
|''1068.42105''
|1073.6842
|1075.6796
|1080.70175
|1080.70175
1080.85225
|1080.85225
|1085.9945
|1088.2591
|1092.26
|''1105.2632''
|-
|-
| rowspan="2" |29
| rowspan="2" |29
|D#^
|D#^
|''S#^''
|''S#^''
| rowspan="2" |1093.85965
| rowspan="2" |1097.7264
| rowspan="2" |1102.2632
| rowspan="2" |''1106.57895''
| rowspan="2" |1112.0301
| rowspan="2" |1114.0967
| rowspan="2" |1119.29825
| rowspan="2" |1119.29825
1119.4541
| rowspan="2" |1119.4541
| rowspan="2" |1124.78
| rowspan="2" |1127.1255
| rowspan="2" |1131.26935
| rowspan="2" |''1144.7368''
|-
|-
| colspan="2" |Ev
| colspan="2" |Ev
Line 491: Line 198:
|30
|30
| colspan="2" |E
| colspan="2" |E
|1131.57895
|1135.1008
|1143.3757
|''1144.7368''
|1150.3759
|1152.5138
|1157.8947
|1157.8947
1158.0559
|1158.0559
|1163.5655
|1165.9919
|1170.2786
|''1184.2105''
|-
|-
|31
|31
| colspan="2" |E^/Fbv
| colspan="2" |E^/Fbv
|1169.29825
|1172.9375
|1181.4882
|''1182.8947''
|1188.7218
|1190.93095
|1196.4912
|1196.4912
1196.6578
|1196.6578
|1202.3511
|1204.8583
|1209.2879
|''1224.2105''
|-
|-
|32
|32
| colspan="2" |Fb
| colspan="2" |Fb
|1207.0715
|1210.7742
|1219.6007
|''1221.0526''
|1227.0677
|1229.3481
|1235.0877
|1235.0877
1235.2567
|1235.2567
|1241.1366
|1243.7247
|1248.2972
|''1263.1579''
|-
|-
|33
|33
| colspan="2" |Fb^/E#v
| colspan="2" |Fb^/E#v
|1244.7368
|1248.6109
|1257.71235
|''1259.2105''
|1265.4135
|1267.7652
|1273.68425
|1273.68425
1273.8616
|1273.8616
|1279.9221
|1282.5911
|1287.3065
|''1302.6316''
|-
|-
|34
|34
| colspan="2" |E#
| colspan="2" |E#
|1282.4561
|1286.4476
|1295.8258
|''1297.3684''
|1303.7594
|1306.1823
|1312.2807
|1312.2807
1312.4634
|1312.4634
|1318.7076
|1321.4575
|1326.3158
|''1342.1053''
|-
|-
|35
|35
| colspan="2" |E#^/Fv
| colspan="2" |E#^/Fv
|1320.1754
|1350.8772
|1324.2843
|1351.0654
|1333.9383
|''1335.5263''
|1342.1053
|1344.59945
|1380.8772
1531.0654
|1357.4931
|1360.3239
|1365.3251
|''1381.57895''
|-
|-
|36
|36
| colspan="2" |F
| colspan="2" |F
|1357.8947
|1326.121
|1372.0508
|''1373.6842''
|1380.4511
|1383.0166
|1389.4737
|1389.4737
1389.6672
|1389.6672
|1396.27865
|1399.1903
|1404.3343
|''1421.0526''
|-
|-
|37
|37
| colspan="2" |F^/Gv
| colspan="2" |F^/Gv
|1395.614
|1399.9577
|1410.1633
|''1411.8421''
|1418.797
|1421.4337
|1428.0702
|1428.0702
1428.269
|1428.269
|1435.0642
|1438.0567
|1443.34365
|''1460.5263''
|-
|-
|38
|38
| colspan="2" |G
| colspan="2" |G
|1433.3333
|1466.{{Overline|6}}
|1437.79435
|1466.8709
|1448.2759
|''1450''
|1457.1429
|1459.8508
|1466.6617
1466.8709
|1473.8497
|1476.9231
|1482.3529
|''1500''
|}
|}
== Harmonics ==
{{Harmonics in equal
| steps = 38
| num = 7
| denom = 3
| intervals = prime
}}
{{Harmonics in equal
| steps = 38
| num = 7
| denom = 3
| start = 12
| collapsed = 1
| intervals = prime
}}

Latest revision as of 07:54, 5 October 2024

← 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.

While 38ed7/3 fails to accurately represent low primes, it provides great approximations of the 13th, 17th, 19th, and a multitude of higher prime harmonics, 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 7/3.5/3.11/9.13.17.19, 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 stretch 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.

Intervals

Degrees Enneatonic ed11\9~ed7/3
1 G^ 38.5965 38.6019
Jbv Abv
2 Jb Ab 77.193 77.2037
3 Jb^ Ab^ 115.7895 115.8056
G#v
4 G# 154.386 154.4075
5 G#^ 192.98245 193.0093
Jv Av
6 J A 231.57895 231.6112
7 J^/Av A^/Bv 270.1754 270.2131
8 A B 308.7719 308.8149
9 A^/Bbv B^/Cbv 347.3684 347.4168
10 Bb Cb 385.9649 386.0187
11 Bb^/A#v Cb^/B#v 424.5614 424.6205
12 A# B# 463.1579 463.2224
13 A#^/Bv B#^/Cv 501.7544 502.6667
14 B C 540.3509 540.4261
15 B^/Cv C^/Qv 578.9474 579.028
16 C Q 617.5439 617.6299
17 C^/Qbv Q^/Dbv 656.14035 656.2317
18 Qb Db 694.7368 694.8336
19 Qb^/C#v Db^/Q#v 733.3 733.43545
20 C# Q# 771.9298 772.0373
21 C#^/Qv Q#/Dv 810.5263 810.6392
22 Q D 849.1228 849.24105
23 Q^/Dv D^/Sv 887.7193 887.8429
24 D S 926.3158 926.4448
25 D^ S^ 964.9123 965.04665
Ebv
26 Eb 1003.5088 1003.6485
27 Eb^ 1042.1053 1042.2504
D#v S#v
28 D# S# 1080.70175 1080.85225
29 D#^ S#^ 1119.29825 1119.4541
Ev
30 E 1157.8947 1158.0559
31 E^/Fbv 1196.4912 1196.6578
32 Fb 1235.0877 1235.2567
33 Fb^/E#v 1273.68425 1273.8616
34 E# 1312.2807 1312.4634
35 E#^/Fv 1350.8772 1351.0654
36 F 1389.4737 1389.6672
37 F^/Gv 1428.0702 1428.269
38 G 1466.6 1466.8709

Harmonics

Approximation of prime harmonics in 38ed7/3
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -3.3 -10.5 -7.0 -10.5 +17.7 -1.3 -2.5 -2.1 +14.6 -0.7 -0.3
Relative (%) -8.7 -27.1 -18.1 -27.1 +45.8 -3.4 -6.5 -5.4 +37.8 -1.8 -0.9
Steps
(reduced) 		31
(31) 		49
(11) 		72
(34) 		87
(11) 		108
(32) 		115
(1) 		127
(13) 		132
(18) 		141
(27) 		151
(37) 		154
(2)
Approximation of prime harmonics in 38ed7/3
Harmonic 37 41 43 47 53 59 61 67 71 73 79
Error Absolute (¢) +2.2 +17.4 +12.2 +12.6 -2.4 +5.0 -14.1 +16.4 -6.7 -16.2 +1.4
Relative (%) +5.6 +45.2 +31.6 +32.7 -6.1 +12.9 -36.6 +42.6 -17.5 -42.0 +3.7
Steps
(reduced) 		162
(10) 		167
(15) 		169
(17) 		173
(21) 		178
(26) 		183
(31) 		184
(32) 		189
(37) 		191
(1) 		192
(2) 		196
(6)
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