5L 3s: Difference between revisions

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5L 3s refers to the structure of moment of symmetry scales with generators ranging from 2\5 (two degrees of [[5edo]] = 480¢) to 3\8 (three degrees of [[8edo]] = 450¢). In the case of 8edo, L and s are the same size; in the case of 5edo, s becomes so small it disappears (and all that remains are the five equal L's). The spectrum looks like this:
{{Interwiki
| en = 5L 3s
| de =
| es =
| ja =
| ko = 5L3s (Korean)
}}
{{Infobox MOS
| Neutral = 2L 6s
}}
: ''For the tritave-equivalent MOS structure with the same step pattern, see [[5L 3s (3/1-equivalent)]].''
{{MOS intro}}
5L 3s can be seen as a [[Warped diatonic|warped diatonic scale]], because it has one extra small step compared to diatonic ([[5L 2s]]).


{| class="wikitable" style="text-align:center;"
== Name ==
|-
{{TAMNAMS name}} 'Oneiro' is sometimes used as a shortened form.
! colspan="5" | generator
! | tetrachord
! | g in cents
! | 2g
! | 3g
! | 4g
! | Comments
|-
| | 2\5
| |
| |
| |
| |
| | 1 0 1
|  | 480.000
|  | 960.000
|  | 240.00
|  | 720.000
|  |
|-
| | 21\53
| |
| |
| |
| |
|  | 10 1 10
|  | 475.472
|  | 950.943
|  | 226.415
|  | 701.887
|  | Vulture/Buzzard is around here
|-
| | 19\48
| |
| |
| |
| |
|  | 9 1 9
|  | 475
|  | 950
|  | 225
|  | 700
|  |
|-
| | 17\43
| |
| |
| |
| |
|  | 8 1 8
|  | 474.419
|  | 948.837
|  | 223.256
|  | 697.674
|  |
|-
| | 15\38
| |
| |
| |
| |
|  | 7 1 7
|  | 473.684
|  | 947.368
|  | 221.053
|  | 694.737
|  |
|-
| | 13\33
| |
| |
| |
| |
|  | 6 1 6
|  | 472.727
|  | 945.455
|  | 218.181
|  | 690.909
|  |
|-
| | 11\28
| |
| |
| |
| |
|  | 5 1 5
|  | 471.429
|  | 942.857
|  | 214.286
|  | 685.714
|  |
|-
| | 9\23
| |
| |
| |
| |
|  | 4 1 4
|  | 469.565
|  | 939.130
|  | 208.696
|  | 678.261
|  | L/s = 4
|-
| |
| |
| |
| |
| |
|  | pi 1 pi
|  | 467.171
|  | 934.3425
|  | 201.514
|  | 668.685
|  | L/s = pi
|-
| | 7\18
| |
| |
| |
| |
|  | 3 1 3
|  | 466.667
|  | 933.333
|  | 200.000
|  | 666.667
|  | L/s = 3
|-
| |
| |
| |
| |
| |
|  | e 1 e
|  | 465.535
|  | 931.069
|  | 196.604
|  | 662.139
|  | L/s = e
|-
| |
| | 19\49
| |
| |
| |
|  | 8 3 8
|  | 465.306
|  | 930.612
|  | 195.918
|  | 661.2245
| |
|-
| |
| |
| | 50\129
| |
| |
|  | 21 8 21
|  | 465.116
|  | 930.233
|  | 195.349
|  | 660.465
| |
|-
| |
| |
| |
| | 131\338
| |
|  | 55 21 55
|  | 465.089
|  | 930.1775
|  | 195.266
|  | 660.335
| |
|-
| |
| |
| |
| |
| | 212\547
|  | 89 34 89
|  | 465.082
|  | 930.1645
|  | 195.247
|  | 660.329
| |
|-
| |
| |
| |
| | 81\209
| |
|  | 34 13 34
|  | 465.072
|  | 930.1435
|  | 195.215
|  | 660.287
| |
|-
| |
| |
| | 31\80
| |
| |
|  | 13 5 13
|  | 465
|  | 930
|  | 195
|  | 660
| |
|-
| |
| | 12\31
| |
| |
| |
|  | 5 2 5
|  | 464.516
|  | 929.032
|  | 193.549
|  | 658.065
|  |
|-
| | 5\13
| |
| |
| |
| |
|  | 2 1 2
|  | 461.538
|  | 923.077
|  | 184.615
|  | 646.154
|  |
|-
| |
| |
| |
| |
| |
|  | √3 1 √3
|  | 459.417
|  | 918.8345
|  | 178.252
|  | 637.669
| |
|-
| |
| | 13\34
| |
| |
| |
|  | 5 3 5
|  | 458.824
|  | 917.647
|  | 176.471
|  | 635.294
|  |
|-
| |
| |
| | 34\89
| |
| |
|  | 13 8 13
|  | 458.427
|  | 916.854
|  | 175.281
|  | 633.708
|  |
|-
| |
| |
| |
| | 89\233
| |
|  | 34 21 34
|  | 458.369
|  | 916.738
|  | 175.107
|  | 633.473
|  |
|-
| |
| |
| |
| |
| | 233\610
|  | 89 55 89
|  | 458.361
|  | 916.721
|  | 175.082
|  | 633.443
|  | Golden father
|-
| |
| |
| |
| | 144\377
| |
|  | 55 34 55
|  | 458.355
|  | 916.711
|  | 175.066
|  | 633.422
|  |
|-
| |
| |
| | 55\144
| |
| |
|  | 21 13 21
|  | 458.333
|  | 916.666
|  | 175
|  | 633.333
|  |
|-
| |
| | 21\55
| |
| |
| |
|  | 8 5 8
|  | 458.182
|  | 916.364
|  | 174.545
|  | 632.727
|  |
|-
| |
| |
| |
| |
| |
|  | pi 2 pi
|  | 457.883
|  | 915.777
|  | 173.665
|  | 631.553
| |
|-
| | 8\21
| |
| |
| |
| |
|  | 3 2 3
|  | 457.143
|  | 914.286
|  | 171.429
|  | 628.571
|  | Optimum rank range (L/s=3/2) father
|-
| | 11\29
| |
| |
| |
| |
|  | 4 3 4
|  | 455.172
|  | 910.345
|  | 165.517
|  | 620.690
|  |
|-
| | 14\37
| |
| |
| |
| |
|  | 5 4 5
|  | 454.054
|  | 908.108
|  | 162.162
|  | 616.216
| |
|-
| | 17\45
| |
| |
| |
| |
|  | 6 5 6
|  | 453.333
|  | 906.667
|  | 160
|  | 613.333
| |
|-
| | 20\53
| |
| |
| |
| |
|  | 7 6 7
|  | 452.83
|  | 905.66
|  | 158.491
|  | 611.321
| |
|-
| | 23\61
| |
| |
| |
| |
|  | 8 7 8
|  | 452.459
|  | 904.918
|  | 157.377
|  | 609.836
| |
|-
| | 26\69
| |
| |
| |
| |
|  | 9 8 9
|  | 452.174
|  | 904.348
|  | 156.522
|  | 608.696
| |
|-
| | 29\77
| |
| |
| |
| |
|  | 10 9 10
|  | 451.948
|  | 903.896
|  | 155.844
|  | 607.792
| |
|-
| | 3\8
| |
| |
| |
| |
|  | 1 1 1
|  | 450.000
|  | 900.000
|  | 150.000
|  | 600.000
|  |
|}
The only notable harmonic entropy minimum is Vulture/[[Hemifamity_temperaments|Buzzard]], in which four generators make a 3/1 (and three generators approximate an octave plus 8/7). The rest of this region is a kind of wasteland in terms of harmonious MOSes.


By a weird coincidence, the other generator for this MOS will generate the same pattern within a tritave equivalence. By yet another weird coincidence, this MOS belongs to a temperament which has [[Bohlen-Pierce|Bohlen-Pierce]] as its index-2 subtemperament. In addition to being harmonious, this tuning of the MOS gives an L/s ratio between 3/1 and 3/2, which is squarely in the middle of the range, being thus neither too exaggerated nor too equalized to be recognizable as such, unlike in octaves, where the only notable harmonic entropy minimum is near a greatly exaggerated 10/1 L/s ratio.
'Father' is sometimes also used to denote 5L 3s, but it's a misnomer, as [[father]] is technically an abstract [[regular temperament]] (although a very inaccurate one), not a generator range. There are father tunings which generate 3L 5s. A more correct but still not quite correct name would be 'father[8]' or 'father octatonic'. "Father" is also vague regarding the number of notes, because optimal generators for it also generate [[3L 2s]].


{| class="wikitable" style="text-align:center;"
== Scale properties ==
|-
! colspan="5" |\
! | tetrachord
! | g in cents
! | 2g
! | 3g
! | 4g
! | Comments
|-
|  | 2\5
|  |
|  |
|  |
|  |
|  | 1 0 1
|  | 760.782
|  | 1521.564
|  | 380.391
|  | 1141.173
|  |
|-
|  | 27\68
|  |
|  |
|  |
|  |
|  | 13 1 13
|  | 755.188
|  | 1510.376
|  | 363.609
|  | 1118.797
|  | 2g=12/5 minus quarter comma near here
|-
|  | 25\63
|  |
|  |
|  |
|  |
|  | 12 1 12
|  | 754.744
|  | 1509.488
|  | 362.277
|  | 1117.021
|  |
|-
|  | 23\58
|  |
|  |
|  |
|  |
|  | 11 1 11
|  | 754.2235
|  | 1508.447
|  | 360.716
|  | 1114.939
|  |
|-
|  | 21\53
|  |
|  |
|  |
|  |
|  | 10 1 10
|  | 753.605
|  | 1507.21
|  | 358.859
|  | 1112.464
|  |
|-
|  | 19\48
|  |
|  |
|  |
|  |
|  | 9 1 9
|  | 752.857
|  | 1505.714
|  | 356.617
|  | 1109.474
|  |
|-
|  | 17\43
|  |
|  |
|  |
|  |
|  | 8 1 8
|  | 751.936
|  | 1503.871
|  | 353.852
|  | 1105.788
|  |
|-
|  | 15\38
|  |
|  |
|  |
|  |
|  | 7 1 7
|  | 750.771
|  | 1501.543
|  | 350.36
|  | 1101.132
|  |
|-
|  |
|  | 28/71
|  |
|  |
|  |
|  | 13 2 13
|  | 750.067
|  | 1500.1335
|  | 348.245
|  | 1098.312
|  |
|-
|  |
|  | 41\104
|  |
|  |
|  |
|  | 19 3 19
|  | 749.809
|  | 1499.618
|  | 347.4725
|  | 1097.282
|  | 3g=11/3 near here
|-
|  | 13\33
|  |
|  |
|  |
|  |
|  | 6 1 6
|  | 749.255
|  | 1498.51
|  | 345.81
|  | 1095.065
|  |
|-
|  |
|  | 24\61
|  |
|  |
|  |
|  | 11 2 11
|  | 748.31
|  | 1496.62
|  | 342.976
|  | 1091.286
|  |
|-
|  |
|  | 35\89
|  |
|  |
|  |
|  | 16 3 16
|  | 747.96
|  | 1495.92
|  | 341.924
|  | 1089.884
|  |
|-
|  |
|  | 46\117
|  |
|  |
|  |
|  | 21 4 21
|  | 747.777
|  | 1495.554
|  | 341.377
|  | 1089.154
|  |
|-
|  |
|  | 57\145
|  |
|  |
|  |
|  | 26 5 26
|  | 747.665
|  | 1495.33
|  | 341.04
|  | 1088.705
|  | 4g=45/8 near here
|-
|  |
|  |
|  |
|  |
|  |
|  | 5+√29 2 5+√29
|  | 747.648
|  | 1495.297
|  | 340.99
|  | 1088.638
|  |
|-
|  | 11\28
|  |
|  |
|  |
|  |
|  | 5 1 5
|  | 747.197
|  | 1494.393
|  | 339.635
|  | 1086.831
|  |
|-
|  |
|  | 20\51
|  |
|  |
|  |
|  | 9 2 9
|  | 745.865
|  | 1491.729
|  | 335.639
|  | 1081.50
|  |
|-
|  |
|  | 29\74
|  |
|  |
|  |
|  | 13 3 13
|  | 745.361
|  | 1490.721
|  | 334.127
|  | 1079.488
|  |
|-
|  |
|  | 38/97
|  |
|  |
|  |
|  | 17 4 17
|  | 745.096
|  | 1490.192
|  | 333.332
|  | 1078.428
|  |
|-
|  |
|  |
|  |
|  |
|  |
|  | 2+√5 1 2+√5
|  | 754.051
|  | 1490.101
|  | 333.197
|  | 1078.247
|  |
|-
|  |
|  | 47\120
|  |
|  |
|  |
|  | 21 5 21
|  | 744.932
|  | 1489.865
|  | 332.842
|  | 1077.7745
|  |
|-
|  | 9\23
|  |
|  |
|  |
|  |
|  | 4 1 4
|  | 744.243
|  | 1488.487
|  | 330.775
|  | 1075.018
|  | L/s = 4
|-
|  |
|  | 34\87
|  |
|  |
|  |
|  | 15 4 15
|  | 743.293
|  | 1486.586
|  | 327.923
|  | 1071.216
|  | 4g=39/7 near here
|-
|  |
|  | 25\64
|  |
|  |
|  |
|  | 11 3 11
|  | 742.951
|  | 1485.902
|  | 326.899
|  | 1069.85
|  |
|-
|  |
|  | 16\41
|  |
|  |
|  |
|  | 7 2 7
|  | 742.226
|  | 1484.453
|  | 324.724
|  | 1066.95
|  |
|-
|  |
|  | 23\59
|  |
|  |
|  |
|  | 10 3 10
|  | 741.44
|  | 1482.88
|  | 322.365
|  | 1063.805
|  |
|-
|  |
|  |
|  |
|  |
|  |
|  | 3+√13 2 3+√13
|  | 741.289
|  | 1482.577
|  | 321.911
|  | 1063.20
|  |
|-
|  |
|  | 30\77
|  |
|  |
|  |
|  | 13 4 13
|  | 741.021
|  | 1482.043
|  | 321.109
|  | 1062.131
|  |
|-
|  |
|  |
|  |
|  |
|  |
|  | pi 1 pi
|  | 740.449
|  | 1480.898
|  | 319.392
|  | 1056.841
|  | L/s = pi
|-
|  | 7\18
|  |
|  |
|  |
|  |
|  | 3 1 3
|  | 739.649
|  | 1479.298
|  | 316.992
|  | 1056.642
|  | L/s = 3
|-
|  |
|  | 89\229
|  |
|  |
|  |
|  | 38 13 38
|  | 739.188
|  | 1478.376
|  | 315.608
|  | 1054.796
|  | 3g=18/5 near here
|-
|  |
|  | 82\211
|  |
|  |
|  |
|  | 35 12 35
|  | 739.148
|  | 1478.297
|  | 315.49
|  | 1054.639
|  |
|-
|  |
|  | 75\193
|  |
|  |
|  |
|  | 32 11 32
|  | 739.102
|  | 1478.203
|  | 315.35
|  | 1054.452
|  |
|-
|  |
|  | 68\175
|  |
|  |
|  |
|  | 29 10 29
|  | 739.045
|  | 1478.091
|  | 315.181
|  | 1054.227
|  |
|-
|  |
|  | 61/157
|  |
|  |
|  |
|  | 26 9 26
|  | 738.976
|  | 1477.952
|  | 314.973
|  | 1053.949
|  |
|-
|  |
|  | 54\139
|  |
|  |
|  |
|  | 23 8 23
|  | 738.889
|  | 1477.778
|  | 314.712
|  | 1053.601
|  |
|-
|  |
|  | 47\121
|  |
|  |
|  |
|  | 20 7 20
|  | 738.776
|  | 1477.552
|  | 314.373
|  | 1053.149
|  |
|-
|  |
|  | 40\103
|  |
|  |
|  |
|  | 17 6 17
|  | 738.623
|  | 1477.247
|  | 313.915
|  | 1052.538
|  |
|-
|  |
|  | 33\85
|  |
|  |
|  |
|  | 14 5 14
|  | 738.406
|  | 1476.812
|  | 313.263
|  | 1051.669
|  |
|-
|  |
|  | 26\67
|  |
|  |
|  |
|  | 11 4 11
|  | 738.072
|  | 1476.144
|  | 312.261
|  | 1050.333
|  |
|-
|  |
|  |
|  |
|  |
|  |
|  | e 1 e
|  | 737.855
|  | 1478.71
|  | 311.61
|  | 1049.465
|  | L/s = e
|-
|  |
|  | 19\49
|  |
|  |
|  |
|  | 8 3 8
|  | 737.493
|  | 1474.986
|  | 310.523
|  | 1048.016
|  | 3g=18/5 minus quarter comma near here
|-
|  |
|  |
|  | 50\129
|  |
|  |
|  | 21 8 21
|  | 737.192
|  | 1474.384
|  | 309.621
|  | 1046.812
|  |
|-
|  |
|  |
|  |
|  | 131\338
|  |
|  | 55 21 55
|  | 737.148
|  | 1474.296
|  | 309.49
|  | 1046.638
|  |
|-
|  |
|  |
|  |
|  |
|  | 212\547
|  | 89 34 89
|  | 737.138
|  | 1474.276
|  | 309.459
|  | 1046.597
|  |
|-
|  |
|  |
|  |
|  | 81\209
|  |
|  | 34 13 34
|  | 737.121
|  | 1474.243
|  | 309.409
|  | 1046.53
|  |
|-
|  |
|  |
|  | 31\80
|  |
|  |
|  | 13 5 13
|  | 737.008
|  | 1474.015
|  | 309.068
|  | 1046.075
|  |
|-
|  |
|  | 12\31
|  |
|  |
|  |
|  | 5 2 5
|  | 736.241
|  | 1472.481
|  | 306.767
|  | 1043.007
|  |
|-
|  |
|  |
|  |
|  |
|  |
|  | 1+√2 1 1+√2
|  | 735.542
|  | 1471.084
|  | 304.6715
|  | 1040.214
|  | Silver false father
|-
|  |
|  | 17\44
|  |
|  |
|  |
|  | 7 3 7
|  | 734.846
|  | 1469.693
|  | 302.584
|  | 1037.41
|  |
|-
|  |
|  | 22\57
|  |
|  |
|  |
|  | 9 4 9
|  | 734.088
|  | 1468.176
|  | 300.309
|  | 1034.397
|  |
|-
|  |
|  | 27\70
|  |
|  |
|  |
|  | 11 5 11
|  | 733.611
|  | 1467.222
|  | 298.879
|  | 1032.49
|  |
|-
|  |
|  |
|  | 59\153
|  |
|  |
|  | 24 11 24
|  | 733.434
|  | 1466.867
|  | 298.346
|  | 1031.779
|  |
|-
|  |
|  | 32\83
|  |
|  |
|  |
|  | 13 6 13
|  | 733.284
|  | 1466.568
|  | 297.897
|  | 1031.181
|  | 2g=7/3 near here
|-
|  | 5\13
|  |
|  |
|  |
|  |
|  | 2 1 2
|  | 731.521
|  | 1463.042
|  | 292.609
|  | 1024.13
|  |
|-
|  |
|  | 48\125
|  |
|  |
|  |
|  | 19 10 19
|  | 730.35
|  | 1460.701
|  | 289.097
|  | 1019.448
|  | 3g=39/11 near here
|-
|  |
|  | 43\112
|  |
|  |
|  |
|  | 17 9 17
|  | 730.215
|  | 1460.43
|  | 288.69
|  | 1018.905
|  |
|-
|  |
|  | 38\99
|  |
|  |
|  |
|  | 15 8 15
|  | 730.043
|  | 1460.087
|  | 288.175
|  | 1018.218
|  |
|-
|  |
|  | 33\86
|  |
|  |
|  |
|  | 13 7 13
|  | 729.82
|  | 1459.64
|  | 287.505
|  | 1017.325
|  | 4g=27/5 near here
|-
|  |
|  | 28\73
|  |
|  |
|  |
|  | 11 6 11
|  | 729.547
|  | 1459.034
|  | 286.596
|  | 1016.113
|  |
|-
|  |
|  | 23\60
|  |
|  |
|  |
|  | 9 5 9
|  | 729.083
|  | 1458.1655
|  | 285.293
|  | 1014.376
|  |
|-
|  |
|  |
|  | 59\154
|  |
|  |
|  | 23 13 23
|  | 728.671
|  | 1457.342
|  | 284.058
|  | 1012.729
|  | 3g=99/28 near here
|-
|  |
|  |
|  | 77\201
|  |
|  |
|  | 30 17 30
|  | 728.61
|  | 1457.219
|  | 283.874
|  | 1012.483
|  |
|-
|  |
|  |
|  | 95\248
|  |
|  |
|  | 37 21 37
|  | 728.5715
|  | 1457.143
|  | 283.7595
|  | 1012.331
|  | Golden BP is index-2 near here
|-
|  |
|  | 18\47
|  |
|  |
|  |
|  | 7 4 7
|  | 728.408
|  | 1456.817
|  | 283.27
|  | 1011.678
|  |
|-
|  |
|  |
|  |
|  |
|  |
|  | √3 1 √3
|  | 728.159
|  | 1456.318
|  | 282.522
|  | 1010.6815
|  | 4g=27/5 minus third comma near here
|-
|  |
|  |
|  | 31\81
|  |
|  |
|  | 12 7 12
|  | 727.909
|  | 1455.817
|  | 281.771
|  | 1009.68
|  |
|-
|  |
|  | 13\34
|  |
|  |
|  |
|  | 5 3 5
|  | 727.218
|  | 1454.436
|  | 279.699
|  | 1006.917
|  |
|-
|  |
|  |
|  | 34\89
|  |
|  |
|  | 13 8 13
|  | 726.59
|  | 1453.179
|  | 277.814
|  | 1004.403
|  |
|-
|  |
|  |
|  |
|  | 89\233
|  |
|  | 34 21 34
|  | 726.498
|  | 1452.996
|  | 277.538
|  | 1004.036
|  |
|-
|  |
|  |
|  |
|  |
|  | 233\610
|  | 89 55 89
|  | 726.4845
|  | 1452.969
|  | 277.4985
|  | 1003.983
|  | Golden false father
|-
|  |
|  |
|  |
|  | 144\377
|  |
|  | 55 34 55
|  | 726.476
|  | 1452.952
|  | 277.473
|  | 1003.95
|  |
|-
|  |
|  |
|  | 55\144
|  |
|  |
|  | 21 13 21
|  | 726.441
|  | 1452.882
|  | 277.368
|  | 1003.809
|  |
|-
|  |
|  | 21\55
|  |
|  |
|  |
|  | 8 5 8
|  | 726.201
|  | 1452.402
|  | 276.468
|  | 1002.849
|  |
|-
|  |
|  |
|  |
|  |
|  |
|  | pi 2 pi
|  | 725.736
|  | 1451.472
|  | 275.252
|  | 1000.988
|  |
|-
|  | 8\21
|  |
|  |
|  |
|  |
|  | 3 2 3
|  | 724.554
|  | 1449.109
|  | 271.708
|  | 996.226
|  | Optimum rank range (L/s=3/2) false father
4g=16/3 near here
|-
|  |
|  | 27\71
|  |
|  |
|  |
|  | 10 7 10
|  | 723.279
|  | 1446.557
|  | 267.881
|  | 991.16
|  |
|-
|  |
|  |
|  | 46\121
|  |
|  |
|  | 17 12 17
|  | 723.057
|  | 1446.115
|  | 267.217
|  | 990.274
|  |
|-
|  |
|  |
|  | 65\171
|  |
|  |
|  | 24 17 24
|  | 722.965
|  | 1445.931
|  | 266.941
|  | 989.907
|  | 3g=7/2 near here
|-
|  |
|  | 19\50
|  |
|  |
|  |
|  | 7 5 7
|  | 722.743
|  | 1445.486
|  | 266.274
|  | 989.017
|  |
|-
|  | 11\29
|  |
|  |
|  |
|  |
|  | 4 3 4
|  | 721.431
|  | 1442.862
|  | 262.338
|  | 983.77
|  |
|-
|  |
|  | 25\66
|  |
|  |
|  |
|  | 9 7 9
|  | 720.4375
|  | 1440.875
|  | 259.3575
|  | 979.795
|  |
|-
|  |
|  |
|  | 64\169
|  |
|  |
|  | 23 18 23
|  | 720.267
|  | 1440.534
|  | 258.848
|  | 979.113
|  |
|-
|  |
|  |
|  |
|  | 167\441
|  |
|  | 60 47 60
|  | 720.2415
|  | 1440.483
|  | 258.7965
|  | 979.001
|  |
|-
|  |
|  |
|  |
|  |
|  | 437\1154
|  | 157 123 157
|  | 720.238
|  | 1440.475
|  | 258.758
|  | 978.996
|  |
|-
|  |
|  |
|  |
|  | 270\713
|  |
|  | 97 76 97
|  | 720.235
|  | 1440.471
|  | 258.751
|  | 978.987
|  |
|-
|  |
|  |
|  | 103\272
|  |
|  |
|  | 37 29 37
|  | 720.226
|  | 1440.451
|  | 258.722
|  | 978.947
|  |
|-
|  |
|  | 39\103
|  |
|  |
|  |
|  | 14 11 14
|  | 720.158
|  | 1440.315
|  | 258.518
|  | 978.676
|  |
|-
|  | 14\37
|  |
|  |
|  |
|  |
|  | 5 4 5
|  | 719.659
|  | 1439.317
|  | 257.021
|  | 976.679
|  |
|-
|  |
|  | 31\82
|  |
|  |
|  |
|  | 11 9 11
|  | 719.032
|  | 1438.064
|  | 255.14
|  | 974.172
|  |
|-
|  |
|  |
|  | 79\209
|  |
|  |
|  | 28 23 28
|  | 718.921
|  | 1437.842
|  | 254.807
|  | 973.728
|  |
|-
|  |
|  |
|  |
|  | 206\545
|  |
|  | 73 60 73
|  | 718.904
|  | 1437.808
|  | 254.757
|  | 973.661
|  |
|-
|  |
|  |
|  |
|  |
|  | 539\1426
|  | 191 117 191
|  | 718.902
|  | 1437.803
|  | 254.75
|  | 973.652
|  |
|-
|  |
|  |
|  |
|  | 333\881
|  |
|  | 118 97 118
|  | 718.90
|  | 1437.80
|  | 254.745
|  | 973.6455
|  |
|-
|  |
|  |
|  | 127\336
|  |
|  |
|  | 45 37 45
|  | 718.893
|  | 1437.787
|  | 254.726
|  | 973.619
|  |
|-
|  |
|  | 48\127
|  |
|  |
|  |
|  | 17 14 17
|  | 718.849
|  | 1437.698
|  | 254.592
|  | 973.441
|  |
|-
|  | 17\45
|  |
|  |
|  |
|  |
|  | 6 5 6
|  | 718.516
|  | 1437.032
|  | 253.549
|  | 972.11
|  |
|-
|  | 20\53
|  |
|  |
|  |
|  |
|  | 7 6 7
|  | 717.719
|  | 1435.438
|  | 251.202
|  | 968.9205
|  |4g=21/4 near here
|-
|  | 23\61
|  |
|  |
|  |
|  |
|  | 8 7 8
|  | 717.131
|  | 1434.261
|  | 249.437
|  | 966.567
|  |
|-
|
|49\130
|
|
|
|17 15 17
|716.891
|1433.7815
|248.717
|965.608
|4g=quarter-comma meantone 21/4 near here


6g=12 near here
=== Intervals ===
|-
{{MOS intervals}}
|  | 26\69
|  |
|  |
|  |
|  |
|  | 9 8 9
|  | 716.679
|  | 1433.357
|  | 248.081
|  | 964.76
|  |
|-
|  | 29\77
|  |
|  |
|  |
|  |
|  | 10 9 10
|  | 716.321
|  | 1432.641
|  | 247.007
|  | 963.328
|  |
|-
|  | 32\85
|  |
|  |
|  |
|  |
|  | 11 10 11
|  | 716.03
|  | 1432.06
|  | 246.135
|  | 962.1655
|  |
|-
|  | 35\93
|  |
|  |
|  |
|  |
|  | 12 11 12
|  | 715.7895
|  | 1431.759
|  | 245.4135
|  | 961.203
|  |
|-
|  | 38/101
|  |
|  |
|  |
|  |
|  | 13 12 13
|  | 715.587
|  | 1431.174
|  | 244.806
|  | 960.393
|  | 2g=16\7 near here
|-
|  | 3\8
|  |
|  |
|  |
|  |
|  | 1 1 1
|  | 713.233
|  | 1426.466
|  | 237.744
|  | 950.9775
|  |
|}


[[Category:mos]]
=== Generator chain ===
{{MOS genchain}}
 
=== Modes ===
{{MOS mode degrees}}
 
==== Proposed mode names ====
The following names have been proposed for the modes of 5L 3s, and are named after cities in the Dreamlands.
{{MOS modes
| Mode Names=
Dylathian $
Ilarnekian $
Celephaïsian $
Ultharian $
Mnarian $
Kadathian $
Hlanithian $
Sarnathian $
| Collapsed=1
}}
 
== Tunings==
=== Simple tunings ===
The simplest tuning for 5L 3s correspond to 13edo, 18edo, and 21edo, with step ratios 2:1, 3:1, and 3:2, respectively.
 
{{MOS tunings|JI Ratios=Int Limit: 30; Prime Limit: 19; Tenney Height: 7.7}}
 
=== Hypohard tunings ===
[[Hypohard]] oneirotonic tunings have step ratios between 2:1 and 3:1 and can be considered "meantone oneirotonic", sharing the following features with [[meantone]] diatonic tunings:
* The large step is a "meantone", around the range of [[10/9]] to [[9/8]].
* The major 2-mosstep is a [[meantone]]- to [[flattone]]-sized major third, thus is a stand-in for the classical diatonic major third.
 
With step ratios between 5:2 and 2:1, the minor 2-mosstep is close to [[7/6]].
 
EDOs that are in the hypohard range include [[13edo]], [[18edo]], and [[31edo]], and are associated with [[5L 3s/Temperaments#A-Team|A-Team]] temperament.
* 13edo has characteristically small 1-mossteps of about 185{{c}}. It is uniformly compressed 12edo, so it has distorted versions of non-diatonic 12edo scales. It essentially has the best [[11/8]] out of all hypohard tunings.
* 18edo can be used for a large step ratio of 3, (thus 18edo oneirotonic is distorted 17edo diatonic, or for its nearly pure 9/8 and 7/6. It also makes rising fifths (733.3{{c}}, a perfect 5-mosstep) and falling fifths (666.7{{c}}, a major 4-mosstep) almost equally off from a just perfect fifth. 18edo is also more suited for conventionally jazz styles due to its 6-fold symmetry.
* 31edo can be used to make the major 2-mosstep a near-just 5/4.
* [[44edo]] (generator {{nowrap|17\44 {{=}} 463.64{{c}}}}), [[57edo]] (generator {{nowrap|22\57 {{=}} 463.16{{c}}}}), and [[70edo]] (generator 27\70 {{=}} 462.857{{c}}}}) offer a compromise between 31edo's major third and 13edo's 11/8 and 13/8. In particular, 70edo has an essentially pure 13/8.
 
{{MOS tunings|Step Ratios=Hypohard|JI Ratios=Subgroup: 2.5.9.21; Int Limit:40; Complements Only: 1|Tolerance=15}}
 
=== Hyposoft tunings ===
[[Hyposoft]] oneirotonic tunings have step ratios between 3:2 and 2:1, which remains relatively unexplored. In these tunings,
* The large step of oneirotonic tends to be intermediate in size between [[10/9]] and [[11/10]]; the small step size is a semitone close to [[17/16]], about 92{{c}} to 114{{c}}.
* The major 2-mosstep (made of two large steps) in these tunings tends to be more of a neutral third, ranging from 6\21 (342{{c}}) to 4\13 (369{{c}}).
 
* [[21edo]]'s P1-L1ms-L2ms-L4ms approximates 9:10:11:13 better than the corresponding 13edo chord does. 21edo will serve those who like the combination of neogothic minor thirds (285.71{{c}}) and Baroque diatonic semitones (114.29{{c}}, close to quarter-comma meantone's 117.11{{c}}).
* [[34edo]]'s 9:10:11:13 is even better.
 
This set of JI identifications is associated with [[5L 3s/Temperaments#Petrtri|petrtri]] temperament. (P1-M1ms-P3ms could be said to approximate 5:11:13 in all soft-of-basic tunings, which is what "basic" [[petrtri]] temperament is.)
 
{{MOS tunings
| Step Ratios = Hyposoft
| JI Ratios =
1/1;
16/15;
10/9; 11/10;
13/11; 20/17;
11/9;
5/4;
13/10;
18/13; 32/23;
13/9; 23/16;
20/13;
8/5;
18/11;
22/13; 17/10;
9/5;
15/8;
2/1
}}
 
=== Parasoft and ultrasoft tunings ===
The range of oneirotonic tunings of step ratio between 6:5 and 3:2 is closely related to [[porcupine]] temperament; these tunings equate three oneirotonic large steps to a diatonic perfect fourth, i.e. they equate the oneirotonic large step to a [[porcupine]] generator. The chord 10:11:13 is very well approximated in 29edo.
 
{{MOS tunings
| Step Ratios = 6/5; 3/2; 4/3
| JI Ratios =
1/1;
14/13;
11/10;
9/8;
15/13;
13/11;
14/11;
13/10;
4/3;
15/11;
7/5;
10/7;
22/15;
3/2;
20/13;
11/7;
22/13;
26/15;
16/9;
20/11;
13/7;
2/1
}}
 
=== Parahard tunings ===
23edo oneiro combines the sound of neogothic tunings like [[46edo]] and the sounds of "superpyth" and "semaphore" scales. This is because 23edo oneirotonic has a large step of 208.7¢, same as [[46edo]]'s neogothic major second, and is both a warped [[22edo]] [[superpyth]] [[diatonic]] and a warped [[24edo]] [[semaphore]] [[semiquartal]] (and both nearby scales are [[superhard]] MOSes).
 
{{MOS tunings
| JI Ratios =
1/1;
21/17;
17/16;
14/11;
6/5;
21/16;
21/17;
34/21;
32/21;
5/3;
11/7;
32/17;
34/21;
2/1
| Step Ratios = 4/1
}}
 
=== Ultrahard tunings ===
{{Main|5L 3s/Temperaments#Buzzard}}
 
[[Buzzard]] is a rank-2 temperament in the [[Step ratio|pseudocollapsed]] range. It represents the only [[harmonic entropy]] minimum of the oneirotonic spectrum.
 
In the broad sense, Buzzard can be viewed as any tuning that divides the 3rd harmonic into 4 equal parts. [[23edo]], [[28edo]] and [[33edo]] can nominally be viewed as supporting it, but are still very flat and in an ambiguous zone between 18edo and true Buzzard in terms of harmonies. [[38edo]] & [[43edo]] are good compromises between melodic utility and harmonic accuracy, as the small step is still large enough to be obvious to the untrained ear, but [[48edo]] is where it really comes into its own in terms of harmonies, providing not only an excellent [[3/2]], but also [[7/4]] and [[The_Archipelago|archipelago]] harmonies, as by dividing the 5th in 4 it obviously also divides it in two as well.
 
Beyond that, it's a question of which intervals you want to favor. [[53edo]] has an essentially perfect [[3/2]], [[58edo]] gives the lowest overall error for the Barbados triads 10:13:15 and 26:30:39, while [[63edo]] does the same for the basic 4:6:7 triad. You could in theory go up to [[83edo]] if you want to favor the [[7/4]] above everything else, but beyond that, general accuracy drops off rapidly and you might as well be playing equal pentatonic.
 
{{MOS tunings
| JI Ratios =
1/1;
8/7;
13/10;
21/16;
3/2;
12/7, 22/13;
26/15;
49/25, 160/81;
2/1
| Step Ratios = 7/1; 10/1; 12/1
| Tolerance = 30
}}
 
== Approaches ==
* [[5L 3s/Temperaments]]
 
== Samples ==
[[File:The Angels' Library edo.mp3]] [[:File:The Angels' Library edo.mp3|The Angels' Library]] by [[Inthar]] in the Sarnathian (23233233) mode of 21edo oneirotonic ([[:File:The Angels' Library Score.pdf|score]])
 
[[File:13edo Prelude in J Oneirominor.mp3]]
 
[[WT13C]] [[:File:13edo Prelude in J Oneirominor.mp3|Prelude II (J Oneirominor)]] ([[:File:13edo Prelude in J Oneirominor Score.pdf|score]]) – Simple two-part Baroque piece. It stays in oneirotonic even though it modulates to other keys a little.
 
[[File:13edo_1MC.mp3]]
 
(13edo, first 30 seconds is in J Celephaïsian)
 
[[File:A Moment of Respite.mp3]]
 
(13edo, L Ilarnekian)
 
[[File:Lunar Approach.mp3]]
 
(by [[Igliashon Jones]], 13edo, J Celephaïsian)
 
=== 13edo Oneirotonic Modal Studies ===
* [[File:Inthar-13edo Oneirotonic Studies 1 Dylathian.mp3]]: Tonal Study in Dylathian
* [[File:Inthar-13edo Oneirotonic Studies 2 Ultharian.mp3]]: Tonal Study in Ultharian
* [[File:Inthar-13edo Oneirotonic Studies 3 Hlanithian.mp3]]: Tonal Study in Hlanithian
* [[File:Inthar-13edo Oneirotonic Studies 4 Illarnekian.mp3]]: Tonal Study in Ilarnekian
* [[File:Inthar-13edo Oneirotonic Studies 5 Mnarian.mp3]]: Tonal Study in Mnarian
* [[File:Inthar-13edo Oneirotonic Studies 6 Sarnathian.mp3]]: Tonal Study in Sarnathian
* [[File:Inthar-13edo Oneirotonic Studies 7 Celephaisian.mp3]]: Tonal Study in Celephaïsian
* [[File:Inthar-13edo Oneirotonic Studies 8 Kadathian.mp3]]: Tonal Study in Kadathian
 
== Scale tree ==
{{MOS tuning spectrum
| 13/8 = Golden oneirotonic (458.3592{{c}})
| 13/5 = Golden A-Team (465.0841{{c}})
}}
 
[[Category:Oneirotonic| ]] <!-- sort order in category: this page shows above A -->
[[Category:Pages with internal sound examples]]

Latest revision as of 13:59, 5 May 2025

↖ 4L 2s ↑ 5L 2s 6L 2s ↗
← 4L 3s 5L 3s 6L 3s →
↙ 4L 4s ↓ 5L 4s 6L 4s ↘ 
┌╥╥┬╥╥┬╥┬┐
│║║│║║│║││
││││││││││
└┴┴┴┴┴┴┴┴┘
Scale structure
Step pattern LLsLLsLs
sLsLLsLL
Equave 2/1 (1200.0 ¢)
Period 2/1 (1200.0 ¢)
Generator size
Bright 3\8 to 2\5 (450.0 ¢ to 480.0 ¢)
Dark 3\5 to 5\8 (720.0 ¢ to 750.0 ¢)
TAMNAMS information
Name oneirotonic
Prefix oneiro-
Abbrev. onei
Related MOS scales
Parent 3L 2s
Sister 3L 5s
Daughters 8L 5s, 5L 8s
Neutralized 2L 6s
2-Flought 13L 3s, 5L 11s
Equal tunings
Equalized (L:s = 1:1) 3\8 (450.0 ¢)
Supersoft (L:s = 4:3) 11\29 (455.2 ¢)
Soft (L:s = 3:2) 8\21 (457.1 ¢)
Semisoft (L:s = 5:3) 13\34 (458.8 ¢)
Basic (L:s = 2:1) 5\13 (461.5 ¢)
Semihard (L:s = 5:2) 12\31 (464.5 ¢)
Hard (L:s = 3:1) 7\18 (466.7 ¢)
Superhard (L:s = 4:1) 9\23 (469.6 ¢)
Collapsed (L:s = 1:0) 2\5 (480.0 ¢)
For the tritave-equivalent MOS structure with the same step pattern, see 5L 3s (3/1-equivalent).

5L 3s, named oneirotonic in TAMNAMS, is a 2/1-equivalent (octave-equivalent) moment of symmetry scale containing 5 large steps and 3 small steps, repeating every octave. Generators that produce this scale range from 450 ¢ to 480 ¢, or from 720 ¢ to 750 ¢. 5L 3s can be seen as a warped diatonic scale, because it has one extra small step compared to diatonic (5L 2s).

Name

TAMNAMS suggests the temperament-agnostic name oneirotonic as the name of 5L 3s. The name was originally used as a name for the 5L 3s scale in 13edo. 'Oneiro' is sometimes used as a shortened form.

'Father' is sometimes also used to denote 5L 3s, but it's a misnomer, as father is technically an abstract regular temperament (although a very inaccurate one), not a generator range. There are father tunings which generate 3L 5s. A more correct but still not quite correct name would be 'father[8]' or 'father octatonic'. "Father" is also vague regarding the number of notes, because optimal generators for it also generate 3L 2s.

Scale properties

Intervals

Intervals of 5L 3s
Intervals Steps
subtended 		Range in cents
Generic Specific Abbrev.
0-oneirostep Perfect 0-oneirostep P0oneis 0 0.0 ¢
1-oneirostep Minor 1-oneirostep m1oneis s 0.0 ¢ to 150.0 ¢
Major 1-oneirostep M1oneis L 150.0 ¢ to 240.0 ¢
2-oneirostep Minor 2-oneirostep m2oneis L + s 240.0 ¢ to 300.0 ¢
Major 2-oneirostep M2oneis 2L 300.0 ¢ to 480.0 ¢
3-oneirostep Diminished 3-oneirostep d3oneis L + 2s 240.0 ¢ to 450.0 ¢
Perfect 3-oneirostep P3oneis 2L + s 450.0 ¢ to 480.0 ¢
4-oneirostep Minor 4-oneirostep m4oneis 2L + 2s 480.0 ¢ to 600.0 ¢
Major 4-oneirostep M4oneis 3L + s 600.0 ¢ to 720.0 ¢
5-oneirostep Perfect 5-oneirostep P5oneis 3L + 2s 720.0 ¢ to 750.0 ¢
Augmented 5-oneirostep A5oneis 4L + s 750.0 ¢ to 960.0 ¢
6-oneirostep Minor 6-oneirostep m6oneis 3L + 3s 720.0 ¢ to 900.0 ¢
Major 6-oneirostep M6oneis 4L + 2s 900.0 ¢ to 960.0 ¢
7-oneirostep Minor 7-oneirostep m7oneis 4L + 3s 960.0 ¢ to 1050.0 ¢
Major 7-oneirostep M7oneis 5L + 2s 1050.0 ¢ to 1200.0 ¢
8-oneirostep Perfect 8-oneirostep P8oneis 5L + 3s 1200.0 ¢

Generator chain

Generator chain of 5L 3s
Bright gens Scale degree Abbrev.
12 Augmented 4-oneirodegree A4oneid
11 Augmented 1-oneirodegree A1oneid
10 Augmented 6-oneirodegree A6oneid
9 Augmented 3-oneirodegree A3oneid
8 Augmented 0-oneirodegree A0oneid
7 Augmented 5-oneirodegree A5oneid
6 Major 2-oneirodegree M2oneid
5 Major 7-oneirodegree M7oneid
4 Major 4-oneirodegree M4oneid
3 Major 1-oneirodegree M1oneid
2 Major 6-oneirodegree M6oneid
1 Perfect 3-oneirodegree P3oneid
0 Perfect 0-oneirodegree
Perfect 8-oneirodegree		 P0oneid
P8oneid
−1 Perfect 5-oneirodegree P5oneid
−2 Minor 2-oneirodegree m2oneid
−3 Minor 7-oneirodegree m7oneid
−4 Minor 4-oneirodegree m4oneid
−5 Minor 1-oneirodegree m1oneid
−6 Minor 6-oneirodegree m6oneid
−7 Diminished 3-oneirodegree d3oneid
−8 Diminished 8-oneirodegree d8oneid
−9 Diminished 5-oneirodegree d5oneid
−10 Diminished 2-oneirodegree d2oneid
−11 Diminished 7-oneirodegree d7oneid
−12 Diminished 4-oneirodegree d4oneid

Modes

Scale degrees of the modes of 5L 3s
UDP Cyclic
order 		Step
pattern 		Scale degree (oneirodegree)
0 1 2 3 4 5 6 7 8
7|0 1 LLsLLsLs Perf. Maj. Maj. Perf. Maj. Aug. Maj. Maj. Perf.
6|1 4 LLsLsLLs Perf. Maj. Maj. Perf. Maj. Perf. Maj. Maj. Perf.
5|2 7 LsLLsLLs Perf. Maj. Min. Perf. Maj. Perf. Maj. Maj. Perf.
4|3 2 LsLLsLsL Perf. Maj. Min. Perf. Maj. Perf. Maj. Min. Perf.
3|4 5 LsLsLLsL Perf. Maj. Min. Perf. Min. Perf. Maj. Min. Perf.
2|5 8 sLLsLLsL Perf. Min. Min. Perf. Min. Perf. Maj. Min. Perf.
1|6 3 sLLsLsLL Perf. Min. Min. Perf. Min. Perf. Min. Min. Perf.
0|7 6 sLsLLsLL Perf. Min. Min. Dim. Min. Perf. Min. Min. Perf.

Proposed mode names

The following names have been proposed for the modes of 5L 3s, and are named after cities in the Dreamlands.

Modes of 5L 3s
UDP Cyclic
order		 Step
pattern		 Mode names
7|0 1 LLsLLsLs Dylathian
6|1 4 LLsLsLLs Ilarnekian
5|2 7 LsLLsLLs Celephaïsian
4|3 2 LsLLsLsL Ultharian
3|4 5 LsLsLLsL Mnarian
2|5 8 sLLsLLsL Kadathian
1|6 3 sLLsLsLL Hlanithian
0|7 6 sLsLLsLL Sarnathian

Tunings

Simple tunings

The simplest tuning for 5L 3s correspond to 13edo, 18edo, and 21edo, with step ratios 2:1, 3:1, and 3:2, respectively.


Simple Tunings of 5L 3s
Scale degree Abbrev. Basic (2:1)
13edo 		Hard (3:1)
18edo 		Soft (3:2)
21edo
Steps ¢ Steps ¢ Steps ¢
Perfect 0-oneirodegree P0oneid 0\13 0.0 0\18 0.0 0\21 0.0
Minor 1-oneirodegree m1oneid 1\13 92.3 1\18 66.7 2\21 114.3
Major 1-oneirodegree M1oneid 2\13 184.6 3\18 200.0 3\21 171.4
Minor 2-oneirodegree m2oneid 3\13 276.9 4\18 266.7 5\21 285.7
Major 2-oneirodegree M2oneid 4\13 369.2 6\18 400.0 6\21 342.9
Diminished 3-oneirodegree d3oneid 4\13 369.2 5\18 333.3 7\21 400.0
Perfect 3-oneirodegree P3oneid 5\13 461.5 7\18 466.7 8\21 457.1
Minor 4-oneirodegree m4oneid 6\13 553.8 8\18 533.3 10\21 571.4
Major 4-oneirodegree M4oneid 7\13 646.2 10\18 666.7 11\21 628.6
Perfect 5-oneirodegree P5oneid 8\13 738.5 11\18 733.3 13\21 742.9
Augmented 5-oneirodegree A5oneid 9\13 830.8 13\18 866.7 14\21 800.0
Minor 6-oneirodegree m6oneid 9\13 830.8 12\18 800.0 15\21 857.1
Major 6-oneirodegree M6oneid 10\13 923.1 14\18 933.3 16\21 914.3
Minor 7-oneirodegree m7oneid 11\13 1015.4 15\18 1000.0 18\21 1028.6
Major 7-oneirodegree M7oneid 12\13 1107.7 17\18 1133.3 19\21 1085.7
Perfect 8-oneirodegree P8oneid 13\13 1200.0 18\18 1200.0 21\21 1200.0

Hypohard tunings

Hypohard oneirotonic tunings have step ratios between 2:1 and 3:1 and can be considered "meantone oneirotonic", sharing the following features with meantone diatonic tunings:

  • The large step is a "meantone", around the range of 10/9 to 9/8.
  • The major 2-mosstep is a meantone- to flattone-sized major third, thus is a stand-in for the classical diatonic major third.

With step ratios between 5:2 and 2:1, the minor 2-mosstep is close to 7/6.

EDOs that are in the hypohard range include 13edo, 18edo, and 31edo, and are associated with A-Team temperament.

  • 13edo has characteristically small 1-mossteps of about 185 ¢. It is uniformly compressed 12edo, so it has distorted versions of non-diatonic 12edo scales. It essentially has the best 11/8 out of all hypohard tunings.
  • 18edo can be used for a large step ratio of 3, (thus 18edo oneirotonic is distorted 17edo diatonic, or for its nearly pure 9/8 and 7/6. It also makes rising fifths (733.3 ¢, a perfect 5-mosstep) and falling fifths (666.7 ¢, a major 4-mosstep) almost equally off from a just perfect fifth. 18edo is also more suited for conventionally jazz styles due to its 6-fold symmetry.
  • 31edo can be used to make the major 2-mosstep a near-just 5/4.
  • 44edo (generator 17\44 = 463.64 ¢), 57edo (generator 22\57 = 463.16 ¢), and 70edo (generator 27\70 = 462.857 ¢}}) offer a compromise between 31edo's major third and 13edo's 11/8 and 13/8. In particular, 70edo has an essentially pure 13/8.


Hypohard Tunings of 5L 3s
Scale degree Abbrev. Basic (2:1)
13edo 		Semihard (5:2)
31edo 		Hard (3:1)
18edo
Steps ¢ Steps ¢ Steps ¢
Perfect 0-oneirodegree P0oneid 0\13 0.0 0\31 0.0 0\18 0.0
Minor 1-oneirodegree m1oneid 1\13 92.3 2\31 77.4 1\18 66.7
Major 1-oneirodegree M1oneid 2\13 184.6 5\31 193.5 3\18 200.0
Minor 2-oneirodegree m2oneid 3\13 276.9 7\31 271.0 4\18 266.7
Major 2-oneirodegree M2oneid 4\13 369.2 10\31 387.1 6\18 400.0
Diminished 3-oneirodegree d3oneid 4\13 369.2 9\31 348.4 5\18 333.3
Perfect 3-oneirodegree P3oneid 5\13 461.5 12\31 464.5 7\18 466.7
Minor 4-oneirodegree m4oneid 6\13 553.8 14\31 541.9 8\18 533.3
Major 4-oneirodegree M4oneid 7\13 646.2 17\31 658.1 10\18 666.7
Perfect 5-oneirodegree P5oneid 8\13 738.5 19\31 735.5 11\18 733.3
Augmented 5-oneirodegree A5oneid 9\13 830.8 22\31 851.6 13\18 866.7
Minor 6-oneirodegree m6oneid 9\13 830.8 21\31 812.9 12\18 800.0
Major 6-oneirodegree M6oneid 10\13 923.1 24\31 929.0 14\18 933.3
Minor 7-oneirodegree m7oneid 11\13 1015.4 26\31 1006.5 15\18 1000.0
Major 7-oneirodegree M7oneid 12\13 1107.7 29\31 1122.6 17\18 1133.3
Perfect 8-oneirodegree P8oneid 13\13 1200.0 31\31 1200.0 18\18 1200.0

Hyposoft tunings

Hyposoft oneirotonic tunings have step ratios between 3:2 and 2:1, which remains relatively unexplored. In these tunings,

  • The large step of oneirotonic tends to be intermediate in size between 10/9 and 11/10; the small step size is a semitone close to 17/16, about 92 ¢ to 114 ¢.
  • The major 2-mosstep (made of two large steps) in these tunings tends to be more of a neutral third, ranging from 6\21 (342 ¢) to 4\13 (369 ¢).
  • 21edo's P1-L1ms-L2ms-L4ms approximates 9:10:11:13 better than the corresponding 13edo chord does. 21edo will serve those who like the combination of neogothic minor thirds (285.71 ¢) and Baroque diatonic semitones (114.29 ¢, close to quarter-comma meantone's 117.11 ¢).
  • 34edo's 9:10:11:13 is even better.

This set of JI identifications is associated with petrtri temperament. (P1-M1ms-P3ms could be said to approximate 5:11:13 in all soft-of-basic tunings, which is what "basic" petrtri temperament is.)


Hyposoft Tunings of 5L 3s
Scale degree Abbrev. Soft (3:2)
21edo 		Semisoft (5:3)
34edo 		Basic (2:1)
13edo
Steps ¢ Steps ¢ Steps ¢
Perfect 0-oneirodegree P0oneid 0\21 0.0 0\34 0.0 0\13 0.0
Minor 1-oneirodegree m1oneid 2\21 114.3 3\34 105.9 1\13 92.3
Major 1-oneirodegree M1oneid 3\21 171.4 5\34 176.5 2\13 184.6
Minor 2-oneirodegree m2oneid 5\21 285.7 8\34 282.4 3\13 276.9
Major 2-oneirodegree M2oneid 6\21 342.9 10\34 352.9 4\13 369.2
Diminished 3-oneirodegree d3oneid 7\21 400.0 11\34 388.2 4\13 369.2
Perfect 3-oneirodegree P3oneid 8\21 457.1 13\34 458.8 5\13 461.5
Minor 4-oneirodegree m4oneid 10\21 571.4 16\34 564.7 6\13 553.8
Major 4-oneirodegree M4oneid 11\21 628.6 18\34 635.3 7\13 646.2
Perfect 5-oneirodegree P5oneid 13\21 742.9 21\34 741.2 8\13 738.5
Augmented 5-oneirodegree A5oneid 14\21 800.0 23\34 811.8 9\13 830.8
Minor 6-oneirodegree m6oneid 15\21 857.1 24\34 847.1 9\13 830.8
Major 6-oneirodegree M6oneid 16\21 914.3 26\34 917.6 10\13 923.1
Minor 7-oneirodegree m7oneid 18\21 1028.6 29\34 1023.5 11\13 1015.4
Major 7-oneirodegree M7oneid 19\21 1085.7 31\34 1094.1 12\13 1107.7
Perfect 8-oneirodegree P8oneid 21\21 1200.0 34\34 1200.0 13\13 1200.0

Parasoft and ultrasoft tunings

The range of oneirotonic tunings of step ratio between 6:5 and 3:2 is closely related to porcupine temperament; these tunings equate three oneirotonic large steps to a diatonic perfect fourth, i.e. they equate the oneirotonic large step to a porcupine generator. The chord 10:11:13 is very well approximated in 29edo.


Soft Tunings of 5L 3s
Scale degree Abbrev. 6:5
45edo 		Supersoft (4:3)
29edo 		Soft (3:2)
21edo
Steps ¢ Steps ¢ Steps ¢
Perfect 0-oneirodegree P0oneid 0\45 0.0 0\29 0.0 0\21 0.0
Minor 1-oneirodegree m1oneid 5\45 133.3 3\29 124.1 2\21 114.3
Major 1-oneirodegree M1oneid 6\45 160.0 4\29 165.5 3\21 171.4
Minor 2-oneirodegree m2oneid 11\45 293.3 7\29 289.7 5\21 285.7
Major 2-oneirodegree M2oneid 12\45 320.0 8\29 331.0 6\21 342.9
Diminished 3-oneirodegree d3oneid 16\45 426.7 10\29 413.8 7\21 400.0
Perfect 3-oneirodegree P3oneid 17\45 453.3 11\29 455.2 8\21 457.1
Minor 4-oneirodegree m4oneid 22\45 586.7 14\29 579.3 10\21 571.4
Major 4-oneirodegree M4oneid 23\45 613.3 15\29 620.7 11\21 628.6
Perfect 5-oneirodegree P5oneid 28\45 746.7 18\29 744.8 13\21 742.9
Augmented 5-oneirodegree A5oneid 29\45 773.3 19\29 786.2 14\21 800.0
Minor 6-oneirodegree m6oneid 33\45 880.0 21\29 869.0 15\21 857.1
Major 6-oneirodegree M6oneid 34\45 906.7 22\29 910.3 16\21 914.3
Minor 7-oneirodegree m7oneid 39\45 1040.0 25\29 1034.5 18\21 1028.6
Major 7-oneirodegree M7oneid 40\45 1066.7 26\29 1075.9 19\21 1085.7
Perfect 8-oneirodegree P8oneid 45\45 1200.0 29\29 1200.0 21\21 1200.0

Parahard tunings

23edo oneiro combines the sound of neogothic tunings like 46edo and the sounds of "superpyth" and "semaphore" scales. This is because 23edo oneirotonic has a large step of 208.7¢, same as 46edo's neogothic major second, and is both a warped 22edo superpyth diatonic and a warped 24edo semaphore semiquartal (and both nearby scales are superhard MOSes).


Superhard Tuning of 5L 3s
Scale degree Abbrev. Superhard (4:1)
23edo
Steps ¢
Perfect 0-oneirodegree P0oneid 0\23 0.0
Minor 1-oneirodegree m1oneid 1\23 52.2
Major 1-oneirodegree M1oneid 4\23 208.7
Minor 2-oneirodegree m2oneid 5\23 260.9
Major 2-oneirodegree M2oneid 8\23 417.4
Diminished 3-oneirodegree d3oneid 6\23 313.0
Perfect 3-oneirodegree P3oneid 9\23 469.6
Minor 4-oneirodegree m4oneid 10\23 521.7
Major 4-oneirodegree M4oneid 13\23 678.3
Perfect 5-oneirodegree P5oneid 14\23 730.4
Augmented 5-oneirodegree A5oneid 17\23 887.0
Minor 6-oneirodegree m6oneid 15\23 782.6
Major 6-oneirodegree M6oneid 18\23 939.1
Minor 7-oneirodegree m7oneid 19\23 991.3
Major 7-oneirodegree M7oneid 22\23 1147.8
Perfect 8-oneirodegree P8oneid 23\23 1200.0

Ultrahard tunings

Main article: 5L 3s/Temperaments § Buzzard

Buzzard is a rank-2 temperament in the pseudocollapsed range. It represents the only harmonic entropy minimum of the oneirotonic spectrum.

In the broad sense, Buzzard can be viewed as any tuning that divides the 3rd harmonic into 4 equal parts. 23edo, 28edo and 33edo can nominally be viewed as supporting it, but are still very flat and in an ambiguous zone between 18edo and true Buzzard in terms of harmonies. 38edo & 43edo are good compromises between melodic utility and harmonic accuracy, as the small step is still large enough to be obvious to the untrained ear, but 48edo is where it really comes into its own in terms of harmonies, providing not only an excellent 3/2, but also 7/4 and archipelago harmonies, as by dividing the 5th in 4 it obviously also divides it in two as well.

Beyond that, it's a question of which intervals you want to favor. 53edo has an essentially perfect 3/2, 58edo gives the lowest overall error for the Barbados triads 10:13:15 and 26:30:39, while 63edo does the same for the basic 4:6:7 triad. You could in theory go up to 83edo if you want to favor the 7/4 above everything else, but beyond that, general accuracy drops off rapidly and you might as well be playing equal pentatonic.


Ultrahard Tunings of 5L 3s
Scale degree Abbrev. 7:1
38edo 		10:1
53edo 		12:1
63edo
Steps ¢ Steps ¢ Steps ¢
Perfect 0-oneirodegree P0oneid 0\38 0.0 0\53 0.0 0\63 0.0
Minor 1-oneirodegree m1oneid 1\38 31.6 1\53 22.6 1\63 19.0
Major 1-oneirodegree M1oneid 7\38 221.1 10\53 226.4 12\63 228.6
Minor 2-oneirodegree m2oneid 8\38 252.6 11\53 249.1 13\63 247.6
Major 2-oneirodegree M2oneid 14\38 442.1 20\53 452.8 24\63 457.1
Diminished 3-oneirodegree d3oneid 9\38 284.2 12\53 271.7 14\63 266.7
Perfect 3-oneirodegree P3oneid 15\38 473.7 21\53 475.5 25\63 476.2
Minor 4-oneirodegree m4oneid 16\38 505.3 22\53 498.1 26\63 495.2
Major 4-oneirodegree M4oneid 22\38 694.7 31\53 701.9 37\63 704.8
Perfect 5-oneirodegree P5oneid 23\38 726.3 32\53 724.5 38\63 723.8
Augmented 5-oneirodegree A5oneid 29\38 915.8 41\53 928.3 49\63 933.3
Minor 6-oneirodegree m6oneid 24\38 757.9 33\53 747.2 39\63 742.9
Major 6-oneirodegree M6oneid 30\38 947.4 42\53 950.9 50\63 952.4
Minor 7-oneirodegree m7oneid 31\38 978.9 43\53 973.6 51\63 971.4
Major 7-oneirodegree M7oneid 37\38 1168.4 52\53 1177.4 62\63 1181.0
Perfect 8-oneirodegree P8oneid 38\38 1200.0 53\53 1200.0 63\63 1200.0

Approaches

Samples

The Angels' Library by Inthar in the Sarnathian (23233233) mode of 21edo oneirotonic (score)

WT13C Prelude II (J Oneirominor) (score) – Simple two-part Baroque piece. It stays in oneirotonic even though it modulates to other keys a little.

(13edo, first 30 seconds is in J Celephaïsian)

(13edo, L Ilarnekian)

(by Igliashon Jones, 13edo, J Celephaïsian)

13edo Oneirotonic Modal Studies

Scale tree

Scale tree and tuning spectrum of 5L 3s
Generator(edo) Cents Step ratio Comments
Bright Dark L:s Hardness
3\8 450.000 750.000 1:1 1.000 Equalized 5L 3s
17\45 453.333 746.667 6:5 1.200
14\37 454.054 745.946 5:4 1.250
25\66 454.545 745.455 9:7 1.286
11\29 455.172 744.828 4:3 1.333 Supersoft 5L 3s
30\79 455.696 744.304 11:8 1.375
19\50 456.000 744.000 7:5 1.400
27\71 456.338 743.662 10:7 1.429
8\21 457.143 742.857 3:2 1.500 Soft 5L 3s
29\76 457.895 742.105 11:7 1.571
21\55 458.182 741.818 8:5 1.600
34\89 458.427 741.573 13:8 1.625 Golden oneirotonic (458.3592 ¢)
13\34 458.824 741.176 5:3 1.667 Semisoft 5L 3s
31\81 459.259 740.741 12:7 1.714
18\47 459.574 740.426 7:4 1.750
23\60 460.000 740.000 9:5 1.800
5\13 461.538 738.462 2:1 2.000 Basic 5L 3s
Scales with tunings softer than this are proper
22\57 463.158 736.842 9:4 2.250
17\44 463.636 736.364 7:3 2.333
29\75 464.000 736.000 12:5 2.400
12\31 464.516 735.484 5:2 2.500 Semihard 5L 3s
31\80 465.000 735.000 13:5 2.600 Golden A-Team (465.0841 ¢)
19\49 465.306 734.694 8:3 2.667
26\67 465.672 734.328 11:4 2.750
7\18 466.667 733.333 3:1 3.000 Hard 5L 3s
23\59 467.797 732.203 10:3 3.333
16\41 468.293 731.707 7:2 3.500
25\64 468.750 731.250 11:3 3.667
9\23 469.565 730.435 4:1 4.000 Superhard 5L 3s
20\51 470.588 729.412 9:2 4.500
11\28 471.429 728.571 5:1 5.000
13\33 472.727 727.273 6:1 6.000
2\5 480.000 720.000 1:0 → ∞ Collapsed 5L 3s
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