Xenharmonic Wiki

5L 3s: Difference between revisions

← Older edit
VisualWikitext
Moremajorthanmajor (talk | contribs)
autopatrolled
3,126 edits
No edit summary
ArrowHead294 (talk | contribs)
14,512 edits
m substitute deprecated template
 
(671 intermediate revisions by 15 users not shown)
Line 1: Line 1:
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
|  |
|-
|  |
|  |
|  |
|  |
|  |
|  | 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
|  |4g=45/8 near here
|-
|  |
|  | 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
|-
|  |
|  | 68\175
|  |
|  |
|  |
|  | 29 10 29
|  | 739.045
|  | 1478.091
|  | 315.181
|  | 1054.227
|  |3g=18/5 near here
|-
|  |
|  | 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
|  |
|-
|  |
|  | 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
|  |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]]
Retrieved from "https://en.xen.wiki/w/5L_3s"