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4L 5s (3/1-equivalent): Difference between revisions

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| Pattern = LsLsLsLss
| Pattern = LsLsLsLss
}}
}}
Suggested for use as a "diatonic scale" when playing Bohlen-Pierce is the 9-note Lambda scale, which is the 4L5s MOS. This can be thought of as an MOS generated by a 3.5.7 rank-2 temperament that eliminates only the comma 245/243, so that 9/7 * 9/7 = 5/3.
This is a very good temperament on the 3.5.7 subgroup, and additionally is supported by many EDT's (and even EDOs!) besides 13-EDT.
Some low-numbered EDOs that support Lambda are 19, 22, 27, 41, and 46, all of which make it possible to play BP music to some reasonable extent. These EDOs contain not only the Lambda BP diatonic scale, but also the 13-note "Lambda chromatic" MOS scale, or Lambda[13], which can be thought of as a "detempered" version of the 13-EDT Bohlen Pierce scale. This scale may be a suitable melodic substitute for the BP chromatic scale, and is basically the same as how 19-EDO and 31-EDO do not contain 12-EDO as a subset, but they do contain the meantone[12] chromatic scale.
When playing this temperament in some EDO, it may be desired to stretch/compress the tuning so that the tritave is pure, rather than the octave being pure - or in general, to minimize the error on the 3.5.7 subgroup while ignoring the error on 2/1.
One can "add" the octave to Lambda temperament by simply creating a new mapping for 2/1. A simple way to do so is to map the 2/1 to +7 of the ~9/7 generators, minus a single tritave. This is Sensi temperament, in essence treating it as a "3.5.7.2 extension" of the original 3.5.7 Lambda temperament.
== List of EDT's supporting Lambda Temperament ==
Below is a list of the equal-temperaments which contain a 4L+5s scale using generators between 422.7 cents and 475.5 cents.
L=1 s=0 4 edt
L=1 s=1 9 edt (5flat40 7sharp18)
L=2 s=1 13 (5flat7 7flat3)
L=3 s=1 17 (5sharp10 7flat12)
L=3 s=2 22 (~14edo)
L=4 s=1 21
L=4 s=3 31
L=5 s=1 25
L=5 s=2 30 (~19edo) (5sharp3 7flat8)
L=5 s=3 35 (~22edo) (5flat14 7sharp0)
L=5 s=4 40
L=6 s=1 29
L=6 s=5 49 (~31EDO) (5sharp8 7sharp8) (Schism*)
L=7 s=1 33
L=7 s=2 38 (~24edo)
L=7 s=3 43 (~27edo) (5sharp0 7flat6)
L=7 s=4 48 (5flat13 7flat0)
L=7 s=5 53
L=7 s=6 58 5sharp1 7sharp10 (Schism*)
*Schism, by which I mean, the most accurate value for 5/3 and-or 7/3 is found outside the 4L+5s MOS.
[Also, the way I see it, as 4edt and 9edt are comparable to 5edo and 7edo, then the "counterparts" of Blackwood and Whitewood would be found in multiples therein and would be octatonic and octadecatonic, e.g. 12edt and 27edt.]
{| class="wikitable" style="text-align:center;"
|-
! colspan="7" | Generator
! | cents
hekts
! | L
! | s
! | notes
|-
| | 1/4
| |
| |
| |
| |
| |
| |
| colspan="2" | 475.489
325
| | 0
| |
|-
| |
| |
| |
| |
| |
| |
| | 8/33
| | 461.08
315.1515
| | 403.445
275.758
| | 57.635
39.394
| |
|-
| |
| |
| |
| |
| |
| | 7/29
| |
| | 459.093
313.793
| | 393.508
268.9655
| | 65.585
44.828
| |
|-
| |
| |
| |
| |
| |
| |
| | 13/54
| | 457.878
312.963
| | 387.435
264.815
| | 70.428
48.148
| |
|-
| |
| |
| |
| |
| | 6/25
| |
| |
| | 456.469
312
| | 380.391
260
| | 76.078
52
| |
|-
| |
| |
| |
| |
| |
| |
| | 17/71
| | 455.398
311.267
| | 375.033
256.338
| | 80.364
54.9295
| |
|-
| |
| |
| |
| |
| |
| | 11/46
| |
| | 454.815
310.8695
| | 372.122
254.348
| | 82.694
56.522
| |
|-
| |
| |
| |
| |
| |
| |
| | 16/67
| | 454.198
310.448
| | 369.036
252.239
| | 85.162
58.209
| |
|-
| |
| |
| |
| | 5/21
| |
| |
| |
| | 452.846
309.524
| | 362.277
247.619
| | 90.569
61.905
| |
|-
| |
| |
| |
| |
| |
| |
| | 19/80
| | 451.714
308.75
| | 356.617
243.75
| | 95.098
65
| |
|-
| |
| |
| |
| |
| |
| | 14/59
| |
| | 451.311
308.4745
| | 354.602
242.373
| | 96.71
66.102
| |
|-
| |
| |
| |
| |
| |
| |
| | 23/97
| | 450.979
308.247
| | 352.94
241.234
| | 98.039
67.01
| |
|-
| |
| |
| |
| |
| | 9/38
| |
| |
| | 450.463
307.895
| | 350.36
239.474
| | 100.103
68.421
| |
|-
| |
| |
| |
| |
| |
| |
| | 22/93
| | 449.925
307.527
| | 347.669
237.634
| | 102.256
69.8925
| |
|-
| |
| |
| |
| |
| |
| | 13/55
| |
| | 449.553
307.273
| | 345.81
236.364
| | 103.743
70.909
| |
|-
| |
| |
| |
| |
| |
| |
| | 17/72
| | 449.073
306.944
| | 343.4085
234.722
| | 105.664
72.222
| |
|-
| |
| |
| |
| |
| |
| |
| |
| | 448.421
306.499
| | 340.148
232.494
| | 108.2725
74.005
| |
|-
| |
| |
| | 4/17
| |
| |
| |
| |
| | 447.518
305.882
| | 335.639
229.412
| | 111.88
76.471
| | Canonical BP scales are between here...
|-
| |
| |
| |
| |
| |
| |
| | 19/81
| | 446.137
304.938
| | 328.733
224.691
| | 117.405
80.247
| |
|-
| |
| |
| |
| |
| |
| | 15/64
| |
| | 445.771
304.6875
| | 326.8985
223.4375
| | 118.872
81.25
| |
|-
| |
| |
| |
| |
| |
| |
| |
| | 445.533
304.525
| | 325.711
222.626
| | 119.822
81.899
| |
|-
| |
| |
| |
| |
| |
| |
| | 26/111
| | 445.503
304.5045
| | 325.559
222.5225
| | 119.943
81.982
| |
|-
| |
| |
| |
| |
| | 11/47
| |
| |
| | 445.138
304.255
| | 323.737
221.277
| | 121.401
82.989
| |
|-
| |
| |
| |
| |
| |
| |
| | 29/124
| | 444.812
304.032
| | 322.105
220.161
| | 122.705
83.871
| | Golden BP is near here
|-
| |
| |
| |
| |
| |
| | 18/77
| |
| | 444.613
303.896
| | 321.109
219.4805
| | 123.5035
84.416
| |
|-
| |
| |
| |
| |
| |
| |
| | 25/107
| | 444.382
303.738
| | 319.955
218.692
| | 120.427
85.047
| |
|-
| |
| |
| |
| | 7/30
| |
| |
| |
| | 443.7895
303.333
| | 316.9925
216.667
| | 126.797
86.667
| |
|-
| |
| |
| |
| |
| |
| |
| | 24/103
| | 443.174
302.913
| | 313.915
214.563
| | 129.26
88.3495
| |
|-
| |
| |
| |
| |
| |
| | 17/73
| |
| | 442.921
302.74
| | 312.65
213.699
| | 130.271
89.041
| |
|-
| |
| |
| |
| |
| |
| |
| | 27/116
| | 442.696
302.586
| | 311.527
212.931
| | 131.169
89.65
| |
|-
| |
| |
| |
| |
| | 10/43
| |
| |
| | 442.315
302.326
| | 309.621
211.628
| | 132.6945
90.698
| |
|-
| |
| |
| |
| |
| |
| |
| | 23/99
| | 441.868
302.02
| | 307.387
210.101
| | 134.482
91.919
| |
|-
| |
| |
| |
| |
| |
| | 13/56
| |
| | 441.525
301.786
| | 305.671
208.929
| | 135.854
92.857
| |
|-
| |
| |
| |
| |
| |
| |
| | 16/69
| | 441.033
301.449
| | 303.21
207.246
| | 137.823
94.203
| |
|-
| |
| | 3/13
| |
| |
| |
| |
| |
| | 438.913
300
| | 292.6085
200
| | 146.304
100
| | ...and here
Boundary of propriety for Lambda scale
|-
| |
| |
| |
| |
| |
| |
| | 17/74
| | 436.935
298.649
| | 282.723
193.243
| | 154.2125
105.405
| |
|-
| |
| |
| |
| |
| |
| | 14/61
| |
| | 436.514
298.361
| | 280.616
191.803
| | 155.897
106.557
| |
|-
| |
| |
| |
| |
| |
| |
| | 25/109
| | 436.228
298.165
| | 279.186
190.826
| | 157.042
207.339
| |
|-
| |
| |
| |
| |
| | 11/48
| |
| |
| | 435.865
297.917
| | 277.368
189.583
| | 158.496
108.333
| |
|-
| |
| |
| |
| |
| |
| |
| | 30/131
| | 435.562
297.71
| | 275.856
188.55
| | 159.706
109.16
| |
|-
| |
| |
| |
| |
| |
| | 19/83
| |
| | 435.387
297.59
| | 274.981
187.952
| | 160.406
109.639
| |
|-
| |
| |
| |
| |
| |
| |
| | 27/118
| | 435.193
297.458
| | 274.0105
187.288
| | 161.183
110.1695
| |
|-
| |
| |
| |
| | 8/35
| |
| |
| |
| | 434.733
297.143
| | 271.707
185.714
| | 163.025
111.429
| |
|-
| |
| |
| |
| |
| |
| |
| | 29/127
| | 434.305
296.85
| | 269.568
184.252
| | 164.736
112.598
| |
|-
| |
| |
| |
| |
| |
| | 21/92
| |
| | 434.142
596.739
| | 268.7545
183.696
| | 165.387
113.0435
| |
|-
| |
| |
| |
| |
| |
| |
| | 34/149
| | 434.003
296.644
| | 268.061
183.2215
| | 165.942
113.423
| | Golden Lambda scale is near here
18\7*30\11=7
|-
| |
| |
| |
| |
| | 13/57
| |
| |
| | 433.779
296.491
| | 266.941
182.456
| | 166.838
114.035
| | 18\7*30\11=7
|-
| |
| |
| |
| |
| |
| |
| | 31/136
| | 433.534
296.3235
| | 265.714
181.618
| | 167.8195
114.706
| |
|-
| |
| |
| |
| |
| |
| | 18/79
| |
| | 433.356
296.2025
| | 264.829
181.013
| | 168.528
115.189
| |
|-
| |
| |
| |
| |
| |
| |
| | 23/101
| | 433.1185
296.04
| | 263.637
180.198
| | 169.484
115.842
| |
|-
| |
| |
| | 5/22
| |
| |
| |
| |
| | 432.2625
295.4545
| | 259.3575
177.273
| | 172.905
118.182
| |
|-
| |
| |
| |
| |
| |
| |
| | 22/97
| | 431.371
294.845
| | 254.901
174.227
| | 176.47
120.619
| |
|-
| |
| |
| |
| |
| |
| | 17/75
| |
| | 431.11
294.667
| | 253.594
173.333
| | 177.516
121.333
| |
|-
| |
| |
| |
| |
| |
| |
| | 29/128
| | 430.912
294.531
| | 252.603
172.626
| | 178.308
121.875
| |
|-
| |
| |
| |
| |
| | 12/53
| |
| |
| | 430.631
294.334
| | 251.202
171.698
| | 179.43
122.6415
| |
|-
| |
| |
| |
| |
| |
| |
| | 31/137
| | 430.369
294.161
| | 249.892
170.803
| | 180.4775
123.358
| |
|-
| |
| |
| |
| |
| |
| | 19/84
| |
| | 430.204
294.048
| | 249.066
170.238
| | 181.139
123.8095
| |
|-
| |
| |
| |
| |
| |
| |
| | 26/115
| | 430.007
293.913
| | 248.081
169.565
| | 181.927
124.348
| |
|-
| |
| |
| |
| | 7/31
| |
| |
| |
| | 429.474
293.548
| | 245.4135
167.742
| | 184.06
125.8065
| |
|-
| |
| |
| |
| |
| |
| |
| | 23/102
| | 428.872
293.137
| | 242.406
165.686
| | 186.466
127.451
| |
|-
| |
| |
| |
| |
| |
| | 16/71
| |
| | 428.6095
292.958
| | 241.093
164.789
| | 187.517
128.169
| |
|-
| |
| |
| |
| |
| |
| |
| | 25/111
| | 428.368
292.793
| | 239.886
163.964
| | 188.482
128.829
| |
|-
| |
| |
| |
| |
| | 9/40
| |
| |
| | 427.94
292.5
| | 237.744
162.5
| | 190.1955
130
| |
|-
| |
| |
| |
| |
| |
| |
| | 20/89
| | 427.406
292.135
| | 235.073
160.674
| | 192.3325
131.461
| |
|-
| |
| |
| |
| |
| |
| | 11/49
| |
| | 426.9695
291.837
| | 232.892
159.184
| | 194.077
132.653
| |
|-
| |
| |
| |
| |
| |
| |
| | 13/58
| | 426.3
291.379
| | 229.546
156.897
| | 196.754
134.483
| |
|-
| | 2/9
| |
| |
| |
| |
| |
| |
| | 422.657
288.889
| colspan="2" | 211.328
144.444
| | Separatrix of Lambda and Anti-Lambda scales
|-
| |
| |
| |
| |
| |
| |
| | 13/59
| | 419.075
286.441
| | 225.656
154.237
| | 193.419
132.203
| |
|-
| |
| |
| |
| |
| |
| | 11/50
| |
| | 418.43
286
| | 228.235
156
| | 190.1955
130
| |
|-
| |
| |
| |
| |
| |
| |
| | 20/91
| | 418.01
285.714
| | 229.907
157.143
| | 188.105
128.571
| |
|-
| |
| |
| |
| |
| | 9/41
| |
| |
| | 417.502
285.365
| | 231.946
158.537
| | 185.557
126.829
| |
|-
| |
| |
| |
| |
| |
| |
| | 25/114
| | 417.095
285.088
| | 233.573
159.649
| | 183.522
125.439
| |
|-
| |
| |
| |
| |
| |
| | 16/73
| |
| | 416.867
284.9315
| | 234.488
160.273
| | 182.379
124.6575
| |
|-
| |
| |
| |
| |
| |
| |
| | 23/105
| | 416.619
284.762
| | 235.48
160.952
| | 181.139
123.8095
| |
|-
| |
| |
| |
| | 7/32
| |
| |
| |
| | 416.053
284.375
| | 237.744
162.5
| | 178.308
121.875
| |
|-
| |
| |
| |
| |
| |
| |
| | 26/119
| | 415.553
284.034
| | 239.742
163.8655
| | 175.811
120.168
| |
|-
| |
| |
| |
| |
| |
| | 19/87
| |
| | 415.3695
283.908
| | 240.477
164.368
| | 174.892
119.54
| |
|-
| |
| |
| |
| |
| |
| |
| | 31/142
| | 415.2155
283.803
| | 241.093
164.789
| | 174.123
119.014
| |
|-
| |
| |
| |
| |
| | 12/55
| |
| |
| | 414.972
283.636
| | 242.067
165.4545
| | 172.905
118.182
| |
|-
| |
| |
| |
| |
| |
| |
| | 29/133
| | 414.712
283.459
| | 243.107
166.165
| | 171.605
117.293
| |
|-
| |
| |
| |
| |
| |
| | 17/78
| |
| | 414.528
283.333
| | 243.84
166.667
| | 170.688
116.667
| |
|-
| |
| |
| |
| |
| |
| |
| | 22/101
| | 414.287
283.168
| | 244.806
167.327
| | 169.481
115.842
| |
|-
| |
| |
| | 5/23
| |
| |
| |
| |
| | 413.4685
282.609
| | 248.081
169.565
| | 165.387
113.0435
| |
|-
| |
| |
| |
| |
| |
| |
| | 23/106
| | 412.688
282.0755
| | 251.202
171.698
| | 161.487
110.377
| |
|-
| |
| |
| |
| |
| |
| | 18/83
| |
| | 412.472
281.928
| | 252.066
172.289
| | 160.406
109.639
| |
|-
| |
| |
| |
| |
| |
| |
| | 31/143
| | 412.312
281.818
| | 252.707
172.727
| | 159.605
109.091
| |
|-
| |
| |
| |
| |
| | 13/60
| |
| |
| | 412.09
281.667
| | 253.594
173.333
| | 158.496
108.333
| |
|-
| |
| |
| |
| |
| |
| |
| | 34/157
| | 411.888
281.529
| | 254.402
173.885
| | 157.487
107.643
| | Golden Anti-Lambda scale is near here
|-
| |
| |
| |
| |
| |
| | 21/97
| |
| | 411.7635
281.443
| | 254.901
174.227
| | 156.862
107.2165
| |
|-
| |
| |
| |
| |
| |
| |
| | 29/134
| | 411.617
281.343
| | 255.4865
174.627
| | 156.131
103.716
| |
|-
| |
| |
| |
| | 8/37
| |
| |
| |
| | 411.2335
281.081.
| | 257.021
175.676
| | 154.2125
105.405
| |
|-
| |
| |
| |
| |
| |
| |
| | 27/125
| | 410.822
280.8
| | 258.666
176.8
| | 152.156
104
| |
|-
| |
| |
| |
| |
| |
| | 19/88
| |
| | 410.649
280.682
| | 259.3575
177.273
| | 151.292
103.409
| |
|-
| |
| |
| |
| |
| |
| |
| | 30/139
| | 410.494
280.5755
| | 259.9795
177.698
| | 150.514
102.878
| |
|-
| |
| |
| |
| |
| | 11/51
| |
| |
| | 410.2255
280.392
| | 261.053
178.431
| | 149.173
101.961
| |
|-
| |
| |
| |
| |
| |
| |
| | 25/116
| | 409.904
280.172
| | 262.339
179.31
| | 147.5655
100.862
| |
|-
| |
| |
| |
| |
| |
| | 14/65
| |
| | 409.652
280
| | 263.348
180
| | 146.304
100
| |
|-
| |
| |
| |
| |
| |
| |
| | 17/79
| | 409.2815
279.747
| | 264.819
181.013
| | 144.452
98.734
| |
|-
| |
| | 3/14
| |
| |
| |
| |
| |
| | 407.562
278.571
| | 271.708
185.714
| | 135.854
92.857
| | Boundary of propriety for Anti-Lambda scale
|-
| |
| |
| |
| |
| |
| |
| | 16/75
| | 405.75
277.333
| | 278.953
190.667
| | 126.797
86.667
| |
|-
| |
| |
| |
| |
| |
| | 13/61
| |
| | 405.335
277.049
| | 280.616
191.803
| | 124.718
85.245
| |
|-
| |
| |
| |
| |
| |
| |
| | 23/108
| | 405.046
276.852
| | 281.771
192.593
| | 123.275
84.259
| |
|-
| |
| |
| |
| |
| | 10/47
| |
| |
| | 404.671
276.596
| | 283.27
193.617
| | 121.401
82.979
| |
|-
| |
| |
| |
| |
| |
| |
| | 27/127
| | 404.353
276.378
| | 284.544
194.488
| | 119.808
81.89
| |
|-
| |
| |
| |
| |
| |
| | 17/80
| |
| | 404.165
276.25
| | 285.293
195
| | 118.873
81.25
| |
|-
| |
| |
| |
| |
| |
| |
| | 24/113
| | 403.955
276.106
| | 286.135
195.575
| | 117.82
80.531
| |
|-
| |
| |
| |
| | 7/33
| |
| |
| |
| | 403.445
275.758
| | 288.175
196.97
| | 115.27
78.788
| |
|-
| |
| |
| |
| |
| |
| |
| | 25/118
| | 402.957
275.424
| | 290.129
198.305
| | 112.828
77.119
| |
|-
| |
| |
| |
| |
| |
| | 18/85
| |
| | 402.767
275.294
| | 290.887
198.8235
| | 111.88
76.471
| |
|-
| |
| |
| |
| |
| |
| |
| | 29/137
| | 402.604
275.1825
| | 291.5405
199.27
| | 111.063
75.912
| |
|-
| |
| |
| |
| |
| | 11/52
| |
| |
| | 402.337
275
| | 292.6085
200
| | 109.728
75
| |
|-
| |
| |
| |
| |
| |
| |
| | 26/123
| | 402.039
274.797
| | 293.79
200.813
| | 108.241
73.984
| |
|-
| |
| |
| |
| |
| |
| |
| |
| | 402.015
274.78
| | 293.896
200.88
| | 108.118
73.9
| |
|-
| |
| |
| |
| |
| |
| | 15/71
| |
| | 401.8215
274.648
| | 294.669
201.4085
| | 107.152
73.239
| |
|-
| |
| |
| |
| |
| |
| |
| | 19/90
| | 401.524
274.444
| | 295.829
202.222
| | 105.664
72.222
| |
|-
| |
| |
| | 4\19
| |
| |
| |
| |
| | 400.412
273.684
| | 300.309
205.263
| | 100.103
68.421
| |
|-
| |
| |
| |
| |
| |
| |
| |
| | 399.692
273.193
| | 303.1855
207.2295
| | 96.507
65.963
| |
|-
| |
| |
| |
| |
| |
| |
| | 17/81
| | 399.176
272.8395
| | 305.252
208.642
| | 93.924
64.1975
| |
|-
| |
| |
| |
| |
| |
| | 13/62
| |
| | 398.797
272.551
| | 306.767
209.677
| | 92.03
62.903
| |
|-
| |
| |
| |
| |
| |
| |
| | 22/105
| | 398.505
272.381
| | 307.936
210.476
| | 90.569
61.905
| |
|-
| |
| |
| |
| |
| | 9/43
| |
| |
| | 398.084
272.093
| | 309.621
211.628
| | 88.463
60.465
| |
|-
| |
| |
| |
| |
| |
| |
| | 23/110
| | 397.6815
271.818
| | 311.229
212.727
| | 86.4525
59.091
| |
|-
| |
| |
| |
| |
| |
| | 14/67
| |
| | 397.423
271.641
| | 312.261
213.433
| | 85.162
58.209
| |
|-
| |
| |
| |
| |
| |
| |
| | 19/91
| | 397.1115
271.429
| | 313.509
214.286
| | 83.602
57.143
| |
|-
| |
| |
| |
| | 5/24
| |
| |
| |
| | 396.241
270.833
| | 316.9925
216.667
| | 79.248
54.167
| |
|-
| |
| |
| |
| |
| |
| |
| | 16/77
| | 395.211
270.13
| | 321.109
219.4805
| | 74.102
50.649
| |
|-
| |
| |
| |
| |
| |
| | 11/53
| |
| | 394.745
269.811
| | 322.9735
220.755
| | 71.772
49.057
| |
|-
| |
| |
| |
| |
| |
| |
| | 17/82
| | 394.308
269.8512
| | 324.724
221.951
| | 69.584
47.561
| |
|-
| |
| |
| |
| |
| | 6/29
| |
| |
| | 393.508
268.9655
| | 327.923
224.138
| | 65.585
44.8275
| |
|-
| |
| |
| |
| |
| |
| |
| | 13/63
| | 392.467
265.254
| | 332.087
226.984
| | 60.3795
41.27
| |
|-
| |
| |
| |
| |
| |
| | 7/34
| |
| | 391.579
267.647
| | 335.639
229.412
| | 55.94
38.235
| |
|-
| |
| |
| |
| |
| |
| |
| | 8/39
| | 390.145
266.667
| | 341.3765
233.333
| | 48.768
33.333
| |
|-
| | 1/5
| |
| |
| |
| |
| |
| |
| colspan="2" | 380.391
260
| | 0
| |
|}


[[Category:Scales]]
[[Category:Scales]]
Retrieved from "https://en.xen.wiki/w/4L_5s_(3/1-equivalent)"