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

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| Pattern = LsLsLsLss
| Pattern = LsLsLsLss
}}
}}
{{MOS intro}}
Suggested for use as a "diatonic scale" when playing Bohlen-Pierce is the 9-note Lambda scale, which is the 4L5s MOS with [[equave]] 3/1. This can be thought of as an MOS generated by a 3.5.7 rank-2 temperament called BPS (Bohlen-Pierce-Stearns) that eliminates only the comma 245/243, so that 9/7 * 9/7 = 5/3.
Suggested for use as a "diatonic scale" when playing Bohlen-Pierce is the 9-note Lambda scale, which is the 4L5s MOS with [[equave]] 3/1. This can be thought of as an MOS generated by a 3.5.7 rank-2 temperament called BPS (Bohlen-Pierce-Stearns) that eliminates only the comma 245/243, so that 9/7 * 9/7 = 5/3.


Line 17: Line 19:


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.
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.
== Modes ==
{{Mode namee}}s


== List of EDT's supporting Lambda Temperament ==
== List of EDT's supporting Lambda Temperament ==
Line 23: Line 28:
Below is a list of the equal-temperaments which contain a 4L+5s scale using generators between 422.7 cents and 475.5 cents.
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
{{Scale tree|depth=7}}
 
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.
*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.]
[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:Nonoctave]]
[[Category:Nonoctave]]
[[Category:9-tone scales]]
[[Category:9-tone scales]]

Revision as of 00:57, 25 June 2023

↖ 3L 4s⟨3/1⟩ ↑ 4L 4s⟨3/1⟩ 5L 4s⟨3/1⟩ ↗
← 3L 5s⟨3/1⟩ 4L 5s (3/1-equivalent) 5L 5s⟨3/1⟩ →
↙ 3L 6s⟨3/1⟩ ↓ 4L 6s⟨3/1⟩ 5L 6s⟨3/1⟩ ↘
Scale structure
Step pattern LsLsLsLss
ssLsLsLsL
Equave 3/1 (1902.0 ¢)
Period 3/1 (1902.0 ¢)
Generator size(edt)
Bright 2\9 to 1\4 (422.7 ¢ to 475.5 ¢)
Dark 3\4 to 7\9 (1426.5 ¢ to 1479.3 ¢)
Related MOS scales
Parent 4L 1s⟨3/1⟩
Sister 5L 4s⟨3/1⟩
Daughters 9L 4s⟨3/1⟩, 4L 9s⟨3/1⟩
Neutralized 8L 1s⟨3/1⟩
2-Flought 13L 5s⟨3/1⟩, 4L 14s⟨3/1⟩
Equal tunings(edt)
Equalized (L:s = 1:1) 2\9 (422.7 ¢)
Supersoft (L:s = 4:3) 7\31 (429.5 ¢)
Soft (L:s = 3:2) 5\22 (432.3 ¢)
Semisoft (L:s = 5:3) 8\35 (434.7 ¢)
Basic (L:s = 2:1) 3\13 (438.9 ¢)
Semihard (L:s = 5:2) 7\30 (443.8 ¢)
Hard (L:s = 3:1) 4\17 (447.5 ¢)
Superhard (L:s = 4:1) 5\21 (452.8 ¢)
Collapsed (L:s = 1:0) 1\4 (475.5 ¢)
View • Talk • Edit

4L 5s⟨3/1⟩ is a 3/1-equivalent (tritave-equivalent) moment of symmetry scale containing 4 large steps and 5 small steps, repeating every interval of 3/1 (1902.0 ¢). Generators that produce this scale range from 422.7 ¢ to 475.5 ¢, or from 1426.5 ¢ to 1479.3 ¢.

Suggested for use as a "diatonic scale" when playing Bohlen-Pierce is the 9-note Lambda scale, which is the 4L5s MOS with equave 3/1. This can be thought of as an MOS generated by a 3.5.7 rank-2 temperament called BPS (Bohlen-Pierce-Stearns) 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.

Modes

Template:Mode namees

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.

Template: Scale tree is deprecated. Please use Template: MOS tuning spectrum instead. Details:
Use of a single Comments parameter has become unmaintainable. Existing scale trees should be migrated to the new template, where comments are entered using a step ratio p/q as a parameter: 
{{MOS tuning spectrum
| 3/2 = Example comment
| 4/3 = Another example comment
}}


The parameters tuning and depth have been replaced with Scale Signature and Depth, respectively.


Scale tree and tuning spectrum of 4L 5s⟨3/1⟩
Generator(edt) Cents Step ratio Comments
Bright Dark L:s Hardness
2\9 422.657 1479.298 1:1 1.000 Equalized 4L 5s⟨3/1⟩
15\67 425.811 1476.144 8:7 1.143
13\58 426.300 1475.655 7:6 1.167
24\107 426.607 1475.348 13:11 1.182
11\49 426.969 1474.986 6:5 1.200
31\138 427.251 1474.704 17:14 1.214
20\89 427.406 1474.549 11:9 1.222
29\129 427.571 1474.384 16:13 1.231
9\40 427.940 1474.015 5:4 1.250
34\151 428.255 1473.700 19:15 1.267
25\111 428.368 1473.587 14:11 1.273
41\182 428.462 1473.493 23:18 1.278
16\71 428.610 1473.345 9:7 1.286
39\173 428.764 1473.191 22:17 1.294
23\102 428.872 1473.083 13:10 1.300
30\133 429.012 1472.943 17:13 1.308
7\31 429.474 1472.481 4:3 1.333 Supersoft 4L 5s⟨3/1⟩
33\146 429.894 1472.061 19:14 1.357
26\115 430.007 1471.948 15:11 1.364
45\199 430.090 1471.865 26:19 1.368
19\84 430.204 1471.751 11:8 1.375
50\221 430.307 1471.648 29:21 1.381
31\137 430.369 1471.586 18:13 1.385
43\190 430.442 1471.513 25:18 1.389
12\53 430.631 1471.324 7:5 1.400
41\181 430.830 1471.125 24:17 1.412
29\128 430.912 1471.043 17:12 1.417
46\203 430.985 1470.970 27:19 1.421
17\75 431.110 1470.845 10:7 1.429
39\172 431.257 1470.698 23:16 1.438
22\97 431.371 1470.584 13:9 1.444
27\119 431.536 1470.419 16:11 1.455
5\22 432.263 1469.693 3:2 1.500 Soft 4L 5s⟨3/1⟩
28\123 432.965 1468.990 17:11 1.545
23\101 433.118 1468.837 14:9 1.556
41\180 433.223 1468.732 25:16 1.562
18\79 433.357 1468.598 11:7 1.571
49\215 433.469 1468.486 30:19 1.579
31\136 433.534 1468.421 19:12 1.583
44\193 433.606 1468.349 27:17 1.588
13\57 433.779 1468.176 8:5 1.600
47\206 433.941 1468.014 29:18 1.611
34\149 434.003 1467.952 21:13 1.615
55\241 434.056 1467.899 34:21 1.619
21\92 434.142 1467.813 13:8 1.625
50\219 434.236 1467.719 31:19 1.632
29\127 434.305 1467.650 18:11 1.636
37\162 434.397 1467.558 23:14 1.643
8\35 434.733 1467.222 5:3 1.667 Semisoft 4L 5s⟨3/1⟩
35\153 435.088 1466.867 22:13 1.692
27\118 435.193 1466.762 17:10 1.700
46\201 435.273 1466.682 29:17 1.706
19\83 435.387 1466.568 12:7 1.714
49\214 435.494 1466.461 31:18 1.722
30\131 435.562 1466.393 19:11 1.727
41\179 435.643 1466.312 26:15 1.733
11\48 435.865 1466.090 7:4 1.750
36\157 436.117 1465.838 23:13 1.769
25\109 436.228 1465.727 16:9 1.778
39\170 436.331 1465.624 25:14 1.786
14\61 436.514 1465.441 9:5 1.800
31\135 436.745 1465.210 20:11 1.818
17\74 436.936 1465.019 11:6 1.833
20\87 437.231 1464.724 13:7 1.857
3\13 438.913 1463.042 2:1 2.000 Basic 4L 5s⟨3/1⟩
Scales with tunings softer than this are proper
19\82 440.697 1461.258 13:6 2.167
16\69 441.033 1460.922 11:5 2.200
29\125 441.254 1460.701 20:9 2.222
13\56 441.525 1460.430 9:4 2.250
36\155 441.744 1460.211 25:11 2.273
23\99 441.868 1460.087 16:7 2.286
33\142 442.004 1459.951 23:10 2.300
10\43 442.315 1459.640 7:3 2.333
37\159 442.593 1459.362 26:11 2.364
27\116 442.696 1459.259 19:8 2.375
44\189 442.783 1459.172 31:13 2.385
17\73 442.921 1459.034 12:5 2.400
41\176 443.069 1458.886 29:12 2.417
24\103 443.174 1458.781 17:7 2.429
31\133 443.313 1458.642 22:9 2.444
7\30 443.790 1458.166 5:2 2.500 Semihard 4L 5s⟨3/1⟩
32\137 444.252 1457.703 23:9 2.556
25\107 444.382 1457.573 18:7 2.571
43\184 444.479 1457.476 31:12 2.583
18\77 444.613 1457.342 13:5 2.600
47\201 444.736 1457.219 34:13 2.615
29\124 444.812 1457.143 21:8 2.625
40\171 444.902 1457.053 29:11 2.636
11\47 445.138 1456.817 8:3 2.667
37\158 445.395 1456.560 27:10 2.700
26\111 445.503 1456.452 19:7 2.714
41\175 445.601 1456.354 30:11 2.727
15\64 445.771 1456.184 11:4 2.750
34\145 445.976 1455.979 25:9 2.778
19\81 446.138 1455.817 14:5 2.800
23\98 446.377 1455.578 17:6 2.833
4\17 447.519 1454.436 3:1 3.000 Hard 4L 5s⟨3/1⟩
21\89 448.776 1453.179 16:5 3.200
17\72 449.073 1452.882 13:4 3.250
30\127 449.281 1452.674 23:7 3.286
13\55 449.553 1452.402 10:3 3.333
35\148 449.787 1452.168 27:8 3.375
22\93 449.925 1452.030 17:5 3.400
31\131 450.081 1451.874 24:7 3.429
9\38 450.463 1451.492 7:2 3.500
32\135 450.834 1451.121 25:7 3.571
23\97 450.979 1450.976 18:5 3.600
37\156 451.105 1450.850 29:8 3.625
14\59 451.311 1450.644 11:3 3.667
33\139 451.543 1450.412 26:7 3.714
19\80 451.714 1450.241 15:4 3.750
24\101 451.950 1450.005 19:5 3.800
5\21 452.846 1449.109 4:1 4.000 Superhard 4L 5s⟨3/1⟩
21\88 453.876 1448.079 17:4 4.250
16\67 454.198 1447.757 13:3 4.333
27\113 454.449 1447.506 22:5 4.400
11\46 454.815 1447.140 9:2 4.500
28\117 455.169 1446.786 23:5 4.600
17\71 455.398 1446.557 14:3 4.667
23\96 455.677 1446.278 19:4 4.750
6\25 456.469 1445.486 5:1 5.000
19\79 457.432 1444.523 16:3 5.333
13\54 457.878 1444.077 11:2 5.500
20\83 458.302 1443.653 17:3 5.667
7\29 459.093 1442.862 6:1 6.000
15\62 460.150 1441.805 13:2 6.500
8\33 461.080 1440.875 7:1 7.000
9\37 462.638 1439.317 8:1 8.000
1\4 475.489 1426.466 1:0 → ∞ Collapsed 4L 5s⟨3/1⟩
  • 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.]

Retrieved from "https://en.xen.wiki/index.php?title=4L_5s_(3/1-equivalent)&oldid=117581"