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5L 2s - "diatonic"


One way of distinguishing the "diatonic" scale is by considering it a moment of symmetry scale produced by a chain of "fifths". This will include 12edo's diatonic scale along with the Pythagorean diatonic scale and meantone systems, while excluding just intonation scales that use more than one size of "tone".

It may be misleading to call 5L 2s "diatonic," since other scales called diatonic can be arrived at different ways (through just intonation procedures for instance, or with tetrachords). Also, a composer working with a 5L 2s scale may choose to do something very different than typical diatonic music.

substituting step sizes


The 5L 2s MOS scale has this generalized form.
L L s L L L s

Insert 2 for L and 1 for s and you'll get the 12edo diatonic of standard practice.
2 2 1 2 2 2 1

When L=3, s=1, you have 17edo:
3 3 1 3 3 3 1

When L=3, s=2, you have 19edo:
3 3 2 3 3 3 2

When L=4, s=1, you have 22edo:
4 4 1 4 4 4 1

When L=4, s=3, you have 26edo:
4 4 3 4 4 4 3

When L=5, s=1, you have 27edo:
5 5 1 5 5 5 1

When L=5, s=2, you have 29edo:
5 5 2 5 5 5 2

When L=5, s=3, you have 31edo:
5 5 3 5 5 5 3

When L=5, s=4, you have 33edo:
5 5 4 5 5 5 4

So you have scales where L and s are nearly equal, which approach 7edo:
1 1 1 1 1 1 1

And you have scales where s becomes so small it approaches zero, which would give us 5edo:
1 1 0 1 1 1 0 or 1 1 1 1 1

a continuum of temperaments


So if 3\7 (three degrees of 7edo) is at one extreme and 2\5 (two degrees of 5edo) is at the other, all other possible 5L 2s scales exist in a continuum between them. You can chop this continuum up by taking "freshman sums" of the two edges - adding together the numerators, then adding together the denominators. Thus, between 3\7 and 2\5 you have (3+2)\(7+5) = 5\12, five degrees of 12edo:

3\7


5\12
2\5


If we carry this freshman-summing out a little further, new, larger edos pop up in our continuum.
generator

in cents
tetrachord




comments
3\7





514.286
1 1 1
239.2945
274.991
307.521
378.193

59\138





513.0435
20 20 19
238.673
274.370
308.142
378.8145

56\131





512.977
19 19 18
238.640
274.337
308.175
378.848

53\124





512.903
18 18 17
238.603
274.300
308.212
378.885

50\117





512.8205
17 17 16
238.562
274.259
308.2535
378.926

47\110





512.727
16 16 15
238.515
274.212
308.300
378.973

44\103





512.621
15 15 14
238.462
274.159
308.353
379.0255

41\96





512.500
14 14 13
238.402
274.098
308.414
379.086

38\89





512.360
13 13 12
238.331
274.028
308.484
379.156

35\82





512.195
12 12 11
238.249
273.946
308.566
379.239

32\75





512.000
11 11 10
238.152
273.848
308.664
379.336

29\68





511.765
10 10 9
238.034
273.731
308.781
379.454

26\61





511.475
9 9 8
237.889
273.586
308.926
379.5985

23\54





511.111
8 8 7
237.707
273.404
309.108
379.781

20\47





510.638
7 7 6
237.471
273.168
309.345
380.017

17\40





510.000
6 6 5
237.152
272.848
309.664
380.336

14\33





509.091
5 5 4
236.697
272.394
310.118
380.791


25\59




508.475
9 9 7
236.389
272.086
310.4265
381.0985

11\26





507.692
4 4 3
235.998
271.695
310.817
381.491


30\71




507.042
11 11 8
235.672
271.3695
311.142
381.846


19\45




506.667
7 7 5
235.485
271.182
311.33
382.003


27\64




506.250
10 10 7
235.277
270.973
311.539
382.211

8\19





505.263
3 3 2
234.783
270.480
312.032
382.705
Optimum rank range (L/s=3/2) diatonic

37\88




504.5455
14 14 9
234.424
270.121
312.391
383.0635
LucyTuning







504.356
pi pi 2
234.329
270.026
312.486
383.158


29\69




504.348
11 11 7
234.3255
270.022
312.490
383.172


21\50




504.000
8 8 5
234.152
269.848
312.664
383.336



55\131



503.817
21 21 13
234.060
269.757
312.755
383.428





144\343

503.790
55 55 34
234.047
269.743
312.769
383.441






233\555
503.784
89 89 55
234.0435
269.740
312.772
383.444
Golden meantone



89\212


503.774
34 34 21
234.038
269.735
312.777
383.449



34\81



503.704
13 13 8
234.003
269.700
312.811
383.485


13\31




503.226
5 5 3
233.7645
269.461
313.051
383.723
Meantone is in this region


31\74



502.703
12 12 7
233.503
269.200
313.312
383.985








502.5135
√3 √3 1
233.408
269.105
313.407
384.079


18\43




502.326
7 7 4
233.314
269.011
313.501
384.183


23\55




501.818
9 9 5
233.061
268.7575
313.754
384.428

5\12





500.000
2 2 1
232.152
267.848
314.664
385.336
Boundary of propriety
(generators larger than this are proper)

42\101




499.010
17 17 8
231.6565
267.353
315.159
385.831


37\89




498.876
15 15 7
231.590
267.287
315,226
385.898


32\77




498.701
13 13 6
231.502
267.199
315.313
385.986


27\65




498.4615
11 11 5
231.382
267.079
315.433
386.105


22\53




498.113
9 9 4
231.208
266.905
315.609
386.278
Pythagorean is around here

17\41




497.591
7 7 3
230.932
266.629
315.883
386.556


29\70




497.143
12 12 5
230.723
266.420
316.092
386.765


12\29




496.552
5 5 2
230.4275
266.124
316.388
387.061



31\75



496.000
13 13 5
230.152
265.848
316.664
387.336






81\196

495.918
34 34 13
230.111
265.808
316.705
387.377







131\317
495.899
55 55 21
230.101
265.798
316.714
387.387





50\121


495.868
21 21 8
230.0855
265.782
316.73
387.402


19\46




495.652
8 8 3
229.978
265.6745
316.837
387.511







495.393
e e 1
229.848
265.545
316.967
387.639
L/s = e

26\63




495.238
11 11 4
229.771
265.4675
317.045
387.717


7\17





494.118
3 3 1
229.210
264.907
317.596
388.286
L/s = 3






493.553
pi pi 1
228.928
264.625
317.887
388.56
L/s = pi

23\56




492.857
10 10 3
228.580
264.277
318.235
388.908


16\39




492.308
7 7 2
228.305
264.002
318.51
389.182


25\61




491.803
11 11 3
228.053
263.750
318.761
389.436

9\22





490.909
4 4 1
227.606
263.303
319.209
389.882
(No-5's) superpyth is in this region
L/s = 4

20\49




489.796
9 9 2
227.050
262.746
319.766
390.438

11\27





488.889
5 5 1
226.596
262.293
320.219
390.892

13\32





487.500
6 6 1
225.9015
261.598
320.914
391.596

15\37





486.4865
7 7 1
225.395
261.092
321.4205
392.093

17\42





485.714
8 8 1
225.009
260.7055
321.807
392.479

19\47





485.106
9 9 1
224.705
260.402
322.111
392.783

21\52





484.615
10 10 1
224.459
260.156
322.356
393.0285

23\57





484.2105
11 11 1
224.257
259.954
322.5585
393.231

25\62





483.871
12 12 1
224.087
259.784
322.728
393.401

27\67





483.582
13 13 1
223.943
259.6395
322.873
393.545

29\72





483.333
14 14 1
223.818
259.515
322.997
393.6695

31\77





483.117
15 15 1
223.710
259.407
323.105
393.778

33\82





482.927
16 16 1
223.615
259.312
323.200
393.873

35\87





482.759
17 17 1
223.531
259.228
323.2845
393.957

37\92





482.609
18 18 1
223.456
259.153
323.539
394.032

39\97





482.474
19 19 1
223.389
259.0855
323.427
394.099

41\102





482.353
20 20 1
223.328
259.025
323.487
394.160

43\107





482.243
21 21 1
223.273
258.970
323.542
394.215

45\112





482.143
22 22 1
223.223
258.920
323.592
394.265

47\117





482.051
23 23 1
223.177
258.874
323.638
394.311

49\122





481.967
24 24 1
223.135
258.832
322.680
394.353

2\5





480.000
1 1 0
222.152
257.848
324.664
395.336


Temperaments above 5\12 on this chart are called "negative temperaments" (as they lessen the size of the fifth) and include meantone systems such as 1/3-comma (close to 8\19) and 1/4-comma (close to 13\31). As these tunings approach 3\7, the majors become flatter and the minors become sharper.

Temperaments below 5\12 on this chart are called "positive temperaments" and they include Pythagorean tuning itself (well approximated by 22\53) as well as superpyth temperaments such as 7\17 and 9\22. As these tunings approach 2\5, the majors become sharper and the minors become flatter. Around 9\22, the thirds fall closer to 7-limit than 5-limit intervals: 7:6 and 9:7 as opposed to 6:5 and 5:4.

5L2s.jpg

5L 2s contains the pentatonic MOS 2L 3s and (with the sole exception of the 5L 2s of 12edo) is itself contained in a dodecaphonic MOS: either 7L 5s or 5L 7s.