5L 2s

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.