5L 2s

One way of distinguishing the "diatonic" scale is by considering it a moment of symmetry scale produced by a chain of "fifths" (or "fourths"). 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 = 1 1 1 1 1

A continuum of temperaments

So if 4\7 (four degrees of 7edo) is at one extreme and 3\5 (three 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 (i. e .adding them together as if you would be adding the complex numbers analogous real and imaginary parts). Thus, between 4\7 and 3\5 you have (4+3)\(7+5) = 7\12, seven degrees of 12edo:

4\7
	7\12
3\5

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

generator																cents	L	s	L/s	comments
4\7																685.714	1	1	1.000
															63\110	687.273	16	15	1.067
														59\103		687.379	15	14	1.071
													55\96			687.500	14	13	1.077
												51\89				687.640	13	12	1.083
											47\82					687.805	12	11	1.091
										43\75						688.000	11	10	1.100
									39\68							688.235	10	9	1.111
								35\61								688.525	9	8	1.125
							31\54									688.889	8	7	1.143
						27\47										689.362	7	6	1.167
					23\40											690.000	6	5	1.200
				19\33												690.909	5	4	1.250
					34\59											691.525	9	7	1.286
			15\26													692.308	4	3	1.333
					41\71											692.958	11	8	1.375
				26\45												693.333	7	5	1.400
					37\64											693.750	10	7	1.429
		11\19														694.737	3	2	1.500	Optimum rank range (L/s=3/2) diatonic
						51\88										695.455	14	9	1.556
																695.644	π	2	1.571	LucyTuning
					40\69											695.652	11	7	1.571
				29\50												696.000	8	5	1.600
						76\131										696.183	21	13	1.615
								199\343								696.210	55	34	1.618
																696.215	φ	1	1.618	Golden meantone
									322\555							696.216	89	55	1.618
							123\212									696.226	34	21	1.619
					47\81											696.296	13	8	1.625
			18\31													696.774	5	3	1.667	Meantone is in this region
					43\74											697.297	12	7	1.714
																697.487	√3	1	1.732
				25\43												697.674	7	4	1.750
					32\55											698.182	9	5	1.800
						39\67										698.507	11	6	1.833
							46\79									698.734	13	7	1.857
								53\91								698.901	15	8	1.875
									60\103							699.029	17	9	1.889
	7\12															700.000	2	1	2.000	Boundary of propriety (generators smaller than this are proper)
									59\101							700.990	17	8	2.125
								52\89								701.124	15	7	2.143
							45\77									701.299	13	6	2.167
						38\65										701.539	11	5	2.200
					31\53											701.887	9	4	2.250
																701.955			2.260	Pythagorean (g = 3/2 ; L=9/8 ; s=256/243)
				24\41												702.409	7	3	2.333
					41\70											702.857	12	5	2.400
			17\29													703.448	5	2	2.500
					44\75											704.000	13	5	2.600
							115\196									704.082	34	13	2.615
								186\317								704.101	55	21	2.619
						71\121										704.132	21	8	2.625
				27\46												704.348	8	3	2.667
																704.607	e	1	2.718
					37\63											704.762	11	4	2.750
						47\80										705.000	14	5	2.800
		10\17														705.882	3	1	3.000
																706.447	π	1	3.142
					33\56											707.143	10	3	3.333
				23\39												707.692	7	2	3.500
					36\61											708.197	11	3	3.667
			13\22													709.091	4	1	4.000	(No-5's) superpyth is in this region
					29\49											710.204	9	2	4.500
				16\27												711.111	5	1	5.000
					19\32											712.500	6	1	6.000
						22\37										713.514	7	1	7.000
							25\42									714.286	8	1	8.000
								28\47								714.894	9	1	9.000
									31\52							715.385	10	1	10.000
										34\57						715.790	11	1	11.000
											37\62					716.129	12	1	12.000
												40\67				716.418	13	1	13.000
													43\72			716.667	14	1	14.000
														46\77		716.883	15	1	15.000
															49\82	717.073	16	1	16.000
3\5																720.000	1	0	-> inf

Temperaments above 7\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 11\19) and 1/4-comma (close to 18\31). As these tunings approach 4\7, the majors become flatter and the minors become sharper.

Temperaments below 7\12 on this chart are called "positive temperaments" and they include Pythagorean tuning itself (well approximated by 31\53) as well as superpyth temperaments such as 10\17 and 13\22. As these tunings approach 3\5, the majors become sharper and the minors become flatter. Around 13\22 through 16\27, the thirds fall closer to 7-limit than 5-limit intervals: 7:6 and 9:7 as opposed to 6:5 and 5:4.

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.