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") with the step combination of 5L 2s. Among the most well-known variants of this MOS proper are 12EDO's diatonic scale along with both the Pythagorean diatonic scale and the various meantone systems. Other similar scales referred to by the term "diatonic" can be arrived at different ways – for example, through just intonation procedures, or with tetrachords. However, it should be noted that at least the majority of the other scales that fall under this category – such as the just intonation scales that use more than one size of whole tone – are actually JI detemperings or tempered approximations of them that both closely resemble and are derived from this MOS.

On the term diatonic

In TAMNAMS (which is the convention on all pages on scale patterns on the wiki), diatonic exclusively refers to 5L 2s. Other diatonic-based scales (specifically with 3 step sizes or more), such as Zarlino, blackdye and diasem, are called detempered (if the philosophy is RTT-based) or deregularized (RTT-agnostic) diatonic scales. The adjectives diatonic-like or diatonic-based may also be used to refer to diatonic-based scales, depending on what's contextually the most appropriate.

Substituting step sizes

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

L L L s L L s

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

2 2 2 1 2 2 1

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

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

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

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

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

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

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

When L=5, s=4, you have 33EDO: 5 5 5 4 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 1 0 1 1 0 = 1 1 1 1 1

Tuning ranges

Parasoft to ultrasoft

"Flattone" systems, such as 26EDO.

Hyposoft

"Meantone" (more properly "septimal meantone") systems, such as 31EDO.

Hypohard

The near-just part of the region is of interest mainly for those interested in Pythagorean tuning and large, accurate EDO systems based on close-to-Pythagorean fifths, such as 41EDO and 53EDO. This class of tunings is called schismic temperament; these tunings can approximate 5-limit harmonies very accurately by tempering out a small comma called the schisma. (Technically, 12EDO tempers out the schisma and thus is a schismic tuning, but it is nowhere near as accurate as schismic tunings can be.)

The sharp-of-just part of this range includes so-called "neogothic" or "parapyth" systems, which tune the diatonic major third slightly sharply of 81/64 (around 414 to 423 cents) and the diatonic minor third slightly flatly of 32/27 (around 282 to 290 cents). Good neogothic EDOs include 29EDO and 46EDO. 17EDO is often considered the sharper end of the neogothic spectrum; its major third at 423 cents is considerably more discordant than in flatter neogothic tunings.

Parahard to ultrahard

"Archy" systems such as 17EDO, 22EDO, and 27EDO.

Modes

Diatonic modes have standard names from classical music theory:

Modes of 5L 2s
UDP	Step pattern	Mode names
6\|0	LLLsLLs	Lydian
5\|1	LLsLLLs	Ionian (major)
4\|2	LLsLLsL	Mixolydian
3\|3	LsLLLsL	Dorian
2\|4	LsLLsLL	Aeolian (minor)
1\|5	sLLLsLL	Phrygian
0\|6	sLLsLLL	Locrian

Scales

Meantone7 – 19edo and 31edo tunings
Nestoria7 – 171edo tuning
Pythagorean7 – Pythagorean tuning
Garibaldi7 – 94edo tuning
Cotoneum7 – 217edo tuning
Pepperoni7 – 271edo tuning
Supra7 – 56edo tuning
Archy7 – 472edo tuning

Scale tree

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.

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

Generator ranges:

Bright generator: 685.7143 cents (4\7) to 720 cents (3\5)
Dark generator: 480 cents (2\5) to 514.2857 cents (3\7)

Bright generator							Cents	L	s	L/s	Comments
4\7							685.714	1	1	1.000
						27\47	689.362	7	6	1.167
					23\40		690.000	6	5	1.200
						42\73	690.411	11	9	1.222
				19\33			690.909	5	4	1.250
						53\92	691.304	14	11	1.273
					34\59		691.525	9	7	1.286
						49\85	691.765	13	10	1.300
			15\26				692.308	4	3	1.333
						56\97	692.784	15	11	1.364
					41\71		692.958	11	8	1.375
						67\116	693.103	18	13	1.385
				26\45			693.333	7	5	1.400	Flattone is in this region
						63\109	693.578	17	12	1.417
					37\64		693.750	10	7	1.429
						48\83	693.976	13	9	1.444
		11\19					694.737	3	2	1.500
						51\88	695.455	14	9	1.556
					40\69		695.652	11	7	1.571
						69\119	695.798	19	12	1.583
				29\50			696.000	8	5	1.600
						66\131	696.183	21	13	1.615	Golden meantone (696.2145¢)
					47\81		696.296	13	8	1.625
						65\112	696.429	18	11	1.636
			18\31				696.774	5	3	1.667	Meantone is in this region
						61\105	697.143	17	10	1.700
					43\74		697.297	12	7	1.714
						68\117	697.436	19	11	1.727
				25\43			697.674	7	4	1.750
						57\98	697.959	16	9	1.778
					32\55		698.182	9	5	1.800
						39\67	698.507	11	6	1.833
	7\12						700.000	2	1	2.000	Basic diatonic (Generators smaller than this are proper)
						38\65	701.539	11	5	2.200
					31\53		701.887	9	4	2.250	The generator closest to a just 3/2 for EDOs less than 200
						55\94	702.128	16	7	2.286	Garibaldi / Cassandra
				24\41			702.409	7	3	2.333
						65\111	702.703	19	8	2.375
					41\70		702.857	12	5	2.400
						58\99	703.030	17	7	2.428
			17\29				703.448	5	2	2.500
						61\104	703.846	18	7	2.571
					44\75		704.000	13	5	2.600
						71\121	704.132	21	8	2.625	Golden neogothic (704.0956¢)
				27\46			704.348	8	3	2.667	Neogothic is in this region
						64\109	704.587	19	7	2.714
					37\63		704.762	11	4	2.750
						47\80	705.000	14	5	2.800
		10\17					705.882	3	1	3.000
						43\73	706.849	13	4	3.250
					33\56		707.143	10	3	3.333
						56\95	707.368	17	5	3.400
				23\39			707.692	7	2	3.500
						59\100	708.000	18	5	3.600
					36\61		708.197	11	3	3.667
						49\83	708.434	15	4	3.750
			13\22				709.091	4	1	4.000	Archy is in this region
						42\71	709.859	13	3	4.333
					29\49		710.204	9	2	4.500
						45\76	710.526	14	3	4.667
				16\27			711.111	5	1	5.000
						35\59	711.864	11	2	5.500
					19\32		712.500	6	1	6.000
						22\37	713.514	7	1	7.000
3\5							720.000	1	0	→ inf

Tunings above 7\12 on this chart are called "negative tunings" (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.

Tunings below 7\12 on this chart are called "positive tunings" and they include Pythagorean tuning itself (well approximated by 31\53) as well as superpyth tunings 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, depending on whether the fifth is flatter than or sharper than 7\12 (700¢).

Related Scales

Main article: 5L 2s MODMOSes and 5L 2s Muddles

Because the diatonic scale is so widely used, it should be no surprise that there are a number of noteworthy scales of different sorts related to this MOS.

Rank-2 temperaments

Main article: 5L 2s/Temperaments

Approaches to Functional Harmony

See also: Diatonic functional harmony