5L 2s

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Scale structure
Step pattern	LLLsLLs sLLsLLL
Equave	2/1 (1200.0 ¢)
Period	2/1 (1200.0 ¢)
Generator size
Bright	4\7 to 3\5 (685.7 ¢ to 720.0 ¢)
Dark	2\5 to 3\7 (480.0 ¢ to 514.3 ¢)
TAMNAMS information
Name	diatonic
Prefix	dia-
Abbrev.	dia
Related MOS scales
Parent	2L 3s
Sister	2L 5s
Daughters	7L 5s, 5L 7s
Neutralized	3L 4s
2-Flought	12L 2s, 5L 9s
Equal tunings
Equalized (L:s = 1:1)	4\7 (685.7 ¢)
Supersoft (L:s = 4:3)	15\26 (692.3 ¢)
Soft (L:s = 3:2)	11\19 (694.7 ¢)
Semisoft (L:s = 5:3)	18\31 (696.8 ¢)
Basic (L:s = 2:1)	7\12 (700.0 ¢)
Semihard (L:s = 5:2)	17\29 (703.4 ¢)
Hard (L:s = 3:1)	10\17 (705.9 ¢)
Superhard (L:s = 4:1)	13\22 (709.1 ¢)
Collapsed (L:s = 1:0)	3\5 (720.0 ¢)
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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	Cyclic order	Step pattern	Mode names
6|0	1	LLLsLLs	Lydian
5|1	5	LLsLLLs	Ionian (major)
4|2	2	LLsLLsL	Mixolydian
3|3	6	LsLLLsL	Dorian
2|4	3	LsLLsLL	Aeolian (minor)
1|5	7	sLLLsLL	Phrygian
0|6	4	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)

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 5L 2s

Generator(edo)								Cents		Step ratio		Comments
								Bright	Dark	L:s	Hardness
4\7								685.714	514.286	1:1	1.000	Equalized 5L 2s
							31\54	688.889	511.111	8:7	1.143
						27\47		689.362	510.638	7:6	1.167
							50\87	689.655	510.345	13:11	1.182
					23\40			690.000	510.000	6:5	1.200
							65\113	690.265	509.735	17:14	1.214
						42\73		690.411	509.589	11:9	1.222
							61\106	690.566	509.434	16:13	1.231
				19\33				690.909	509.091	5:4	1.250
							72\125	691.200	508.800	19:15	1.267
						53\92		691.304	508.696	14:11	1.273
							87\151	691.391	508.609	23:18	1.278
					34\59			691.525	508.475	9:7	1.286
							83\144	691.667	508.333	22:17	1.294
						49\85		691.765	508.235	13:10	1.300
							64\111	691.892	508.108	17:13	1.308
			15\26					692.308	507.692	4:3	1.333	Supersoft 5L 2s
							71\123	692.683	507.317	19:14	1.357
						56\97		692.784	507.216	15:11	1.364
							97\168	692.857	507.143	26:19	1.368
					41\71			692.958	507.042	11:8	1.375
							108\187	693.048	506.952	29:21	1.381
						67\116		693.103	506.897	18:13	1.385
							93\161	693.168	506.832	25:18	1.389
				26\45				693.333	506.667	7:5	1.400
							89\154	693.506	506.494	24:17	1.412
						63\109		693.578	506.422	17:12	1.417
							100\173	693.642	506.358	27:19	1.421
					37\64			693.750	506.250	10:7	1.429
							85\147	693.878	506.122	23:16	1.438
						48\83		693.976	506.024	13:9	1.444
							59\102	694.118	505.882	16:11	1.455
		11\19						694.737	505.263	3:2	1.500	Soft 5L 2s
							62\107	695.327	504.673	17:11	1.545
						51\88		695.455	504.545	14:9	1.556
							91\157	695.541	504.459	25:16	1.562
					40\69			695.652	504.348	11:7	1.571
							109\188	695.745	504.255	30:19	1.579
						69\119		695.798	504.202	19:12	1.583
							98\169	695.858	504.142	27:17	1.588
				29\50				696.000	504.000	8:5	1.600
							105\181	696.133	503.867	29:18	1.611
						76\131		696.183	503.817	21:13	1.615
							123\212	696.226	503.774	34:21	1.619
					47\81			696.296	503.704	13:8	1.625
							112\193	696.373	503.627	31:19	1.632
						65\112		696.429	503.571	18:11	1.636
							83\143	696.503	503.497	23:14	1.643
			18\31					696.774	503.226	5:3	1.667	Semisoft 5L 2s
							79\136	697.059	502.941	22:13	1.692
						61\105		697.143	502.857	17:10	1.700
							104\179	697.207	502.793	29:17	1.706
					43\74			697.297	502.703	12:7	1.714
							111\191	697.382	502.618	31:18	1.722
						68\117		697.436	502.564	19:11	1.727
							93\160	697.500	502.500	26:15	1.733
				25\43				697.674	502.326	7:4	1.750
							82\141	697.872	502.128	23:13	1.769
						57\98		697.959	502.041	16:9	1.778
							89\153	698.039	501.961	25:14	1.786
					32\55			698.182	501.818	9:5	1.800
							71\122	698.361	501.639	20:11	1.818
						39\67		698.507	501.493	11:6	1.833
							46\79	698.734	501.266	13:7	1.857
	7\12							700.000	500.000	2:1	2.000	Basic 5L 2s Scales with tunings softer than this are proper
							45\77	701.299	498.701	13:6	2.167
						38\65		701.538	498.462	11:5	2.200
							69\118	701.695	498.305	20:9	2.222
					31\53			701.887	498.113	9:4	2.250
							86\147	702.041	497.959	25:11	2.273
						55\94		702.128	497.872	16:7	2.286
							79\135	702.222	497.778	23:10	2.300
				24\41				702.439	497.561	7:3	2.333
							89\152	702.632	497.368	26:11	2.364
						65\111		702.703	497.297	19:8	2.375
							106\181	702.762	497.238	31:13	2.385
					41\70			702.857	497.143	12:5	2.400
							99\169	702.959	497.041	29:12	2.417
						58\99		703.030	496.970	17:7	2.429
							75\128	703.125	496.875	22:9	2.444
			17\29					703.448	496.552	5:2	2.500	Semihard 5L 2s
							78\133	703.759	496.241	23:9	2.556
						61\104		703.846	496.154	18:7	2.571
							105\179	703.911	496.089	31:12	2.583
					44\75			704.000	496.000	13:5	2.600
							115\196	704.082	495.918	34:13	2.615
						71\121		704.132	495.868	21:8	2.625
							98\167	704.192	495.808	29:11	2.636
				27\46				704.348	495.652	8:3	2.667
							91\155	704.516	495.484	27:10	2.700
						64\109		704.587	495.413	19:7	2.714
							101\172	704.651	495.349	30:11	2.727
					37\63			704.762	495.238	11:4	2.750
							84\143	704.895	495.105	25:9	2.778
						47\80		705.000	495.000	14:5	2.800
							57\97	705.155	494.845	17:6	2.833
		10\17						705.882	494.118	3:1	3.000	Hard 5L 2s
							53\90	706.667	493.333	16:5	3.200
						43\73		706.849	493.151	13:4	3.250
							76\129	706.977	493.023	23:7	3.286
					33\56			707.143	492.857	10:3	3.333
							89\151	707.285	492.715	27:8	3.375
						56\95		707.368	492.632	17:5	3.400
							79\134	707.463	492.537	24:7	3.429
				23\39				707.692	492.308	7:2	3.500
							82\139	707.914	492.086	25:7	3.571
						59\100		708.000	492.000	18:5	3.600
							95\161	708.075	491.925	29:8	3.625
					36\61			708.197	491.803	11:3	3.667
							85\144	708.333	491.667	26:7	3.714
						49\83		708.434	491.566	15:4	3.750
							62\105	708.571	491.429	19:5	3.800
			13\22					709.091	490.909	4:1	4.000	Superhard 5L 2s
							55\93	709.677	490.323	17:4	4.250
						42\71		709.859	490.141	13:3	4.333
							71\120	710.000	490.000	22:5	4.400
					29\49			710.204	489.796	9:2	4.500
							74\125	710.400	489.600	23:5	4.600
						45\76		710.526	489.474	14:3	4.667
							61\103	710.680	489.320	19:4	4.750
				16\27				711.111	488.889	5:1	5.000
							51\86	711.628	488.372	16:3	5.333
						35\59		711.864	488.136	11:2	5.500
							54\91	712.088	487.912	17:3	5.667
					19\32			712.500	487.500	6:1	6.000
							41\69	713.043	486.957	13:2	6.500
						22\37		713.514	486.486	7:1	7.000
							25\42	714.286	485.714	8:1	8.000
3\5								720.000	480.000	1:0	→ ∞	Collapsed 5L 2s

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