User:Moremajorthanmajor/5L 2s (major sixth-equivalent)

One way of distinguishing the 3/4 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 12EDS'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 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 17EDS: 3 3 1 3 3 3 1

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

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

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

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

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

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

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

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

1 1 1 1 1 1 1

And you have scales where s becomes so small it approaches zero, which would give us 5EDS:

1 1 0 1 1 1 0 = 1 1 1 1 1

Tuning ranges

Parasoft to ultrasoft

"3/4 Flattone" systems, such as 26EDS.

Hyposoft

"3/4 Meantone" (more properly "3/4 septimal meantone") systems, such as 31EDS.

Hypohard

The near-just part of the region is of interest mainly for those interested in “3/4” Pythagorean tuning and large, accurate eds systems based on close-to-Pythagorean fifths, such as 41EDS and 53EDS. This class of tunings is called trischismic temperament; these tunings can approximate 5^3/4-limit harmonies very accurately by tempering out a small comma called the schisma. (Technically, 12EDS 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 “3/4 neogothic" or "3/4 parapyth" systems, which tune the diatonic major third slightly flatly of 6/5 and the diatonic minor third slightly sharply of 12/11. Good 3/4 neogothic EDSs include 29EDS and 46EDS. 17EDS is often considered the sharper end of the 3/4 neogothic spectrum; its major third at 313 cents (417 śata) is considerably more concordant than in flatter neogothic tunings.

Parahard to ultrahard

"3/4 Archy" systems such as 17EDS, 22EDS, and 27EDS.

Modes

Diatonic modes have standard names from classical music theory:

Mode	UDP	Name
LLLsLLs	6\|0	Lydian
LLsLLLs	5\|1	Ionian
LLsLLsL	4\|2	Mixolydian
LsLLLsL	3\|3	Dorian
LsLLsLL	2\|4	Aeolian
sLLLsLL	1\|5	Phrygian
sLLsLLL	0\|6	Locrian

Scales

Scale tree

If 4\7 (four degrees of 7EDS) is at one extreme and 3\5 (three degrees of 5EDS) 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 12EDS:

4\7
	7\12
3\5

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

Generator							Cents	Śata	L	s	L/s	Comments
4\7							533.333	711.111	1	1	1.000
						27\47	531.1475	708.196	7	6	1.167
					23\40		530.769	707.692	6	5	1.200
						42\73	530.526	707.368	11	9	1.222
				19\33			530.232	706.977	5	4	1.250
						53\92	530.000	706.667	14	11	1.273
					34\59		529.870	706.4935	9	7	1.286
						49\85	529.730	706.306	13	10	1.300
			15\26				529.411	705.882	4	3	1.333
						56\97	529.134	705.512	15	11	1.364
					41\71		529.032	705.376	11	8	1.375
						67\116	528.947	705.263	18	13	1.385
				26\45			528.814	705.085	7	5	1.400
						63\109	528.617	704.895	17	12	1.417
					37\64		528.571	704.762	10	7	1.429
						48\83	528.440	704.587	13	9	1.444
		11\19					528.000	704.000	3	2	1.500	L/s = 3/2
						51\88	527.586	703.448	14	9	1.556
					40\69		527.4725	703.297	11	7	1.571
						69\119	527.3885	703.185	19	12	1.583
				29\50			527.272	703.030	8	5	1.600
						76\131	527.168	702.890	21	13	1.615	Golden 3/4 meantone
					47\81		527.103	702.804	13	8	1.625
						65\112	527.027	702.703	18	11	1.636
			18\31				526.829	702.409	5	3	1.667	3/4 Meantone is in this region
						61\105	526.619	702.158	17	10	1.700
					43\74		526.531	702.041	12	7	1.714
						68\117	526.452	701.9355	19	11	1.727	The generator closest to 3/4 of a just 3/2 for EDSs less than 200
				25\43			526.316	701.754	7	4	1.750
						57\98	526.154	701.5385	16	9	1.778
					32\55		526.028	701.370	9	5	1.800
						39\67	525.843	701.124	11	6	1.833
	7\12						525.000	700.000	2	1	2.000	Basic 3/4 diatonic (Generators larger than this are proper)
						38\65	524.138	698.851	11	5	2.200
					31\53		523.944	698.5915	9	4	2.250
						55\94	523.8095	698.413	16	7	2.286
				24\41			523.636	698.182	7	3	2.333
						65\111	523.490	697.987	19	8	2.375
					41\70		523.404	697.872	12	5	2.400
						58\99	523.308	697.744	17	7	2.428
			17\29				523.077	697.436	5	2	2.500
						61\104	522.857	697.142	18	7	2.571
					44\75		522.772	697.030	13	5	2.600
						71\121	522.699	696.9325	21	8	2.625	Golden 3/4 neogothic
				27\46			522.581	696.774	8	3	2.667	3/4 Neogothic is in this region
						64\109	522.449	696.599	19	7	2.714
					37\63		522.352	696.4705	11	4	2.750
						47\80	522.222	696.296	14	5	2.800
		10\17					521.739	695.652	3	1	3.000	L/s = 3/1
						43\73	521.212	694.9495	13	4	3.250
					33\56		521.053	694.737	10	3	3.333
						56\95	520.930	694.574	17	5	3.400
				23\39			520.755	694.340	7	2	3.500
						59\100	520.588	694.118	18	5	3.600
					36\61		520.482	693.976	11	3	3.667
						49\83	520.354	693.805	15	4	3.750
			13\22				520.000	693.333	4	1	4.000	3/4 Archy is in this region
						42\71	519.588	692.7835	13	3	4.333
					29\49		519.403	692.537	9	2	4.500
						45\76	519.231	692.308	14	3	4.667
				16\27			518.919	691.892	5	1	5.000
						35\59	518.5185	691.358	11	2	5.500
					19\32		518.182	690.909	6	1	6.000
						22\37	517.647	690.196	7	1	7.000
3\5							514.286	685.714	1	0	→ inf

Tunings above 7\12 on this chart are called "positive tunings" (as they greaten the size of the fifth) and include 3/4 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 "negative tunings" and they include 3/4 Pythagorean tuning itself (well approximated by 31\53) as well as 3/4 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 10\17 through 13\22, the thirds fall closer to 5-limit than 7-limit intervals: 6:5 as opposed to 7:6.

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 (675¢, 700$).

Related Scales

Main article: 5L 2s (major sixth equivalent) MODMOSes
and 5L 2s (major sixth equivalent) 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 (major sixth equivalent)/Temperaments