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	L	s	L/s	Comments
4\7	533.333¢	1	1	1.000
27\47	531.148¢	7	6	1.167
23\40	530.769¢	6	5	1.200
19\33	530.232¢	5	4	1.250
15\26	529.411¢	4	3	1.333
11\19	528.000¢	3	2	1.500	L/s = 3/2
29\50	527.272¢	8	5	1.600
18\31	526.829¢	5	3	1.667	3/4 Meantone is in this region
25\43	526.316¢	7	4	1.750
32\55	526.028¢	9	5	1.800
39\67	525.843¢	11	6	1.833
7\12	525.000¢	2	1	2.000	Basic 3/4 diatonic (Generators larger than this are proper)
38\65	524.138¢	11	5	2.200
31\53	523.944¢	9	4	2.250
24\41	523.636¢	7	3	2.333
17\29	523.077¢	5	2	2.500
27\46	522.581¢	8	3	2.667	3/4 Neogothic is in this region
10\17	521.739¢	3	1	3.000	L/s = 3/1
13\22	520.000¢	4	1	4.000	3/4 Archy is in this region
29\49	519.403¢	9	2	4.500
16\27	518.919¢	5	1	5.000
19\32	518.182¢	6	1	6.000
22\37	517.647¢	7	1	7.000
3\5	514.286¢	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