User:Moremajorthanmajor/5L 2s (5/3-equivalent)

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↖ 4L 1s⟨5/3⟩↑ 5L 1s⟨5/3⟩ 6L 1s⟨5/3⟩ ↗
← 4L 2s⟨5/3⟩5L 2s<5/3>6L 2s⟨5/3⟩ →
↙ 4L 3s⟨5/3⟩↓ 5L 3s⟨5/3⟩ 6L 3s⟨5/3⟩ ↘
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Scale structure
Step pattern LLLsLLs
sLLsLLL
Equave 5/3 (884.4¢)
Period 5/3 (884.4¢)
Generator size(ed5/3)
Bright 4\7 to 3\5 (505.3¢ to 530.6¢)
Dark 2\5 to 3\7 (353.7¢ to 379.0¢)
Related MOS scales
Parent 2L 3s⟨5/3⟩
Sister 2L 5s⟨5/3⟩
Daughters 7L 5s⟨5/3⟩, 5L 7s⟨5/3⟩
Neutralized 3L 4s⟨5/3⟩
2-Flought 12L 2s⟨5/3⟩, 5L 9s⟨5/3⟩
Equal tunings(ed5/3)
Equalized (L:s = 1:1) 4\7 (505.3¢)
Supersoft (L:s = 4:3) 15\26 (510.2¢)
Soft (L:s = 3:2) 11\19 (512.0¢)
Semisoft (L:s = 5:3) 18\31 (513.5¢)
Basic (L:s = 2:1) 7\12 (515.9¢)
Semihard (L:s = 5:2) 17\29 (518.4¢)
Hard (L:s = 3:1) 10\17 (520.2¢)
Superhard (L:s = 4:1) 13\22 (522.6¢)
Collapsed (L:s = 1:0) 3\5 (530.6¢)

5L 2s⟨5/3⟩ is a 5/3-equivalent (non-octave) moment of symmetry scale containing 5 large steps and 2 small steps, repeating every interval of 5/3 (884.4¢). Generators that produce this scale range from 505.3¢ to 530.6¢, or from 353.7¢ to 379¢.Among the most well-known variants of this 3/4 diatonic MOS proper are 12ED5/3'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 12ed5/3 diatonic of standard practice.

  • 2 2 1 2 2 2 1

When L=3, s=1, you have 17ED5/3: 3 3 1 3 3 3 1

When L=3, s=2, you have 19ED5/3: 3 3 2 3 3 3 2

When L=4, s=1, you have 22ED5/3: 4 4 1 4 4 4 1

When L=4, s=3, you have 26ED5/3: 4 4 3 4 4 4 3

When L=5, s=1, you have 27ED5/3: 5 5 1 5 5 5 1

When L=5, s=2, you have 29ED5/3: 5 5 2 5 5 5 2

When L=5, s=3, you have 31ED5/3: 5 5 3 5 5 5 3

When L=5, s=4, you have 33ED5/3: 5 5 4 5 5 5 4

So you have scales where L and s are nearly equal, which approach 7ED5/3:

  • 1 1 1 1 1 1 1

And you have scales where s becomes so small it approaches zero, which would give us 5ED5/3:

  • 1 1 0 1 1 1 0 = 1 1 1 1 1

Tuning ranges

Parasoft to ultrasoft

"3/4 Flattone" systems, such as 26ED5/3.

Hyposoft

"3/4 Meantone" (more properly "3/4 septimal meantone") systems, such as 31ED5/3.

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 41ED5/3 and 53ED5/3. This class of tunings is called trischismic temperament; these tunings can approximate 53/4-limit harmonies very accurately by tempering out a small comma called the schisma. (Technically, 12ED5/3 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 29ED5/3 and 46ED5/3. 17ED5/3 is often considered the sharper end of the 3/4 neogothic spectrum; its major third at 312 cents (416 śata) is considerably more concordant than in flatter neogothic tunings.

Parahard to ultrahard

"3/4 Archy" systems such as 17ED5/3, 22ED5/3, and 27ED5/3.

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 7ED5/3) is at one extreme and 3\5 (three degrees of 5ED5/3) 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 12ED5/3:

4\7
7\12
3\5

If we carry this freshman-summing out a little further, new, larger ED5/3s pop up in our continuum.

Scale Tree and Tuning Spectrum of 5L 2s⟨5/3⟩
Generator(ed5/3) Cents Step ratio Comments
Bright Dark L:s Hardness
4\7 505.348 379.011 1:1 1.000 Equalized 5L 2s⟨5/3⟩
23\40 508.506 375.852 6:5 1.200
19\33 509.176 375.182 5:4 1.250
34\59 509.630 374.728 9:7 1.286
15\26 510.207 374.152 4:3 1.333 Supersoft 5L 2s⟨5/3⟩
41\71 510.686 373.673 11:8 1.375
26\45 510.963 373.396 7:5 1.400
37\64 511.270 373.089 10:7 1.429
11\19 511.997 372.362 3:2 1.500 Soft 5L 2s⟨5/3⟩
40\69 512.672 371.687 11:7 1.571
29\50 512.928 371.431 8:5 1.600
47\81 513.146 371.212 13:8 1.625
18\31 513.499 370.860 5:3 1.667 Semisoft 5L 2s⟨5/3⟩
43\74 513.884 370.475 12:7 1.714
25\43 514.162 370.197 7:4 1.750
32\55 514.536 369.823 9:5 1.800
7\12 515.876 368.483 2:1 2.000 Basic 5L 2s⟨5/3⟩
Scales with tunings softer than this are proper
31\53 517.266 367.092 9:4 2.250
24\41 517.673 366.685 7:3 2.333
41\70 517.982 366.377 12:5 2.400
17\29 518.417 365.942 5:2 2.500 Semihard 5L 2s⟨5/3⟩
44\75 518.824 365.535 13:5 2.600
27\46 519.080 365.279 8:3 2.667
37\63 519.385 364.973 11:4 2.750
10\17 520.211 364.148 3:1 3.000 Hard 5L 2s⟨5/3⟩
33\56 521.140 363.219 10:3 3.333
23\39 521.545 362.814 7:2 3.500
36\61 521.917 362.442 11:3 3.667
13\22 522.576 361.783 4:1 4.000 Superhard 5L 2s⟨5/3⟩
29\49 523.396 360.963 9:2 4.500
16\27 524.064 360.294 5:1 5.000
19\32 525.088 359.271 6:1 6.000
3\5 530.615 353.743 1:0 → ∞ Collapsed 5L 2s⟨5/3⟩

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.

5L2s.jpg

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

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