User:Moremajorthanmajor/5L 2s (5/3-equivalent)
| ↖ 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⟩ ↘ |
┌╥╥╥┬╥╥┬┐ │║║║│║║││ │││││││││ └┴┴┴┴┴┴┴┘
sLLsLLL
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.
| 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.
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
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.