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
Parasoft to ultrasoft
"Flattone" systems, such as 26EDO.
"Meantone" (more properly "septimal meantone") systems, such as 31EDO.
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
Diatonic modes have standard names from classical music theory:
- Meantone7 – 19edo and 31edo tunings
- Pythagorean7 – Pythagorean tuning
- Garibaldi7 – 94edo tuning
- Cotoneum7 – 217edo tuning
- Pepperoni7 – 271edo tuning
- Supra7 – 56edo tuning
- Archy7 – 472edo tuning
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.
- Chroma-positive generator: 685.7143 cents (4\7) to 720 cents (3\5)
- Chroma-negative generator: 480 cents (2\5) to 514.2857 cents (3\7)
|26\45||693.333||7||5||1.400||Flattone is in this region|
|66\131||696.183||21||13||1.615||Golden meantone (696.2145¢)|
|18\31||696.774||5||3||1.667||Meantone is in this region|
|7\12||700.000||2||1||2.000||Basic diatonic |
(Generators smaller than this are proper)
|31\53||701.887||9||4||2.250||The generator closest to a just 3/2 for EDOs less than 200|
|55\94||702.128||16||7||2.286||Garibaldi / Cassandra|
|71\121||704.132||21||8||2.625||Golden neogothic (704.0956¢)|
|27\46||704.348||8||3||2.667||Neogothic is in this region|
|13\22||709.091||4||1||4.000||Archy is in this region|
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¢).
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.
- Main article: 5L 2s/Temperaments
Approaches to Functional Harmony
- See also: Diatonic functional harmony