What's significant about diatonic scales? Are there equivalents to the diatonic scales in smaller divisions of the octave (e.g. 19-EDO, 31-EDO, etc.)?

Question

Wikipedia defines a diatonic scale like so:

In music theory, a diatonic scale is a heptatonic scale that includes five whole steps and two half steps in each octave, in which the two half steps are separated from each other by either two or three whole steps, depending on their position in the scale. This pattern ensures that, in a diatonic scale spanning more than one octave, all the half steps are maximally separated from each other (i.e. separated by at least two whole steps).

This is pretty clear. A diatonic scale is a subset of the twelve notes of the octave (in the system of 12-EDO). But what I want to know is why this particular pattern? Why heptatonic? Why must semitones be maximally separated? Why are only two allowed? Is there anything special about this configuration? I guess what I'm asking is where the diatonic falls out of something more fundamental, or is it just a particular choice? Or perhaps I'm looking at it backwards, is the diatonic in some sense prior to the chromatic scale?

The reason why I ask is that I want to know, if the diatonic scale is a consequence of something more fundamental, whether there are analogues of it in other divisions of the octave, for example 19, 22, or 31 EDO?

Answer 1

I'll confine this answer to meantone temperaments because building heptatonic scales in other temperaments is more complicated.

In a meantone temperament, all tones are the same size. The octave may be divided into five tones and two minor seconds (diatonic semitones) to make a heptatonic scale.

On each of the seven degrees of this scale, a triad may be built. If the degrees of the scale are numbered (upwards, starting on the tonic) 1, 2, 3, 4, 5, 6, 7, then the triads on those respective degrees are 135, 246, 357, 461, 572, 613, 724. In each case I have listed the degrees in this order: the triad's root, third and fifth. A triad will sound relatively concordant and pleasant if it is major or minor, but relatively discordant and unpleasant otherwise. A triad is major if its third and fifth are a major third and perfect fifth above the root; it is minor if they are a minor third and perfect fifth above the root.

Abbreviating "tone" to T and "semitone" to S, TT makes a major third; TS, a minor third; and TTTS, a perfect fifth.

A diatonic scale is a heptatonic scale where the intervals are in the order TTSTTTS or one of its cyclic permutations. Why are these scales significant? If the intervals are arranged in such a way, then the triads built on six of the seven degrees of the scale are either major or minor. The other triad is a diminished triad, so that triad sounds more discordant. For example, a major scale has intervals in the order TTSTTTS. The triad 724 on the 7th degree is diminished. For example, in C major, that triad is B D F.

There are other heptatonic scales. For example, the five tones and two semitones may be arranged in a different way. Or a heptatonic scale may be built out of a different combination of intervals, for example, an augmented second, three tones and three minor seconds. But then there are not so many major or minor triads among the scale's seven triads.

All this applies regardless of the relative sizes of tone and semitone, and so applies to every meantone temperament, including 12-, 19- and 31-equal temperaments. (But not 22-equal, because that is not meantone.)

Answer 2

Diatonic scales have per se nothing to do with 12-edo tuning, or any other temperament. They are first of all defined in just intonation. In fact there exist at least two different diatonic scales (modulo modes), namely the Pythagorean scale and Ptolemaic scale. I discussed the construction of these scales quite extensively here and here. As for practical use, a rough wrap-up: Pythagorean is the default for melodies and is what modulation theory is derived from, but instruments playing in harmony must lower major thirds down and minor thirds up to sound properly consonant; this gives you more or less the Ptolemaic tuning.

One interesting thing is that the Ptolemaic scale consist thus not simply of full steps and half steps, but distinguishes between major and minor tones. In a major scale, the frequency ratio between Ⅰ and ⅱ is 9/8, but the ratio between ⅱ and ⅲ is only 10/9.

In any sufficiently fine-grained equal-tempered scale, you will be able to achieve some approximation of these “true” diatonic scales. Essentially, you just pick for each scale degree the note whose frequency is closest. Most tunings don't give very satisfying approximations; 12-edo is the one with the fewest steps that gives acceptable results – in quite some sense, it is just the cheapest available compromise. Other scales that approximate diatonic well include 17-edo, 19-edo, 22-edo, 31-edo and 34-edo. Of these, 19-edo and 31-edo are meantone temperaments. That means, they give you an approximation of the Ptolemaic scale, but the Ⅰ-ⅱ-ⅲ degrees are quantised so the two intervals between are equal. Only this actually gives you a well-defined notion of major second! This property is heavily exploited for many of the modulation tropes found in western music, thus 22-edo and 34-edo are tricky (in other words: interesting) to use for existing music. OTOH, you can play any tonal piece 31-edo, just enharmonics don't quite work the way we're used to and you may end up with some comma pumps.