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.

Answer 3

The diatonic scale is one of many scales called rank two temperaments, a very important concept in the way we tune instruments. What this means is that two intervals are used to generate the entire scale, one used as the period, which is the octave. What this means is that a high D and a low D, or any other versions of the same note, will both be included in the scale if one of them is, the scale repeats.

The other is the generator, which generates every other interval of the scale, and is stopped after however many notes is desired. Now this is three variables that can be changed to whatever we want, so how should we choose? Well, the first two are intervals, and some intervals are much more consonant than others, the two most consonant generally considered to be the octave, or 2/1, and the perfect fifth, or 3/2. Using the first as the period and second as the generator creates a scale that maximizes the number of perfect fifths, and as such maximizes consonance to a certain degree. No matter how many notes are used, each note but one will have a perfect fifth above it.

So how many notes should be used? If we want a scale with only two kinds of steps, the two options would generally be 5 notes and 7 notes. 5 notes has the advantage of including little to no dissonance, and is known as the Pentatonic scale. However, the 7 note version has a different strength, and it lies in the fact that perfect fifths can generate other intervals. Two more intervals that are well known consonances are 5/4, or the major third, and 6/5, or the minor third. These intervals sound consonant yet colorful, unlike the largely colorless octave and fifth. When you use pure perfect fifths to generate the scale, which is known as the Pythagorean scale, four fifths approximate 5/4 decently well and three upside down fifths going down approximate 6/5 decently well. 7 notes allows every note to have either a major third or a minor third above it, giving each mode of the scale its own flavor.

The pythagorean scale, however, has not been in common use since the medieval era, as we have found that flattening each perfect fifth every so slightly can improve the thirds to a great degree, with the amount differing depending on how one would desire the thirds to sound, collectively called Meantone temperaments. Quarter-Comma Meantone, for instance, tunes the major third purely. 12 tone equal temperament includes a fifth that is flat by an unnoticeable amount, so the fifths all sound good and the thirds are better than pythagorean, though other equal temperaments include different fifths, and as such, different diatonic scales. Fifths of size between 4 steps of 7-ET and 3 steps of 5-ET generate diatonic scales, but fifths that are flatter or sharper can generate interesting scales as well. A fifth around 9 steps of 16-ET, for instance, generates mavila, with similarly tuned thirds but wildly different step sizes. You can explore the scales of other equal temperaments on your own, microtonality really is a very interesting world.

Rank two temperaments, however, are not the only type of temperament, and one very significant variant of the diatonic scale is based in Just Intonation instead, where every note is based on an exact frequency ratio. This is called the Intense Diatonic scale, Zarlino scale, or Ptolemy scale. It has more than two types of step sizes, and only five notes have a perfect fifth above them, as opposed to six, but the other ratios such as major and minor thirds are tuned purely without compromising the majority of the fifths.

This system is hard to expand to an entire 12-tone system, though certain tuning schemes such as the Asymmetric Scale, Centaur, and Kirnberger I attempt it, and instruments such as the Kalimba that often only include seven notes work quite well with the Ptolemy scale. ET systems such as 15 and 53 tone equal temperament include approximations to this Ptolemy scale, but not a meantone one (though 53 contains Pythagorean), so they take advantage of its alternative approach to the diatonic scale. These scales are not intrinsically connected to an equal temperament however, and they are just a few out of many ways to map the tonal space, so try some other temperaments out! Some of my personal favorites are Orwell9, Blackwood, and Mavila.