Scales as subsets

The size of the problem

Twelve notes, choose seven. That is C(12,7) = 792 possible scales. Most of those are the same shape starting on a different note, and if you treat transposition as irrelevant, which musically it is, the twelve rotations collapse them into 66 genuinely different seven-note scales.

Sixty-six is a small number. Western music has spent most of its history on essentially one of them, with two or three others in regular use. That is a strong enough preference to demand an explanation, and unlike most questions in this subject it is one where you can simply examine all the candidates.

Three properties worth testing for

The figure below computes four things about whatever set you select, and none of them involves recognising it. Three are worth introducing properly.

Generated. Can the whole set be built by starting somewhere and repeatedly stepping by one fixed interval? Stack fifths from F and you get F, C, G, D, A, E, B, which is the C major scale. Most sets cannot be built this way at all.

Myhill's property. Take any generic interval, say "a third", meaning two steps up the scale whatever those steps happen to be. In the major scale a third is always either 3 or 4 semitones, never a third thing. The same is true of seconds, fourths, fifths and the rest: exactly two sizes each. This is the reason a word like "third" is usable at all, and it is much rarer than it sounds.

Deep scale. Count how many pairs in the set are separated by each interval class. For the major scale the answer is 2, 5, 4, 3, 6, 1, and the interesting part is that all six numbers are different. No two intervals appear equally often, so the interval content of any subset tells you unambiguously what you are looking at.

What happens when you check all 66

Since there are only 66, there is no need to argue. Enumerate them and apply the tests.

Exactly two of the 66 are generated by a single interval. Exactly two have Myhill's property. Exactly two are deep. And they are the same two sets, every time.

One of them is seven adjacent semitones, which is a cluster rather than a scale and is generated trivially by stepping in ones. The other is the diatonic set.

So those three properties are not three coincidences that happen to land on the same object. Over seven notes in twelve they are the same property described from three directions, and one non-trivial set has it.

Two exceptions, and why they matter

The obvious thing to try in that figure is moving a single note, and the obvious expectation is that the properties collapse. Mostly they do. There are 35 ways to move one note of the diatonic set somewhere else, and 33 of them break at least one of the three tests.

The other two land on another diatonic set. They are the notes at the ends of the chain of fifths: in C major, F and B. Move F up a semitone and the chain extends one fifth in one direction; move B down a semitone and it extends in the other. Both give you a different key.

That is worth pausing on, because it is the mechanism behind everything in the next lesson. Neighbouring keys differ by exactly one note, and they do so because a diatonic set is a chain, and a chain has two ends.

The evenness story, told accurately

You will often read that the major scale is the maximally even way to spread seven notes through twelve, and that is true, but it is easy to state in a way that is not.

Seven notes cannot be spread perfectly evenly through twelve, because 12/7 is not a whole number. The best available compromise uses five whole tones and two semitones, and there is no way to do better. But three of the 66 use exactly that mixture: the major scale, the melodic minor, and one unnamed set. On step sizes alone they tie.

What separates them is where the two semitones go. The major scale puts them as far apart as the arithmetic allows, at positions 3 and 7 of the pattern. Melodic minor puts them closer together, and the unnamed set puts them adjacent. Maximal evenness is a claim about that placement, not merely about the step sizes, and only the diatonic set satisfies it.

The figure's evenness readout is deliberately the weaker measure, the plain spread of step sizes, and it will report the same number for all three of those scales. That is not an oversight. A figure that quietly used the strong test and presented it as obviously-evenness would be doing the thing this course keeps complaining about, which is asserting a result and dressing it as a derivation.

What this does and does not prove

It proves the diatonic set is mathematically distinguished among seven-note scales, uniquely and in several equivalent ways. That is a real result and it is not obvious in advance.

It does not prove the major scale sounds good, and it certainly does not prove other scales sound worse. Those are different claims and this argument does not reach them. The honest version is narrower and more interesting: if you want a scale where every interval name has a consistent meaning, where the whole thing can be generated by one interval, and where interval content identifies position unambiguously, then in twelve-note equal temperament you have exactly one non-trivial choice.

Those turn out to be very useful properties to have, because they are what makes keys, modulation and functional harmony possible at all. The next three lessons are about cashing them in. But the preference came first and the explanation came centuries later, and pretending otherwise would be tidy rather than true.

Worth noting what else scores well. The pentatonic set has Myhill and is generated by fifths too, which is a decent hint about why it turns up independently in so many musical traditions. The whole-tone scale is perfectly even, with six identical steps, and pays for it: its interval vector is full of zeros and it is so symmetric that no note in it can be a root, which is the augmented triad's problem from lesson 1 written six notes long.