Two sizes of every step, which is why the names work

Every musician is taught, early and without comment, that C to E is a third and D to F is also a third. One is four semitones and the other is three. They are given the same name, the difference is handled by adding “major” or “minor”, and nobody explains why one word covers two distances.

The reason is a property of the scale rather than a convention of the naming, and it does not hold for most seven-note selections.

Two sizes for the seconds — the tone and the semitone. Two for the thirds. Two for the fourths, fifths, sixths and sevenths. That is not a fact about the intervals; it is a fact about the set, and it has a name.

Generic and specific #

The distinction the figure rests on is worth stating carefully, because ordinary musical language runs the two together.

A generic interval counts scale degrees: a third is two steps up the scale whatever those steps are. A specific interval counts semitones: a third is three or four. Ordinary notation writes the generic interval — the notehead’s position on the stave says which degree, and nothing else — and leaves the specific one to be inferred from the key signature and any accidentals.

Notation gets away with recording only the generic interval because the specific one is nearly determined by it. If a third could be three, four or five semitones there would be no way to read a stave without an accidental on every note, and the name “third” would be doing no work at all.

The property, and how rare it is #

A scale in which every generic interval takes exactly two specific sizes is said to have Myhill’s property, after the mathematician John Myhill; the term entered music theory through John Clough and Gerald Myerson’s work on diatonic set theory in the mid-1980s.

Counting how many seven-note sets have it is a small computation. Fixing the tonic and choosing six more notes from the remaining eleven gives 462 sets. Of those, fourteen have Myhill’s property — three per cent.

The fourteen are not fourteen different scales. They fall into two rotation classes, which is to say two step patterns each appearing in its seven rotations:

• 1·2·2·1·2·2·2 — the diatonic scale, in its seven modes.
• 1·1·1·1·1·1·6 — six consecutive semitones followed by one large leap.

The second is a genuine solution and a degenerate one. A chromatic run of seven adjacent notes trivially has two sizes for each generic interval, because every interval within the run is one size and every interval that wraps round the gap is another. It passes the test and is useless.

What separates them is that both are generated: each is produced by stacking one interval repeatedly and folding the result into an octave. The diatonic scale is seven consecutive fifths, and that is where the pentatonic comes from too. The chromatic run is seven consecutive semitones. A scale that is generated and has Myhill’s property is called well-formed, and both of these are — which is the honest statement, rather than the tidier one that the diatonic scale is the unique solution.

The difference between them is the generator. Stacking fifths produces a set whose two step sizes are a tone and a semitone, in a ratio near enough to 2:1 that they read as two different-sized steps of the same kind. Stacking semitones produces six steps of one and a leap of six, which is not a scale anybody would use for melody.

What the property buys #

Three things follow from two sizes, and each is something musicians rely on without naming it.

Interval names mean something. A generic interval has two possible sizes, so qualifying it with one bit — major or minor, perfect or diminished — pins it down completely. That is why the naming system has exactly the shape it does.

A pattern is recognisable after transposition. Starting the same seven notes at a different degree gives a different mode, and a listener tracks it as a rearrangement of familiar material rather than as a new set of distances. That works because the inventory of distances did not change.

A chord type is countable, and this turns out to be the same property rather than a consequence of it. Stacking alternate degrees gives a triad, and because each generic third has two sizes and each generic fifth has two, a triad has at most four qualities. The diatonic scale realises three of the four — major, minor and diminished, with the fourth combination, a major third under a tritone, never arising — and their distribution across the seven degrees is fixed.

Counting triad qualities over all 462 sets rather than over the one gives an equivalence:

No set has fewer than three, and the fourteen that reach three are exactly the fourteen with Myhill’s property. Not a subset of them and not an overlap — the same fourteen sets, arrived at by counting chords instead of by counting interval sizes. So “every generic interval has two sizes” and “the scale has three kinds of triad” are one statement about a seven-note set in the twelve, and the second is the form a musician would recognise.

The shape of the rest of the table is the argument for why that matters. The median seven-note set gives six different triad qualities on its seven degrees, and 140 of the 462 give seven — a different chord type on every degree, which is the same as having no chord types at all. In such a scale there is nothing to learn: a triad is a fact about the degree it sits on, no two are alike, and there is no vocabulary of qualities to name.

That also disposes of a tidier claim than the one made above. The three qualities are not three because somebody chose well; three is the floor for any seven-note set in the twelve, and reaching the floor is exactly what being well formed is. The chromatic run reaches it too, on the same fourteen-set list, and gives its own three qualities — which is another way of saying the property is arithmetic and the usefulness is not.

There is a smaller observation in the fourth row that is worth keeping, because it is the only place the count of four appears. Forty-nine sets give four triad qualities, which is the largest number two sizes of third and two of fifth could produce — and none of them has Myhill’s property, so they are getting four qualities out of more than two sizes somewhere and happening to land on four distinct combinations. A count alone cannot tell those from a well-formed scale that used all four of its available slots, and no seven-note set in the twelve does that. Whether some other universe has one is the kind of question the census across universes is for, and it is not asked there.

Why two rather than one #

There is an obvious candidate for a better scale: one where every generic interval has exactly one size. That would make the naming perfectly unambiguous and remove the need for “major” and “minor” entirely.

Such scales exist and they are all useless in the same way. A scale with one size for every generic interval is one whose steps are all equal — the whole-tone scale, the diminished-seventh arpeggio, the chromatic scale itself — and equal steps destroy the thing a scale is for.

Equal steps mean the pattern maps onto itself under rotation, so no degree is picked out and there is no tonic. Debussy’s use of the whole-tone scale is a use of precisely that property — it is a way of suspending the sense of a home note — and it works as an effect because it is heard against a background of scales that do have one.

So the two sizes are not a defect that a better scale would fix. They are the minimum irregularity required for a scale to have a shape at all, and Myhill’s property is the statement that the diatonic set has exactly the minimum and no more. That is the sense in which it is the tidy answer to a question with a floor under it.

Where the property fails, and what happens then #

The minor scales are the interesting failure, because they are in constant use.

The natural minor is a mode of the major and has the property. The harmonic minor does not: raising the seventh degree creates an augmented second, which gives the generic second three sizes rather than two — one, two and three semitones.

The cost is real and audible. The augmented second is the sound of the harmonic minor, it is why the scale is often described as exotic, and it is why melodic practice avoids it by raising the sixth degree on the way up. What the raised seventh buys is a major triad on the dominant, and the whole of the minor mode’s harmonic apparatus is a negotiation between that gain and this loss.

Non-Western scale systems supply failures of a different kind. A set with three step sizes is perfectly usable melodically; what it does not support is a compact interval vocabulary of the European sort, and traditions that use such sets tend to name their intervals individually rather than generically.

The octatonic scale fails in a third way, and it is the instructive one because it fails by being too regular rather than too irregular. Alternating semitones and tones gives eight notes, and running the same computation over it returns sizes of 2, 1, 2, 1, 2, 1, 2 for its seven generic intervals — every other generic interval has only one size. That is not Myhill’s property, which requires exactly two everywhere; it is something stricter and worse, because a generic interval with a single size carries no information at all.

The cause is symmetry. The octatonic pattern repeats every three semitones and so maps onto itself under that rotation, which is also why it has no tonic: three of its eight degrees are indistinguishable from the first. It appears in Rimsky-Korsakov, Stravinsky and Messiaen as a colour rather than as a key, and the reason is visible in that row of ones.

So the property is a floor and a ceiling at once. Fewer than two sizes for some interval means symmetry and no home note; more than two for any interval means the naming breaks down. The diatonic set is one of very few sets that sits exactly between.

The three properties, and how they line up #

Myhill’s property is one of a small family of statements about the diatonic set that keep turning out to describe the same seven notes from different directions, and it is worth setting them beside each other because the coincidence is the interesting part.

Maximal evenness. Of all seven-note subsets of twelve, the diatonic set is the one whose notes are spread as evenly as the arithmetic allows — the same construction that makes a Euclidean rhythm out of k onsets in n steps, applied to pitch instead of time. It arrives at the diatonic scale without mentioning intervals at all.

Generation by a single interval. Seven consecutive fifths, folded into an octave, give the same seven notes. That arrives at them without mentioning evenness.

Myhill’s property. Two sizes for every generic interval. That arrives at them without mentioning either.

Three descriptions, three different starting points, one answer. It is tempting to treat that as evidence of something deep, and the more careful statement is that they are not independent: a scale generated by an interval close to a simple ratio, folded into an octave, is close to maximally even for arithmetic reasons, and a maximally even set has at most two sizes of each generic interval for the same reasons. The convergence is a theorem rather than a coincidence, and Clough and Douthett proved the connection in 1991.

What is not a theorem is that the generator should be the fifth. That comes from acoustics — the fifth is the simplest ratio after the octave, so a chain of them produces notes that are consonant with each other — and it is the one place in this argument where a fact about the world enters a chain of otherwise purely combinatorial reasoning.

The model, and its limits #

The computation assumes twelve equal divisions of the octave. That is a strong assumption and it is the one doing the most work, because “specific size” is measured in semitones and semitones only exist if the octave has been divided into twelve of them.

The property generalises. In any equal division of n steps, one can ask which subsets have two sizes for every generic interval, and the answer is always the well-formed scales generated by a single interval — in 19, 31 or 53 divisions as much as in 12. What changes is which generator produces a useful pattern.

It does not generalise to systems without a fixed division. In a tradition where a degree’s size is a matter of practice and varies within a range, counting distinct specific sizes is not a well-defined operation, and the question this essay asks does not arise.

Two further limits are worth naming. The property is about a scale sitting still, and says nothing about which of its degrees are used or stressed — a scale can have it and be melodically useless. And the count of fourteen is a count of sets, treating all rotations as distinct; counting step-pattern classes instead gives two, which is the number that actually matters and is the one the figure reports.

What a reader can check by ear #

The property is combinatorial and the consequences are audible, which is unusual enough to be worth using.

Play the seven thirds of the major scale in order — C–E, D–F, E–G, F–A, G–B, A–C, B–D — and what arrives is two sounds alternating in an irregular pattern: major, minor, minor, major, major, minor, minor. Two qualities, seven positions, and the pattern of which is which is the scale’s fingerprint. Do the same on the harmonic minor and a third sound appears, in one place only, and it is immediately conspicuous.

The seconds are the same experiment at closer range. Five of the diatonic seconds are tones and two are semitones, and the two semitones are what a listener uses to work out which mode is in force — they are the only asymmetry available. In a scale with three step sizes there would be more information and, paradoxically, less orientation, because the pattern would be harder to hold.

That is what the property is worth in practice. Two sizes is few enough to be memorable and more than one, which is the narrow band in which a scale is both learnable and directional. Everything above is a computation; this is the reason it is worth computing.

What the picture cannot show #

A table of interval sizes cannot show the asymmetry between the two step sizes in use. The tone and the semitone are not merely two sizes: the semitone is where the melodic tension is, and its position is what distinguishes the modes from each other. The property guarantees that there are two sizes and says nothing about which of them a piece will lean on.

It also says nothing about how far either step can be moved before it stops being itself. That is a question about how wide an interval category is, and the answer — about a hundred cents, against a discrimination limit of five — is why a scale’s two step sizes survive every temperament that has ever been applied to them.

It also cannot show that the property is invisible from inside. A musician who has only ever used well-formed scales has no reason to notice that a third comes in two sizes rather than three, because there has never been a case where it did — which is exactly what a successful piece of structure looks like from the inside.

The ladder continues to what happens when the octave itself is given up, and whether a scale can be well-formed against an equave that is not a 2:1.