Why seven

Five rungs of this ladder have each established one thing about the diatonic set, and each of them turned out to be a property something else also has.

It is a chain of six fifths, and so is the pentatonic set, and so is every other length of the same chain. Every generic interval comes in exactly two sizes, and one other seven-note shape manages that too — the run of seven adjacent semitones, and no third. It is spread as evenly as seven things in twelve can be, and so are the pentatonic and octatonic collections. Its six interval counts are all different, and so are a chromatic cluster’s.

Four properties, four sets of holders, and no single argument that arrives at the object. So take all four at once and count.

The four tests #

Each is a definition rather than a fit, and each is computed here rather than looked up.

A chain of one interval. Is the set reachable by starting somewhere and repeatedly adding a fixed number of semitones? The diatonic set is a chain of the fifth; the chromatic cluster is a chain of the semitone; most sets are a chain of nothing.

Two sizes of every step. Take every generic interval — every second, every third, every fourth, round the octave — and ask how many distinct specific sizes it has. Myhill’s property is that the answer is two, every time. It is what lets one word do for two distances.

As even as possible. Is the set the one Clough and Douthett’s construction returns for that size, up to rotation? For seven in twelve that is the diatonic set and for five it is the pentatonic.

Deep. Are the six interval counts all different, so that no two transpositions share the same number of notes with the original?

The tests are independent in the sense that no one of them implies another, which is what makes the census worth running rather than reasoning about.

What each size fails on #

The interesting rows are the ones that pass three tests.

Five notes fail on deepness, and cannot pass. Six distinct non-negative counts must total at least fifteen and five notes make only ten intervals, so no five-note set anywhere is deep. The pentatonic collection passes the other three — it is a chain, it is well formed, it is maximally even — and the fourth is arithmetically closed to it. It is not a near miss; it is an impossibility, and it is the reason the pentatonic and the diatonic are structurally different objects rather than the same object at two sizes.

Six notes fail worse than any other size. Only one hexachord in the twelve is well formed at all — the run of six semitones — and only one is maximally even, and they are different sets. Six in twelve divides exactly, so the evenest hexachord is the whole-tone scale, which maps onto itself under a rotation of two semitones; that symmetry costs it deepness and well-formedness together.

Meanwhile the chain of six fifths, which is the diatonic set with a note taken away, is deep and is neither well formed nor even. It has steps 2 2 3 2 2 1, and the two occurrences of “a third” in it are three and four semitones and five semitones, which is three sizes where Myhill’s property allows two.

That is the sharpest failure in the table. At six notes no shape passes more than two tests, and the two hexachords that get that far do it by opposite routes — one by being a cluster and one by being a chain.

Eight notes and above fail on deepness, and not for the reason it is tempting to give. Twenty-eight intervals in six classes do not force a repeat — 8, 7, 6, 4, 2, 1 are six distinct counts summing to twenty-eight — so the pigeonhole argument that works at five notes does not work here. What does the work is the complement. A set and its complement have interval vectors differing by the same constant in every class, so a set is deep exactly when its complement is; an eight-note set’s complement has four notes, and no four-note set is deep, because six distinct counts need fifteen intervals and four notes make six. The impossibility at eight is the impossibility at four, seen through the complement, and the same argument rules out nine, ten and eleven. Only one eight-note shape passes as many as two tests, and it is the chromatic run again. The octatonic collection is maximally even and is nothing else: it is not a chain of any interval, it is not well formed, and its symmetry rules out deepness — one property of the four.

Seven, and the two shapes that reach it #

At seven notes the four tests come down to two candidates, both of which pass three.

The chromatic heptachord — steps 1 1 1 1 1 1 6 — is a chain of the semitone, is well formed, and is deep. It fails only on evenness, and it fails on it enormously: its largest step is six times its smallest.

The diatonic set — steps 2 2 1 2 2 2 1 — is a chain of the fifth, is well formed, is deep, and is maximally even.

Printing the whole census rather than describing it makes the shape of the result visible, and the table is small:

Three hundred and forty-nine shapes across the sizes from two to eleven, and the single row that passes everything is the seven-note one with steps 1 2 2 1 2 2 2.

Seven is the only size at which all four counts are non-zero at once. Deepness is available only at six and seven; Myhill’s property has at most three holders at any size and exactly two at seven; being a chain has at most six; and exactly one shape of every size is maximally even. Four columns, and seven is where the last of them stops being zero and the first of them has not yet run out.

That is a tighter statement than a survivor and a rival. The census does not merely produce one survivor; it produces a survivor and a single rival that is eliminated by the one test that measures spread. The question “why seven” is answered by the row above and the row below, and the question “which seven” is answered by the one property that distinguishes a scale from a cluster.

Symmetry is what the census is really measuring #

Reading the failures together, one property keeps doing the eliminating and it is not on the list of four.

The whole-tone scale maps onto itself when transposed by two semitones. The octatonic maps onto itself at three. The diminished seventh maps onto itself at three, and the augmented triad at four. Every one of them is eliminated, and every one is eliminated by deepness or by well-formedness or by both.

The reason is direct. If a set maps onto itself under transposition by k, then it shares all its notes with that transposition, so the interval count at k equals the size of the set — and any other transposition sharing that many notes would have to be a symmetry too. Symmetry forces the counts to repeat, and repeated counts are the negation of deepness.

So the census is largely a test for asymmetry, and the diatonic set’s central structural virtue is that it has none. No rotation but the identity maps it onto itself, which is what makes every one of its transpositions distinguishable from every other and what makes the twenty-four keys twenty-four distinct objects rather than a smaller number of objects with several names.

That reframes the four criteria as three plus a consequence, and it also explains why the symmetrical sets are used for what they are used for. A composer reaching for the whole-tone or octatonic collection is reaching for exactly the property this census penalises: a set with no orientation, in which no transposition is far from any other.

What a single survivor establishes #

This is the part where the result has to be handled carefully, because a census with one answer invites a stronger conclusion than it supports.

What it establishes is that the diatonic set is not an arbitrary selection. Four independent structural criteria, each with several holders, intersect in exactly one shape out of three hundred and forty-nine, and that shape is the one European music has used for six hundred years. A coincidence of that specificity is worth reporting.

What it does not establish is that the diatonic set was chosen for these reasons, or that these are the reasons it works, or that any listener perceives any of the four properties. None of the four was known before the twentieth century and the scale is far older than any of them.

And it does not establish that the properties are the right ones. Four criteria were selected — by theorists, over thirty years, with the diatonic set in view — and a set of criteria chosen with an answer already in mind will tend to select that answer. The honest test of the census is whether the four are independently motivated, and they mostly are: well-formedness is motivated by naming, deepness by locating a modulation, evenness by construction, generation by the acoustics of the fifth. But the motivation was found after the object.

The site has made the same caution once before in this ladder about a different explanation, and the two cautions are different in a way worth naming. There, a physical model was asked to select a scale and could not, because it was answering a different kind of question. Here, a set of combinatorial criteria selects the scale decisively, and the doubt is about where the criteria came from.

What the survivor does with its properties #

It is worth putting the four together in the form a musician would meet them, because taken separately they read as combinatorics and taken together they read as a description of how a key works.

The set is a chain, so it can be reached from anywhere by one interval, which is why the circle of fifths is a circle of keys rather than a list of them.

Every generic interval has two sizes, so the seven degrees can be named once and counted, and a third is a third whether it spans three semitones or four.

Nothing is symmetrical, so each of the twelve transpositions is a distinguishable object and a modulation has a direction and a distance.

And the counts are all different, so the distance is recoverable from the overlap alone.

Those four together are a set that can be moved, named, told apart from its own transpositions, and located after being moved. That is not a list of curiosities; it is the minimum a collection needs in order to support a music in which changing key is a structural event, and every one of them fails for at least one of the alternatives.

Whose music, and when #

All four properties come from the American music-theory literature between about 1965 and 1995: the deep-scale theorem in the mid-1960s, Myhill’s property and well-formedness in the 1980s, maximal evenness in 1991. They were developed as a body of work about the diatonic set, in a discipline that had it as its central object.

The music they describe is European tonal practice, and the twelve-note chromatic set they are computed inside is itself a European object of the last few centuries. A tradition with twenty-two shrutis, or seven near-equal steps, or a fixed drone, is not described by any row of this table — the requirements that produce a twelve-note chromatic set are not universal, and neither is the question this census answers.

And the pentatonic result cuts across the frame. Five-note collections built on the same chain are used almost everywhere, and the census says they cannot be deep. Whatever it is that makes a pentatonic collection work — and it works, in more traditions than the diatonic set does — is not on the list of four, which is a useful reminder that the list was assembled to explain one object.

The property none of the four is about #

The census is silent on the one thing a musician would name first, and the silence is worth marking rather than glossing.

Nothing in the four tests mentions triads. The diatonic set happens to yield seven triads of three qualities in a fixed pattern — three major, three minor, one diminished — and that pattern is most of what tonal harmony is built on, and no criterion here comes near it.

It is not an accident that the survivor has it: a set with two step sizes arranged evenly will stack thirds into a small number of qualities, so the harmonic result follows from the structural ones. But it follows as a consequence rather than being a test, and a reader who came to this ladder wanting to know why the major scale supports the chords it does has been answered obliquely.

That is the property the census would have been designed around if the theorists had been harmonists rather than set theorists. There is a second one they are silent about too, and unlike the triads it can be run through the same machinery.

Where the model stops #

The census is over set classes and ignores which note is home. All seven modes of the diatonic set are one row here, and the difference between them is most of what a listener responds to. A mode is the same seven notes with a different one treated as home, and nothing in this table can see the difference.

It is over pitch-class sets, so octave and order are gone. A scale used in music is a register-spanning ordered resource with weighted degrees, and this is a set of seven numbers modulo twelve.

And four is a small number of tests. Other structural properties exist — cardinality equals variety, structure implies multiplicity, the various coherence conditions — and adding them would not change the survivor, since the diatonic set has those too. Adding a property the diatonic set lacks would change everything, and no such property has been proposed, which is itself evidence about how the list was assembled.

What the picture cannot show #

It cannot show that the tests are independent. They are logically independent — each has holders that fail the others — and they are not statistically independent, since chains tend to be well formed and even sets tend to be chains. The table’s single survivor would be less surprising if the correlations were drawn, and they are not drawn.

It cannot show the near misses’ distances. A shape that fails evenness by having one step of six is drawn identically to one that fails by having a step of three, and the difference between those two is large.

And it cannot show what the twelve are. Every row of this table is a selection from a chromatic set whose own construction — twelve fifths that do not quite close — is the subject of a whole other ladder on this site, and the census takes it as given.

Where this ladder ends #

Six rungs. The first three described what the diatonic set is — a selection of seven from twelve, a chain of fifths, a set with two sizes of every step. These three asked what else has each property, and found that the answer is always something, and that the intersection of all of them is one shape in the whole twelve-note universe. That is the strongest structural result this field has, and its limitation is stated in the same breath: four criteria assembled by people who already knew the answer will find it, and the achievement is that they are independent of each other rather than that they are independent of the object.