Every interval a different number of times

A seven-note scale contains twenty-one intervals, sorted into six sizes. For the major scale the counts are:

Six numbers, all different. There is no reason a seven-note set should have that property — 448 of the 462 selections available do not — and the consequence is one of the strongest structural facts about the scale. It is also, unusually for this site, a fact with no acoustics in it at all: nothing below mentions a frequency, a partial or a listener’s threshold, and the whole argument is counting.

What the property is for #

Two keys a fifth apart share six of their seven notes, and that fact generates the circle of fifths and the key signatures. The interval counts say something considerably stronger.

Transpose the major scale up by k semitones and count how many notes it keeps. The answer, for k from one to six, is 2, 5, 4, 3, 6, 1 — the interval vector itself, which is not a coincidence: a note is shared with the transposition by k exactly when the original contains a pair of notes k apart, so the common-tone count is the interval count.

Since the six counts are all different, the number of shared notes determines the distance uniquely. Six shared notes means a fifth; five means a whole tone; one means a tritone. A listener who registers nothing but how much of the previous key survives can still recover exactly where the music has gone.

The two shapes, and why only one is a scale #

Fourteen selections have the property, and they are seven rotations each of two shapes.

The first is the diatonic set, with steps 2 2 1 2 2 2 1 and counts 2, 5, 4, 3, 6, 1.

The second is seven consecutive semitones — C C♯ D E♭ E F F♯ — with steps 1 1 1 1 1 1 6 and counts 6, 5, 4, 3, 2, 1. It is deep by a much more obvious route: in a run of seven adjacent notes there are six pairs a semitone apart, five a tone apart, four a minor third apart, and so on down to one pair six apart. The counts are all different because they are the numbers one to six.

Both shapes have the property and one of them is a chromatic cluster with a gap in it. So deepness is not sufficient, and what distinguishes the two is nothing to do with interval counting: it is that the diatonic set’s steps are 2 and 1 and the cluster’s are 1 and 6.

That is the honest form of the result and it is more useful than the usual statement. The diatonic set is not the only deep seven-note set; it is the only one that is deep and spread out, and the two requirements come from completely different arguments.

Why five notes cannot have it #

The pentatonic scale’s counts are 0, 3, 2, 1, 4, 0. Two of the six are zero, so two are equal and the set is not deep.

That is not a failure of the pentatonic scale; it is arithmetic. Six distinct non-negative counts must total at least 0 + 1 + 2 + 3 + 4 + 5 = 15, and five notes make only ten intervals. No five-note set in any tuning system can be deep, and the same argument rules out four notes and three.

So the property has a floor at six, it is achievable at six and seven, and above seven nothing in the twelve is deep at all. The window is two cardinalities wide, and the diatonic set sits inside it.

The ceiling does not have the same reason as the floor, and it is worth getting right because the obvious reason does not work. Eight notes make twenty-eight intervals, six distinct counts need only sum to fifteen, and twenty-eight is comfortably more than fifteen — so the counting bound that rules out five notes does not rule out eight. It rules out three, four and five and nothing above them.

What rules out eight is the same bound applied to the complement. A set and its complement have interval vectors that differ by a constant: for a set of k notes in twelve, every count except the tritone’s is larger by 2k − 12, and the tritone’s by half that. The offsets are checkable in one line and come out exactly — 4, 4, 4, 4, 4, 2 for an eight-note set against its four-note complement.

A constant added to five numbers preserves their distinctness. So an eight-note set’s five non-tritone counts are distinct exactly when its four-note complement’s are — and five distinct non-negative counts must sum to at least 0 + 1 + 2 + 3 + 4 = 10, while a four-note set has only six intervals in total. It is impossible, and the same argument kills nine notes and ten more easily still. The window is closed at both ends by one counting bound, applied to the set at the bottom and to its complement at the top.

Which makes the pentatonic the reason the diatonic is deep #

Run that backwards at seven notes and something falls out that the two scales are never put together for.

The diatonic set’s complement is the pentatonic. For the diatonic’s five non-tritone counts to be distinct, the pentatonic’s five must be distinct — and five distinct non-negative counts sum to at least ten, while a five-note set has exactly ten intervals. There is no slack at all. The pentatonic’s five non-tritone counts must be precisely 0, 1, 2, 3 and 4 in some order, and it must contain no tritone whatever.

They are, and it does. The pentatonic’s vector is 0, 3, 2, 1, 4, 0: the numbers nought to four across the first five entries, and a zero in the tritone column.

So the property this essay is about belongs to the diatonic set because the pentatonic saturates a counting bound exactly. The scale that is not deep is the reason the scale that is deep can be, and the mechanism is the tritone entry — which is offset by half as much as the others, so the pentatonic’s second zero, the thing that disqualifies it, becomes a 1 in the diatonic and the only count that occurs once.

That also explains the shape of the six-note case. At six notes the offset is zero in every entry, which is the hexachord theorem: a hexachord and its complement have identical interval vectors. So deep hexachords come in complementary pairs by construction, and the twenty-four the search returns are closed under complementation — which is a property the seven-note case cannot have, since a seven-note set’s complement is not the same size.

And the window generalises, which is the check that the argument is about counting rather than about twelve. Enumerating every subset of every even division from ten to twenty:

It is always exactly two sizes, and they are always half the division and one more. The floor bites just below half and the complement’s floor bites just above it, so the window is pinned to the middle of the universe wherever the universe is. That puts the diatonic set in the only place it could have been: seven of twelve is half the chromatic plus one, which is the larger of the two sizes at which any set in twelve can carry a distinct count for every interval. Choosing seven notes was not a decision that happened to land on a deep set — it was a decision to work at the top of the only window there is.

That gives the pentatonic and the diatonic genuinely different structural characters, which is worth stating because they are usually presented as the same object at two sizes. The pentatonic is the longest chain of fifths with no semitone in it and it is not deep; the diatonic is two links longer, has semitones, and is. A modulation between pentatonic collections cannot be located by counting common tones and a modulation between keys can.

The property drawn as a shape #

The counts are easier to believe as a picture than as a row of numbers, because what they describe is an asymmetry.

That is the general principle behind the whole property. A symmetrical set cannot be deep, because symmetry means two different transpositions produce the same overlap. The diatonic set has no symmetry at all — no rotation but the identity maps it onto itself — and its lopsidedness is what makes every distance measurable.

The site has already said the selection is lopsided on purpose, and this is the sharpest statement of what the lopsidedness buys. It is not that an even selection would sound worse. It is that an even selection would make every modulation indistinguishable from every other.

What a listener would have to be doing #

The claim that a distance is recoverable from a common-tone count is a claim about information, not about a mechanism, and the two should be kept apart.

What the arithmetic establishes is that the information is there — that a device counting overlaps has enough to identify the modulation uniquely. Whether a listener does anything of the kind is a separate question with its own evidence, and the site has some of it.

Krumhansl and Kessler’s probe-tone experiment measured how well each of the twelve notes fits after a passage in a key, and the answers fall into four groups with no overlap: the tonic, the rest of the tonic triad, the rest of the scale, and everything else. That is a listener registering set membership, which is the raw material a common-tone count would need. And key-finding algorithms built on those profiles work by correlating a passage’s pitch distribution against the profile of each of the twenty-four keys, which is a weighted version of the same operation.

So the mechanism is plausible and unproven, and the arithmetic is exact. The distinction matters here because deepness is often offered as an explanation of why keys feel near or far, and what it actually shows is that they could be told apart, not that they are.

The one place the count is used explicitly #

There is a piece of ordinary musical machinery that is a common-tone count and is never described as one: the key signature.

A key signature records how far round the circle of fifths a key is, in accidentals. One sharp is one step, two sharps two, and the number of accidentals is exactly seven minus the number of notes shared with C major. The notation records the overlap, in a form a reader decodes without thinking about it.

That works because the counts are distinct. If two different distances shared a common-tone count, they would need the same number of accidentals and the signature would be ambiguous — which is what happens in the one place the system does break down, at six sharps against six flats. There the two spellings name the same set of pitch classes, the overlap with C major is one either way, and the choice between them is settled by convenience rather than by distance.

Whose music, and when #

The deep-scale property was named in the 1960s in the American music-theory literature, alongside the theorem that a scale is deep if and only if it is generated by an interval that shares no common factor with the size of the chromatic set. That theorem explains both shapes at once: seven links of the fifth gives the diatonic set and seven links of the semitone gives the cluster, and 5 and 1 are the generators coprime with 12.

The musical practice it describes is European tonal music of roughly 1650 to 1900, in which modulation between keys is a structural device and the distances between keys are treated as meaningful. The large shape of a classical movement is a path on that key lattice, and the lattice is exactly the object the common-tone counts measure distances on.

And it is a property of the twelve-note system, not of scales in general. In a division of the octave into a different number of parts the coprimality condition picks out different generators, and in a tradition where the tonic is fixed by a drone the whole notion of a distance between keys does not arise.

What the search refused #

The essay was slated on the assumption that the diatonic set is the only deep seven-note shape, which is how the property is usually introduced, and the search returned fourteen sets rather than seven.

The extra seven are the rotations of the chromatic cluster, and they are not an edge case or an artefact of the counting: the cluster is deep by the most direct route there is, and it is also well formed and generated. It passes three of the four tests this ladder cares about and fails only evenness.

Two things follow, and both are better than the claim the essay set out to make.

The property has to be stated with a companion. “The diatonic set is the unique deep heptachord” is false; “the unique deep heptachord that is not a cluster” is true and is an awkward sentence, which is a sign that the real statement needs a second property rather than a qualifier. It gets one at the next rung but one.

And the counterexample is informative rather than annoying. A run of seven semitones has counts 6, 5, 4, 3, 2, 1 — the numbers one to six, in order — which is deepness in its purest form and is completely useless, because a cluster’s transpositions are all nearly the same set. The property is necessary and not sufficient, and having the useless example in hand is what makes that legible.

Where the model stops #

The property is about a set and music is about an order. Counting intervals discards which note came first, which is most of what makes a passage in a key sound like that key. Two ragas with identical pitch sets are different ragas, and no interval vector can tell them apart.

The common-tone count assumes both keys are complete. A modulation happens over a few bars in which neither collection has been fully stated, and a listener hearing four notes of the new key has an ambiguous overlap. The clean result is about two complete sets.

And the distinctness of six numbers is fragile. Change one note of the scale and the counts change: the harmonic minor’s are 3, 3, 4, 4, 4, 2 — three pairs of equal values — so the raised seventh that repairs the dominant chord destroys deepness entirely. The property belongs to the diatonic set and not to the minor mode as it is actually used.

What the picture cannot show #

It cannot show which notes are shared. A count of six is drawn as a six, and whether the six shared notes include the tonic is the difference between a modulation that feels like a step and one that feels like a lurch.

It cannot show duplication across octaves. Every count here treats a scale as seven pitch classes, and a real passage sounds them in several octaves at once, which multiplies some intervals and not others.

It cannot show the tritone. The one interval that occurs exactly once is the one that does most of the work in tonal harmony, and its uniqueness is what makes a dominant seventh point at one key. In this figure it is a 1 in the last column.

Where the ladder goes next #

Deepness is one of four properties the diatonic set has, and it is the one with the fewest other holders. The next rung takes a different property — that the set is spread as evenly as seven things in twelve can be — and finds that the algorithm producing it is one this site has been running since its first phase on a completely different object.