Pick seven of the twelve notes the way the major scale does, and a strange bookkeeping falls out: every size of interval appears, and no two of them appear the same number of times. Almost no scale manages that.
There are 212 = 4096 ways to choose a subset of the twelve pitch classes, and music has spent centuries circling a tiny handful of them. The major scale — the white keys, {0,2,4,5,7,9,11} — is the most circled of all. There are many ways to say why it's special; here is one that is purely combinatorial and that you can check by counting on your fingers.
Take the scale and tally its intervals by class: how many pairs of notes sit a semitone apart (class 1), how many a whole tone (class 2), and so on up to the tritone (class 6). For the major scale the tally is ⟨2, 5, 4, 3, 6, 1⟩ — two semitones, five whole tones, four minor thirds, three major thirds, six perfect fourths, one tritone. Look at those six numbers: 1, 2, 3, 4, 5, 6, each used exactly once. No interval class is as common as any other. A scale with this property — every interval class occurring a different number of times — is called deep (the term is Terry Winograd's, 1966; the theory is Carlton Gamer's, 1967). The major scale is the deep scale everyone knows. Here is the instrument; the major scale is loaded. Move the notes and watch the tally.
There is a second way to see deepness, and the instrument shows it on the strip below the histogram. Slide the whole scale up by t semitones and ask how many of its notes land back on notes it already had — the common tones with its own transposition. For the major scale this count is 2 for a one-semitone shift, 5 for two, and so on: the very same numbers 2, 5, 4, 3, 6. That is no accident — the number of common tones under a shift of t is the interval-class tally at t (each shared pair of notes is exactly an interval of size t). So a deep scale is one you can identify by its shadow: every transposition overlaps it by a different amount, so each of the eleven other keys is a different distance from home. Gamer's word for it was deep; Richmond Browne (1981) called this the diatonic's position-finding power — the reason a few notes are enough to tell you what key you're in.
It is tempting to think deepness is just another name for the major scale being evenly spread. It isn't. The black keys — the pentatonic {0,2,4,7,9} — are spread as evenly as five notes in twelve can be (they are a maximally even set, the sibling fact this ground's Euclidean-rhythms layer turns into a drum). Yet their tally is ⟨0,3,2,1,4,0⟩: two classes tie at zero, so the pentatonic is not deep. The whole-tone scale {0,2,4,6,8,10}, more even still, tallies ⟨0,6,0,6,0,3⟩ — riddled with ties. Maximal evenness and deepness are different kinds of specialness, and the major scale is rare in holding both. (Load the presets and watch the bars: a tie turns a bar dark.)
Once the property is stated for twelve notes it asks to be stated for any number. Work in Zn — n equally spaced slots on a circle — and call a subset deep when its ⌊n/2⌋ interval-class counts are all different. Now a flat combinatorial question: in each universe of size n, how many deep scales are there? We counted — exactly, by walking every one of the 2n subsets — and got the sequence below. The instrument will recompute any single term in front of you; then we will tell you what the sequence actually is, because the honest answer surprised us.
Two sequences, depending on whether you count a scale and its transpositions as the same object. Counting every subset (sequence A), and counting up to transposition — i.e. as a necklace on the circle (sequence C):
We searched the On-Line Encyclopedia of Integer Sequences for both sequences, in several windows, and neither came back — so the first version of this page reported them as absent from OEIS, a small discovery. That was true to the letter and wrong to the spirit, and the bottom row of the table above is the giveaway.
Line up C(n) — the deep scales up to transposition — against φ(n), Euler's totient, the count of numbers below n that share no factor with it. From n = 7 on they are the same sequence, term for term, with no exception we can find or that the proof allows:
C(n) = φ(n) (deep scales up to transposition) · A(n) = n·φ(n) (counting every subset), for all n ≥ 7.
So the "uncatalogued" sequences are two of the most catalogued objects in mathematics wearing a disguise: C is OEIS A000010 (the totient) and A is A002618 (Pillai's n·φ(n)). The search returned "no results" only because the verbatim strings carry two odd small terms — C(4)=4 where φ(4)=2, and C(6)=3 where φ(6)=2 — and those off-key heads were enough to slip past an exact-match search. The first pass even saw the totient flicker by in the fuzzy matches and waved it off: "C(4)=4 ≠ φ(4)=2, so C is not the totient." But two small exceptions are not a refutation; they are the two small exceptions. The rule is the totient.
The reason is a theorem that was already in the literature — and that the first pass missed. Up to transposition, every deep scale is a "generated" scale (a chain of a single interval, the way the major scale is a chain of fifths F C G D A E B) with one lone exception: {0,1,2,4} ⊂ Z6, tally ⟨2,3,1⟩ — deep, yet built by stacking no single interval. This is not our observation; it is a proved classification (Winograd, 1966; independently Clough, Engebretsen & Kochavi, 1999; stated cleanly with the Z6 exception named by Demaine and colleagues, 2005). The non-generated deep scale isn't merely "a counterexample to a slogan" — it is the only one, in any universe, forever.
Granting that classification, the count falls out. A generated deep scale is a chain {0, g, 2g, …} of length ⌊n/2⌋ or ⌊n/2⌋+1 whose step g is coprime to n — there are φ(n) such steps, and two lengths, giving 2φ(n) chains. Each scale is counted twice: a chain and its mirror, g and n−g, are the same necklace. And nothing collapses them further, because the generator is recoverable from the scale — in a deep scale's tally the generator's interval class is the unique tallest bar (try it: load the major scale and the longest bar, the six perfect fourths, sits at interval class 5, the fifth that generates it). So 2φ(n) chains fold exactly in half: C(n) = φ(n). Every deep scale turns out to be aperiodic, so each necklace stands for n distinct subsets, and A(n) = n·φ(n). The full argument, machine-checked over every deep scale through n = 34, is in research/deep-scales/census-theorem.md.
The old "prime law" is just this seen at a prime: for p ≥ 5 every nonzero step is coprime to p, so φ(p) = p − 1 deep scales up to transposition and p(p−1) in all — a special case of C = φ, not a separate fact.
Every deep scale is its own mirror. Reflect any deep scale — turn it inside out, the way inversion does in music — and you get back the same scale, up to transposition, always (checked for every deep scale through n = 34). This is why counting up to transposition already counts up to reflection too: the two columns would be identical. It also drives the proof above — the mirror of a chain on g is the chain on n−g, which is why the 2φ(n) chains fold cleanly to φ(n).
And the small print is worth keeping. The clean C = φ holds for n ≥ 7; the universes n = 4 and n = 6 are the only ones that break it, and they break it in opposite directions — n = 6 runs one over the totient, and that one extra is exactly the sporadic {0,1,2,4} (it is in the instrument as a preset); n = 4 runs over because two small scales there are deep without being coprime chains. Two anomalies, both named, both understood. The honest correction is the whole point: the sequence was never new — it was the totient, and saying so is truer than keeping the disguise.