Nothing in the census knows which note is home

This ladder has now made six structural claims about one object. The diatonic set is seven of the twelve chosen unevenly; it is a chain of fifths two notes longer than the pentatonic; it has two sizes of every step and every interval a different number of times; it is the output of the algorithm that makes the tresillo; and it is the only shape of the 349 with all four of those properties at once, at a size and with a generator that the size of the universe forces.

Every one of those claims is false of nothing and true of the same object seven times over. Here is what that means.

The measurement #

The claim is checkable in a line of arithmetic and it is worth checking rather than asserting, because a property that looked rotation-invariant and was not would be the interesting case.

Take the diatonic set. Rotate it to each of its seven degrees; transpose each rotation to each of the twelve pitch classes. That is eighty-four sets. Compute all four properties on each, and the interval vector besides.

One answer, eighty-four times. Generated by five, well formed, maximally even, deep, vector ⟨2, 5, 4, 3, 6, 1⟩. Not four answers or seven; one.

The reason is structural rather than lucky. All four properties are defined on the multiset of distances between the notes — the step pattern up to rotation, the interval vector, the generating interval, the evenness of the spacing. Rotating a set does not change any distance; it changes which note was written down first. Transposing does not change any distance either. So a function of distances is a function of the set class, and the set class is what the census enumerates.

That is why the previous rungs kept saying 349 shapes and never 4,096 sets. There are 4,096 subsets of the twelve; there are 349 shapes; and the difference between those two numbers is exactly the information this ladder throws away, on purpose, at the start.

What is thrown away is most of the subject #

The discarded information is not a technicality. It is the tonic, and the tonic is what a great deal of music theory is about.

A major scale and its relative minor share every structural property and share no repertoire. Ionian and Dorian are the same shape and the difference between them is the whole of modal practice. The set does not merely fail to name the tonic; the set is compatible with seven different answers and cannot be used to prefer any of them.

This is where the ladder’s own argument turns on it. Rung four is about the deep property, and it says the number of common tones between a key and a transposition of itself identifies the distance uniquely — which is true and useful and is a statement about keys, an object with a tonic. The property that makes the identification possible does not know what a key is. It works on the shape; the listener supplies the rest.

An objection worth answering #

There is a natural objection to all of this, and it is that the properties were simply chosen badly. If a set cannot name a tonic, add a property that can — something about where the semitones fall relative to the first note, say, or which degree has a fifth above it.

The objection fails, and it fails in a way that is more interesting than the objection.

Anything of that kind is a property of an ordered set, so adding it changes the object under study: the census would no longer be over 349 shapes but over the 4,096 subsets, or the 462 seven-note selections with a marked first note, which is 462 × 7 rotations again. That is a perfectly good census and it is not this one. Its results would not transfer between keys or modes and would have to be restated for each.

More decisively, the added property would not settle anything, because there is no shortage of notes that could be home. Six of the diatonic set’s seven degrees have a perfect fifth above them; three carry a major triad and three a minor. A property that rules out Locrian leaves six candidates, and no property computed from the notes will get below that, because the six really are all available — every one of them has been used as a final in some repertoire.

The tonic is not underdetermined by the set because the properties are weak. It is underdetermined because it is decided by something else entirely, and the something else is time.

What does supply it, briefly #

Something has to, and this site has already measured most of it on other ladders.

The profile is not derivable from anything in this ladder. It has to be measured, it was measured, and it is roughly reproducible from how long each note is sounded in actual pieces — which is to say from time, a dimension the census does not have. A pitch-class set has no durations, no order, no metre and no beginning. Every candidate mechanism for finding a tonic uses at least one of those.

The sharpest form of this is arithmetic rather than experimental, and it is worth stating because it is the cleanest possible demonstration that a set is the wrong object. The standard key-finding algorithm correlates a weighted histogram of the twelve pitch classes against rotated copies of that profile. Give it a pitch-class set — every member weighted 1, every non-member 0 — and it returns the parent key, identically, for all seven rotations. Give it a completely flat histogram and the correlation is undefined, because Pearson’s r divides by a standard deviation and a flat vector has none.

The algorithm cannot be run on the object this ladder is about. What it takes to run it, and what that costs in seconds, is the next ladder’s third rung.

What the discard costs, in bits #

The essay has been saying most of the subject and costs everything without a size, and the size is available by counting.

There are 792 seven-note subsets of the twelve and they fall into 66 shapes. Seven is coprime to twelve, so no seven-note set maps to itself under any transposition — all 66 checked, none symmetric — which means every shape has exactly twelve transpositions and seven rotations, and the figure below is uniform rather than an average.

The ladder discards 6.392 bits per scale and its entire result identifies 6.044. Naming the diatonic set out of the 66 is worth six bits; the tonic and the key it is thrown away with are worth six and a half. The two are within six per cent of each other, which is a coincidence of the numbers and is also the cleanest possible statement of what the closing section is about: the census is not a small simplification of a large subject, it is one half of a subject that happens to be almost exactly halved.

The split inside the discarded half is the part with a musical statement in it. Transposition is 3.585 bits and rotation is 2.807, so the dimension nobody minds losing is the larger of the two — twelve keys against seven modes. That is worth saying because the essay’s own rhetoric runs the other way: rotation costs everything and transposition costs nothing, and yet transposition is the bigger number. What makes rotation expensive is not how much of it there is but that it is not recoverable from anything else, while a transposition is recovered the moment anybody names a pitch.

One number from the wider census belongs here too, because it says how special seven is. Across every size from one to eleven there are 350 shapes and only 15 of them have a transpositional symmetry at all — the whole-tone scale, the diminished collections, the augmented triad and their relatives. Every other shape sits in an orbit of exactly twelve, and every seven-note shape does. So the arithmetic of this section is not a property of the diatonic set: it is what a census costs for almost any scale anyone would build.

The two dimensions that are left over #

It is worth being precise about what a set class discards, because there are exactly two things and they are not the same kind of thing.

Transposition is discarded and nobody minds — though it is worth seeing what it costs before waving it away.

Seven neighbouring major keys against the chromatic scale are seven transpositions of one shape, and the census sees one object where a musician sees seven keys, six sharps and a modulation plan. That is the same blindness as the one this essay is about, moved sideways: a property of a set cannot distinguish things that differ only in where the set is.

The mode is its tritone. What decides which of the seven a passage is in is never a single degree and is always the same interval, which is as close as a census of sets can come to naming a home — and it is not close, because it names a pair of degrees rather than the one a listener would point at.

C major and D major are different keys and no theory claims they are different scales; the difference is a pitch standard and, before equal temperament, a set of measurably different interval sizes. In equal temperament the difference is pitch height and nothing else, so throwing it away costs nothing structural.

Rotation is discarded and it costs everything. There is no sense in which Ionian and Locrian are the same scale, and the census cannot see any difference at all.

So the honest summary of the whole ladder is: it is a theory of scale shape, and a scale shape is not a scale. The word “scale” in ordinary use means an ordered thing with a first note. What the six rungs before this one describe is the unordered thing underneath, and the reason to describe it is that it is the part that transfers — to another key, to another mode, to another tradition, to a rhythm.

The same blindness, put to work #

An invariance is usually written down as a limitation, and this one is also the reason the ladder’s results are worth anything.

A property that changed under rotation would be a property of a scale-with-a-tonic, and it would have to be restated seven times over, once per mode, and again in every key. A property that does not change is stated once and holds for all eighty-four. So the invariance is what makes “the diatonic set has two sizes of every step” a sentence about Ionian, Dorian, Phrygian, Lydian, Mixolydian, Aeolian and Locrian at once, in all twelve keys, without a word of extra work.

It also makes the results portable in a direction that has nothing to do with pitch. The maximal-evenness rung found that the same construction produces the diatonic set and the tresillo, and the reason the transfer works is exactly this invariance: a rhythm on a cycle has no first onset either, as its own ladder says in as many words. Two fields that both throw away orientation can share their arithmetic. A field that kept it could not.

So the thing this ladder cannot say is the price of everything it can. That is not a consolation; it is the design, and the six earlier rungs are all cashing the same cheque.

What the picture cannot show #

Invariance is not a limitation that better properties would remove. Any property computed from the distances inside a set is rotation-invariant, and every property this literature has proposed is of that kind. Getting a tonic out of a set requires adding information that is not in the set, and once it is added the result is not a property of the set.

The eighty-four-way check is a check, not a proof. It covers the diatonic set and the twelve. The general statement — a function of a set’s distance multiset is constant on its rotations — is a two-line argument rather than a computation, and the computation is here because a two-line argument that has never been run against the code is how the ticks bug survived twenty-four gates for a month.

The check is on one shape. Eighty-four sets is the diatonic set’s own orbit. Nothing here rules out some other shape whose properties differ between its rotations, and nothing could, because the argument that forbids it is the general one above rather than this computation. What the computation does is confirm that the general argument has been implemented correctly, on the one shape the ladder cares about.

And there is one thing rotation does change that this essay has not measured: the ear’s own preference. Given seven notes with no other cue, listeners do not choose a tonic at random — there is a bias toward the note that appeared first, the note that is longest, and the note at the bottom. Those are order and duration effects rather than set effects, so they sit outside this ladder in every direction; they are named here so that “the set cannot decide” is not misread as “nothing can”.

The anchor closes here #

This is the ninth rung and the last. The criterion is the one consonance closed on and room-acoustics closed on: not that no further rung could be written, but that the model the ladder built has been bounded in every dimension the model has.

The model is a subset of an equally divided universe, considered up to rotation and transposition. Its dimensions and the rungs that bound them:

• Which positions, and how unevenly — rung 1, the two semitones and where they sit.
• How the set is generated — rung 2, the chain of fifths and the prefix that is the pentatonic.
• Step-size variety — rung 3, two sizes of every generic interval.
• Interval multiplicity — rung 4, six counts, no two alike.
• Evenness as a construction — rung 5, the J-function and Bjorklund’s algorithm proved identical.
• Cardinality — rung 6, the census over all 349 shapes at every size from two to eleven.
• The series of sizes the generator admits — rung 7, the moment-of-symmetry cardinalities and their identity with the comma’s convergents.
• The size of the universe — rung 8, the same census in every n from four to thirty.
• Orientation — this rung: there is none, and the absence is a theorem rather than a gap.

The test is the one the previous closures used and it is checkable: name a variable the census’s answer moves with, and it is on that list. Change which positions are filled and rung 1 or 6 covers it; change how many, rung 6 or 7; change the universe, rung 8; rotate or transpose and nothing moves at all, which is rung 9.

Every remaining question has somewhere to go, and each of them is a question about something the set is not:

• Which note is home goes to modes, whose whole subject is rotation, and to tonal-expectation, which is where the profile lives.
• How keys relate to each other goes to key-relations, which needs a tonic before it can start.
• What the notes are tuned to goes to the-comma, where a scale degree is a frequency ratio rather than a position.
• Why the generator is the fifth goes to consonance, and it is the only part of the account that involves an ear.
• What happens outside an equal division goes to beyond-twelve, where there is no n and the census cannot be run at all.

Unlike room-acoustics, which closed with one question that had nowhere to go, this ladder ends with every remaining question filed — and the reason is not virtue. It is that the model is a piece of combinatorics with a small number of dimensions, and the things it cannot say are so completely outside it that each one is obviously somebody else’s.

Where the ladder went #

Nine rungs, one object, and the thing worth carrying is the last one rather than the first. The diatonic set is the most heavily determined object in this subject: fix twelve positions and four defensible properties and it is the only survivor, at a size and with a generator that are forced. That is a strong result and it is also strictly less than it sounds, because the same filter has a survivor in every universe of the right shape, and because the survivor it names is not a scale until somebody decides which of its notes is first.

The structure is forced and the music is in what the structure cannot say. That is not a disappointment; it is the division of labour that lets the next four ladders exist.