The set with fewer modes than notes

There are seven modes because there are seven notes. Every essay in this ladder has said so, and none of them has ever said why the two numbers should be the same one.

They usually are, and the usually is doing all the work. Start the same seven notes on a different degree and a different set of pitch classes results, seven times over, and the seven are all different. Do it to the whole-tone scale and the same six pitch classes come back — every time, all six times. Six notes, one mode.

This rung is about which collections behave that way, how many of them there are, and what happens to the previous rung’s question when they do.

The number of modes is a quotient #

Take a set, transpose it by t semitones, and ask whether the result is the same set. Usually it is not, for any t. Occasionally it is, and then the smallest such t divides twelve.

Call the order of that symmetry s, meaning 12/t. Rotating a set of size k gives k rotations, but the ones separated by t semitones are the same pitch classes, so the distinct rotations number k/s. That is the whole rule, and it is arithmetic that has never heard of a tonic:

The reason five, seven and eleven are empty is the neatest thing in the census. A symmetry of order s cuts the set into s identical pieces, so s has to divide k as well as twelve — and 5, 7 and 11 share no factor with 12. Every seven-note scale has exactly seven modes, and that is a fact about the number seven rather than about music. The diatonic set’s sevenfold family, the harmonic minor’s, the melodic minor’s, and the seven rotations of any of the other 789 seven-note sets in the universe: all guaranteed in advance.

That is also, in retrospect, why the harmonic minor’s rotations turned out not to be a family the way the diatonic modes are. The count was never in question. What was in question was whether the seven behave alike, and they do not.

Sixteen patterns, and everybody already knows them #

Seventy-five sets is small. Sixteen step patterns is a list.

They are: the tritone; the augmented triad; the diminished seventh; two four-note patterns of order two; the whole-tone scale; the hexatonic; four more six-note patterns of order two; two eight-note patterns; one nine-note; one ten-note; and the chromatic scale. Every collection that gets called symmetric in a harmony class is on that list, and the list has nothing on it that does not.

One distinct mode rather than six. Transposing the whole-tone scale by two semitones returns the same set, so five of its six rotations are the collection itself and there is nothing for a deciding set to decide. The census cannot find a mode here because there is only one, and that is a property of the collection rather than a failure of the method.

The musical consequence is the one composers reached for. A collection with a symmetry is a collection without a preferred starting point, and that is exactly what somebody wanting to write without a tonic wants. Debussy’s whole-tone passages and Messiaen’s modes of limited transposition are not exotic scales; they are the sixteen, chosen for the property this census measures.

One number, two consequences, and Messiaen named the other one #

The symmetry order does two things at once, and music theory has a separate name for each.

Rotating the set gives k/s modes — that is this ladder’s question. Transposing the set gives 12/s distinct results, which is somebody else’s. The whole-tone scale has two transpositions, not twelve: start it on C and it contains C, D, E, F♯, G♯, A♯; start it on C♯ and it contains the other six; start it on D and the first one is back. The octatonic has three, the hexatonic four, the diminished seventh three.

That is exactly the property Messiaen listed in 1944 as modes à transpositions limitées — limited transposition — and it is the same number as the mode shortage, arriving from the other side. A collection that is short of modes is short of keys by the same factor. One quotient, two symptoms, and the two are always taught apart.

The consequence for a composer is that the symmetric collections are cheap to move around and expensive to establish. There are only two whole-tone scales in existence, so any whole-tone passage is one of two objects and modulating between them changes every note; but no rotation of either is distinguishable from any other, so nothing inside a whole-tone passage can function as an arrival. The absence of a tonic and the shortage of keys are the same fact.

Two distinct modes rather than eight, because transposing by three semitones returns the collection. So the symmetric sets are not all alike either: the whole-tone scale collapses to one mode and the octatonic to two, and the number of modes is the size divided by the order of the symmetry rather than anything to do with how the notes sound.

Symmetry is the opposite of deepness #

There is a property this site has already proved about the diatonic set which turns out to be the exact complement of this one.

A set is deep when every interval class occurs a different number of times, and the diatonic set is deep — its interval vector is 2 5 4 3 6 1, six different numbers. The whole-tone scale’s is 0 6 0 6 0 3, and the octatonic’s is 4 4 8 4 4 4. Both repeat, heavily.

That is not a coincidence and it is not quite an implication either, but the direction is one-way and worth stating precisely. If a set has a symmetry of order s, its notes fall into s identical groups, so every interval count is a multiple of s and cannot be six different numbers unless s is one. Every deep set is asymmetric, and therefore every deep set has as many modes as notes. The converse fails badly — the section below counts how badly — but the useful half holds: deepness, which is what makes a set’s rotations tell each other apart, forbids the symmetry that would have collapsed them.

So the two rungs are one argument seen twice. Deepness gives the diatonic set a coordinate system for its rotations; symmetry takes one away from the whole-tone scale; and the census that found only one set with all four structural properties at once was, without saying so, working in the region of the universe where the mode count is guaranteed to be the note count.

How rare deepness is, counted #

“Most asymmetric sets are not deep” is doing a lot of work in that paragraph and it understates the case by two orders of magnitude.

Of the 4,095 non-empty subsets of the twelve, 75 are symmetric and 4,020 are not. Forty-eight of those 4,020 are deep — 1.2 per cent. And the forty-eight are twelve transpositions each of only four distinct collections:

Two of the four are chromatic clusters, which are deep for a reason that has nothing to do with music: a run of adjacent semitones contains every interval class a decreasing number of times by construction, so its vector counts down and its counts are trivially all different. Discard those and there are exactly two deep collections in the twelve-note universe, and one is the diatonic set and the other is the diatonic set with a note removed.

That is a much stronger statement than the essay was making. Deepness was introduced here as the complement of symmetry — the property that makes a set’s rotations tell each other apart — and it turns out to be so restrictive that at twelve it has essentially one non-trivial answer. The seven modes are not merely guaranteed to number seven; the collection they rotate is one of a pair, and its partner is its own subset.

And the rule in the other divisions #

The last figure claims the rule holds in any equal division. It does, and two more things hold with it.

Enumerating every subset of every division from six to twenty-four:

The right-hand column is zero everywhere, which is the one-way implication above verified across nine universes rather than argued in one. Every deep set in every division tested is asymmetric, so in every division a deep collection has as many modes as it has notes.

A prime division has exactly one symmetric subset and it is the whole thing. That is the claim made about nineteen, and it holds for seven, eleven, thirteen and seventeen too, for the reason the essay gives about five and seven: a symmetry of order s needs s to divide both the set size and the universe, and a prime universe offers no s but itself.

And the symmetric collections thin out as the universe grows. They are 14.3 per cent of the subsets of six, 5.9 per cent of eight, 1.8 of twelve, 0.4 of sixteen and 0.1 of twenty. The count of symmetric sets rises — 75 at twelve, 1,035 at twenty — but the number of subsets rises faster, so a composer working in a larger division has proportionally fewer places to go for a collection with no preferred starting point. Twelve is not the richest division in symmetric sets; it is a division small enough that its seventy-five are a list somebody can learn and large enough that sixteen distinct patterns exist to be learned.

Both collections are as evenly spread as their size allows, and only one of them is symmetric in the relevant way — eight into twelve leaves a pattern that repeats every three semitones, six into twelve one that repeats every two. Even spacing and transposition symmetry are not the same property, and it is the second that costs a collection its modes.

What happens to the previous rung’s question #

The rung before this one asked how much of a scale has to be heard before the mode is settled. It enumerated every set of degrees lying inside at least one of the seven modes, found 284 that leave exactly one standing, and reported fifteen minimal deciding sets — every one a pair, and every one a tritone or a tritone with a degree beside it.

Run the identical enumeration on the whole-tone scale and it returns something that is not an answer to the question.

That is the shape of a question dissolving rather than being answered, and it is worth naming, because the previous rung’s machinery reports it as a triumph — a hundred per cent decided on zero evidence. A measurement that returns a perfect score on a degenerate case is measuring its own assumptions.

More modes is not harder to tell apart #

The obvious guess, once the count varies, is that a collection with more modes takes more hearing to identify. It does not.

The octatonic is the sharp case. Two modes, and eight of the twelve pitch classes decide between them on their own — one note, no pair, no tritone. The reason is that its two modes differ in which semitone comes first, so any degree not shared by both settles it instantly, and four semitones apart there are a lot of those.

And the pentatonic makes the reverse case: five modes, no symmetry at all, and yet a single degree decides it — the major third above the tonic, or the minor sixth. Five modes, one note, because the pentatonic’s rotations are unusually unlike each other.

Which computation produced the numbers #

The symmetry order is a direct test: for each of the 4,095 non-empty subsets and each t dividing twelve, whether transposing by t returns the set. The step patterns are canonicalised by taking the lexicographically smallest rotation, so a pattern is counted once however it is spelled.

The deepness and cross-division counts are the same enumeration with the universe size as a parameter, using the site’s own isDeepIn — all ⌊n/2⌋ interval counts distinct — rather than a twelve-note special case, and the twelve-note row reproduces the figures the figure draws, which is what makes the other rows worth quoting. Sets of fewer than two notes are excluded from the deep count, because a set with no intervals in it satisfies the condition vacuously and would otherwise dominate every total.

Twenty-four is the largest division enumerated; beyond it the subset count passes sixteen million and the sweep stops being a thing to run while reading. Nothing in the pattern changes over the range covered, and the two claims that matter — that a prime division has one symmetric subset, and that no deep set is symmetric — are provable rather than merely observed, so the enumeration is a check on the argument rather than the evidence for it.

The determination census is the same function the previous rung used, called with a different parent. modeDetermination walks all 4,096 subsets of the twelve, keeps the ones lying inside at least one rotation, counts how many rotations survive, and calls a deciding set minimal when no proper subset of it decides. Nothing in it was written for the diatonic set; it takes a list of modes.

The arrival statistic is exhaustive over every ordering of a mode’s own degrees, with the tonic given, averaged over modes as well as orders. That last part matters and is a change of convention worth stating: the earlier rung reported 5.67 for Ionian, and the average over all seven diatonic modes is 5.38, because Lydian and Locrian settle at 4.67 while the other five settle at 5.67.

Whose music, and when #

The seven modes as a family are a mediaeval and Renaissance European construction, revived as a compositional resource around 1900 and taught as a fixture since. The symmetric collections are the later half of that same revival: whole-tone writing from the 1880s, octatonic writing running from Rimsky-Korsakov through Stravinsky and Bartók, and Messiaen’s modes à transpositions limitées of 1944, which is a published list of most of the sixteen.

What the census adds is that the list was not a discovery about music. It is a list of the subsets of a twelve-element cyclic group with a non-trivial stabiliser, and it would be the same list for a tradition that had never heard a fifth. What is musical is the use — that a collection with no preferred rotation is what a composer reaches for when a tonic is what they want to avoid.

What the picture cannot show #

A rotation is not a mode until something makes it one. Everything counted here is a set of pitch classes. Whether a listener hears one degree as home is a different question and this ladder has already found the answer is not in the notes: nothing in the census knows which note is home, and what a tonic costs is measured in seconds of emphasis rather than in set theory.

The determination census assumes the tonic is given. Every subset it scores includes degree zero by construction. A listener facing an unfamiliar collection does not know where zero is, and the joint problem — find the tonic and the rotation together — is strictly harder than either.

Twelve is assumed throughout, and the whole result is arithmetic about twelve. In nineteen equal divisions the symmetric sizes would be different, and 19 being prime, there would be no symmetric subsets at all except the whole thing — which is one of the reasons the other equal divisions do not simply inherit this repertoire.

And nothing here is weighted by use. The census treats the 924 six-note sets alike; music does not. Two of the sixteen symmetric patterns account for nearly all the symmetric writing anybody has done, and the census cannot see that. The same caution applies with more force to the deep collections: finding that only two non-trivial ones exist at twelve, and that both are diatonic, is a fact about a definition rather than a discovery about music, and it would be a different fact under any other definition of what makes a set’s rotations distinguishable.

The ladder ends here #

modes closes at nine rungs. The model is a collection of pitch classes, a chosen starting degree, and what that choice does — and the ladder now bounds it in every variable it has: what rotation changes (one), what a collection can be altered to repair (two), how long the choice takes to establish (three), which rotations are structurally disqualified (four), which one has a dominant (five), whether a second collection’s rotations behave like the first’s (six), how the family is ordered (seven), how much has to be heard to decide (eight), and now over which collections any of that is even defined.

Name a variable the model has, and it is on that list. What is not on it is not a further rung: which seven of the twelve to take in the first place is the-diatonic-set, and what happens outside twelve is beyond-twelve.

All of this is true inside twelve and not outside it. The symmetries that collapse these collections exist because twelve has divisors — a collection can repeat at 2, 3, 4 or 6 semitones — and in a universe of a prime number of steps no subset can be symmetric at all. The whole phenomenon this essay describes is a consequence of twelve being highly composite.