The one note that decides the mode

Two of the seven notes of a diatonic collection are a tritone apart, and only two. That is the least interesting fact about the diatonic set until it is asked to do something.

This essay asks it to identify a mode, and it turns out to be sufficient — not as a heuristic, but as the answer to an exhaustive count.

The question, put so it can be counted #

A key signature says which seven pitch classes are in play and not which of them is home. The page has no way to say the second, because all seven rotations of one set are the same seven staff positions; and a listener with no page has the same problem, which this collection has measured as a number of seconds.

So suppose the tonic is settled — a drone, a repeated bass, a phrase that has already landed — and the question is which of the seven modes the music is in. What the listener has is a growing set of degrees above that tonic, and each new degree is either consistent with a mode or rules it out.

That makes the question finite. Enumerate every subset of the twelve pitch classes; keep the ones that lie inside at least one mode; count how many modes each one leaves standing.

Five hundred and ten subsets lie inside at least one mode. Two hundred and eighty-four of them leave exactly one standing.

That framing is worth defending for a moment, because it is a strong idealisation and the strength is deliberate. It gives the listener a hard membership test — a degree either belongs to the mode or does not — where a real listener has a graded expectation and hears wrong notes, passing notes and inflections all the time. What that buys is a bound: whatever a listener is doing, they cannot be doing better than a perfect filter over subsets, so any number this census produces is a floor on how much has to be heard. A floor is the thing worth computing, because it is the part that does not depend on the listener.

What the minimal ones are #

A subset that decides is not interesting on its own — the whole mode decides it. What is interesting is the minimal deciding sets: the ones where removing any member leaves more than one mode possible.

There are fifteen, and every one of them has exactly two members.

Every minimal deciding pair for those five modes is a tritone, and it is that mode’s own tritone — the unique pair of its degrees six semitones apart.

Once seen, the reason is immediate and it is worth stating because it makes the result structural rather than numerical. A diatonic set contains exactly one tritone. A mode is a rotation of the set relative to a tonic. So the position of the tritone relative to the tonic is the rotation, and naming both of its members names the mode.

The two that need three #

Lydian and Locrian each have their tritone against the tonic: Lydian’s augmented fourth is the tonic and the sharp fourth, and Locrian’s diminished fifth is the tonic and the flat fifth. Both therefore have the pair (0, 6), and the tonic is already known, so hearing the sixth semitone above the tonic narrows the field to those two and no further.

They need one more degree, and the figure shows all five ways of supplying it — the six paired with anything that separates them.

That gives the census a clean shape:

Five modes are settled by one interval. Both members of the tritone settle it, and no smaller set does.

Two modes are settled by the tritone and one other degree, and they are exactly the two whose tritone touches the tonic — which is also, and not coincidentally, the pair a great deal of modal theory treats as the exceptions. Locrian has no perfect fifth above the tonic; Lydian has an augmented fourth against it. The reason is one fact stated twice.

Drawn as a necklace with the single tritone marked, the set has one such pair and rotating the ring moves which degrees it lands on without moving the notes — which is the whole mechanism in one picture.

What the pedagogy says, and what it assumes #

Every teaching account of the modes names a “characteristic note” — Dorian’s raised sixth, Phrygian’s flattened second, Lydian’s sharpened fourth, Mixolydian’s flattened seventh. That is the same information and it is offered under a different assumption.

Set it against the census. Dorian’s characteristic note is the sixth; the census says Dorian is settled by the minor third and the sixth. The pedagogy has dropped one of the two, and it can, because it has already established that Dorian is a minor mode — the minor third is assumed before the comparison starts.

So the characteristic note is one member of the tritone, and the other member is smuggled in as “minor” or “major”. The census, which assumes nothing, needs both. That is not a criticism of the pedagogy; it is a statement of what the pedagogy is doing, and it explains why the characteristic notes always come in a list that is grouped into major-ish and minor-ish first.

And no degree does it alone #

Two negative results fall out of the same enumeration and they are worth stating because both are easy to assume the other way.

No degree belongs to exactly one mode. Every one of the twelve pitch classes that any mode contains is contained by at least two, so there is no note whose mere presence settles anything. The sharp fourth is in Lydian and in Locrian; the flat second is in Phrygian and in Locrian.

Only the tonic is in all seven. Not the fifth — Locrian has none. Not the fourth — Lydian has none. The intersection of the seven modes is a single note, which is a much smaller common core than the usual talk of “the same seven notes” suggests, once the seven are measured from a fixed tonic rather than as an unordered set.

And the diatonic set has exactly one tritone for an arithmetical reason: it is seven adjacent positions on the chain of fifths, and six semitones is the interval between the two ends of a run of six fifths — a run of seven contains exactly one such pair. An evenly spaced seven-note set contains three.

The link is worth making explicit because it turns a curiosity into a consequence. Every interval a different number of times is one of the four properties the diatonic set has and almost nothing else does; deepness, as it is called, means the interval content is a set of distinct multiplicities. The tritone’s multiplicity is one. A set with a unique interval has a coordinate for its rotations, and a set without one does not.

How long it actually takes #

A minimal pair is a lower bound and not a prediction. A listener does not get to choose which degrees arrive; they arrive in whatever order the tune supplies, and the deciding pair is settled when its second member turns up.

That is an expectation and it can be computed exactly, over all 720 orders in which the six non-tonic degrees could arrive.

Averaged over every order, 5.67 of the seven degrees have to have been heard before the mode is determined, for five of the modes; 4.67 for Lydian and Locrian, which are quicker on average because the sharp fourth narrows the field to two as soon as it appears. The minimum in every case is three — the tonic and two others — and the best case happens in 48 of the 720 orders for the five, which is one time in fifteen.

So the tritone is a sufficient pair that arrives late. Nearly the whole scale has to be heard before the mode is pinned down, and the reason is not that the information is thin but that it is concentrated in one pair of notes and most orders do not deliver both of them early.

The distribution, which is a worse story than the mean #

An average over 720 orders hides its shape, and here the shape is the point. Counting how many of the 720 finish at each length:

For five of the seven modes the distribution is a rising ramp — exactly 48, 96, 144, 192, 240, which is 48 times one, two, three, four, five — so the single commonest outcome is the worst one. A third of all orders require every degree of the scale to have been heard, and only one in fifteen achieves the three-note minimum. The mean of 5.67 is not a typical case with a spread around it; it is the centre of a distribution that leans hard toward the far end.

And the two exceptional modes are not merely quicker, they are a different shape. Lydian and Locrian finish at the minimum in 240 of 720 orders — a third, against one fifteenth — and are flat at 120 thereafter. Their mean is lower because their distribution is front-loaded, not because they are uniformly faster. That is what having the tritone against the tonic buys: the sharp fourth arrives at some point in every order, and whenever it arrives early the field collapses to two immediately.

So there are two regimes rather than one number with two values. Five modes are identified late and almost never early; two are identified early a third of the time and otherwise no faster than the rest.

What a prior over modes actually buys #

The last caveat below says a listener with a prior needs less evidence than this census, because no listener treats all seven modes as live candidates. That is worth running, since the census takes a list of modes and does not care how long it is.

A prior buys almost nothing until it is almost total. Discarding Locrian, which is the mode nobody writes, saves a tenth of a degree. Discarding Lydian as well saves another tenth. Cutting the field nearly in half, to the four modes that account for the overwhelming majority of modal writing, saves three tenths of one degree out of five and a third.

The cliff is at two. A listener who has already decided the music is either major or minor needs 2.75 degrees — the tonic and rather less than two others — because with two candidates the third alone very nearly settles it and the census collapses to the question the pedagogy was answering all along.

That is the same finding as the section on characteristic notes, arriving from the other end. The pedagogy’s shortcut is not a small saving on the census; it is the entire saving, and it comes from the one step the census refuses to take. Narrowing seven candidates to five is nearly free of benefit; narrowing five to two is worth more than half the evidence. The information in a mode identification is not spread over the field of candidates — it is almost all in the major-minor split, and everything after that split costs nearly as much as the whole problem did.

What the census leaves out is that a listener is not equally likely to hear any two degrees: the probe-tone profile weights the tonic, then its triad, then the rest of the scale, so the two-note sets a real passage supplies are not drawn uniformly from the 510 the enumeration counts.

The difference between the two framings is the difference between a bound and a behaviour. The census says five modes cannot be told apart in fewer than three notes and gives the three; a profile model says how confident a listener is after each bar and never reaches certainty. Both are worth having and the first is the one that can be proved.

Why the tritone, and not something with a better reputation #

The result is a little surprising because the tritone spends most of its life on this site being the interval that has to be resolved. It is the one that inverts to itself, it is what makes the dominant seventh the only one of its kind in a key, and it was avoided in medieval theory for long enough to acquire a nickname.

All of those are the same property from different sides. There is one tritone in a diatonic set; it is the least stable interval in it; it is therefore the one whose resolution defines a direction; and it is the one whose position identifies the rotation. A set with one of anything has that one thing as its coordinate, and everything the tritone does in tonal theory follows from being the diatonic set’s unique element.

Across six scales, how many semitone sizes each interval name has to cover is two for the diatonic set and three for the harmonic and melodic minors — which is the property the tritone count is a special case of, and the reason the minor scales are harder to identify from two notes.

Which computation produced the numbers #

The census enumerates all 4,096 subsets of the twelve pitch classes, keeps the 510 contained in at least one of the seven modes, and counts survivors for each. A subset is contained in a mode if every pitch class in it is a degree of that mode; nothing is weighted, nothing is ordered, and the tonic is not required to be present.

Minimality is checked directly: a deciding set is minimal if removing any one of its members leaves more than one mode standing. That is what produces fifteen rather than two hundred and eighty-four.

The expected number of degrees is exhaustive rather than sampled — all 720 permutations of the six non-tonic degrees, for each of the seven modes, with the tonic counted as the first arrival. Five modes give 5.667 and two give 4.667, and the two figures are exactly 34/6 and 28/6.

The full distribution comes from the same enumeration by recording each order’s stopping length rather than only its mean, so it is the same 720 counts read one step earlier. The ramp is exact — 48, 96, 144, 192, 240 — and the fact that it is exactly 48 times the integers is a consequence of the deciding pair being a pair: an order stops at length k when the second member of the tritone arrives at position k, and the number of orders placing it there grows linearly.

The candidate-list sweep is the identical function called with shorter lists. It takes the modes as an argument and always did, which is why the sweep needed no new code and is a check on the census rather than a second implementation of it; the seven-mode row reproduces the 5.381 the ninth rung reports as the average over all seven.

The claim that all fifteen minimal sets involve the tritone is checked in the figure’s own code rather than asserted in the caption: the drawing tests whether every minimal set is either six semitones wide or contains the sixth semitone, and says so either way.

What the picture cannot show #

It assumes the tonic is known, which is the harder half of the real problem and is an accumulation with its own time constant. A listener deciding tonic and mode at once is solving a joint problem that this enumeration has cut in half.

It has no frequencies in it. Which degrees are heard is treated as a set, so a passage that dwells on the sixth for eight bars and touches the third once counts the same as one that does the reverse. The tonal hierarchy is a weighted thing and this census is unweighted.

It has no time in it either, so nothing here distinguishes hearing the tritone’s two notes together from hearing them four phrases apart, and those are not the same event for a listener.

It treats all seven modes as live candidates, which no listener does. Locrian is vanishingly rare and Lydian is uncommon, so a listener with a prior over modes needs less evidence than this census says. The section above computes how much less and the answer is: almost none, until the prior is strong enough to leave two candidates. This bullet previously added that the two modes a prior would discount hardest are the two the census says take longest to confirm; that contradicts the body of the essay, where Lydian and Locrian come out at 4.67 against the others’ 5.67. They are the quickest, not the slowest, and a prior that discards them therefore removes the two easy cases and leaves the field of five that is hardest.

And a mode is not only a set. A raga is a set with rules about direction, and the seven Western modes have their own conventions about which degrees are approached how. Everything in this essay is about the set, and the set is the part that a census can reach.

The ladder from here #

This rung asked how much of a mode has to be heard and got a two-note answer that arrives late. What the ladder still lacks is the same question asked of a listener who does not have the tonic — the joint identification, which is the real one — and a treatment of the modes whose sets are not the diatonic, where the unique tritone that makes all of this work is not available.