Three is the largest agreeable number

Western harmony is built out of three-note chords. That is usually presented as a historical fact with a mechanism attached — thirds were stacked because thirds became consonant in the fifteenth century, and stacking two of them gives three notes — which is a description of what happened rather than a reason it should have.

The reason can be looked for directly. There are only fifty-five three-note chords available from the twelve pitch classes if the root is held fixed, which is few enough to examine all of them, and there are two independent things worth measuring about each one.

The two axes measure genuinely different things, and the figure’s shape is the argument: the chords are spread over both, and the interesting corner is empty except for one point.

The two measures #

Roughness is the Plomp–Levelt sum: for each pair of notes in the chord, the interference between every pair of their partials, added up. It is a sensory quantity with a published model behind it, and it has a strong bias that has to be stated. Two partials stop being rough once they are far enough apart, so roughness rewards spacing. A chord whose notes are widely separated scores well almost regardless of what the notes are.

Ratio simplicity is the classical measure and it is exact arithmetic. Take each interval at its 5-limit just ratio, put the three over a common denominator, cancel, and read off the integers. The major triad is 4:5:6. The minor triad is 10:12:15. Stacked fourths are 15:20:27. There is no tolerance anywhere in that computation, which is the reason to do it this way.

The two are close to independent, and it is easy to be badly served by either alone.

What each measure says on its own #

The smoothest three-note chord available is 0–5–10 — stacked fourths, C–F–B♭. Its pairs are five, five and ten semitones, all comfortably wide, and nothing in it beats. Written as ratios it is 15:20:27, which is not a chord any tradition treats as a place to arrive at.

Going the other way is no better. The simplest integers among three-note chords that are not triads belong to 0–2–4, which is 8:9:10 — three consecutive harmonics, about as simple as a ratio set gets, and one of the roughest chords in the collection. Three notes a tone apart grate, and no amount of arithmetic elegance changes that.

So each measure alone endorses something absurd. The interesting question is what survives both.

Exactly one chord survives both #

Taking the best eighteen per cent on each axis and asking what is in the intersection gives, out of fifty-five, one chord: 3:4:5. The eighteen is swept below and the answer does not depend on it.

That is the major triad. It appears here as 0–5–9 rather than 0–4–7 because the census holds the root fixed at C and the arrangement in which the triad’s ratios come out as 3:4:5 is the one with C on top — F–A–C, the major triad in first inversion. The set is the same object under a different name, and the fact that its simplest integer form is an inversion rather than root position is itself worth noticing: the arrangement that fits a harmonic series most cheaply is not the one theory calls fundamental.

The minor triad is not in the intersection at eighteen per cent, and that is a real result rather than a defect. As 10:12:15 it is markedly more complicated than 4:5:6, and on the roughness axis it is a little worse than several chords nobody uses. The asymmetry between the two triads is genuine, it has been noticed for centuries, and every attempt to make the minor triad come out as simple as the major one has required assuming what it set out to prove — usually by inverting the harmonic series and calling the result a subharmonic series, which is not a thing a physical object produces.

Why two measures rather than one #

It would be tidier to have a single number. Several have been proposed, and the reason none of them is used here is worth setting out, because it explains what the two axes are actually doing.

The oldest single measure is Euler’s gradus suavitatis of 1739, which takes the least common multiple of a chord’s integers and scores it by the sum of its prime factors. It is elegant, it is exact, and it is entirely a statement about arithmetic: it knows nothing about registers, spectra or ears, so it says the same thing about a chord in the bass as in the treble, and it says the same thing about a chord played on a clarinet as on a bell. Every one of those is a case where the sound genuinely differs.

The modern single measure runs the other way. A sensory model computes roughness from the actual spectra at the actual frequencies, so it knows all the things Euler’s does not — and it has no notion of a ratio at all. What it reports is a continuous surface with shallow local minima, and it will happily rate a chord of six notes spread across four octaves as the smoothest thing in the collection, because it is.

The two are not competitors and they are not the same claim measured twice. One is about the relationship between the notes, and the other is about the sound the mixture makes. A chord that scores well on both is one whose notes are related simply and which also happens not to grate at the register it is played in, and there is no reason in advance for those to coincide. The finding is that at three notes, exactly one chord manages it.

There is a third thing neither measures, and it is the reason the argument stops here rather than claiming too much. A chord’s fusion — whether the ear reports one object or several — is a separate perceptual fact, closer to what makes a missing fundamental audible than to either axis above. It correlates with ratio simplicity without being identical to it, and adding it as a third axis would not change which chord is in the corner at three notes.

Where the criterion stops working #

The double criterion is only interesting if it discriminates. Running the same census at other chord sizes shows where it stops.

At two notes the answer is the fifth, and there is nothing surprising in it. At three notes the answer is the major triad, which is the result this essay exists for. At four notes it falls apart.

Eight winners at four notes, twenty at five. The criterion does not fail in the sense of returning nonsense — the eight include the dominant seventh, the minor seventh and the half-diminished, which are exactly the four-note chords common practice does use — but it stops picking one, and a criterion that admits eight equally good answers is not the reason a tradition chose any of them.

The eighteen per cent, swept #

Eighteen per cent is a stated choice and the whole result rests on it, so it is worth turning. Sweeping the cut from four per cent to a half and counting the intersection at each chord size:

The three-note answer holds over a five-fold range of the threshold and eighteen per cent sits inside it rather than at its edge. That is the audit the number needed: it is not tuned to produce one winner, because any cut from a twenty-fifth of the field to a fifth produces the same one. The two-note answer is broader still, which is what a result nobody would dispute ought to look like.

The four-note row is where the finding has to be restated, and the restatement is sharper than the original. It is not that four notes admits eight winners; it is that four notes admits a different number of winners at every threshold — one at four per cent, two at six, three at eight, five from ten to sixteen, eight at eighteen, nine at twenty. The count never settles, so there is no plateau to read an answer off. At two and three notes the criterion has an answer and the threshold is a detail; at four the threshold is the answer.

One detail of that row is worth a sentence on its own. The chord that wins at the tightest cut is the major seventh, not the dominant seventh — 8:10:12:15, which is a major triad with a major third stacked on it, and the four-note chord common practice treats as a dissonance requiring preparation while later repertoires treat it as a place to stop. So the double criterion at four notes, pushed to its sharpest, names a chord whose history runs the opposite way to the two it names at two and three, which is another reason not to read it as the reason a tradition chose anything.

That is the sense in which three is the largest agreeable number. It is the largest chord size at which smoothness and simplicity, taken together, still name a single object. Above it, harmony has to be decided by something else — by voice leading, by function, by convention — and, historically, that is exactly when those things arrive.

The two triads, and the asymmetry nobody has explained away #

The minor triad deserves more than a note in passing, because the census makes its awkwardness quantitative and the awkwardness is old.

In just intonation the major triad is 4:5:6 and the minor is 10:12:15. Those are not close. The largest integer more than doubles, and the ratios between adjacent notes go from 5:4 and 6:5 to 6:5 and 5:4 — the same two intervals in the other order, which sounds as though it should cost nothing and costs a great deal in this measure. The reason is that 4:5:6 has all three notes as low harmonics of a single fundamental one octave and a fifth below the root, and 10:12:15 does not: the smallest fundamental that fits it is three octaves and more below, which is to say the chord does not present itself as one object nearly as readily.

Rameau in 1722 gave the major triad a derivation from the harmonic series and could not do the same for the minor one; the corps sonore produces 4:5:6 and produces nothing resembling 10:12:15. Riemann’s answer, in the nineteenth century, was to posit an undertone series — the harmonic series reflected downwards — in which the minor triad is as simple as the major. It is a beautiful symmetry and there is no physical object that produces it. A string vibrates in halves and thirds; nothing vibrates in doubles.

So the asymmetry stands, and the honest statement is that the minor triad is a genuinely more complicated object which musical practice uses as freely as the simpler one. That is a fact about music rather than a hole in the account: whatever makes a chord usable is evidently not exhausted by being smooth and simple, which is the same conclusion the four-note census reaches by a different route.

The model, and what it assumes #

Three assumptions are doing work here and each is a place the answer could change.

The roughness axis assumes a string spectrum. Change it and the whole census moves, which is not a weakness of the method but its most testable prediction. The consonant intervals for a spectrum are a property of that spectrum, and a census run on an inharmonic timbre puts different chords in the corner.

The integer axis assumes just intonation, which no keyboard has. The same computation on tempered frequencies has no exact answer at all: it has to fit a fundamental to the chord within some tolerance, and the ranking then depends on the tolerance in a way that makes it useless. At sixteen cents of forgiveness the tempered major triad is the second-best chord of the fifty-five; at twelve cents — a difference nobody can hear — it is the thirtieth. A measure that unstable is not measuring the chord, and stating the assumption is the price of a measure that is stable.

The threshold of eighteen per cent is a choice. It is not tuned to produce this answer: the figure reports whatever is in the intersection for whatever fraction it is given, and the shape of the result — one at two notes, one at three, many at four — holds across a range of thresholds rather than at one.

Whose harmony, and when #

The claim being tested is a claim about a repertoire, and it is worth naming which.

Three-note chords as the unit of harmony belong to European music from roughly the fifteenth century to the nineteenth. Before that, the vertical sonorities of medieval polyphony are largely fourths, fifths and octaves — the intervals the two-note census picks out — and thirds are treated as things that pass rather than things to arrive on. The change is documented and datable: English practice of the fourteenth and fifteenth centuries, fauxbourdon, and the theorists who followed it in codifying the third as a consonance rather than a dissonance.

That history fits the census in a way worth noticing. If the double criterion were describing something universal, the medieval practice would be inexplicable. If it were describing nothing, the convergence on triads would be a coincidence. What it seems to describe is a constraint: the triad is available to be found, on the spectra of the instruments and voices in use, and finding it took several centuries of practice rather than a derivation.

Outside that repertoire the census’s assumptions stop holding one by one. Gamelan tunings sit on genuinely inharmonic spectra, and the ratio axis has nothing to say about a metallophone whose partials are not whole-number multiples of anything. Much of the world’s music is monophonic or heterophonic, where the question of which simultaneous chord to prefer does not arise. And a great deal of twentieth-century Western music is written specifically to avoid the corner of the figure this essay is about, which is a use of the same knowledge rather than a refutation of it.

What the picture cannot show #

The larger limit is what the two axes are measuring. Both are computations over a spectrum, and the cross-cultural evidence says preference is the half that is learned — so the census identifies the chord a roughness model and an integer model jointly single out, which is a real and surprising result, and not a demonstration that anybody had to like it.

The census treats a chord as a set of pitch classes, which throws away everything about voicing. A major triad spread over three octaves and the same triad squeezed into five semitones are one point here and two quite different sounds — and the difference is precisely the roughness the model is supposed to capture, appearing at a level of detail the pitch-class abstraction has discarded.

It has nothing to say about register, either, which is the variable that decides whether a given interval is rough at all and which the census fixes at one root.

It also has nothing to say about what a chord does. Being smooth and simple is a property a chord has sitting still, and most of what harmony is about is motion: which chord follows which, and what the smallest move between two of them costs. A chord that scores badly here can be indispensable for what it resolves to, and the tritone is the standing example — an interval nothing recommends on either axis, and the engine of the entire tonal system.

The next rung up this ladder is that motion. Having found which chords are worth resting on, the question becomes how a piece gets from one to another, and why the shortest path is not always the one that sounds final.