The series is not a chord

There is an argument about the major triad that turns up wherever anybody wants music theory to rest on physics. It goes: a vibrating string produces a harmonic series; the first six partials of that series are a root, an octave, a fifth, another octave, a major third and another fifth; those are the notes of a major triad; therefore the major triad is given by nature.

Every step of it is true. The conclusion does not follow, and the reason is in the phrase the first six.

Every other stopping point #

The series does not stop. Choosing to consider six of it is a choice, and the argument would be a different argument at any other number.

Reading that table as a sequence of derivations:

Stopping at four derives medieval organum. Stopping at six derives the Renaissance. Stopping at seven derives the blues. Each is as much “given by nature” as the others, and the series has no opinion about which.

The criterion that was never stated #

The obvious defence is that six is not arbitrary — that there is a reason to stop there. Two reasons are usually offered and neither survives.

“The higher partials are too quiet to matter.” This is a claim about a particular spectrum rather than about the series, and it is false for many instruments: a bowed string’s partials fall as one over n, so the seventh is a seventh of the fundamental’s amplitude and perfectly audible, and a clarinet or a reed organ stop has a good deal of energy well above the sixth. If loudness were the criterion the cut would fall in a different place for every instrument, and the triad would be a property of the violin rather than of nature.

“The higher partials are too rough.” This is checkable and the check is in the figure. At the fundamental the figure is drawn on, roughness per pair is lowest at the first four partials and climbs steadily; there is no knee at six, no discontinuity, and nothing distinguishing the sixth from the fifth or the seventh. If smoothness were the criterion the answer would be four, and the derivation would produce a chord with no third — which is what medieval theory in fact had, for four centuries, while looking at exactly the same series.

And the roughness criterion does not even give a stable answer #

That is generous to the roughness defence, because it grants it a determinate answer to be wrong about. It does not have one. Recomputing the same per-pair roughness at five fundamentals:

The answer runs from three to ten depending on where the series is played, and at the bottom of the range the ordering reverses outright: roughness falls with every partial added, so the smoothest prefix is the longest one tried and a roughness criterion would derive an ever-larger chord.

The mechanism is the one the whole site’s roughness machinery rests on. A critical band is a fixed fraction of a frequency, not of a ratio, so at a low fundamental the first few partials sit close together relative to the band and are rough, and adding higher partials — which are widely separated relative to their own bands — dilutes the average. At a high fundamental even the octave is comfortably resolved, so every prefix starts smooth and each addition can only make it worse.

Which means the caption below is half wrong, and the half it gets wrong is the load-bearing half. Pitch-class content really is a property of the ratios: four partials give two classes, five and six give three, seven gives four, ten gives five, sixteen gives eight, at any fundamental whatever. Roughness is not, and the second prefix figure is offered as a check that the argument is about the series rather than about the note it was drawn on. It is not that check. Drawn an octave down it would show the opposite ranking.

None of that rescues the six-partial derivation — it makes the case against it stronger, since a criterion that gives three answers at three registers cannot select any of them. What it costs is the tidy secondary claim that the answer would have been four. The answer would have been four at C3, three above it and ten below.

So the stopping point is chosen by the answer. That is not a scandal; it is how a great many derivations work. It is only a problem when the result is presented as having been derived rather than selected.

What the seventh partial actually is #

The reason nobody stops at seven is worth making concrete, because it is the most interesting number in the series.

Thirty-one cents is not a subtlety. It is a syntonic comma and a half, eight times the smallest audible pitch difference at that frequency, and unmistakable to anybody. A seventh partial played against a keyboard’s minor seventh beats hard and continuously.

The natural seventh is as periodic as any of the simple ratios — seven against four repeats after four cycles of the lower note — and its consonance is not in doubt: it is smoother than the equal-tempered minor seventh, which is why barbershop singers and brass ensembles find it and hold it. The seventh partial is not excluded because it is dissonant; it is excluded because it is unavailable. A theory that derives its chords from the series and its pitches from a twelve-note keyboard has to stop before the two disagree, and they first disagree at the seventh.

That is the crux. The seventh partial is not excluded because it is dissonant; it is excluded because it is unavailable. A theory that derives its chords from the series and its pitches from a twelve-note keyboard has to stop before the two disagree, and they first disagree at the seventh.

What the exclusion has cost #

The traditions that did not have the keyboard constraint did not make the exclusion, and the differences are audible.

Barbershop harmony tunes its dominant seventh chords with a natural seventh and the characteristic “ring” of the style is the resulting beatlessness. Brass ensembles playing without valves — where the available notes are the series itself — produce it as a matter of course. Several West African and Southeast Asian traditions use intervals in that region. And the blue seventh of African-American vocal practice sits in the same territory, which is not a claim that it is the seventh partial but is a reason to be careful about calling it a flattened note.

And the roughness curve for a harmonic spectrum puts its wells at the fourth, the fifth and the octave — the intervals the low partials supply — with nothing at the seventh’s ratio, because the curve is computed against a twelve-note sweep that never visits it.

The minor triad, which the derivation never gets to #

There is a failure of the six-partial argument that is more damaging than the choice of stopping point, and it is the one its own proponents noticed first.

The series contains a major triad. It does not contain a minor one anywhere — not in the first six partials, not in the first sixteen, not at any prefix. Partials 10, 12 and 15 are a minor triad in pitch-class terms, but they are the tenth, twelfth and fifteenth, arriving after five other pitch classes have already appeared, and no principle that selects six as a stopping point can reach them.

Every attempted repair has the same shape: an additional principle, introduced for this case, which the series does not supply. Rameau’s own was that the minor triad arises from the same fundamental heard differently; the dualists’ was the undertone series; twentieth-century versions appeal to voice leading or to symmetry. All of them are arguments that the major triad is prior and the minor a modification, which is a claim about a repertoire and not about physics.

The series is not a chord in either direction #

There is a second half to this that is easy to miss.

Reading the series downward — root over k rather than root times k — gives the undertone series, which nineteenth-century theory proposed as the physical basis of the minor triad. That proposal was examined here and the objection to it is not that the arithmetic fails but that no physical system produces it: a string vibrating in k parts is a real mode with k−1 nodes, and a string vibrating in one k-th of a part is not a thing.

So the series is not a chord going up, because where to stop is unstated; and it is not a chord going down, because nothing produces it. What it is, in both directions, is a ratio table — and a ratio table is a fine thing to build a theory of intervals on and a poor thing to derive a specific chord from.

Read as evidence, it points somewhere else #

If the derivation is a selection rather than a deduction, the question is what the series is evidence for, and the answer is better than the received one.

That is an explanation with no free parameter in it, and it produces the interval hierarchy every tradition uses. The step it does not take is from intervals to a particular three-note chord — and that step is where the stopping point has to be introduced, because a chord is a selection and the series is not.

What the series does explain #

Setting all of this out makes the harmonic series sound like a red herring, and it is nothing of the sort. It explains a great deal; it simply does not explain the triad by prefix-taking.

It explains why the octave, fifth and fourth are the intervals every tradition treats as structural: they are the low-order ratios, they are where the roughness curve has its deepest wells, and they are the coincidences between two harmonic spectra.

It explains why a spectrum and a scale go together — a change of spectrum moves the wells, and the intervals a tradition uses follow.

It explains why a note has a pitch at all, and why a bell does not.

And, one register up, it explains what a valveless brass instrument can play — which is the same object read as a set of melodic steps rather than as a chord, with the opposite conclusion. The partials that make the series unusable vertically are the ones that make it usable horizontally, because a melody visits a note and leaves it, and a chord holds it against others.

Same sixteen numbers, two directions of reading, two opposite verdicts. That is the strongest reason to treat the series as material rather than as an explanation.

The one number that does single out six #

For completeness, there is a criterion that picks six out, and it is worth naming because it is the honest form of the received argument.

Six is the largest prefix whose pitch classes are three in number. Four gives two, five and six give three, seven gives four, ten gives five. If the thing being derived is a triad — a three-note chord — then six is the last stopping point compatible with the target, and the derivation is: assume the answer has three notes, and take the longest prefix consistent with that.

That is a perfectly respectable piece of reasoning and it is not a derivation of the triad from the series. It is a demonstration that if the chord has three notes, the series names which three, which is a real and non-trivial result — the series could have given a diminished triad or a stack of fourths and it does not.

What the series supplies is which three, given three. What it does not supply is three.

And that formulation survives the register sweep, which the roughness argument does not — because the pitch-class count is arithmetic on ratios and carries no frequency anywhere. Four partials give two classes and seven give four at C2, at C6 and on an instrument nobody has built. So the one criterion that does pick six out is also the only one in this essay that is a property of the series rather than of where the series was played, which is a better reason to prefer it than the fact that it gives the received answer.

Which computation produced the numbers #

The deviations are 1200 log₂(n) minus the nearest multiple of 100, evaluated for each partial. Nothing is quoted.

The roughness of each prefix is the Plomp–Levelt model this site uses everywhere, summed over every pair of partials in the prefix and divided by the number of pairs. The pure-tone form is used because the partials of a harmonic series are pure tones — treating each as though it had a spectrum of its own would be counting the same energy twice.

Dividing by the number of pairs is the load-bearing choice and it should be stated. The raw total rises with the prefix simply because there are more pairs; per pair is the honest comparison, and it is what makes the minimum a real minimum rather than an artefact of counting.

It is not, however, what makes the minimum land at four. That is the fundamental, and the register sweep above is the same per-pair computation run at five of them. The choice of C3 for the main figure is not defended anywhere and was presumably made because it is a comfortable place to draw a series; had it been drawn at C4 the essay would have reported a smoothest prefix of three, and at C2 it would have reported that roughness falls with every partial added and never turns round. Both of those are the same model on the same series.

Whose theory, and when #

The six-partial derivation is European and its canonical statement is Rameau’s, from 1722, though the observation that a string sounds its own partials is much older. Its authority in modern teaching comes largely from nineteenth-century acoustics — Helmholtz gave it a physical mechanism, and a physical mechanism is what a theory of harmony had been looking for.

The medieval stopping point is not a hypothetical. European practice from roughly 900 to 1300 treated the octave, fifth and fourth as the consonances and the third as a dissonance requiring resolution, which is exactly the theory that a four-partial prefix would give. The change was not caused by anybody looking further up the series; it happened in practice first, in England and then across the continent, and the theory followed.

That sequence is the point. The series did not tell anybody where to stop; the music did, and the series was consulted afterwards.

What the picture cannot show #

Real spectra are not the series. A piano string is stiff and its partials are progressively sharp; a bell’s are not harmonic at all; a struck bar’s are wildly inharmonic. Every number here is about the ideal series, and the ideal series is an idealisation of a flexible string and an open pipe and of nothing else.

Roughness is one model among several. The Plomp–Levelt curve is a good model with known limits, and this collection has already found it insufficient to select a scale. Using it here to show that no roughness argument selects six is a weaker use than using it to select something, which is why the conclusion is negative — and the register sweep makes that use weaker still in one direction and stronger in the other. Weaker, because a model whose answer moves by a factor of three across the compass is not a model to draw a conclusion from. Stronger, because the conclusion being drawn is that it selects nothing, and a model that selects three different things is a model that selects nothing rather more emphatically than one that selects the wrong thing consistently.

And a chord is not a sum of pairs. Everything above scores a prefix by summing over pairs, which has no term for the fact that a chord has a root, an inversion and a function. The subharmonic matching that finds a root is a different computation and it does distinguish the prefixes — a point this essay does not develop and which is the obvious next rung.

The ladder from here #

This anchor has three rungs and is the thinnest ladder in the collection relative to how much of the site rests on it: the harmonic series is the object teal is reserved for, it appears in a third of the essays here, and it has had three arguments made about it.

The two it most obviously wants: what happens to all of this when the spectrum is not harmonic, which is a question every percussion essay here answers piecemeal and none answers directly; and the root question above — score each prefix by how strongly it implies a fundamental rather than by how rough it is, and see whether that criterion selects six. If it does, the received derivation has a defence it has never been given.