The root an ear supplies

A triad has a root, and the root is not simply the lowest note. It survives inversion — which is the whole reason the idea is useful and the reason the previous rung’s prohibition is so hard to explain — so whatever fixes it is a property of the set of notes rather than of the arrangement.

The best available account is one this collection has already built the machinery for. A set of partials with no fundamental still has a pitch: the ear finds the fundamental those partials would belong to, and reports it. A triad is a set of tones rather than a set of partials, but nothing in the mechanism cares about the distinction, so the same matcher can be handed one.

The major triad, which behaves #

The answer is 4:5:6, the fundamental is two octaves below the bass, and there are no empty slots — the notes are consecutive terms of the series. That is as clean a result as this kind of matching produces.

Tune the triad justly and it becomes exact.

Partials four, five and six of a string at 65.4 hertz are 262, 327 and 392 — a major triad in root position, two octaves up — so the chord the matcher finds a fundamental for is the chord a single note already contains.

So the major triad is a slice of a harmonic series, its root is the fundamental of that series, and the root survives inversion because moving a partial by an octave does not change which series it belongs to.

The minor triad, which does not #

Run the same matcher on a minor triad and something different happens.

That is the ambiguity, and it is worth being precise about what kind of thing it is. It is not that the model has no answer. It is that the model’s two criteria — completeness of the series and accuracy of the fit — point at different notes, and there is no principled way to weigh one against the other.

In just intonation the ambiguity takes a different and cleaner form.

The just minor triad is exactly 10:12:15 and nothing else fits. So there is a unique answer, and the answer is a fundamental more than three octaves below the chord’s lowest note, reached by skipping the eleventh, thirteenth and fourteenth partials. Compare the major triad’s 4:5:6 — two octaves down, consecutive.

The asymmetry is real, it is arithmetical, and it has nothing to do with anybody’s preference. A minor triad is a perfectly good chord and is not less consonant by the roughness measure in any interesting degree. What it lacks is a low, complete, unambiguous position in a harmonic series, and a listener’s pitch-inference machinery is therefore working with much weaker evidence about where its root is.

The partials the matcher was not given, which turn out to change nothing #

The last caveat below records that the matcher is handed three bare tones where a played chord brings its own partials, that the set to be matched is therefore nine or twenty-four components rather than three, and that whether this helps or hinders is not obvious. It is obvious once written down, and the answer is neither.

If the three notes are harmonics n₁, n₂, n₃ of some candidate fundamental, then the kth partial of note i sits at k·nᵢ·f₀ — which is harmonic k·nᵢ of the same fundamental, automatically. And the error in cents is unchanged, because the common factor cancels: 1200·log₂(kf / knf₀) is 1200·log₂(f / nf₀). A candidate that explains the three fundamentals to within some number of cents explains every partial of every note to within exactly the same number.

Running it confirms the algebra with nothing left over. Scoring each candidate against the growing component set:

Not a digit moves. The partials are exactly redundant with respect to accuracy, so the minor triad’s ambiguity is not an artefact of having stripped the chord down to three sine tones — it survives giving the chord back everything it has.

Where they are not redundant is the harmonic numbers, and that is the parameter the caveat should have named. The sixth partial of the G in a minor triad is 2,352 hertz, which against a fundamental of 26 hertz is harmonic ninety. Every model of this kind has a limit on how far up the series it will look — this site’s is sixteen — and with the partials included every candidate is past it, so the matcher returns nothing at all rather than returning something worse.

That is a more useful conclusion than the one the caveat expected. The evidence in a real chord is identical in quality and much more demanding in extent, so what decides whether a triad has a computable root is not how many components it has but how deep a subharmonic template the listener is credited with. The ambiguity between 6:7:9 and 10:12:15 is untouched by realism about the sound and entirely at the mercy of one number in the model.

Two centuries of trying to fix this #

The asymmetry was noticed as soon as anybody tried to derive harmony from the harmonic series, which is to say by Rameau in 1722, and it has never stopped being a problem for that project.

Rameau derived the major triad from the series and was left with the minor one, which he explained at various times by a co-vibrating string an octave and a twelfth below, by the sons harmoniques of a lower fundamental, and — in the Génération harmonique of 1737 — by a set of strings vibrating in sympathy at subharmonic frequencies. He changed his mind more than once, and the changes were not evasions: he could see the problem clearly and none of the available answers worked.

The nineteenth-century answer was bolder. If the major triad is the fourth, fifth and sixth terms of a series counting upward from a fundamental, then the minor triad is the same three terms counting downward, and its generator is its own fifth rather than its own root. Hauptmann, von Oettingen and Riemann built entire systems on that symmetry, and it is called dualism.

The mirror really is perfect, and that is what made it convincing. Every relation in the major system has a dual in the minor one; the two triads are reflections; the whole apparatus of function — tonic, dominant, subdominant — reflects with them, which is why Riemann’s minor keys have their functions upside down.

Why there is no undertone series #

The objection is not aesthetic and it is not about elegance. It is that nothing produces one.

A harmonic series exists because of a boundary condition. A string fixed at both ends can only support standing waves with nodes at the ends, which quantises the wavelength to the length divided by a whole number, which puts the frequencies at whole multiples of the lowest. A stopped pipe has a different boundary condition and gets a different subset of the same ladder — odd multiples only — and that is exactly the sort of variation the mechanism allows.

What no boundary condition permits is a mode at a frequency below the fundamental, because the fundamental is by construction the lowest wave the object can support. There is no way to divide a frequency by asking a string to vibrate in a third of a part.

Sympathetic resonance does not supply one either, which was Rameau’s hope. A string will resonate when driven at one of its own modes, so a low string responds to a high note that happens to be one of its harmonics — but what it produces is its own harmonic, at the driving frequency, not a subharmonic at a third of it.

Period-doubling in strongly nonlinear systems is real and it does produce a component at half the driving frequency; it is why a badly overblown pipe can drop an octave. It produces a division by two, occasionally by three, in specific unstable regimes. It does not produce a ladder, and no instrument in ordinary use is operating there.

So the undertone series is a diagram rather than a spectrum. It describes a symmetry in the arithmetic of intervals, which is real, and asserts a physical origin, which is not there.

And yet the template exists — in the listener #

Here is the part that makes the dualists more than a curiosity, and it is the point this rung exists for.

The matcher used at the top of this essay is doing exactly what the dualists wanted: it is comparing a set of tones against a subharmonic template and asking which fundamental best accounts for them. That is Terhardt’s model of virtual pitch, it is the standard account of the missing fundamental, and the template it uses is a series of subharmonics of each heard component.

And the template the matcher is a model of does exist, in the listener: remove the first three partials of a harmonic complex and the pitch a listener reports is still the fundamental, because the mechanism fits a series to what is present rather than reading off what is lowest. That mechanism is real, it is measured, and it is what the major triad’s fit is a use of.

So the nineteenth century was half right in a way it could not have known. There is something in the argument that works downward from a heard frequency to a lower one; it is a property of the listener rather than of the vibrating body, and it is the same property that supplies a bell’s strike note and a drum’s.

What the correction changes is what the theory can claim. A template in the ear explains why a minor triad has a root at all and why it is ambiguous; it does not license the symmetric system, because the template is applied to whatever is heard and is not a second physics running alongside the first.

The measurement this predicts, and it holds #

If a minor triad’s root is inferred from weaker evidence than a major triad’s, three things should follow, and they are all observable.

Minor triads should be more sensitive to inversion. Support from the bass is a separate measurement, made in the previous rung, and it says so: a root-position major triad scores 1.533 and a root-position minor one 1.333, while the minor triad’s first inversion scores 1.200 — much nearer its root position than the major triad’s first inversion is to its own.

The ambiguity should be larger in the bass. It is: at low frequencies the partials that would resolve the question fall inside a single critical band, and the practical rule every orchestration text gives — do not voice minor thirds low — is the same rule this collection derived from critical bandwidth.

And the minor triad’s root should be fixed by something other than the ear. It is. In practice nobody is in any doubt about which note of a minor triad is the root, because the key, the bass line and the progression settle it long before any inference from the spectrum could — which is the same conclusion the previous rung reached about the six-four, arriving from the other side.

On the Tonnetz a major triad’s triangle points one way and a minor triad’s the other, and no rotation of the plane turns one into the other — the same asymmetry with no arithmetic in it. And ranked over every three-note chord in the system, the major triad is the one chord in the best eighteen per cent on both roughness and integer simplicity, which is the same finding arrived at from the other end.

What this does not settle, and it is the important one #

None of the above says why the minor triad is used. It says why its root is a weaker inference, and those are different claims that get run together constantly.

The temptation, once the asymmetry is established, is to conclude that the minor triad is a defective major one — less natural, borrowed, secondary. That conclusion does not follow from anything here and the arithmetic argues against it.

And a great deal of music is built on collections in which the distinction does not arise at all. A pitch-class set has no orientation — the diatonic set’s whole census is invariant under rotation, so nothing in it can prefer a major triad to a minor one — and the traditions that use neither as a harmonic unit have nothing to explain.

What the asymmetry does explain is narrower and firmer: why two centuries of theorists could derive one triad from a physical fact and not the other, and why the attempted fix had to invent a spectrum. The evidence for a minor triad’s root is genuinely weaker, the model says how much weaker, and what supplies the rest of it is syntax.

What the picture cannot show #

The matcher is a model with three parameters, and they matter. How many harmonics it will consider, how large an error it tolerates, and how it weighs an empty slot are all choices. Sweeping the first two over sixteen combinations — tolerances of 15 to 40 cents against harmonic ceilings of 12 to 32 — settles which of them the answer turns on, and it is not the one recorded here.

The major triad’s winner is 4:5:6 in every one of the sixteen, at a worst error of 10.1 cents. The minor triad’s winner flips with the tolerance: at 15 and 20 cents it is 10:12:15 at 9.9 cents, and at 30 and 40 it is 6:7:9 at 24.2, because 6:7:9’s error only fits inside the wider window and once admitted it wins on completeness. At a tolerance of 15 or 20 with a ceiling of twelve the minor triad has no candidate at all.

So the claim that the difference between the chords is insensitive holds, and holds strongly — one chord’s answer never moves and the other’s moves under a change nobody would think twice about making. What does not hold is “the major triad’s fit is unique”: the number of admissible candidates runs from two to thirty as the parameters widen, for both chords. What is unique is the winner, which is the thing that matters and is not the thing that was said.

Nobody has been asked. The essay computes which fundamental a matcher prefers. Whether listeners report a pitch at that frequency for a triad is a separate empirical question, and the experiments that exist are about isolated complex tones rather than about chords.

And the tuning changes the answer. The equal-tempered minor triad has three candidates and the just one has a single exact fit, so the ambiguity drawn here is partly an artefact of a temperament. That cuts both ways: the temperament is what the chord is played in.

The matcher is handed three sine-like tones and a real chord is not that. Each note of a played triad brings its own partials, so the set to be matched has nine or twenty-four components rather than three. That was recorded here as a correction of unknown sign; the section above computes it and it has no sign, because a partial of a harmonic is a harmonic and the cents error is unchanged by the common factor. What it does change is the harmonic numbers required, and those run into the model’s own ceiling — which is a fact about the model rather than about the chord, and is the parameter this caveat should have been about.

Whose theory, and when #

Dualism is a nineteenth-century German project and its lineage is precise — Hauptmann’s Die Natur der Harmonik und der Metrik in 1853, von Oettingen’s Harmoniesystem in dualer Entwicklung in 1866, and Riemann’s work from 1873 onward. Riemann himself eventually gave up the physical claim while keeping the functional system, which is a good indication that the two were separable.

What survived is the part that never needed a spectrum. The reflection is a real symmetry of the interval arithmetic, it is drawn perfectly well on a lattice, and it is the same symmetry the neo-Riemannian transformations use — where it is treated as a fact about a group acting on a set of chords, with no claim about strings at all.

The undertone series is what happens when a diagram is asked to be a mechanism. It is worth keeping in view precisely because it was proposed by serious people, defended for fifty years, and is wrong for a reason a reader can check in one line: nothing has a mode below its fundamental.

The ladder from here #

Both triads have been treated so far as sets of pitch classes. The next rung asks what the register does — the same three notes voiced two octaves apart are not the same sonority by any measurement here, and the orchestration rule about low thirds falls out of the arithmetic exactly.