The inversion that cannot end a phrase

Move the bottom note of a triad up an octave and every theory of harmony calls the result the same chord. The first rung of this ladder says so, and adds that the claim rests entirely on octave equivalence, and that the practice does not quite believe it.

On a keyboard the move is one key: send the lowest note of C–E–G up an octave and the same three pitch classes are still sounding, which is why theory calls the result the same chord.

The place where the practice does not believe it is specific and it is famous. A second-inversion triad — a six-four, with the fifth in the bass — cannot end a phrase. Root position can, first inversion can at a weak cadence, and the six-four cannot, in the whole of the repertoire that eighteenth- and nineteenth-century theory describes. When it appears, it is either a passing chord, or a chord over a pedal, or the cadential six-four, which is heard as a dissonance over the dominant and resolves before the cadence proper.

That is a very strong asymmetry between three things called the same chord, and it is worth asking what measurement produces it. This site has three, and all three fail.

The first measurement: roughness #

The usual explanation offered is that the fourth above the bass is dissonant, and this collection has an apparatus for computing dissonance: sum the beating between every pair of partials in the whole sonority, and read off a number.

Run it on the three positions and the answer is not that the six-four is rougher.

It is the smoothest of the three, at every register a cadence happens in. With the bass on C below middle C, root position scores 0.616, first inversion 0.597, and the six-four 0.543. An octave up the numbers change and the order does not. An octave down likewise.

That is worth sweeping rather than sampling, because the roughness model is the crudest thing in this essay and it is being asked to carry the first of three arguments. Running all three positions at six timbres and at every semitone of bass from C1 to C7 — 438 cases:

The six-four is smoothest for a string spectrum at every bass from C1 to C7, and for all six timbres at C3. It is also smoothest for an organ spectrum everywhere, which matters because those two are the spectra of the instruments this repertoire was written for.

The exceptions are real and they are outside the box. With pure tones below C2 the ordering reverses, because a sine has no partials and the roughness is then between the fundamentals alone. With the odd-harmonic and inharmonic spectra — clarinet above G5, reed above D6, bell above C♯5 — root position becomes the smoothest, because a sparse spectrum high up has few enough pairs that the exact coincidences of the root-position ratios outweigh the near-misses that a fuller spectrum supplies.

Neither exception touches the argument. A cadential six-four whose bass is above G5 puts the whole sonority above the treble staff, and nothing in the repertoire the prohibition governs is scored for sine tones. What the sweep establishes is not that the result is universal but that it is not an artefact of the one timbre and the one register the essay first tried, which is the thing a crude model most needs checking for.

It is worth noticing what shape the exceptions have, because it is the same shape twice. Both are cases where the number of contributing partial pairs collapses — a sine tone has one pair, and a sparse spectrum two octaves up has most of its partials beyond the range where anything beats. Root position wins wherever the sonority is too thin for near-misses to matter, because its ratios coincide exactly and the six-four’s do not; the six-four wins wherever there are enough pairs for the width of the intervals to dominate. That is not a defect of the model, it is the model saying that the two positions are good for different reasons, and which reason applies is a question about the spectrum rather than about the chord.

The ordering that is not stable is between the other two #

The sweep turns up something the two sampled registers hid. The six-four’s position at the bottom of the ranking never moves; the ranking of the other two does.

Below a bass of about G♯3 root position is the roughest of the three, and above it root position is smoother than first inversion — the two swap at 211 hertz, a little below middle C. The reason is the one this essay already gives about the six-four, applied one step along: root position puts a third at the bottom, first inversion puts a third at the top, and a third low down is the rough interval, so the position with the low third is penalised in the bass and not in the treble.

That sharpens the whole comparison. The one ordering that holds across the entire compass is the one the practice contradicts, and the ordering that is unstable is between the two positions the practice treats as more or less interchangeable — root position closing a phrase and first inversion closing a weak one. If the practice were tracking roughness it would be tracking the unstable pair and ignoring the stable one, which is the opposite of what it does.

The reason is not mysterious. A six-four voicing spreads its notes into a fourth and a third above the bass rather than a third and a fourth, and a third low down is the rough interval — which is the orchestration rule every text gives and which this site has computed. The six-four puts its wide interval at the bottom, where wide intervals belong.

So the roughness account does not merely fail to explain the prohibition; it points the other way.

The second measurement: support from the bass #

There is a second property, distinct from roughness, which the first rung of this ladder asserted from prose: in root position the upper notes match low partials of the bass note’s own sound, and in the inversions they do not.

That is checkable.

And a count of shared partials against the bass says the same thing from the other side: a root-position triad’s notes sit on the bass’s own third and fifth partials, and a six-four’s do not — the bass has a fourth above it, which is the interval a bass note’s own series does not supply low down.

This one works, and it works exactly as far as the first rung claimed and no further. It separates root position from the inversions. It does not separate the inversions from each other.

Both come out at 1.000 — the bass supporting itself and nothing else — because neither a fourth nor a sixth above a bass, and neither a third nor a minor sixth above one, is a low harmonic of it in any octave. So this measurement explains why root position is the stable one, which was never in dispute, and says nothing at all about why one of the other two can close a phrase and the other cannot.

The third measurement: what an ear supplies as the root #

The last available account is the one the next rung of this ladder is about: the ear infers a fundamental that the chord’s notes would be partials of, and the security of that inference might differ between the positions.

It does differ, and again in the wrong direction.

Root position fits 4:5:6, worst error 10.1 cents, nothing missing. First inversion fits 5:6:8, with a gap at the seventh partial, worst error 11.5. The six-four fits 3:4:5 — the lowest three integers of the three — with nothing missing and the smallest error of the set.

That result is not an artefact of the tuning: the same ordering holds in just intonation, where the fits become exact and the six-four’s ratios remain the simplest.

Three measurements, and the practice unmoved #

So the position of the chord that practice singles out as unable to close is:

• the smoothest of the three by roughness, at every register;
• tied with the first inversion for support from its own bass, neither having any;
• and the best fit to a low harmonic series of the three.

Ranked against every three-note chord in the system the six-four is not distinguished at all — it is the same pitch-class set as the root-position triad, so a census over sets cannot see it. That is the point: the object the rule is about does not exist in pitch-class space.

Whatever forbids the six-four is not a property of the sound. It is a rule, and the interesting question is what the rule is about.

The fourth above the bass, and the framework it belongs to #

The rule is stated in every counterpoint text and it is stated as a rule about an interval: the perfect fourth is a dissonance when it is formed against the bass, and a consonance everywhere else.

That is an odd-looking rule and it is exactly the shape of rule that this collection keeps finding when it looks: a rule with a jurisdiction. It is not a claim about two notes. It is a claim about two notes in a particular structural position, and the position is the one the framework treats as the reference.

Species counterpoint is a two-voice discipline before it is anything else. Every interval in it is measured from the lowest sounding voice, because that voice is the cantus against which the exercise is written; consonance and dissonance are properties of an interval with the lower note as its reference. When a fourth is formed against that voice, the upper note is a fourth above the reference — which is the same thing as saying the reference is a fifth below the upper note, and therefore not where the sonority’s root is.

So the prohibition is a statement that the bass must be the root, dressed as a statement about an interval. And a six-four violates it by construction, because in a six-four the bass is not the root.

Every interval and its inversion add to twelve semitones, so a fourth above the bass is a fifth turned over — and the reason a fourth is treated as a dissonance above a bass and a consonance between upper voices is that only the bass supplies a series for the others to fit.

Which makes it a historical claim, and a checkable one #

If the rule is about which voice is the reference, then it should have arrived when the bass became the reference, and not before. That is a date.

In medieval organum and through the fourteenth century the reference voice was the tenor, and the perfect fourth was a consonance — it is one of the primary consonances of early organum, and parallel fourths are the basic texture of a great deal of it. Nothing about the ear changed in the fifteenth century. What changed is that the lowest voice became the structural reference: the contratenor bassus arrived, four-voice writing became normal, and the sonority came to be described from the bottom up.

The fourth’s demotion is contemporaneous with that. It is a consonance in Johannes de Garlandia’s thirteenth-century list; it is a dissonance against the bass in the fifteenth-century treatises and everything after. The interval did not move.

The prediction that follows is testable and is borne out. A repertoire that does not take the lowest voice as its structural reference should have no prohibition of the fourth, and no six-four problem. Medieval organum does not. Fauxbourdon, which is built on parallel six-three sonorities, treats the sixth above the bass as normal and is transitional. And a great deal of twentieth-century music that abandons the bass-as-root convention ends on six-fours freely, which is what quartal harmony makes into a principle.

The cadential six-four, which is the case that proves the reading #

The one place a six-four is common in exactly the repertoire that forbids it is the cadential six-four, and how it is treated settles the matter.

It occurs on a strong beat, over the dominant bass, and it resolves — the sixth falling to the fifth and the fourth to the third — before the dominant itself moves to the tonic. Every eighteenth-century figured-bass description treats those two upper notes as suspensions over the dominant rather than as the notes of a tonic chord in an unusual arrangement.

And written out in four parts, the cadential six-four is a IV–V–I in which the tonic chord arrives twice: once over the dominant bass as a suspension and once in its own right. The rule is about the first of those, which is a delaying gesture rather than a chord.

That is the reading in which the rule and the practice agree completely. There is no six-four chord in a cadence; there is a dominant with two suspensions, and the thing that looked like a tonic triad is a delay. Which is a strong claim about what a chord is — that it is not a set of pitch classes but a set of pitch classes plus an account of which notes are structural — and it is the claim the whole voice-leading ladder rests on.

The one thing that does separate the three, and it is not about chords #

There is a measurement that distinguishes the positions, and it is worth naming because it is the one the practice is actually tracking.

Where the chord can go next. A root-position tonic triad at the end of a phrase is a place the bass has arrived; a six-four is a place the bass is passing through, because the note in the bass is the fifth and the fifth of a key is the one degree with an overwhelming tendency of its own. The dominant is where a tonal piece is most strongly pointed somewhere else, and a six-four puts that degree in the bass while the upper voices spell the tonic.

On the criteria plane the two arrangements sit at the same point, because both criteria are computed on pitch-class sets — which is the same blindness the census has, stated in the ladder’s own terms.

That is a property of the chord’s position in a key, not of its sound, and it is measurable only once a key exists. Which is the honest form of the whole result: the six-four is forbidden at a cadence for reasons that live entirely in the syntax, and every attempt to ground it in the acoustics — this essay tried three — either fails or comes out backwards.

One more consequence is worth drawing out, because it is the reason the prohibition survived the arrival of every later theory. A rule that is about syntax rather than about sound is portable: it does not depend on the timbre of the instruments, on the temperament, on the register or on the hall, all of which changed enormously between 1600 and 1900 while the rule did not. An acoustic rule would have had to move when the tuning moved or when the instruments got louder. This one had nothing to move with.

What the picture cannot show #

Roughness is a crude model and this essay leans on it three times. It is Plomp and Levelt’s curve summed over partials, with no masking, no phase and no account of duration. A better model could conceivably rank the positions differently — although it would have to overturn the register dependence too, which is independently measured, and it would have to do so across six timbres and six octaves at once, which the sweep above is there to make explicit. What the sweep cannot do is defend the model against a different model; it only establishes that this one is not being read at a lucky point.

And the timbres are stated spectra rather than recorded ones. Six fixed amplitude sets with no decay, no attack and no formant. The two that carry the argument — string and organ — are the fullest of the six, so the result leans on the case where the most partial pairs are contributing, which is also the case where a summed model is least likely to be dominated by any single pair.

The subharmonic fit is a fit and not a perception. Whether a listener actually infers a fundamental from a triad, and how strongly, is a psychoacoustic question the next rung takes up; what is computed here is which candidate a least-squares matcher prefers, which is a property of the numbers.

And none of this touches whether inversions are heard as the same chord. They are, and the evidence for it is that listeners name them the same. What is at issue is a much narrower claim — that they are interchangeable at a cadence — and the practice says they are not.

Whose music, and when #

Everything above is about the repertoire from roughly 1600 to 1900 as its own theorists described it, and about the fifteenth-century shift that produced the rule. Outside that, most of it does not apply.

There is no six-four problem in music with no bass-as-root convention, and there is none in music with no chords. The rule is one of the clearest examples this collection has of something taught as though it were acoustics that is in fact a convention with a date, a jurisdiction and a documented predecessor that said the opposite.

What makes it worth three measurements rather than an assertion is that the acoustic explanations for it are still offered, in print, and all three of them are checkable. They come out smoothest, tied, and best fitting.

The ladder from here #

The subharmonic fit used here as a test is the beginning of the next rung’s subject. Run it on a minor triad rather than a major one and the answer becomes ambiguous in a way that occupied two centuries of argument — and produced an invented spectrum to resolve it.