A chord is a register

Pitch-class analysis says a chord is a set of classes. C major is {0, 4, 7} and that is the whole of it — the octave a note is in does not change which class it belongs to, and the theory built on that assumption goes a very long way.

A player cannot use it. Somebody has to decide where the notes go, and the first rung of this ladder noted in passing that the decision is not free and that “spacing has audible consequences that the theory quietly discards”. That paragraph asserted three things without measuring any of them: that low intervals are rough, that the rule about them follows from critical bandwidth, and that traditional voicing imitates the harmonic series.

All three are computable. Two come out as asserted and the third comes out better than asserted.

Five times, across the piano #

The same is true of a whole chord. A major triad in close root position scores 1.031 with its bass at C two octaves below middle C, 0.616 an octave higher, 0.288 at middle C, and 0.068 two octaves above that. Between the bottom of a cello’s range and the top of a violin’s, the identical set of pitch classes changes its measured roughness by a factor of fifteen.

A minor triad behaves the same way and is slightly rougher everywhere, which is worth recording because it is often assumed to be much rougher and it is not: 1.033 against 1.031 at the bottom, 0.298 against 0.288 at middle C. Whatever separates the two triads, and the previous rung says it is the security of the inferred root, it is not this.

Where the rule comes from #

The orchestration rule is stated in every text and it is stated as a table: a list of intervals with the lowest note at which each is acceptable. The tables disagree with each other by a few semitones and none of them says where the numbers come from.

They come from one place.

Solving the crossings gives the numbers. On Glasberg and Moore’s equivalent rectangular bandwidth, a minor third fits inside one critical band below 380 hertz, a major third below 202, a fourth below 138 and a fifth below 85. On Zwicker and Terhardt’s older and wider Bark scale the same crossings are at 844, 514, 385 and 263.

The two models differ by a factor of two and a half, which is honest and is why both are drawn. What they agree on is the ordering and the spacing: every interval has a frequency below which it is inside a band, the frequency is higher for narrower intervals, and the sequence — thirds highest, fifths lowest, octaves lower still — is exactly the sequence the orchestration tables give.

So the rule is a fact about the cochlea with a repertoire’s practice wrapped round it. Which of the two models a text’s numbers agree with is a question about how conservative that text is.

The claim that was asserted, tested #

Here is the third assertion, which is the interesting one: that the traditional voicing — wide at the bottom, close at the top — is the spacing of the harmonic series itself, and that a chord voiced that way is imitating the internal structure of a single note.

That has an exact test. Take a bass note, enumerate every voicing of a major triad above it within three octaves, score each for roughness, and look at what wins.

Twenty-eight semitones above a fundamental is a ratio of 5.04, and thirty-one is 5.99. The best voicing is the bass note with its own fifth and sixth partials.

The second best is the bass’s own third and sixth partials with the third of the chord underneath. The third best is the first, third and fifth partials.

The three voicings against the bass’s own harmonic series say why. Close root position crams its notes into the first fifth of the series, where nothing coincides; the two smooth ones put their upper notes on the fifth and sixth partials and on the third and fifth. What the roughness minimiser found is the harmonic series of the note it was already given — and the first six partials of that bass are exactly the arrangement every orchestration text describes: an enormous gap at the bottom, then a fifth, then a third, closing up as it rises. Nothing about the harmonic series was put into the search; the model knows about critical bandwidth and coincidence and nothing else.

Nothing about the harmonic series was put into the search. The roughness model is Plomp and Levelt’s curve summed over pairs of partials, and it knows about critical bandwidth and about coincidence and about nothing else. Handed a bass and told to place two notes, it puts them where the bass already has energy.

So the assertion is true, and it is true for a reason: a chord voiced at its bass’s own partial frequencies has almost no closely-spaced pairs anywhere, because the harmonic series itself has almost none low down. Imitating a single note is the same thing as minimising roughness, and either description will do.

The same chord, two octaves up, is not the same sonority #

The corollary is the one that should trouble a pitch-class theory, and it is worth stating as flatly as possible.

That is not a small effect. The worst thing a composer can do with a major triad at middle C is measurably better than the best thing available with the same triad two octaves below, so the choice of register dominates the choice of voicing entirely.

What a bass player already does about it #

The rule, once it has a mechanism, predicts a set of practices, and the practices are all in place.

A bass line is mostly roots and fifths. At 65 hertz a major third scores 0.362 and a fifth scores 0.300, and both are high — but the third is worse, and there is a further consideration: the third is the note that decides the chord’s quality, so putting it low costs the most and buys the least. What every convention says instead is to put the third in an upper voice.

A low third is spread rather than removed. Move the third up an octave — a major tenth instead of a major third, sixteen semitones — and the same two notes score 0.178, less than half the roughness of the close version, on a pair whose lower note has not moved. The interval that a Baroque continuo player, a jazz pianist and an arranger all reach for in the bass is the tenth, and this is why.

And a rock guitarist’s low chord has no third in it at all. A power chord is a root and a fifth, played low and loud through a distorting amplifier — which multiplies the partials together and so multiplies the roughness with them. The absence of the third is not a stylistic minimalism; it is the only interval that survives the treatment.

The test the account could fail #

Everything above depends on the chord having partials. If the register dependence were about the fundamentals alone — two low notes beating against each other — then it would be the same for any timbre, and it is not.

The numbers give the prediction a sharper form. Across five octaves, a triad of pure tones changes its roughness by a factor of thirty-seven; a triad of string-like tones by fifteen; a clarinet-like one by twenty-one. The register dependence is strongest for the simplest spectrum, because with one partial each there is nothing but the fundamentals to beat, and their separation in hertz is what changes with register.

That is the falsifiable part. If low chords were rough because of some property of low frequencies as such, richer spectra would be more affected rather than less. They are less, because their upper partials are up where the bandwidth is narrow and are resolved.

How much the ranking really moves #

“The ranking is not the same” and “the ordering is robust across the timbres in the site’s table” are both claimed in this essay, in different sections, and they cannot both be right as stated. Running the twenty-seven voicings under all six of the site’s spectra and correlating each ranking against the string’s settles it:

The ordering is robust and the winner is not, which is why both sentences felt true. A correlation of 0.96 means the clarinet ranking is very nearly the string’s; what changes is the top of it, and the top is the part a figure draws. Six spectra, five different degrees of agreement, and one thing that does not move at all: close root position is the worst of the twenty-seven under every one of the six, pure tones included.

That last fact is stronger than anything this essay has claimed and it belongs to the low-interval rule rather than to the harmonic series. If the reason a close low triad is bad were that its notes are crowded into the first fifth of the bass’s series, then a spectrum with no series to be crowded into would not care. Pure tones have one partial each and close root position is still last of twenty-seven, because three fundamentals within seven semitones at 65 hertz are three fundamentals inside one critical band whatever else is or is not present.

And the two families of winner are the mechanism, checked twice. The three spectra with strong even partials — string, reed, organ — all put the upper notes on the bass’s fifth and sixth partials. The two with weak or absent even partials — the clarinet’s and the bell’s — both put them on the third and fifth instead, which is the prediction the stopped-pipe figure above makes and which the bell, an inharmonic spectrum nobody chose for this, confirms independently.

The pure-tone case is the one that cannot join in, and it is instructive. Its winner is a voicing whose lowest note is the chord’s third and whose upper notes sit at four and six times the nominal root — but nothing is landing on a partial, because there are no partials. It is the answer to a different question: with one component each, the only way to be smooth is to be far apart, so the search returns the widest arrangement it is allowed rather than a resembling one. Imitating a single note and maximising separation give the same answer for a harmonic spectrum and different answers for a pure one, which is as clean a demonstration as this ladder has that the first description is the operative one.

It also puts a limit on the section above it. The harmonic-series reading of the winning voicing is a reading of one winner, and it survives being asked for under five different spectra only because four of the five have a series to land on. What is general is the ranking and the loser; what is spectrum-specific is which partials the winner picks out. An orchestrator reading this essay would be right to take the first and would be reading a string quartet’s answer if they took the second.

Which is a problem for octave equivalence, and a small one #

Octave equivalence is the assumption that makes pitch-class theory work: two notes an octave apart belong to the same class, so a chord is the set of classes and the arrangement is a performance detail.

This site has already found the assumption to be approximate rather than exact on the perceptual side — listeners’ subjective octaves are stretched, and the amount depends on register. What this rung adds is a different objection, and it is worth separating.

The perceptual objection is that an octave is not exactly two to one. That is a matter of a few cents and it does not disturb the classification.

The objection here is that the classification throws away a factor of fifteen. Nothing about the pitch classes changes across five octaves, and something the ear cares about a great deal does. The assumption is not false; it is silent about the largest single variable in how a chord sounds.

That is a reasonable trade for an analytic theory and it is stated as a trade. Where it becomes a mistake is when the theory’s results are read back as claims about sound — as when a set-class relation between two chords in different registers is treated as an audible resemblance.

A second reading of the same numbers #

There is a way of putting the whole result that makes it less about chords and more about what a chord is for, and it is worth setting down because it connects two ladders.

A sonority is heard as one thing or as several. What makes two partials one note is that they are harmonically related and behave alike, and the same machinery decides whether three simultaneous notes fuse into a chord or stay separate as three voices. A voicing that puts its notes on the bass’s own partials is a voicing that maximises fusion — it is asking the ear to hear one complex tone — and a voicing that crams them into the first fifth of the series is one the ear will resolve into parts, and will find rough while doing it.

So the register question and the question of whether a chord has a root are the same question asked twice. The arrangement that makes a triad smoothest is the one that makes its root most evident, because both are the arrangement that most resembles a single note.

That reading also says where it stops. A texture built to be heard as independent lines rather than as chords has no reason to want fusion, and a great deal of counterpoint spends its time keeping two voices from becoming one — which is the opposite instruction, from the same mechanism.

Whose orchestration, and when #

The rule is a European orchestral rule and the tables are nineteenth- and twentieth-century. Rimsky-Korsakov’s Principles of Orchestration, Piston’s Orchestration and Adler’s Study of Orchestration all give a version of it; they disagree in detail and agree completely in shape.

The practice is older than the tables. The spacing of a bass, a fifth and a third is the normal disposition of a Renaissance final sonority, and it is where a keyboard continuo player’s left hand goes without being told. What the nineteenth century added was not the practice but the statement of it, and what the twentieth century added was the mechanism.

Two repertoires make the jurisdiction visible by declining it.

Organ mixture stops deliberately stack close intervals high — a mixture sounds the twelfth, fifteenth, nineteenth and twenty-second above every key played — and the reason it works is exactly the reason drawn above: the roughness of a close interval collapses with height, so a chord of narrow intervals four octaves up is smooth. The same stop transposed down two octaves would be unusable, and organ builders do not build it.

And a gamelan’s low instruments play a slow melodic skeleton in single notes rather than in chords, so the question does not arise; where instruments do sound simultaneously they are deliberately detuned pairs rather than triads, and the beating is the point rather than the thing to be avoided.

What the picture cannot show #

One roughness model, and it is crude. Everything here is Plomp and Levelt’s dissonance curve summed over partials, with a one-over-n spectrum standing in for an instrument. It has no masking in it, no account of level — although the site has measured that separately, and it makes low chords worse — and no account of duration.

The voicing enumeration is close-position only, over three octaves, with each pitch class once. Real orchestration doubles, spreads over four octaves, and mixes timbres, and the doubling question — which note to double, and where — is a different rung’s.

And a real bass note is not one over n. A double bass, a bassoon and a tuba have very different spectra at the same pitch, and the roughness of a chord depends on the spectra of the instruments playing it as much as on the notes. The measurement here is of an idealised string; the ordering is robust across the six spectra the site carries, at rank correlations of 0.84 to 0.99, and the numbers and the winner are not. Six model spectra are not six instruments, and none of them is measured from a bassoon.

The ladder from here #

Every chord so far has been thirds stacked on thirds. The next rung asks what happens when the stack is built from a different interval — every constant-interval stack, scored for roughness and for whether an ear can find a root in it — and finds one that is smoother than the triad and has no root at all.