Which player on which note

Which instrument is underneath put two different spectra on one interval and found that the roughness sum is no longer symmetric: a clarinet under a violin and a violin under a clarinet are not the same sound, and can differ by a factor of five. That rung ended by naming what the asymmetry makes available and nothing had taken.

Two notes have two orderings and one of them is a relabelling. Three notes have six, and none of them is.

Six arrangements of one chord, and a spread of forty-two per cent in a quantity nobody has a name for on a score. A chord symbol cannot express it. A figured bass cannot. A voicing — which note is where — is a different variable and is held fixed in every bar of the figure above.

The number is worth being careful about before anything is built on it. It is a roughness ratio, not a loudness ratio and not a preference, and this collection has said many times that roughness is one component of consonance and not the whole of it. What the ratio establishes is that the six arrangements are different objects to the same measurement — which is the claim that has to be true before any of the rest is worth asking.

Why there is anything to compute #

The reason the six differ is the one the previous rung established, and it is worth restating because everything here rests on it.

Roughness is computed between partials, not between notes. Two notes are rough in proportion to how many pairs of their partials fall within a critical band of each other, weighted by how loud those partials are. So the roughness of a pair depends on both spectra — and the spectra of real instruments are not scaled copies of one another, because the body is a fixed filter that does not move when the note does.

A clarinet on the bottom of a close triad contributes almost nothing at the second partial, so the octave-plus-a-bit region where the middle voice’s fundamental lives is quiet. An oboe on the same note fills that region. Whether that helps or hurts depends on what the middle voice is doing there, which is why the answer is an enumeration rather than a rule.

The rule that nearly falls out, and does not #

The obvious candidate rule is the one every orchestration treatise gives: put the brightest instrument on top. It is good advice and this computation half agrees with it.

The trouble is that “brightest” is not a property an instrument has independently of the note it is playing. An instrument is not one timbre: a fixed filter over a moving fundamental means the same oboe is bright at one pitch and much less so at another, and the ordering of three instruments by brightness is therefore itself a function of register.

Three different orderings are optimal somewhere in six registers, on one chord with one voicing. That is the sense in which the treatises’ rule is a rule of thumb: it is a good summary of a surface, quoted as though it were a constant.

How good a summary, in numbers #

“Half agrees with it” is a phrase and not a measurement, and the rule is testable. Running every trio drawn from the five radiators this site can build, on five chord types, at nine registers — 450 assignment problems, six arrangements each — and scoring the arrangement the rule picks against the six that were available:

The rule identifies the smoothest arrangement no more often than a coin does — 18 per cent against a chance rate of 17 — and it is nevertheless worth having. Those two facts are not in tension, and the second column is where the difference lives. A blind pick lands 22 per cent above the best available; following the rule lands 13.5 per cent above it; ignoring it deliberately lands 31.5 per cent above. The rule recovers about two fifths of the gap between guessing and optimising, while being useless at optimising.

That is exactly what a heuristic on a surface with three different optima should look like, and it is a better defence of the treatises than the essay’s own hedge was. A rule that moved the mean and could also find the maximum would be a rule that a surface did not need; a rule that moves the mean and cannot find the maximum is a rule for a quantity whose maximum moves. Ranking the instruments once at a fixed pitch, rather than at the notes they are playing, does almost as well — 17 per cent and 1.168 — which says the register dependence the surface figure draws is worth about three per cent of the total, and that most of what the rule buys is available without knowing the register at all.

One number from the same sweep is worth carrying past this section, because the essay’s headline is a single case. The mean spread across all 450 problems is 1.49, not the 1.42 of the figure at the top — so the G minor triad drawn there is a slightly gentler example than the average, and the typical chord has half again as much roughness in its worst arrangement as in its best.

And the spread is not the stable part. Across the 450 problems it runs from 1.04 to 6.22 — a clarinet, a violin and a voice on stacked fourths at C5, where the worst arrangement is six times the roughness of the best, against a clarinet, an oboe and a flue pipe on a diminished triad at C3, where the six arrangements are within four per cent of each other and the choice is genuinely free. So the honest scale is not one number: an assignment is usually worth a fifth of a register change, and in its extreme case it is worth more than moving the chord across the whole instrument. Which case a passage is in is itself a computation, and it is the one an orchestrator would actually want.

Four players, and where the enumeration stops being one #

The three-note case is the smallest complete one. It is not the common one.

Adding players makes the spread smaller and the enumeration larger, which is an unpromising combination. Six players over six notes is 720 arrangements with fifteen pairs each; twelve is 479 million. Somewhere in there the question stops being an enumeration and becomes an optimisation with a heuristic, and the heuristic would be the treatises’ rule.

That is a fair account of why the treatises say what they say. A rule of thumb is what remains when the exhaustive answer is unavailable, and it is right in the middle of the surface and wrong at its edges — which is exactly the shape of the register result above.

What the spread is worth, in the units the collection already uses #

A factor of 1.42 in roughness needs a scale to be worth anything, and the roughness ladder has one.

A chord is a register found that one triad’s roughness moves by a factor of about five between the bottom of a piano and the middle. A third is rougher in the bass found the critical band crossing that produces it. Against that, an assignment is worth a fifth to a third of a register change — which is to say, choosing who plays which note is worth a few semitones of moving the chord.

That comparison is the honest framing. Voicing is the large variable, register is the large variable, and assignment is a smaller one that has the unusual property of being free: it costs nothing to write, changes no pitch, and is not visible in any reduction of the score.

It also says something about a fact of orchestral practice that is usually explained by convenience. Wind sections are written in pairs and the second player is nearly always given the lower of two notes; string sections divide with the firsts above the seconds. Those are conventions about ordering by seniority, and the arithmetic here says that ordering an ensemble by anything fixed — seniority, part number, seating — is a decision about roughness taken for a reason that has nothing to do with roughness, and that the decision it takes is right in part of the range.

Which computation produced the numbers #

The chord’s frequencies come from the root and the semitone offsets. Each player’s partial list at its own note comes from radiatedPartials, which is a source spectrum this site already had multiplied by a filter this site already had — a string through the measured violin body, a glottal pulse through a published vowel, the site’s own clarinet and reed lists cut off above the computed tone-hole lattice cutoff. Nothing in the list was chosen to make this work.

Each pair of notes is scored by the same Plomp–Levelt roughness kernel the whole consonance ladder uses, summed over every pair of partials. A chord’s total is the sum over its three note-pairs, which is what the roughness model has always done for a chord of one timbre; the only change is that each note carries its own list.

The six assignments are then an enumeration, not a search. Three players over three notes is 3! = 6, which is small enough to be exhaustive and is the reason this is the smallest complete form of the orchestration question. Four players over four notes is 24 and still exhaustive; a real orchestral chord of twelve parts is 479 million, and at that size the question stops being an enumeration and becomes an optimisation this essay does not attempt.

The register surface is the same computation repeated at six roots and reduced to a rank, which is a deliberate loss of information: the ratios at each register are on different scales — a low chord is rougher in absolute terms than a high one, by the factor the register rung measured — so plotting the raw totals across registers would draw that much larger effect and hide this one entirely. The rank is what survives the normalisation.

The amplitudes are normalised to each spectrum’s strongest partial before scoring, because roughness is quadratic in amplitude and an unnormalised comparison would be reporting which instrument was set louder.

Where the model stops #

Roughness is one component of consonance and this collection has said so repeatedly. Roughness cannot choose a scale and consonance is half learned; the six arrangements above differ in roughness and there is no claim that the smoothest is the one anybody should write.

The balance is fixed and is the larger variable. Every note in every figure is at the same level, and a real orchestrator’s first decision is dynamics — which is a decision about amplitude, which enters roughness quadratically. A note played a third as loud contributes a ninth as much roughness with any partner, so a balance change swamps an assignment change. That is the same gap the triad ladder recorded when it closed, and it is still open.

And the spectra do not change with dynamic. The site has the machinery to say they do — a harder blow is a brighter note — and using it here would make the assignment problem and the balance problem the same problem, which is probably what it is.

Whose music, and where the choice is actually made #

The claim about roughness is about a listener and is not about any repertoire. The claim that assignment is a free variable is about a practice, and the practice is European orchestral and chamber writing from roughly the eighteenth century onward, where a composer writes the notes for each named instrument and the assignment is therefore made at the desk.

It is worth naming the traditions where the choice does not exist. In a four-part chorale the assignment is fixed by the compass of the voices — why the exercise is in four parts is about exactly that constraint — so a soprano cannot take the bass note and the six arrangements collapse to one. In a gamelan the instruments’ registers are fixed by the set. In most keyboard music there is one player and one spectrum, so the question is empty.

The choice exists precisely where the ensemble has overlapping compasses and independent parts, which is a narrow slice of the world’s music and the slice that produced the treatises. That the treatises’ rule is a summary of a surface is then not a criticism of them: they were describing a practice within a range, and inside that range the summary is usually right.

What the picture cannot show #

Whether any of it is audible. A factor of 1.42 in a roughness model is not a factor of 1.42 in anything a listener reports, and the model has no calibration to a report. What can be said is that the same model’s factor of five across registers is audible and well documented, so a fifth of that is plausibly at the edge.

The critical band is a model with two versions and they differ in the bass. The width of the band is where every roughness number on this site comes from, and the two published models for it differ by a factor of three at the bottom of the range — which is exactly the register where the assignment spread is largest. The ordering survives the choice; the size does not.

And the figure has no attack in it. Three instruments entering a chord together do not enter together — a clarinet and a voice are heard tens of milliseconds apart — so the sonority the assignment is chosen for takes a moment to exist at all.

The thing that is genuinely new here #

It is worth separating what is new from what is a restatement.

That two spectra on one interval do not commute is the previous rung’s, and it is the harder result: it needed the asymmetry to be established at all. That three players can be arranged six ways is arithmetic anybody could have written down.

What is new is the register surface — that the ordering of the six is not stable, that three different arrangements are optimal within two octaves, and that the instability comes from the one property this ladder spent four rungs establishing, which is that a fixed filter under a moving fundamental makes an instrument’s spectrum a function of its note. Every treatise rule about who goes on top is stated as a property of the instruments. It is a property of the instruments at a pitch, and there is no pitch at which all of them hold.

Where this ladder goes next #

Six rungs. A spectrum is a list; a mouth filters it; a body filters it; a fixed filter under a moving note makes the list a function of pitch; two lists together do not commute; and now, three lists over three notes have six arrangements that differ by up to nearly a factor of two, with the best one changing across the range.

The rung after it is the one the balance gap names, and it is the one that would turn this from an enumeration into a real orchestration model. Amplitude enters roughness quadratically and enters the radiated spectrum through the excitation nonlinearity, so a dynamic marking changes both how much a note contributes and what it contributes. Putting the two together means the assignment problem and the balance problem are one problem with two kinds of variable — a discrete choice of who plays what and a continuous choice of how loud — and the site has both halves and has never joined them.