Two players on one note

Which player on which note took three instruments and three notes and found six arrangements differing by nearly a factor of two, with the best one changing across the range. It ended by naming the joint problem with dynamics in it, and that was written — which leaves this ladder holding a much simpler omission that every one of its six rungs shares.

Every figure on it puts exactly one instrument on each note. Orchestration does not work that way. The commonest single thing an orchestrator does is put two different instruments on the same note, and the reason is that the pair sounds like neither of them.

The composite is a power sum #

Two players are independent sources. Their phases are unrelated and drifting, so their partials add in power rather than in amplitude: the composite’s nth partial is the root of the sum of the squares of theirs.

That is the same assumption the choir rung makes about sixteen sopranos, and for the same reason. It is what makes a doubling a spectral operation rather than an interference one.

The immediate consequence is about brightness. A spectral centroid is a power-weighted mean frequency, so the composite’s centroid is not the average of the two instruments’. A flue pipe’s is 710 hertz and a voice’s 1,072; together they give 917, not 891. Put a violin at 432 against a brass instrument at 1,657 and the composite is 1,461 — a fifth of the way from the brass to the string rather than half.

The pull is not the same at every pitch, because an instrument is not one timbre: a fixed body resonance or a fixed formant under a moving fundamental changes each player’s own centroid up the range, so the composite’s ownership can change hands with pitch at a fixed balance. That is the mechanism the fourth rung built and this one inherits without re-deriving.

A doubling sounds nearly as bright as its brighter member. That is a rule orchestration treatises give and this is where it comes from: the pull is the square in the weighting.

What blending is, and it is not similarity #

The interesting quantity is not the centroid but how far the composite sits from each of its parts, and that turns out to depend almost entirely on the balance.

Measured as a log-spectral distance, the composite of a clarinet and an oboe at equal level is 7.3 decibels from one and 5.3 from the other, on a pair that are 12.6 apart. It belongs to the oboe.

Bring the clarinet up and at −3.4 decibels the ownership changes hands. That crossing is what makes the pair blend: there is a balance a player can actually set at which neither instrument owns the composite.

Now take a flue pipe and a brass instrument. Their crossing is nowhere in twenty-four decibels either way. At every balance a conductor can ask for, the composite is one of them with the other as a colouring. The pair does not blend, and the reason is not that they are dissimilar — a clarinet and a voice are further apart in spectral distance than a violin and a flue pipe, and it is the violin and the flue pipe that refuse to blend.

So this rung can offer a definition rather than a description. A pair blends when the level at which the composite changes owner lies inside the range the players can reach. The clarinet and the oboe cross at −3, the violin and the clarinet at +4, the violin and the oboe at +16 — all reachable, all named as blending pairs in every orchestration manual. The pairs involving a voice cross at −21, which is to say the voice owns the composite until it is twenty decibels down, and pairs involving brass cross at +14 to +20 or never.

That is computed from two spectra and nothing else, and it agrees with the manuals about which pairs blend without having been told.

And then the thing that is not about spectrum at all #

The figure at the top of this essay is the result that changes what the ladder thought it was doing.

Take a flue pipe and an oboe on the same note and mistune them. At zero cents the roughness is 0.0007, which is nothing — every partial of one coincides exactly with a partial of the other and there is nothing to beat. At five cents it is 0.047, which is sixty-six times as much. At ten cents it is 0.087.

Now take the same two instruments a major third apart. At zero cents the roughness is 0.078, and at ten cents out it is 0.079. A change of half a per cent.

Those two numbers cross, and where they cross is the sentence:

A unison ten cents out of tune is rougher than a major third dead in tune.

Why, and it is one line #

Partial n of a note mistuned by c cents is also mistuned by c cents — but that is n times as many hertz. The sixteenth partial of a note five cents flat is five cents from its partner and about two and a half hertz away from it, which is squarely inside the beating region, while the fundamentals are a sixth of a hertz apart and coincide as far as any ear is concerned.

So a mistuned unison is rough at the top of the spectrum first, and the roughness sweeps downward as the mistuning grows. That is why the curve rises from zero so fast and why it does not saturate.

A third has no such mechanism. Its roughness is set by which pairs of partials nearly coincide — the fourth against the fifth, and so on up — and a few cents does not change which pairs those are, only where they sit inside an already-wide interval. The curve is flat because the geometry is unchanged.

Intonation, as a roughness problem, is a unison problem. Every other interval in an orchestra is forgiving to within a comma and a unison is forgiving to within about two cents.

What it costs to be out of tune, in the units the ladder uses #

There is a way of putting the size of the effect that connects it to the rest of this ladder, and it is worth doing because a bare roughness number means nothing on its own.

Which player on which note found that the six arrangements of three players over a triad differ in total roughness by up to a factor of 1.9, and treated that as a large number worth an essay — which it is, because it is the difference between a good scoring and a bad one made with the same notes and the same instruments.

A single doubled unison five cents out multiplies its own roughness by sixty-six. Ten cents out, that one pair contributes more roughness than an entire in-tune major third does.

So the intonation of one doubling is worth more than the whole assignment problem. An orchestrator agonising over which instrument takes which note is optimising a quantity that a violinist five cents sharp destroys.

That is not an argument against the previous six rungs. It is the ordering nobody had, and it says which of the two things a rehearsal should spend its time on — which is the thing every rehearsal already does, in the ratio the figure gives.

Which is what orchestral practice says #

This is one of the few places in this collection where a computed result matches a piece of received professional wisdom exactly and without hedging.

Every orchestra rehearses unisons. The standard complaint is that a doubled line “isn’t together”, meaning it is not in tune, and the standard remedy is to have one of the two play alone and the other match. Nobody rehearses the intonation of a major third with anything like the same care, and the reason usually given — that a third is a wider interval so the ear is less sensitive — is not the reason. The ear is not less sensitive; there is nothing there to be sensitive to.

It also explains why the trouble scales with the brightness of the instruments. A pair with strong high partials has more to go wrong: the roughness of a mistuned unison is dominated by the top of the spectrum, so two oboes are harder to tune together than two flutes, and a doubled line with a brass instrument in it is harder still.

And it explains the octave. An octave doubling is the other commonest thing an orchestrator does, and it has the same structure: the upper instrument’s partials coincide with the lower’s even ones, so an octave doubling is rough in exactly the same way and a fifth is nearly as bad. The curve for a fifth on the figure at the top rises to about half the unison’s, which is the ordering everybody uses and nobody states.

The case where two on one note is not two sources #

There is one instrument in this collection that has already been through this and got a different answer, and the difference is instructive.

A piano’s unison is three strings joined at the bridge, and that rung found that they are not independent: the pair has normal modes, one of which drives the bridge and dies quickly while the other cancels at the bridge and rings for twenty seconds. Detuning them a couple of cents is not a mistake there, it is the design — it is what gives a piano its long, complex decay.

Nothing in this essay applies to that case, and the reason is the assumption in its second paragraph. Two orchestral players are independent sources whose powers add. Three piano strings are one coupled oscillator whose modes interfere. The same physical arrangement — two vibrating objects at the same pitch — is two completely different problems depending on whether they are connected, and the connection is a bridge.

Which is also why a piano tuner counts beats at a unison and an orchestra does not. The tuner is setting the coupling; the orchestra is trying to abolish the beating altogether.

Which computation produced the numbers #

The five radiators are the ladder’s own: a violin with its body resonance, a clarinet and an oboe with their radiation cutoffs, an organ flue pipe, a voice as three formants, and a brass instrument with a high-pass at its bell’s cutoff. Each returns a list of partial amplitudes at a given fundamental.

The composite is the root of the sum of squares, partial by partial, over sixteen partials, with the second instrument scaled by the stated balance.

The spectral centroid is the power-weighted mean frequency. The log-spectral distance is the root-mean-square difference in decibels between two spectra, each normalised to its own total power, which makes it a comparison of shapes rather than of levels.

The roughness is dissonancePair, unchanged: every partial of one against every partial of the other, through the Plomp–Levelt curve scaled to the critical bandwidth. The unison curve is that with the second note moved by a stated number of cents.

Where the model stops #

Two players are not two spectra. A real doubling of a flute and a clarinet is two people breathing, articulating and vibrating independently, and the fluctuation that produces is what the choir rung is about and is not here. A doubled unison with two vibratos in it is rough in a way that changes at six hertz.

The mistuning is static. A real unison is never out by a fixed number of cents; it wanders, and what a listener hears is a beating that comes and goes. The static curve is the average of that and it loses the thing musicians actually complain about.

Blend is defined here and not measured. The definition — that the ownership crossing lies inside reach — is a construction of this essay’s, and its agreement with the manuals is a check rather than a validation. Nobody has asked listeners to rank these fifteen pairs.

And the balance range is asserted. Twenty-four decibels either way is a generous estimate of what two players can do against each other in an ensemble, and a narrower one would make fewer pairs blend.

What the picture cannot show #

It cannot show the room. A doubling heard at ten metres has been through the same reverberation twice, and the correlation between the two sources at the listener’s ear is not zero — which is the assumption the power sum rests on.

Nor can it show attack. The strongest cue that two instruments are two instruments is that they do not start together, and the onset ladder puts the difference between a doubled flute and a doubled violin at tens of milliseconds. A blend is a blend of steady states here.

It cannot show what a player does about it. A wind player hearing a rough unison lips toward it and a string player moves a finger, so a real doubling is a servo with the roughness as its error signal — and how quickly it converges is the quantity that separates a good section from a bad one and is nowhere here.

It cannot show the number of players. Two firsts and two seconds on one line is not the same as one of each, and a section doubling a solo instrument is the commonest case of all.

It cannot show the bass. A third is rougher in the bass because the critical band is a fixed width in hertz, and the same is true of a mistuned unison the other way round: low down, the partials of a five-cent mistuning are further inside the band than they are in the treble, so the effect computed here at G4 is a different size two octaves below it. That sweep is one line and it is not on this page.

And it cannot show what the composite is for. An orchestrator doubles to get a colour, to get a level, to cover a weak register, or to make a line carry — and this figure prices one of the four.

Whose instruments, and when #

The radiators are generic and the levels are arbitrary, so nothing here is a measurement of an orchestra.

The historical claim available is a narrow one about the practice. Doubling in the sense used here — two different instruments on one line, chosen for the composite colour — is a nineteenth-century preoccupation; eighteenth-century orchestration doubles far more often at the octave and for weight. The blend table’s top entries are the wind pairs the nineteenth-century treatises single out, and its bottom entries are the pairs those treatises tell a student to avoid, which is consistent and is not evidence.

Where this ladder goes next #

Seven 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; three lists over three notes have six arrangements; and now two lists on one note, which is the case the first six all excluded.

What is owed after this is the third player. Everything above is a pair, and the definition of blend it produces — that ownership of the composite changes hands at a reachable level — has a shape that does not obviously survive being extended: three instruments have three pairwise crossings and there is no reason for them to be consistent, so a trio may have a balance at which no one instrument owns the composite, or none at all, or a region rather than a point. The assignment machinery on this ladder already enumerates trios and the balance machinery already sweeps levels, and the thing they make together is the question of whether a section has a colour or only a loudest member.