The fourth player is a spectrum, not a decision

Every figure on the six rungs before this one puts exactly one instrument on each note of the chord. That is not a convention, it is a constraint the machinery insists on: an arrangement is a permutation of players onto notes, and the enumeration refuses a chord and a player list of different lengths.

Orchestration does not work that way. The commonest single operation in the craft is a doubling, and a doubled note has two players on it, so the ensemble is larger than the chord and an arrangement stops being a permutation. Four players on three notes have thirty-six arrangements rather than six, and there is no reason in advance to expect the extra choice to be small.

It is worth saying what kind of object thirty-six is. Four players onto three notes is a surjection, and the count is thirty-six rather than the eighty-one of all maps because a note left empty is a different chord and not a different orchestration. The enumeration checks its own total against the closed form for the number of surjections, because an arrangement that quietly dropped a note would score smoother than every honest one and would win the drawing.

Thirty-six instead of six, and one of them wins by very little #

The enumeration is the anchor’s own with the permutation constraint lifted. Every arrangement holds the same chord, the same four players and the same total loudness in sones, and the two players sharing a note split that note’s power so that nothing is won by being louder. What is left to compare is the roughness, which is the objective the first rung set — and holding the total loudness equal is not a nicety, since the ranking survives the dynamic and the chord does not showed that a scoring compared at a different level is being compared on a different set of audible partials.

The spread is a factor of 2.21 from best to worst. That is a slightly larger number than the six-permutation problem produced on three players, which is what one would expect from six times as many candidates, and it is the wrong number to look at on its own. The interesting question is not how wide the range is but which choice the range is made of.

The fourth player is the smallest thing on the page #

Which note gets the doubling accounts for seven per cent of the variance across the thirty-six arrangements. Which players end up on the bass accounts for eighty-nine.

That is a strong result and it is worth stating in the form a reader can check against their own experience. An orchestrator choosing whether to double the root, the third or the fifth of a triad is making a decision that moves this objective by a few per cent. An orchestrator choosing whether the bass is a flue pipe or a clarinet is making one that moves it by nearly the whole range. The received advice runs the other way round: doubling is discussed at length in every treatise and the choice of bass instrument is usually taken as given by the scoring. It is also the opposite emphasis from the one which player on which note arrived at with three players and three notes, where no single position dominated — adding a fourth player does not spread the decision out, it concentrates it.

It is also, on reflection, what the arithmetic had to say. Roughness is dominated by the pairs of partials that fall inside a critical band, and a critical band is a fixed width in hertz, so the low notes of a chord contribute most of it. Whichever instrument is at the bottom supplies one member of nearly every rough pair in the scoring. A doubling at the third or the fifth adds a spectrum where the band is already wide.

The ranking of the bass instruments themselves is worth reading off, because it is the part an orchestrator could act on. Over the thirty-six, a flue pipe on the bass averages the smoothest scoring, a violin next, then an oboe, and a clarinet is comfortably the roughest — and the reason is the clarinet’s own partial list rather than anything about the chord. An odd-dominant spectrum puts strong energy at the third and fifth partials with almost nothing between, and those land where the other parts’ low partials are, which is the definition of a rough pair. The same instrument three notes higher does no such damage, which is why the ordering here is a statement about a clarinet on the bass of this chord and not about clarinets.

There is a second reading of the seven per cent that is worth separating from the first, because they are different claims. One is that the doubling decision is small. The other is that it is small relative to a decision made earlier, and an orchestrator does not usually get to make them in that order — the instruments available are given, and what is being chosen is where the spare player goes. Read that way the figure says something more useful: given four fixed players, the arrangement that matters is which of them takes the bass, and the doubling is what happens to the other three.

A doubled note priced as one spectrum #

Which suggests a reduction, and the spectrum ladder is where it comes from. Two players on one note found that a composite belongs to one of its two members — it is nearer that member’s shape than the other’s over most of the balance range — and a section has a loudest member found that ownership across all six of the collection’s radiators is a total order.

So there is a prediction available: replace a doubled pair by whichever of the two owns their composite, and the four-player problem becomes a three-player one that the existing machinery already solves.

It works, at a rank correlation of 0.85 with the full enumeration, and it picks the same winner.

The control, which is the whole of the test #

A rank correlation of 0.85 proves nothing on its own, because four spectra resemble three whatever substitution is made. The question is whether the owner is doing the work, and the way to ask it is to run the same reduction with the prediction removed: replace the pair by the member that does not own the composite.

That gives 0.64. The two are different, the owner is the better predictor, and the reduction has said something.

This is the shape of test the collection uses everywhere and it is worth naming why: a substitution that improves a computation is not evidence about the mechanism unless the wrong substitution is worse. Reporting 0.85 without 0.64 beside it would have been reporting that a doubled note is a bit like a single note, which nobody needed.

And the ensemble that refuses it #

Put a trumpet in the quartet and the whole thing inverts. The owner reduction falls to 0.66 and the control rises to 0.83 — the member that does not own the composite is the better predictor, and it is the one that picks the winning arrangement.

That is a refusal rather than noise, and the mechanism is legible in two partial lists.

A trumpet at this pitch radiates a nearly flat list — 0.62, 0.94, 1.00, 0.96, 0.88, 0.81, 0.71, 0.63 — because its bell is a high pass and the partials are all above the cutoff. A clarinet at the same pitch radiates 1.00, 0.04, 0.50, 0.03, 0.19, 0.01, 0.07: a huge fundamental and almost nothing even-numbered. In log-spectral distance the composite of the two is 1.3 decibels from the trumpet and 14.9 from the clarinet, so the trumpet owns it, decisively and correctly.

But the clarinet’s fundamental is the strongest single partial in the pair, and it is a low partial, which is where the critical bands are narrow and where the roughness against the rest of the chord is made. Ownership is a statement about spectral shape and roughness is a level-weighted sum inside bands. Replacing the pair by its owner keeps the shape and discards the one partial that was doing the work.

So the honest version of the reduction is not replace a pair by its owner. It is replace a pair by whichever member supplies the partials the roughness sum actually uses, and that is the owner only when the two members’ low partials are comparable. It is a genuinely different criterion, it agrees with ownership on three of the four ensembles tried, and the case where it disagrees is the one with the most extreme filter in the collection in it.

What this leaves the assignment problem #

Three things, and the third one is a correction of something the ladder was on the way to assuming.

The doubling is a small decision inside a large one. Seven per cent against eighty-nine on the first quartet, three against forty-eight on the trumpet one. An orchestrator who has chosen the bass has already made most of this choice.

A pair can be priced as one player, with a caveat about which. That keeps the joint problem the size it was: the continuous half still solves a level per part, but the discrete half does not have to enumerate surjections if the reduction is used, and the reduction is right about the winner more often than not.

And the intonation of the doubling does not dominate, which was the obvious guess and is wrong. The spectrum ladder found that two players on one note five cents apart are sixty-six times rougher than the same two exactly in tune, and it is very easy to read that as saying the doubling’s tuning swamps the whole assignment problem. Put the mistuning into the scoring and it does not: five cents adds 4.1 per cent to the scoring’s total roughness, ten cents adds 7.8 and twenty cents adds 14.2, against the 121 per cent that separates the best arrangement from the worst.

The factor of sixty-six was a ratio measured on the pair alone, where a perfect unison has essentially no roughness at all and any mistuning is therefore an enormous multiple of nothing. Inside a chord the pair sits among every other pair of parts, and the same absolute increase is a few per cent of a much larger total. Both numbers are right and only one of them is about orchestration.

Which computation produced the numbers #

The arrangements are every map from the players onto the notes that leaves no note empty, which for four players on three notes is thirty-six and is checked against the closed form for the number of surjections rather than trusted to the enumeration.

Each arrangement is put at one total loudness by the anchor’s own bisection, so that two candidates are compared at equal loudness rather than one being quieter. The two players sharing a note are each given three decibels below the note’s nominal, so that the note’s total power matches a single player’s.

The roughness is scoringRoughness unchanged: every partial of every part against every partial of every other part, through the Plomp–Levelt curve scaled to the critical bandwidth, quadratic in pressure. Two players on one note are two parts, so the pair between them is in the sum — at an exact unison it contributes almost nothing, since their partials coincide.

The variance shares are the between-group sum of squares over the total, for three different groupings of the same thirty-six numbers. They overlap and do not sum to one, because choosing which two players are the pair partly chooses who is left for the other notes.

The reduction replaces the doubled pair with a single player at the note’s full nominal level and re-solves for the common gain, so the reduced scoring is at the same loudness as the full one. The rank correlation is Spearman’s over all thirty-six.

Where the model stops #

Two on a note is not a section. Sixteen first violins doubling eight seconds is the real case, and what a choir does that a soloist cannot is that the fluctuation between nominally identical players becomes the sound. Nothing here has been asked at more than two.

Four ensembles is not a survey. The variance shares run from three to twenty per cent for the doubled note across the quartets tried, and the ordering — doubling smallest, bass largest — held in each, but four quartets on three chords is a handful.

The balance is fixed at an equal split. The two players on a note are given equal levels, which is one point on the whole line the spectrum ladder sweeps, and the composite at the crossing is a different spectrum from the composite at equal level. That is a real gap rather than a caveat: the balance is the one continuous variable the joint solve exists to set, and here it is held at a value chosen for tidiness.

And every one of these numbers is at one pitch. The blend table has a row for every note is the standing correction to everything computed from two spectra in this collection, and it applies to the reduction directly — the owner of a pair is a different instrument at a different note, so which member carries the roughness is a function of register too. The two quartets here are at pitches a fifth apart, which is not enough to separate the effect of the trumpet from the effect of the note.

And the chord is held still. Three rungs of this anchor made the objective a functional over time and this one is back to one chord, which is a real retreat and is where the ladder now owes something.

What the picture cannot show #

It cannot show why an orchestrator doubles. Weight, reach, covering a weak register, freeing a player, or a colour — this prices the last of the five and pretends the other four are not the reason.

Nor can it show the octave. Doubling at the octave is at least as common as doubling at the unison and is a different problem, because the upper instrument’s partials coincide with the lower’s even ones rather than with all of them.

It cannot show the attack. Two players on a note do not start together, and a low note cannot start on time — which is the strongest cue that a doubling is a doubling, and it is a steady state here.

And it cannot show the room. Every part arrives at a listener through the same reverberation, so the assumption that two players are independent sources is weakest exactly where an orchestra is heard.

Whose music, and when #

The players are generic and the chord is generic, so nothing here measures a repertoire.

The observation that fits a practice is about the bass. The result that which instrument sits at the bottom carries most of the objective is the one thing on this page that orchestration teaching does not emphasise and orchestration practice obviously does: the bass line is scored first, it is scored conservatively, and the instruments that take it are a short list that changes very little across two centuries. That is consistent with the arithmetic and it is not caused by it — a bass part is also the hardest to hear, the easiest to muddle and the one a conductor complains about.

Where this ladder goes next #

Seven rungs. Who plays what and how loud is one question; the ranking survives the dynamic and the chord does not; a dynamic mark changes what a note is; a subito piano is a rate rather than a level; the other quantity succession has arrives whole; the mask arrives with it and pushes the same way; and now more players than notes, which turns out to be the smallest decision in the problem.

What is owed after this is the doubling over time. Three of the rungs above made this anchor’s objective a functional over a passage — a running loudness, a roughness window, a mask — and this one has quietly gone back to holding one chord still, which is the retreat every enumeration makes when the enumeration gets large. A doubling is not a property of a chord: an orchestrator doubles a line, across a phrase, and the pair that carries a note is usually the pair that carried the one before it. What that changes is a question with a plausible answer already visible — a doubled part has one attack rather than two and therefore a different onset, and the mask reaches a chord through whichever instrument was loudest in the one before — and the machinery for it is on this ladder in pieces. It needs arithmetic and a passage, and no listener.