A section has a loudest member, not a colour

Two players on one note offered a definition rather than a description. Two independent sources add in power, so their composite is neither of them; one of the two owns it in the sense of being the spectrum the composite most resembles; and the balance at which ownership changes hands is a number. A pair blends when that number lies inside the range the players can reach. A clarinet and an oboe cross at −3.4 decibels, a violin and a clarinet at +3.5, a violin and an oboe at +16.3 — all reachable, all named as blending pairs in every orchestration manual, and none of them told so.

Everything on that rung is a pair, and an orchestra is not made of pairs. Three players on one note have three pairwise crossings, and nothing about the pair result says they have to agree with each other. The machinery for asking is already here: which player on which note enumerates trios, and the balance sweep the pair rung built runs over levels. What the two of them make together is the question of whether a section has a colour or only a loudest member.

What a third source does to the arithmetic, which is nothing #

The power sum extends without a decision. Three independent sources have unrelated, drifting phases, so their partials add in power and the composite’s nth partial is the root of the sum of three squares rather than two. Nothing about that step is a choice, and it is the only assumption the rung inherits.

It is worth being clear about what is and is not being reused. The partial lists are the ones the first rung of this ladder argues the ear actually receives, filtered by the mechanisms the rungs after it built — a mouth, a body, a bell. Nothing about the third player asks any of them for anything new.

What it costs is a dimension. With two players the balance is one number, the decibels of the second against the first, and ownership is a question about a line: the composite is nearer to one spectrum below the crossing and nearer to the other above it. With three players the balance is two numbers, and ownership is a question about a plane — a partition of it into three regions, one per player, with boundaries where two distances are equal and a single point where all three are.

That point is what a section colour would have to be. Everywhere else on the plane the composite resembles one player more than the other two, which is the same sentence the pair rung ended on with a third instrument added to it.

The geometry is worth a moment because it decides what kind of answer is available. Three regions in a plane have three boundaries, and three boundaries generically meet at one point — not along a curve, not over an area. So a trio’s colour, if it has one, is a setting and not a range: move either dial from it in any direction and one of the three players takes ownership back. Nothing about that is special to spectra; it is what happens whenever three quantities compete to be smallest over two free variables.

The question the pair result cannot answer #

Here is the shape of the thing that would break the extension, and it is worth stating before computing it because it is a real possibility rather than a rhetorical one.

Suppose the composite of a violin and a clarinet belongs to the violin; that of a clarinet and an oboe belongs to the clarinet; and that of an oboe and a violin belongs to the oboe. Nothing in the construction forbids it. Each of the three statements is about a different pair of spectra, and each is decided by a separate comparison of two log-spectral distances. But taken together they say that in the trio there is no strongest member — that ownership goes round in a circle, and that the question which instrument does this section sound like has no answer even in principle.

That would be a genuinely interesting result, and it is the one the ladder was pointed at. It is also checkable in one pass: read the ownership of every pair, and look for a cycle.

And it does not happen #

Fifteen pairs, twenty triples, no cycle. The relation is a total order: a voice over a trumpet over a violin over an oboe over a clarinet over an organ flue pipe, and the composite of any two of them belongs to whichever is higher on that list.

The win counts are the cleanest way to see it. A tournament on six players is a total order exactly when the six win counts are 5, 4, 3, 2, 1 and 0 with none repeated, and that is what they are. It is also exactly the statement that no three of them form a cycle, and the two checks are made separately here because a single one could be wrong in a way that looked tidy.

The order itself is not the one an intuition about brightness would give. An instrument is not one timbre, and ownership is not brightness: a trumpet is far brighter than a voice on this set of radiators and sits below it, while a violin — the dullest of the six by spectral centroid — sits third. What decides ownership is how much of the sixteen partials a spectrum covers, because the composite is an upper envelope and an envelope belongs to whatever fills more of it.

The temptation is to conclude that a cycle is impossible, and it is not. The composite of two spectra is their upper envelope in all but name — the root of a sum of squares is within three decibels of the larger term — so ownership asks which of two shapes covers the other, and covering is not a scalar comparison. A can cover B over the low partials, B can cover C over the middle ones and C can cover A at the top, and the three comparisons then come back in a circle.

Handing the same arithmetic sixteen random partial amplitudes per instrument, drawn independently over eighty decibels, produces an intransitive triple 4.65 per cent of the time — 930 in twenty thousand. So one triple in twenty of arbitrary spectra has no strongest member, and a cycle is not some measure-zero curiosity.

What removes it is that instrument spectra are not arbitrary. Every radiator in this collection is a monotone rolloff with one resonance or one cutoff imposed on it, and two shapes of that kind cross each other at most a couple of times, which is not enough crossings to build a circle out of. Generating twenty thousand triples of that kind — a random rolloff exponent, a random resonance frequency, a random sharpness — gives 8 intransitive triples in twenty thousand, one in twenty-five hundred. The instruments are not special; the family is.

Where the balance belonging to nobody actually is #

So every trio has a strongest member, and the interesting question moves to the other half of the pair result: the crossing. For two players it is a level. For three it is a point in a plane, and the drawing above says where it is for each of the twenty.

It exists in every trio and it is reachable in four. The clarinet, oboe and voice cross at −0.3 and −19.1 decibels, which is a real instruction — hold the two woodwinds level and put the voice nineteen decibels under them — and at that balance the composite sits 15.27, 15.28 and 15.25 log-spectral decibels from the three of them. Nobody owns it. That is a section colour in the only sense this ladder can define one.

The other sixteen put it somewhere no ensemble can go. The violin, oboe and flue pipe cross at −9.2 and +22.2, which sounds inside the range until the third comparison is made: the oboe and the flue pipe would then be 31.4 decibels apart, and the ladder’s own estimate of what two players can do against each other is twenty-four. The trio has a colour that no arrangement of three human beings can produce.

And the four that are reachable are not a random four. Every one of them contains the voice, and the reason is legible in the pair rung’s own table: the voice’s crossings against everything else sit near −21 decibels, which is unusually deep and unusually consistent, so the three boundaries in a trio containing it are crowded into the same corner of the plane and meet inside it. A trio blends when one of its members is the voice, on this set of radiators, and that is a statement about the voice rather than about trios.

What is left of blending, which is most of it #

The pair definition survives, and it survives in a weaker form than it was stated in.

Across the twenty trios the member that owns the most of the reachable square owns a median of 75 per cent of it, with a minimum of 56 and a maximum of 93. So a conductor turning both dials over their whole range is choosing, in the typical case, between a large region where the composite is one instrument with colouring and two smaller ones where it is another. That is not the pair picture, where the crossing splits the line into two comparable halves. Adding a third player does not make the answer more balanced; it makes it more lopsided.

Which is worth stating plainly, because it points the other way from the intuition orchestration teaching runs on. Three instruments are usually described as more fusible than two — a chorus of winds, a family sound — and on the definition this ladder is using they are less. The extra source has to compete for the composite with two others rather than one, and the total order guarantees that one of the three wins nearly everywhere.

What the pairwise order predicts, and where it stops #

There is one more thing the tournament buys, and it is the practical one.

At the ladder’s own working pitch the trio’s owner at equal level is the member the pairwise order ranks highest, in all twenty cases. Which means the expensive computation — three spectra, a power sum, three log-spectral distances — is answered by the cheap one, which needs only the fifteen pair results the rung below already has.

That is a prediction rather than a theorem, and it fails. Asked at middle C the pairwise order gets eighteen of the twenty right, and asked an octave and a fifth above the working pitch it also gets eighteen. The two misses in each case are trios whose members sit adjacent on the order, where the composite of three tips a comparison the composite of two did not.

So the honest form is: the strongest member of a pair predicts the strongest member of a trio nine times in ten, and the composite of three is not the composite of two. The failures are informative because they are all near-ties, which is what a prediction from an incomplete model should look like when it is nearly right.

It also puts a bound on how much a fourth player, or a fifth, could change. Ownership is decided by which spectrum covers the others, and adding a source can only ever raise the envelope — so a member that failed to own a trio cannot be rescued into owning a quartet by anything except a change of balance. The order is stable under addition, which is why a section of any size has a loudest member and why who plays what and how loud can go on treating a doubled note as one part.

Which computation produced the numbers #

The six radiators are the ladder’s own and are unchanged: a violin through its measured body resonances, a clarinet and an oboe with their computed radiation cutoffs, an organ flue pipe, a voice as three formants on a vowel, and a trumpet as a high pass at its bell’s cutoff. Each returns sixteen partial amplitudes at a stated fundamental, normalised to its own strongest.

The composite is the root of the sum of the squares, partial by partial, with each player scaled by its stated balance in decibels.

The distance between two spectra is the log-spectral distance the rung below defines: each normalised to its own total power first, then the root mean square of the difference in decibels over the sixteen partials. Normalising first is what makes it a comparison of shapes; without it the answer would be a restatement of which player had been turned up.

The ownership plane is that distance evaluated on a grid over the two balances, from −24 to +24 decibels in steps of a decibel and a half. The triple point is found by minimising the spread of the three distances over a coarse grid from −48 to +48 and refining twice, deliberately outside the reachable square, so that a trio whose crossing is at +46 decibels is reported as unreachable rather than as absent.

The two intransitivity rates come from the same arithmetic on generated spectra rather than on instruments: twenty thousand triples of sixteen independent amplitudes over eighty decibels, and twenty thousand of a rolloff with a resonance.

Where the model stops #

A section is not three sources. Sixteen first violins are the case an orchestrator actually faces, and what a choir does that a soloist cannot is that the fluctuation between nominally identical players is itself the sound. Three is the smallest number larger than two, and nothing here has been asked at sixteen.

The range is asserted. Twenty-four decibels either way is the rung below’s estimate of what two players can produce against each other, and the count of reachable trios is entirely a function of it. At thirty decibels seven of the twenty would blend; at eighteen, one.

Roughness is a different quantity and a much larger one. Everything above is about which spectrum the composite resembles, and the same chord is harsher when it is louder is about how much beating it produces — the two are computed from the same partial lists and have nothing else in common. A trio that blends beautifully by this definition can be intolerable, and a third is rougher in the bass for reasons no ownership plane can see.

Ownership is not audibility. A composite three log-spectral decibels from the voice and twenty-six from the clarinet is a fact about two vectors, and no listener has been asked whether they hear a voice. The pair rung’s agreement with the manuals is the only external check either rung has, and it is a check on the pairs.

And the balance plane is a steady state. Three players do not begin together, and the strongest cue that a section is a section is that it does not. Nothing here has an attack in it.

What the picture cannot show #

It cannot show intonation. The rung below found that a unison ten cents out is rougher than a major third dead in tune, and three players on one note have three mistunings rather than one. That is a different quantity from ownership and it is much larger.

Nor can it show the room. Three sources heard at ten metres have been through the same reverberation, and the correlation between them at the listener’s ear is not zero — which is the assumption the power sum rests on.

It cannot show what the third player is for. An orchestrator adds an instrument to a doubled line for weight, for reach, for a register the pair cannot cover, or to free a player elsewhere. This prices the colour and none of the other four.

And it cannot show more than one note. Every figure here is a unison. Which player on which note puts three players on three different notes and gets a completely different problem, and the two have not been joined.

Whose instruments, and when #

The radiators are generic and the levels are arbitrary, so nothing here measures an orchestra.

The observation that fits a practice is about the reachable four. Every one contains the voice, and the arrangement in which a solo voice is balanced against two instruments deep enough below it that no single line owns the sound is exactly the texture of accompanied song — a nineteenth-century preoccupation, written with the accompaniment marked far below the vocal line as a matter of course.

That the arithmetic picks out the one texture where the marking is conventional is consistent and is not evidence. It is stated here so that the temptation to make it an explanation is visible.

Where this ladder goes next #

Eight 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; two lists on one note make a composite one of them owns; and now three lists on one note, which make a composite one of them owns rather more emphatically.

What is owed after this is the pitch. Every figure on every rung of this ladder sounds its instruments at one note, and the fourth rung is the reason that is not a detail: every filter in this collection is fixed in frequency and the fundamental is not, so a radiator’s shape is a function of the note it is playing and everything computed from two shapes is a function of it too. The blend table, the crossings, the total order and the twenty trios above are all statements about one note in the middle of the treble staff, presented as statements about instruments. That is one sweep, the machinery already exists, and the reason it is owed rather than optional is that if the answer moves, none of the tables on this ladder means what its caption says.