Who listens to whom when an orchestra tunes

The previous rung prices one match: one player putting their A on another’s, good to about two cents when a beat is available and to about six when it is not.

An orchestra is sixty of those, and sixty matches can be arranged. Every player ends up at the reference plus the accumulated error of whatever path they matched along, and the paths are decided by a convention rather than by anything acoustic. The chain that does not close is the same shape of problem asked about intervals rather than about people. Nobody chose that convention as an error-propagation strategy; it is what the ritual happens to be.

The arithmetic is elementary and it is not in any account of orchestral practice. A player d matches from the giver carries d independent errors, which add in quadrature, so their distance from the reference is the square root of d times the cost of one match.

A chain is the worst thing that could be done and it is not absurd #

Passing the note down a line — each player taking it from the one before — puts the last of sixty players fifty-nine matches from the source, at √59 times 2.0 cents, which is 15.1. The spread across the whole ensemble is four standard deviations, or sixty cents.

Sixty cents is over half a semitone. An ensemble tuned that way would be audibly, comically out of tune, and nobody does it — which makes it worth asking why the arrangement is even worth drawing.

It is worth drawing because it is what happens inside a section that tunes by ear from its neighbours rather than from its principal, and because it is what a chamber group without a designated giver does by default. Four players passing a note around a circle is a chain of three, at 3.4 cents, which is nothing; the arrangement only becomes bad at scale.

That is the whole shape of the argument. At small numbers the topology does not matter and at large numbers it is the only thing that matters, and a convention adopted in a chamber ensemble and carried into an orchestra would be carrying a decision that was free when it was made and is not free any more.

A star does not grow and that is the surprising half #

The flat lines are the result. A player one match from the giver is one match from the giver whether the ensemble is four or a hundred: their error is their own and nobody else’s is added to it.

That is not obvious from inside the ritual. A hundred players tuning feels like a larger operation than four tuning, and in every other respect it is; in this respect it is exactly the same size. An ensemble’s tuning spread has nothing to do with how many players are in it, provided the note goes to each of them directly.

The consequence is a genuinely useful piece of advice with an arithmetic behind it. There is no ensemble too large to tune well, and there is no benefit whatever to a player taking their A from the person beside them rather than from the oboe — the near neighbour is not a better source, it is a source with one more error in it.

Orchestral practice is not the optimum and is within a cent of it #

The arrangement every orchestra uses is neither of the two above. The oboe gives the A, the principals take it, and each section takes it from its principal — which is two matches deep for most of the room.

Two independent errors of 2.0 cents give 2.8, against a pure star’s 2.0. The convention costs 0.8 cents more than the best arrangement available, at every ensemble size, and it is the second-best of the four drawn by a factor of five over the third.

That is a good place for a convention to sit and it is worth asking what it buys for its 0.8 cents, because the ritual is not obviously trying to minimise anything.

It buys time. A star requires sixty players to hear the oboe individually, in sequence, which is sixty tuning notes; a two-level star requires the oboe to sound once for the principals and each principal once for their section, which is a dozen. The convention trades a factor of five in wall-clock for 0.8 cents of precision, and 0.8 cents is a third of the difference between two pitch standards this ladder treats as identical.

There is a second thing it buys and it is the one an orchestra would name first. A section that tunes to its own principal is internally tighter than one that tunes to the oboe: its players share a common error, so they are close to each other even where they are collectively a little off. A unison in a violin section is a unison against the desk beside it, not against the woodwind, and the shared error is invisible where it matters most and visible where it matters least.

Where a chain does happen, and it is a fifth at a time #

There is one place a chain is unavoidable and the collection has already measured what it does.

A violinist tunes the A to the oboe and then the other three strings from each other, a fifth at a time: D from A, G from D, E from A. So the G string is three matches from the oboe and the E is two, and the instrument’s own strings are a small chain whatever the section does.

Three matches at two cents is 3.5 cents, which is small. But the fifths a string player lays are not tuned to the tempered fifth — they are tuned pure, by nulling the beat, which is what makes them fast and accurate, and which is the tuner’s own method — and a chain of pure fifths does not stay inside a temperament. Four pure fifths overshoot the tempered two-octave-plus-major-third by a syntonic comma, which is why a violin’s G is flat against a piano’s and its E is sharp.

That error is systematic and this rung’s errors are random, and the two behave completely differently. Random error averages down across a section: sixteen violins with independent errors of 3.5 cents have a section mean good to 0.9. Systematic error does not average at all — every violin’s G is flat by the same amount for the same reason — so a section of sixteen is exactly as wrong as one player.

The averaging is the whole difference between the two kinds of error, and it means that the arithmetic on this page describes something that gets better with numbers while the comma ladder describes something that does not.

What the criterion does to every bar at once #

The topology multiplies the cost of one match, and the previous rung showed that the cost of one match depends entirely on which criterion is available.

So the two variables are independent and they multiply. An ensemble matching by beats in a two-level star arrives at 2.8 cents; the same ensemble matching by pitch — because the tuning note was too short to beat against — arrives at 8.1, and its band goes from 11 cents to 32.

That gives the ritual’s two decisions a joint ranking that neither has on its own. Holding the A longer is worth more than reorganising who listens to whom, because the criterion is a factor of three and the topology’s remaining slack is a factor of 1.4.

It also says which failure is which. An orchestra that sounds out of tune after tuning has either used the wrong criterion or the wrong graph, and the two are distinguishable by their size: eleven cents is a well-run ritual, thirty is a short tuning note, sixty is a chain.

The ordering does not change, which is what makes the two decisions separable at all. A shorter note scales every arrangement by the same factor, so an orchestra that has decided how to pass the note around does not have to revisit that decision when it decides how long to hold it.

What a fixed instrument does to the graph #

A keyboard cannot match. Its pitch is set before anybody arrives and does not move, so it is not a node in this graph at all — it is the source, and every other player is one match from it.

That turns any ensemble with a piano in it into a star with no choice about it, which is the tightest arrangement there is and is why a rehearsal with a keyboard is a shorter tuning than one without. It also removes the oboe’s job entirely, which is what happens in practice: an orchestra with a concerto soloist tunes to the piano.

The price is that the star’s centre is wherever the instrument happens to be. An organ’s pitch is the length of its pipes, a woodwind cannot be pulled far, and a piano’s is where the tuner left it, and neither is negotiable — so a fixed instrument buys the ensemble a tighter band around a pitch it does not get to choose. This ladder’s fourth rung is about what that cost historically; this one adds that it also buys something.

There is a middle case and it is the commonest one in chamber music. A string quartet has no fixed instrument and no oboe either, and what it does is a star with a rotating centre: whoever has the most exposed part gives the A, or the first violin does by convention. That is the same two cents as an orchestral star and it is available because four players can all hear one note at once, which sixty cannot. The topology an ensemble can use is bounded by how many players fit inside one sounding note, and that is a fact about a room rather than about a limen.

A section of sixteen is not sixteen players #

The averaging point from the fifths section is worth taking on its own, because it is the one place where the arithmetic says something an orchestra could act on.

Sixteen violins playing a unison are not heard as sixteen pitches. They are heard as one, at the section’s mean, with a width — which is the same object the ensemble’s own vibrato and detuning produce and which is most of what a section sounds like as against a soloist.

The mean of sixteen independent errors of 2.8 cents is good to 0.7. So a section is nearly four times better placed than any of its members, and the ensemble’s sections are far closer to each other than its players are. That is what makes an orchestra’s tuning sound better than this arithmetic says it is: what a listener hears is a handful of section means, and the means average away most of what the individual matches got wrong.

Which reverses the practical advice one more time. If the audible object is the section mean, then the thing worth minimising is the error common to a whole section — and a section that tunes to its own principal shares its principal’s error entirely. The convention that makes a section internally tight is the same convention that makes its mean less accurate, and the two cannot be had together: matching everybody to the oboe gives accurate section means and looser unisons, and matching everybody to the principal gives the reverse.

That is a real trade with a name in the practice, and orchestras have chosen the tight unison. Nothing here says they are wrong; roughness inside a unison is audible and half a cent of section mean is not.

What the pictures cannot show #

Every arrangement here is a tree, and a real tuning is not. Players hear more than the person they are matching to: a violinist tuning to a principal is also hearing the section around them, the oboe if it is still sounding, and whatever else is in the room. Every one of those is an extra constraint, and extra constraints narrow the band rather than widening it — so the numbers here are upper bounds, and the real orchestra is tighter than any of these bars.

Nothing here models the order in time. A section that tunes while the oboe is still sounding is making a two-way comparison rather than a chain of one-way ones, and a section that tunes after the oboe has stopped is matching a remembered pitch, which the previous rung says is a worse criterion than either. The ritual’s timing may matter more than its topology and nothing on this page can say.

The four topologies are also four hand-chosen shapes rather than a search over the arrangements an ensemble could use, and a tree’s cost here depends on one number — its depth — so anything with the same depth costs the same whatever else is true of it. That is a real property of quadrature rather than a simplification, but it means the drawing has nothing to say about the many arrangements that are neither a line nor a star: a section that takes the A from two sources and splits the difference is doing something the model has no term for, and averaging two references is better than either.

And the independence assumption is doing heavy work. Two players sitting at one desk almost certainly do not make independent errors — they hear each other, they adjust toward each other, and the correlation is unknown. A correlation of one within a desk would halve the effective number of independent matches in a section and would change none of the ordering.

Whose ritual, and what to measure #

The practice is the orchestral tuning convention of the last century and a half, and the reason it is worth pricing is that it is uniform across traditions that agree about very little else — the same arrangement in Vienna and in Tokyo, with no acoustic argument ever having been offered for it.

What the arithmetic supports is narrow. The convention is second-best of four by a small margin and best-by-far on any account that weighs time; the loss to the optimum is under a cent, which is under half the precision of one match. A convention chosen for convenience has landed one step off an optimum nobody computed, which is the ordinary way conventions land well rather than a coincidence.

The measurement that would test it is the same one the previous rung asked for, run twice: an orchestra’s pitch distribution immediately after its usual ritual, and after a ritual in which every player takes the A from the oboe individually. The prediction is a spread of about eleven cents in the first case and eight in the second, and a null would say the independence assumption is wrong rather than that the arithmetic is.

The cheaper version needs no orchestra at all. Two players, one reference, twenty repetitions each: the distribution of where they land is one match, and it is the number every bar on this page is a multiple of. That measurement would replace the whole of the previous rung’s model with a fact, and nothing in this collection depends on the model rather than on the fact.

The ladder from here #

Twelve rungs, and the last two have taken a standard out of the document and put it into the room. It reaches an ensemble through a limen, which is two cents when a beat is available and six when it is not; and through a graph, which multiplies that by the square root of how many matches deep a player sits.

Both of those describe the moment tuning ends. What happens next is not covered by either, and it is the larger part of the evening: once the oboe has stopped, nobody is matching a reference at all. Each player corrects toward what they hear around them, which is the other players — and a correction toward one’s neighbours has no term in it pulling anybody back to the note they were given. A dynamic like that has a fixed point at every common value, so the players converge on each other while the ensemble as a whole is free to go wherever the noise takes it.