A consensus with nothing to hold it

Three rungs have priced the ritual: a limen, a graph, and the finer criteria the music itself supplies. All three describe the first minute of a rehearsal.

The rest of the evening has a different structure and the difference is not a detail. During the ritual there is a reference — one player sounding a note that nobody is asking to change. During the performance there is not. Each player corrects toward what they hear around them, and what they hear around them is other players, correcting toward them.

That is a consensus dynamic, and a consensus dynamic with no external term has a fixed point at every common value. It has nothing to say about which one.

The disagreement converges and the pitch does not #

The two quantities behave completely differently and the drawing separates them.

The spread — how far the players are from each other — falls, from 2.8 cents at the start to about 0.5 after a few hundred corrections. That is what a consensus does, and it is what an ensemble sounds like getting better in tune with itself as a piece goes on.

The centroid — where the ensemble as a whole is — does not converge on anything, because nothing in the dynamic prefers one value to another. Every correction carries a little noise, the noise averages across the ensemble but does not vanish, and what is left is a random walk of the mean.

An ensemble playing by ear gets more and more precisely in tune with itself, about a pitch that is moving. That sentence is the rung, and it is a shape that a great deal of musical practice takes for granted without stating: a section’s unison is a thing that can be fixed by listening and an ensemble’s pitch is not.

The size is the result, and it is small #

The displacement grows as the square root of the number of corrections, as any random walk does, and the band in the drawing is that growth: 0.84 cents a quarter of the way through the run and 2.01 at the end of it. After four hundred and eighty corrections — a few minutes of playing at a correction every second or two — the root-mean-square displacement is about two cents and the furthest of eight runs is a little over four.

Two cents is nothing. It is under the ensemble’s own matching limen, it is a fifth of the band the tuning ritual produced, and no listener could report it.

That number is the reason this rung is written the way it is, because the mechanism has an obvious application that it will not support. Unaccompanied choirs are famous for going flat, by amounts that are reported in semitones over the course of a piece, and a consensus with no anchor looks exactly like the explanation. It is not: to reach a semitone by random walk would need a hundred thousand corrections, which is more than a choir makes in a lifetime, and the walk has no preferred direction while the observed effect is almost always downward.

So the mechanism is real, and the effect it produces is a hundredth of the effect it would have been invoked to explain. Recording that is more useful than not computing it.

What does explain a choir going flat, and it is not this #

Two candidates remain and this collection has one of them.

A passage tuned justly at every step does not come back to its starting pitch. The gap is a syntonic comma — 21.5 cents — per completed cycle of the relevant progression, it is systematic rather than random, and it always goes the same way for a given progression.

That is the right shape for the observed effect in every respect the random walk is the wrong shape. It accumulates linearly rather than as a square root; twenty cycles of a comma-generating progression is a semitone. It has a direction. And it is a consequence of tuning well, which is why the choirs that do it are the ones with the best intonation — a fact every choral director reports and no account of fatigue predicts.

The other candidate is not acoustic at all: singers tire, the larynx sits lower, and pitch follows. That one is not entirely outside this collection either — the voice ladder computes where the register break sits and a tiring singer’s relationship to it is exactly the quantity the ninth rung of this ladder put on an axis. What is missing is any model of the tiring, which is physiology rather than acoustics. That one is outside this collection entirely and is the explanation most often given.

What this rung contributes is the elimination of the third candidate. Mutual correction without an anchor is a real mechanism, it is present in every unaccompanied ensemble, and it is far too small to be what anybody is hearing.

What one fixed instrument is worth #

A restoring term changes the dynamic completely, and the term has a physical realisation: a player whose pitch cannot move.

A keyboard, an organ, a harp or a set of tuned percussion is not a node in the consensus: it is a term that pulls everybody toward a value it does not itself update. With none at all the displacement after the run is 1.94 cents; with a restoring weight of 0.05 on each correction it is 0.31.

The curve’s shape is the useful part. It is steep at the left, so one fixed instrument among sixteen is worth nearly as much as a great deal of discipline, and the marginal value of a second one is small. That is consistent with the practice: an ensemble either has a keyboard or does not, and nobody has ever suggested that two keyboards hold the pitch better than one.

It also gives an arithmetic to a thing choral directors do that looks superstitious. Giving a choir a chord from a piano halfway through an unaccompanied piece is described as “checking the pitch”, and on this account it is a single application of a restoring term to a system that has none — which resets the walk rather than damping it, and is the correct intervention for a mechanism with no memory.

The walk against the ritual, on one axis #

Setting this rung’s number beside the previous three puts the whole anchor’s last quarter on one scale, and the ordering is not the one the sequence suggests.

The tuning ritual leaves the ensemble spread over about eleven cents. The intervals of the music then tighten that, because a fifth betrays three times what a unison does and a third five times. And the walk moves the whole thing by two or three cents over a movement.

So the three effects are eleven, a tightening of unknown size, and two to three — and the largest of them is the one that happens before a note is played and is corrected within a bar of one being played. The ritual sets a starting condition and the music does the tuning, which is a reordering of the usual account rather than an addition to it.

The one that grows is the walk, and it grows slowly. Doubling the length of a movement multiplies it by 1.4; a concert of four works is two hours and perhaps four cents. That is a quantity nothing in the practice would notice and it is the only one on this page that gets worse rather than better.

The version with a conductor in it #

There is a term the model does not have and the room does: somebody who is listening and is not playing.

A conductor who stops and says “that is flat” is applying a restoring term to a value they hold themselves, which is exactly the anchor of the figure above with a weight applied intermittently rather than continuously. On the numbers, that is worth a great deal: a single application resets the walk to zero, and a walk that is reset has no long-run growth at all.

What the model cannot say is whether the value the conductor holds is any good. A conductor’s own pitch reference is a memory for a convention, which this ladder’s second rung established is rare, dated and drifts with the standard the person learned. So an intermittent anchor bounds the walk and centres it wherever the anchor is, and the anchor may be a semitone off in 1750 terms and nobody would know.

That is the arrangement most unaccompanied singing actually uses — a director with a pitch pipe, applied between movements — and it makes the ensemble’s absolute pitch a property of one person’s memory and one small metal object, with the sixteen players’ consensus doing everything else.

Whether anybody is correcting at all #

The model assumes each player is continuously adjusting, and there is a case where they are not.

A string player’s open strings do not move. A pianist’s do not. A fretted instrument’s do not. Those players are not in the consensus at all, and an ensemble made of them has no dynamic to speak of — its pitch is whatever it was set to, and it changes only when the temperature does, which the tenth rung already priced.

The consensus applies to the players who can move: singers, winds, brass, and strings on stopped notes. Which is most of an orchestra most of the time, and is the entirety of a choir.

That division suggests where the mechanism would be most visible, and it is not the orchestra. An orchestra has a keyboard’s worth of fixed pitch in it almost always — a harp, a celesta, timpani that were tuned before the movement, or simply a large number of open strings — so the restoring term is never quite zero. A choir is the only common ensemble in which it is exactly zero, which is why a choir is where the question was asked and is also, on these numbers, the only place the answer could be measured.

Two mechanisms that could be told apart in one recording #

The two candidate accounts of what happens to an unaccompanied ensemble’s pitch make different predictions in more than their size, and the differences are the kind a recording would settle.

A random walk has no direction, so half of a set of performances should end sharp. Its displacement grows as the square root of time, so a movement twice as long should drift 1.4 times as far. And it does not depend on what is being sung: a piece with no cadences in it drifts exactly as much as one full of them.

A comma has a direction, fixed by the progression rather than by chance. It accumulates linearly, so twice the length is twice the drift. And it depends entirely on the harmony: a passage whose progression does not complete a comma-generating cycle produces none at all, and the polyrhythm ladder’s reading of the same object says exactly which cycles do.

Three differences, all measurable on the same recording: the sign, the exponent, and the dependence on the music. Nothing in this collection has to be believed for the test to work, and the test would settle a question that choral practice has argued about without evidence for a century.

The prediction this rung is willing to make is that the comma wins on all three. What would be genuinely surprising — and would send the whole account back — is a downward drift in a piece whose harmony generates no comma at all, which would mean the mechanism is the singers rather than the tuning.

What the pictures cannot show #

The dynamic is a caricature and its parameters are chosen rather than measured. How much of the disagreement a player removes per correction, how often a correction happens, how many neighbours a player is actually attending to, and how much noise each correction carries — four numbers, none of them known, and the displacement is proportional to the square root of the product of the last two. The shape of the result is robust to all of them, because a consensus with no anchor has a free centroid whatever its coefficients are; the two cents is not.

Nothing here has a piece of music in it. Real corrections are not toward a mean: they are toward whatever interval the harmony asks for, which is the comma ladder’s subject and is exactly the systematic term this model leaves out. A version with the harmony in it would produce both mechanisms at once and would be the honest simulation; it would need a passage, a tuning rule and a decision about who yields to whom, and the third of those is the one nobody has a model of.

The correction rule is also the wrong shape in a way worth naming, because the right shape is harder and is where the interesting behaviour would be. A player does not move toward a mean; they move toward whichever of two notes they are hearing a beat against, and a beat is a symmetric measurement that says two notes disagree without saying which is wrong. So a real correction is a negotiation, and a negotiation between two people who both yield is a different dynamic from one in which the junior yields — the first converges to their mean and the second converges to the senior’s value, which is an anchor of exactly the kind this rung says an ensemble does not have. A convention about who gives way would be a restoring term made of nothing but seating, and orchestras have very strong conventions about that.

And the players are on a ring, each listening to three neighbours either side. A real ensemble’s listening graph is nothing like that — a section hears its own principal, everybody hears the loudest part, and a player at the back hears the room rather than any individual. The graph decides how fast the spread converges and, on any graph, does not decide that the centroid is free.

Closing the anchor #

pitch-standard closes at fourteen rungs. The model is a convention, a number, and the physical things that have to change when the number does — and every variable it has now has a rung.

What was the number, historically? Rungs one and three: four hundred years, 296 cents, and a documented climb.

What is it a specification of? Rung four: a length and a tension, with a material constant under a convention.

What does moving it cost each family? Rungs five to nine: the strings, the winds, the organ that cannot be moved, and the voice, which has no length and pays linearly.

What width does it have in practice? Rungs ten to twelve: the room, which is systematic; the matching limen, which is random; and the graph, which multiplies the second.

And what holds it once it has been set? Rungs thirteen and fourteen: intervals sharper than the one it arrived on, and — when nothing in the room is fixed — nothing at all.

What is not on that list belongs elsewhere. Why the chain of pure intervals does not close is the comma; how finely a listener can hear any of it is pitch acuity; what a temperament does with the error is a fraction of a comma. Those are different models rather than further rungs.

The one debt this anchor closes owing is the measurement its last four rungs all ask for and none can make: the distribution of where an orchestra’s players actually are, in cents, immediately after tuning and again ten minutes later. Every number on the last four pages is a prediction about that distribution — eleven cents wide, narrowing, with a mean that wanders by a cent or two — and it needs one recording, one analyser and one afternoon.