The quartet settles at two pitches, not four

The essay that gave the quartet five fixed attractors gave a quartet five open strings on one Pythagorean chain, made each a weak attractor for every stopped note sharing its pitch class, and found that the ensemble settles flat in every major key — by 0.9 cents in E, 2.5 in C and 2.6 in A♭.

Its closing paragraph named what that hides. There is no such thing as “the quartet’s G”. The violin, the viola and the cello each have a G string, each tuned it down their own chain of fifths by ear, and the essay that priced a fifth set by ear established that tuning down a chain by ear is not exact and gets worse the lower it goes. So a stopped G pulls toward whichever of three strings the player hears — which is their own, under their ear — the sympathetic ring the tuning a quartet cannot change identified as the one fixed reference the ensemble has — and the three do not agree.

How far apart the three G strings are #

The scatter follows from the second essay and needs nothing new. A fifth is set by nulling the beat between the third partial of the lower note and the second of the upper, and that beat runs at a rate proportional to frequency. A player who holds the beat under one cycle in two seconds sets the fifth to about $1731/(3fT)$ cents — a third of a cent on a violin’s A–E and nearly two on a cello’s C–G. That asymmetry is the second essay’s central finding: the string whose tuning is most wrong is the string whose tuning is least certain.

The A is given rather than set, so it carries no error at all. Every other string is reached from it one fifth at a time and the uncertainties accumulate.

The violin’s G sits at −3.9 cents with a standard deviation of 1.77; the viola’s and the cello’s C sit at −5.9 with 2.83. Two independent draws from those give expected separations of 2.5 cents between the three G strings and 4.0 between the two C strings. Those are not small next to the 21.5 cents of a syntonic comma, and they are certainly not small next to the 0.9 to 2.6 cents the third essay’s whole effect amounted to.

One detail in the figure is worth pausing on because it is not symmetric and looks as though it should be. The violin’s E, one fifth above the A, is set to 0.44 cents; its D, one fifth below, to 0.98. The same interval, set by the same method, with more than twice the precision going up — because the beat that is being nulled is at three times the lower note, and the lower note of an upward fifth is the A itself.

And how little difference they make #

Put each player’s own strings into the drift model, let each be pulled only by their own, and draw each player’s tuning error afresh on each run. The prediction from the paragraph above is that the ensemble should scatter.

It does not. The quartet’s settled pitch in C moves from −2.29 cents with one shared set of strings to −2.41 with four private ones, and the spread between the four players’ pitches moves from 0.90 cents to 0.91.

A hundredth of a cent. The reason is arithmetic rather than subtle: each player’s tuning error is an independent draw with a mean of zero, and the consensus term — each player correcting toward the others — is seven times stronger than the strongest open-string pull considered. Four independent errors averaged by a process that is mostly averaging come out at their mean, and their mean is nothing.

So the third essay’s simplification was safe, and this essay’s first result is that it was safe for a reason that can be stated: a shared error moves an ensemble and independent errors do not. The cellist’s 5.7-cent uncertainty mattered in the earlier essay because it was the whole quartet’s reference; four such uncertainties, one per player, matter a great deal less.

What does differ, and it is not the tuning #

There is a real difference between the four players and it has nothing to do with how well anybody tuned. It is in the instruments.

A violin’s open strings are G, D, A and E. A viola’s and a cello’s are C, G, D and A. The violins have an E and no C; the lower instruments have a C and no E. So in a passage in C the viola and the cello are pulled by a string sitting 5.9 cents flat and the violins are pulled by nothing at that pitch class at all; in a passage in E the violins are pulled by a string 2.0 cents sharp and the lower instruments are not.

That difference is systematic. It does not average away over runs, because it is not an error — it is what the instruments are.

The two violins land on each other to within two hundredths of a cent in every key, and the viola and cello land on each other to within three. The two groups land 0.125 cents apart in C at the pull the third essay used, 0.40 at four times that pull and 0.80 at ten times it. The lower instruments are always the flatter pair.

So a quartet settles at two pitches rather than four, and the split is by section. It is not the answer the third essay’s closing paragraph expected, which was four players at four slightly different pitches; the four collapse into two, and the two are the two string lists.

Two kinds of error, and only one of them survives an ensemble #

The pair of results above is worth stating as one thing, because the distinction generalises past quartets.

An ensemble averaging its members is a low-pass filter on their disagreements. An error that each player draws independently is averaged down by the square root of the number of players and then flattened further by the correction, which is why four private tuning errors of two or three cents come out at a hundredth. An error every player shares — a reference pitch that is wrong, a hall that flattens everybody, a single chain everyone tuned to — the five-key Pythagorean keyboard the first essay named — is not averaged at all, because averaging a constant returns the constant.

So the question to ask of any intonation effect is not how large it is but whether it is common to the players. The third essay’s open-string pull is common: every player’s stopped notes are drawn toward the same Pythagorean positions, so the whole quartet goes flat together and the effect survives at its full size. This essay’s tuning scatter is private, and it does not survive at all.

That is also why the systematic difference between the string lists is the only thing left standing. It is neither quite common nor quite private: it is shared within a section and differs between sections, so the averaging works inside each pair and not across them. An effect with that structure survives at the strength of the difference between the groups rather than at the strength of the effect itself, which is how a two-and-a-half-cent disagreement between three G strings comes out as a tenth of a cent between two sections.

Whether a tenth of a cent is anything #

It is not, and saying so is most of what this essay establishes.

A tenth of a cent is a five-hundredth of a semitone. The finest pitch discrimination anybody reports for a sustained tone is a few cents; the third essay’s own effect, which it took seriously, is 0.9 to 2.6 cents; a cellist’s tuning of their own C string is uncertain by 5.7. Nothing in this collection’s account of hearing would let a listener detect a systematic difference of 0.125 cents between two sections of a quartet.

At a pull ten times stronger the split reaches 0.8 cents, which is within a factor of three of audible — and that pull is far outside what the third essay argued for. The honest statement is that the effect is real, systematic, in a direction the instruments predict, and below the threshold of anything.

That matters for what the model is for. A quantity that is systematic and tiny is still worth computing, because the next question is whether anything makes it larger — and this essay has found the two things that would. A stronger pull, which would come from a more resonant instrument or a player attending harder to sympathetic ring. And a weaker consensus, which is what an ensemble has when it is spread out, or reading, or recording in separate booths.

Where it would be larger #

The key dependence survives intact and it is the larger effect by a factor of twenty. A quartet in C settles 2.4 cents flat and one in E settles 1.2, because C major puts more of its weight on the flat end of the chain. That is the third essay’s finding and nothing here disturbs it, and it is the same key colour the quartet’s five fixed pitches produce by being a five-key Pythagorean keyboard.

What this essay adds is that the key dependence is not quite the same for the two sections, and the reason is legible: C major’s tonic is an open string for the viola and cello and not for the violins, and E major’s tonic is an open string for the violins and not for the others. The asymmetry is largest exactly where the key’s tonic is on one section’s instrument and not the other’s — which is C and E, and which is what the figure shows.

An ensemble with a weaker consensus term would show it. A string orchestra, where a player corrects toward a section rather than toward three individuals, has a longer path for a correction to travel and a louder set of sympathetic strings under each ear. Whether that is enough to bring a tenth of a cent up to a cent is a question the same model answers with a different gain, and it is the one this essay would extend to next.

What the sections would have to do to hear it #

There is one arrangement in which a tenth of a cent stops being a rounding error, and it is worth naming because it is an ordinary piece of quartet writing.

Two sections settling a tenth of a cent apart is inaudible as pitch. Two sections a tenth of a cent apart sounding the same note is a beat, and a beat has a rate rather than a size: a tenth of a cent on a note at 440 hertz is 0.025 hertz, which is one cycle in forty seconds. Nothing survives forty seconds.

But the split grows with the pull, and at the stronger settings it reaches 0.8 cents — 0.2 hertz at 440, one cycle in five seconds, which is a slow undulation on a held unison and is exactly the kind of thing players describe when they say a chord will not settle. So the mechanism has a signature and the signature is in unisons and octaves between the sections rather than in the pitch of either.

Whether it would ever be attributed to the instruments is another matter, and the arithmetic says why not: the split is a function of the key, so a chord that undulates in C settles in E, and a player experiencing that would reach for the chord rather than for the tuning. The prediction is nonetheless specific — the sections should disagree most in the keys whose tonic is an open string for one of them and not the other — and it is a prediction about which chords in which keys are hard to settle, which is the sort of thing a quartet would recognise or flatly deny.

Which computation produced the numbers #

Each instrument’s open strings are its standard four, at the Pythagorean position its chain of fifths puts them relative to the given A: a pure fifth is 701.96 cents and a tempered one is 700, so each step from the A displaces a string by 1.96 cents, and a string $k$ fifths away sits at $1.96k$.

The uncertainty on each step is $1731/(3 f T)$ cents, with $f$ the lower note of that fifth and $T$ two seconds — the second essay’s own criterion, that a player sets a fifth until the beat is slower than one cycle in the time they give it. Steps are independent and the variances add, so a string $k$ fifths from the A carries the root sum of its own chain’s uncertainties.

The drift model is the third essay’s: each player corrects toward the mean of the others at a stated gain, is pulled toward an open string whenever the note they are playing shares its pitch class, and carries a small noise. The only change is that the open strings are now per player, drawn once at the top of each run from that player’s own scatter, and that each player is pulled only by their own.

The pitch classes a passage visits are drawn from the major-key probe-tone profile as a stand-in for how often each degree sounds, which is the third essay’s assumption and is carried unchanged so that the two sets of numbers can be compared.

Where the model stops #

Each player is pulled only by their own strings and that is too strong. A cellist’s open G is audible to a violinist and a violinist’s open E rings under everybody. The truth is somewhere between this essay’s model and the third essay’s, and the two bracket it: one shared set of strings and four private ones give answers a hundredth of a cent apart, so anything between them does too.

The tuning is drawn once and held. Players retune between movements and adjust pegs during rests, and a string that has been tuned twice is not two independent draws — the second is a correction of the first.

And the consensus gain is not measured. It is the parameter the third essay swept and could not fix, and everything here inherits that. What can be said is that the finding is not sensitive to it: the sections split by the same tenth of a cent whether the gain is 0.35 or 0.05.

What the picture cannot show #

It cannot show the viola in the middle. A viola shares a C and a G with the cello and a G, D and A with the violins, so if a player is pulled by what they hear rather than by what is under their own chin, the viola is pulled from both directions and the two violins are not. That is a plausible model and it is a different one.

Nor a real passage. Pitch classes are drawn from a profile, so every player is playing the same distribution of degrees. Four parts do not do that: a cello plays roots and fifths and a first violin plays the melody, so the share of notes landing on each open pitch class differs by part in a way this model averages over.

And it cannot show the open strings actually sounding. Everything here treats an open string as an attractor for a stopped note of the same class. A quartet also plays its open strings, which is where a guitar’s whole problem begins, and an open string played is not pulled by anything — it is where the peg left it, which is the one pitch in the ensemble that cannot be corrected — and which, on a guitar, is the whole of the tuning problem rather than a corner of it.

Still open: whether the parts pull differently #

The simplification with the most in it is the last one. Each player is given the same distribution of pitch classes, and a quartet’s four parts have systematically different ones: a cello line is weighted to the tonic and the dominant, a first violin line to the third and the upper degrees, and an inner part to whatever is left.

That interacts with the finding directly. The lower instruments are the ones with a C string and they are also the ones whose parts sit on the tonic most often — so in a passage in C the cello is pulled toward its flat C string by a larger share of its own notes than the model allows, and the split between the sections should be larger than a tenth of a cent by whatever that share is.

Running it needs one thing the essays here do not have: the degree distribution of each part, by part, over a repertoire. That is a corpus measurement rather than an arithmetic one, and it is a small one — four histograms over a few dozen movements — and it would turn the profile every essay of the account here has leaned on into four profiles. It would also say something the account has not been able to ask: whether the quartet’s tendency to settle flat is shared by its players or is mostly the cello’s, carried to everyone else by the consensus.