An open string pulls the quartet flat

A consensus with nothing to hold it followed an ensemble’s pitch after the tuning note has stopped. Each player corrects toward what they hear around them, which is other players correcting toward them, so the players converge on each other and the ensemble as a whole random-walks. One fixed instrument changed that completely: a keyboard pulling toward the pitch it was given stops the walk almost as well as any amount of discipline.

A string quartet has no keyboard, and it is not without fixed pitches either. The tuning a string quartet cannot change found that its open strings are five pitch classes on a chain of pure fifths, which put the cello’s C 5.87 cents flat of a keyboard and the violin’s E 1.96 sharp, and the cello cannot hear its own tempering added that the bottom of that chain is set with a scatter nearly as large as its error. Its last paragraph named the question those strings raise. An open string is not only a fixed pitch; a stopped note in tune with it makes it ring in sympathy, and players hear that ring and move toward it.

That makes each open string a weak fixed reference, at its own pitch, for only some of the notes. What five such references do to an ensemble that is otherwise free to walk is arithmetic on the consensus dynamic with one term added, and the answer depends on the key.

Five fixed pitches that disagree #

The dynamic is the consensus essay’s, cut to four players. At every correction each player moves a stated fraction of the way toward the average of the other three and adds a small random error. What is new is a second term: if the note the player is on has the pitch class of an open string, the player also moves a small fraction of the way toward that string’s pitch. That fraction is the pull’s weight, and it is the one quantity nobody has measured, so it is swept.

The open strings’ pitches are the pure-fifth chain from the A: C, three fifths down, 5.87 cents flat of a keyboard; G 3.91 flat; D 1.96 flat; A exact; E 1.96 sharp.

The chain is lopsided, and the lopsidedness is the whole of what follows. Three of the five strings sit below the given A and one above it, and the flattest is three times as far from the keyboard as the sharpest. A quartet pulled toward all five alike would settle flat before any key entered the arithmetic. A note only feels the string that shares its pitch class — the resonance a player hears is an open string ringing at the note’s own pitch or an octave from it — so a stopped B feels nothing, a stopped G feels the G strings and a stopped E the violin’s E.

Which notes are played depends on the key, and the key enters in one place only: how often each degree sounds. The stand-in for that is the probe-tone profile how much evidence a modulation needs reads keys with — Krumhansl and Kessler’s ratings of how well each degree fits a major key, which track how often degrees occur closely enough to weight a draw by.

The fixed point has a closed form. A linear pull toward several targets, each applied to a share of the notes, settles where the targets’ weighted mean is: the profile-weighted average of the offsets of the open strings the key’s notes land on. The simulated runs land there to within a few hundredths of a cent in every key, which is the check that the dynamic is doing what its arithmetic says.

A quartet is a small ensemble for this, and its size matters to the walk before any string pulls. With no pull at all the four players’ average pitch wanders 3.7 cents root-mean-square over 480 corrections, where the consensus essay’s sixteen players wandered 2.0 over the same number. Fewer players average away less of each other’s error, so a quartet walks nearly twice as far as a section — and has correspondingly more to gain from anything fixed.

Every key settles flat #

The first result is that there is no major key in which the quartet settles sharp. The chain has one sharp string and four flat or exact ones, and the one sharp string is E, whose pitch class is the third of C major, the second of D, the fifth of A and the tonic of E. Wherever it is weighted most it is outweighed by G and D or balanced by A, and the nearest any key comes to the pitch it was given is E major, 0.87 cents flat.

The second is which keys settle flattest, and they are not the string keys. C, G and F — the keys that use every open string, and whose open-string thirds the tuning a string quartet cannot change found clash worst — settle 2.3 to 2.5 cents flat. E♭ and A♭ major settle flatter still, at 2.54 and 2.58 in closed form. Their diatonic notes include C and G, the two flattest strings, and D only as a leading note or not at all; the E and the A that would pull back up are not in the key. The flat keys are pulled less often — only 34 and 37 per cent of their notes land on an open string, against 55 in C — and pulled further when they are.

That separates two things the phrase “string keys” runs together. How often an open string pulls is highest in the keys that use all five pitch classes. Where it pulls to is decided by which of the five the key uses, and the keys that use only the bottom of the chain are pulled furthest from the keyboard.

How strong the pull has to be #

The weight of the pull is the assumption, and it is worth seeing how much of the result depends on it.

Where the quartet ends up does not depend on the weight at all, once the movement is long enough: the target is set by the chain and the key. What the weight decides is how fast it gets there and how firmly it is held.

A pull of a hundredth of a correction is enough to beat the walk. At that weight the quartet has moved two thirds of the way to its target in about four hundred corrections — the length of a movement, at a correction every second or two — and its scatter across runs has fallen from 3.1 cents to 1.4. At a weight of 0.02 it reaches two thirds of the way in about sixty corrections, and at 0.05 in thirty. At 0.005 it has not arrived by the end of the movement, and the walk and the pull are still arguing.

The settling times have a closed form of their own, and it is worth checking because it says what the weight means in time. A pull of weight λ applied to a share $p$ of the notes moves the ensemble toward its target with a time constant of about $1/(\lambda p)$ corrections. In C major, where 55 per cent of notes are pulled, that predicts 36 corrections at a weight of 0.05 and 18 at 0.1; the simulated quartet reaches two thirds of the way in 30 and 17. At the weakest weights the prediction and the runs part company — 91 predicted at 0.02 against 62 found, 182 at 0.01 against about 380 — because there the walk is as large as the pull and the two are no longer separable.

That is the same shape one fixed instrument had: steep at the left, so a very little fixedness buys most of the stabilisation. The difference is where the stabilised pitch is. A keyboard holds an ensemble at the pitch it was given. Four open strings on a Pythagorean chain hold it somewhere else, and the somewhere else is flat.

The cellist’s scatter becomes the quartet’s #

The open strings’ pitches above are the chain’s, exactly. The cello’s C is not set exactly: tuned down three fifths by ear for two seconds a fifth, it lands within about 5.65 cents of where it was aimed.

Once the open strings pull, that scatter does not stay on the C string. Wherever the quartet settles is a weighted mean of the strings, so a C string that lands a cent flatter moves the whole ensemble by the C’s share of the weight.

The shares follow from the key’s use of the C. In C major the tonic is the C string’s pitch class and carries 28 per cent of the pull, so the cellist’s ±5.65 cents becomes the quartet’s ±1.56. In D and A major the C is not in the key and only the chromatic C♮ reaches it, and the quartet moves by ±0.6. In A♭ major the C is the mediant and one of the few open pitch classes the key uses at all, and it carries 30 per cent of the pull: the cellist’s scatter moves the whole quartet by ±1.72 cents.

That is a strange place for the largest effect to be, and it is a direct consequence of the flat keys’ being pulled rarely and far. A key with few open-string notes has few strings sharing the weight, and whichever of them it does use carries more of it.

What the model is built from #

The quartet is four players, each correcting a third of the way toward the average of the other three at every step with a random error of a third of a cent, the correction the consensus essay used for a larger ensemble listening to its neighbours. At every step each player’s note is drawn afresh from the major-key probe-tone profile on the stated tonic, and a note whose pitch class is C, G, D, A or E moves a stated fraction of the way toward that open string’s pitch on the pure-fifth chain from A. The settled pitch of a run is its average over the last third of 480 corrections, and each key’s figure is the mean and spread over 24 seeded runs. The closed-form target is the profile weight of each open pitch class the key uses, times that string’s offset, summed and divided by the weights. The cello’s scatter is added to the C string’s offset and the closed form’s response read off.

What the pull leaves out #

That all the strings pull alike. The model gives every open string the same pull on a shared note, and they do not ring alike. A violin’s E string rings loudly in sympathy and a cello’s C barely at all at a violin’s register, and a player hears their own instrument’s open strings much better than their colleagues’. Weighting by who is playing whom would move every target toward the strings nearest the most notes.

That the resonance is only at the pitch class. A stopped note excites an open string through any partial they share, so a stopped E makes the A string ring weakly at its third partial as well as the E string at its first. Adding the weaker coincidences would pull more notes, more weakly, and the targets would move toward the middle of the chain.

That the profile is how often degrees sound. The probe-tone profile measures how well each degree fits a key, which correlates with how often degrees occur without being a count of them. A movement’s actual notes would give different weights, and a slow movement that dwells on its tonic pulls toward that one string far more than the profile says.

And that nobody adjusts on purpose. Players move toward an open string’s ring because it sounds in tune, and some deliberately avoid open strings or play the note stopped for exactly that reason. An orchestra is given a note is the ritual that sets the A; nothing here models a player deciding to hold a pitch against the pull.

What a settled pitch cannot establish #

That this is the flattening ensembles are known for. Unaccompanied ensembles are reported to go flat by far more than this, and somebody has to pay the comma describes the systematic mechanism that can do it — a comma lost per cycle of a progression tuned justly at every step. That drift grows with every cycle. The pull here does the opposite: it stops at a fixed distance below the keyboard and holds the ensemble there, so it can account for a quartet being slightly flat of a piano and never for one sinking through a piece.

That anybody hears a quartet 2.5 cents flat. An ensemble’s pitch moving by two and a half cents over a movement is below any listener’s ability to report it, and the unison is the coarsest thing in the room found a unison matched to no better than a few cents. What would be heard is not the ensemble’s pitch but its disagreement with anything fixed at the keyboard’s pitch — a piano entering after an unaccompanied passage, which would find the quartet a clear two or three cents under it in C and nearly with it in E.

That the pull is the size swept. A weight of a hundredth of a correction is enough for everything above, and nothing measured says whether the real pull is that, a tenth of it, or ten times it. What the figures do establish is that any weight above about a hundredth gives the same destination, so the result stands on the existence of the pull rather than on its size.

And that sympathy is what players move toward. The account in which players hear an open string’s ring and correct toward it is the standard one given by string teachers and a plausible one, and it has not been measured in performance.

Whose quartets, and whose keys #

The string quartet is the case because it is the ensemble with no fixed instrument and several fixed pitches, and because its repertoire spans every key. The orchestral string section is the same case with more players and the same five pitch classes, and with a wind section whose players have no open strings and may pull the other way.

The result has a direct bearing on a piece of performance lore it cannot confirm: that string players prefer the sharp keys — D, A, E — for their resonance and brilliance, and that the flat keys sound veiled. On this arithmetic the sharp keys are exactly the ones in which the quartet’s settled pitch sits nearest the keyboard’s, and the flat keys the ones in which the open strings pull it furthest below. A quartet in A♭ major is pulled rarely and pulled flat; a quartet in E major is pulled often and pulled almost nowhere. Round the circle of keys, the settled pitch is highest at E and A and falls away in both directions, to C and G on one side and to A♭ and E♭ on the other. Whether that is part of what players mean by a key’s colour, or merely coincides with it, is a question for a recording, not for the chain.

Still open: the second violin’s G and the viola’s C are not one string #

Every open pitch class here is one pitch, and it is not. The violin, the viola and the cello each have a G string and each tuned it down the chain by ear, so the quartet’s G is three strings scattered around the Pythagorean G by different amounts — the violin’s within about two cents, the viola’s and the cello’s within several. A stopped G pulls toward whichever of the three rings loudest where it is being played, and the three disagree. Giving each instrument its own open strings, each with the scatter its own tuning left it, and each ringing most for its own player, would say whether the quartet settles at a single pitch in a key at all, or whether its four players settle at four slightly different ones — which is what a quartet whose G strings disagree by three cents would be expected to do, and what the pair tuned apart on purpose would call a chorus.