The cello cannot hear its own tempering

The tuning a string quartet cannot change found that the only fixed pitches in a quartet are four pure fifths long, that they put the cello’s C 5.87 cents below a piano’s, and that every third two open strings can make is a comma from just. The repair is not mysterious. Narrow every fifth by 1.96 cents and the chain becomes a segment of equal temperament: the C meets the piano, the E meets the piano, and the open-string thirds become the keyboard’s rather than Pythagorean ones.

The repair has to be made by the same method that made the problem. A string player sets a fifth by bowing two strings together and listening to the beat between them. A pure fifth is the setting with no beat, and a tempered one is a setting with a slow beat of a particular rate — so tempering by ear means finding a beat that is present but slow, and stopping at the right slowness. Whether that can be done depends entirely on how slow the right beat is, and it is not the same on any two strings.

One narrowing, eight different beats #

A fifth’s beat is between the lower string’s third partial and the upper string’s second, which sit at three times the lower string’s frequency. Narrow the fifth by a fixed number of cents and the upper partial moves by that fraction of three times the lower frequency, so the beat of a tempered fifth is proportional to the frequency of its lower string — the same arithmetic that makes a fifth betray a mistuning three times sooner than a unison, read down a chain instead of across intervals.

The violin’s A–E, with A4 at 440 hertz underneath, beats 1.49 times a second when tempered. Its D–A beats once a second. The viola’s G–D and the violin’s G–D beat 0.66 times a second; the viola’s C–G 0.44; the cello’s D–A 0.50, its G–D 0.33 and its C–G 0.22. From the top of the quartet to the bottom the same tempering slows by a factor of six and two thirds, and the slowest beat, on the fifth that decides where the cello’s C lands, takes four and a half seconds to complete once.

A whole bow at the slow speed tuning uses lasts some seconds, and a player listening for a beat needs at least one cycle of it to know it is there. So on the violin’s top strings a tempered fifth is a beat a player hears within a stroke, and on the cello’s bottom strings it is a beat that may not complete in the stroke at all.

How well a fifth can be set #

The precision of setting a fifth by ear follows from the same criterion. A fifth can be set to within the mistuning whose beat completes one cycle in the time the player listens: set it closer than that and nothing in the stroke distinguishes it from exact. That mistuning is proportional to one over the lower string’s frequency and one over the listening time.

Down a chain the errors add. The cello’s D is set against the A, the G against the D and the C against the G, so the C carries three settings’ errors, and if they are independent they add as the square root of the sum of their squares — the same rule who listens to whom when an orchestra tunes applied to a line of players passing a note along, applied to a line of strings.

With two seconds a fifth, the violin’s E lands within two thirds of a cent of where it was aimed and its G within 1.8. The viola’s C lands within 2.8. The cello’s C lands within 5.65 cents, and its Pythagorean error is 5.87. The string whose pitch is furthest from the keyboard is the string whose pitch is least certain, and by very nearly the same amount.

The viola is the instrument in between, and the arithmetic makes it the more exposed of the two lower ones. Its C3 is exactly as flat as the cello’s C2, and at two seconds a fifth it lands within 2.8 cents rather than 5.7 — so it is the viola’s open C, not the cello’s, that a pianist will most reliably hear as flat, because it is the one whose flatness is larger than its scatter.

That coincidence at the cello is not an accident of the chosen time. Both quantities grow down the chain — the systematic error by 1.96 cents a fifth, the random one by a larger step each fifth because each lower string is slower — and both reach their largest values at the same string. A quartet’s two kinds of tuning error pile up at the same end, and the end is the cello’s C.

How long a fifth has to be listened to #

The coincidence of 5.65 and 5.87 depends on two seconds, and the dependence is simple enough to draw.

For any listening time shorter than about two seconds a fifth, the cello’s C is more uncertain than it is wrong — a cellist tuning pure fifths in a hurry produces a C whose Pythagorean flatness is smaller than the scatter of where it lands. At four seconds a fifth the scatter is 2.8 cents, half the offset. At one second it is 11.3, nearly twice. The viola’s C crosses at a second, and the violin’s G is inside its own 3.9-cent offset from a second up.

So the question whether a cello’s open C is Pythagorean has an answer that depends on patience. Tuned carefully, it is; tuned the way the few seconds between movements allow, it is a random pitch within a range that includes both the Pythagorean C and the keyboard’s.

Patience makes the Pythagorean C real #

Double the listening time and every scatter on the chain halves, because the mistuning a beat can reveal in a stroke is inversely proportional to the length of the stroke.

At four seconds a fifth the cello’s C lands within 2.8 cents of its Pythagorean position, and the Pythagorean position is 5.9 cents from the keyboard’s. The C string has become distinguishably flat: a cellist who spends that long on each fifth produces a string that is reliably Pythagorean, and one that is reliably wrong against a piano. The violin’s G, at ± 0.9, is distinguishably flat by any listening time at all.

That is an uncomfortable conclusion for the idea that care improves tuning. Care improves the precision of tuning pure fifths, and a precise Pythagorean chain is a chain whose disagreement with a keyboard is certain rather than hidden in scatter. A quartet that tunes quickly between movements is less consistently wrong than a quartet that tunes slowly before the concert, and the difference is entirely at the bottom of the ensemble.

The lever is the one a tuner counts beats with. A fifth’s beat multiplies a mistuning of the upper string by two in hertz, as the third sound magnifies cents, not hertz found for every interval’s lowest coincidence, and nothing a cellist does changes that factor except choosing a different interval to tune by.

Pure and tempered are the same to a cellist #

That makes the tempering itself hard to verify at the bottom of the quartet, and the drawing below says how hard.

The C moves three cents for every cent a fifth is narrowed, because three fifths lie between it and the A. Pure, it is 5.87 cents flat; at the equal-tempered narrowing it is exact; narrow each fifth by three cents and it is 3.1 sharp. Across that whole range the keyboard’s C lies inside the band of where the string may land when each fifth is set for two seconds. A cellist tuning down the chain for two seconds a fifth cannot distinguish pure fifths from tempered ones by the result, and any narrowing a cellist believes they have made is a narrowing the method could not have confirmed.

The violin is in the opposite position. Its E is set against the A in one step, at 0.66 cents of precision, and its tempering moves the E by 1.96; a violinist aiming at equal temperament can hit it and hear that it has been hit.

The thirds a tempering buys #

Narrowing the fifths does something besides meeting a keyboard, and it is the thing the open-string thirds needed. Every cent taken off a fifth is four cents off the major third C–E, since four fifths make it.

Narrowed by 1.96 cents, the open thirds become the keyboard’s. Narrowed by 5.38 — a quarter of a syntonic comma, the tempering a fraction of a comma places at the centre of three centuries of meantone tunings — the major third C–E becomes exactly 5:4 and the two minor thirds are only 5.4 cents narrow. A quartet whose open strings were tuned in quarter-comma meantone would have better thirds on its fixed pitches than a keyboard does.

It would also put the cello’s C 10.3 cents sharp of a piano, and a fifth narrowed by 5.38 cents beats 0.61 times a second on the cello’s C–G. That is a beat a cellist could hear within two seconds — but 10.3 cents against a noise of 5.65 is a setting that typically lands between 4.6 and 16 cents sharp, and a meantone quartet tuned by ear would carry most of its intended improvement in the scatter.

A shorter route to the bottom of the quartet #

The chain is not the only way to tune the cello’s C, and the arithmetic says it is the worst one. Setting the C string directly against a keyboard’s C, as a cellist playing with a pianist can, replaces three fifths’ accumulated error with one unison’s — and a unison can be matched at a partial rather than at the fundamental.

A unison compared at the k-th partial beats k times as fast as the fundamentals, so it can be set k times as precisely for the same listening time. The cello’s C matched to a piano’s C by its fourth partial for two seconds is good to 3.3 cents, and by its eighth to 1.7. Down the chain, for the same two seconds a fifth, it was good to 5.65. Tuning the lowest string to the keyboard is three times as precise as tuning it down the fifths, because the chain spends its precision on the slowest beats in the quartet and a unison can spend it on the fastest partial the ear can follow.

That also explains a habit that looks like indecision. String players who tune by fifths when they are alone and by unisons with a piano are choosing the more precise route in each case, since without a reference below them the chain is the only route there is.

A section of cellos is a chorus at its open C #

An orchestral cello section is eight or so players each tuning the same chain, and each C string lands somewhere in the same scatter. The section’s average C is more certain than any one player’s, by the square root of the number of players: eight cellists tuning for two seconds a fifth put the section’s mean C within 2.0 cents of the Pythagorean position.

The players’ Cs are not averaged, though; they sound together. Two cellists tuning independently differ from each other by a scatter of 5.65 cents times the square root of two, about eight cents, and a section of eight open Cs is a cluster of eight pitches spread over that range. The pair tuned apart on purpose found that a piano’s unison strings are detuned by a couple of cents to sustain and that anything past four and a half is in the regime of a deliberate chorus. A cello section’s open C is detuned into that regime by the method it was tuned with, and nobody chose it.

That is a width a section has on its open strings and not on its stopped notes, which each player corrects by ear against the others within a note. The open C is the one pitch in the section that cannot be corrected once the bow is on it.

The assumptions under the scatter #

The beat of a narrowed fifth is exact arithmetic. The precision criterion — a fifth is set to within the mistuning whose beat completes one cycle in the listening time — is a convention, and a stated one: a listener attending to a slow swell may notice half a cycle, or need two. Either choice scales every scatter above by the same factor and moves the crossing times in proportion, and the ordering of the strings does not change.

The errors of successive fifths are taken as independent. A player who systematically hears a slow beat as no beat will narrow every fifth the same way, and a systematic bias adds three times over rather than as a root sum of squares — which would make the cello’s C more wrong and less scattered than drawn. The A is taken as exact, and an orchestra is given a note put its own matching error at about two cents, which every string inherits on top of the chain’s.

What the listening model leaves out #

Harmonics. Cellists commonly tune fifths by touching the strings at nodes and sounding the two partials that coincide on their own — the D string’s third harmonic against the A string’s second, which are the same A. That isolates the pair the beat is made of from the fundamentals underneath it, which makes a slow beat easier to hear without making it any faster: the scatter drawn here is the limit of the method either way, and harmonics may let a player approach it more closely.

Bow noise and the start of a stroke. A beat is heard against a tone that has settled, and the first part of a bowed note is a transient that carries no steady beat at all. A four-and-a-half-second beat is being listened for inside a stroke that spends a fraction of its length getting started.

Vibrato. None is used when tuning, and none is modelled. A player who tunes with vibrato has no stationary beat to null.

And what the players do next. A tuned string is a starting point, and a consensus with nothing to hold it followed what an ensemble’s pitch does once the tuning note has stopped. Its open strings were the ensemble’s restoring term — the pitches that do not move while everything around them corrects toward everything else.

Whose tuning #

The practice is the string quartet’s and the orchestral string section’s, tuning in fifths from a given A, with and without a keyboard. The numbers are for modern pitch at A = 440 and scale with it; at a baroque standard a semitone lower every beat is six per cent slower and every scatter six per cent wider. The cello’s C is the extreme case of the method and not an exotic one — it is the lowest open string in the ensemble and the one whose pitch every bass line in C, F and G returns to.

Still open: whether an open string pulls the ensemble toward itself #

The consensus arithmetic treated an open string as a fixed reference that pulls nothing. A stopped note in tune with an open string makes the open string ring in sympathy, and players hear that resonance and move toward it — so each open string is a weak attractor for every stopped note that shares a partial with it. With four attractors on a Pythagorean chain and a fifth, the C, scattered by 5.7 cents, the question is whether a quartet’s stopped pitches drift during a movement toward the chain’s positions, how strongly, and whether the scatter of the lowest open string becomes the scatter of everything played in its key. That is the consensus dynamic with a pull toward a set of fixed pitches added, and the pull’s size is the one quantity the arithmetic would need to be given.