The other wolf

Somewhere on most cellos, usually between E and F sharp on the G string, there is a note that will not play. It stutters, breaks up, warbles several times a second, and does not respond to being played harder. Cellists call it the wolf.

This site already has a wolf. Twelve perfect fifths overshoot seven octaves by 23.5 cents, and a tuning system that puts the whole error into one interval leaves a fifth that howls. The two have nothing whatever to do with each other, and this essay exists partly to say so.

Two oscillators that stop being two things #

The body is a filter, and a filter is a passive thing that colours what passes through it without being changed by it. That description is a very good approximation and it fails in one place.

The string’s lower end is the bridge, and the bridge stands on a top plate that moves. Where the plate is stiff, the bridge is effectively rigid and the string has a fixed termination. Where the plate has a strong resonance — at the frequency of a corpus mode — the bridge moves substantially, and the string’s end is no longer fixed.

Two oscillators sharing a boundary and exchanging energy are not two oscillators with two frequencies. They are one system with two normal modes, at frequencies either side of where the uncoupled pair sat. The splitting grows with the coupling, and the system’s response has two peaks where the isolated components had one.

That is the whole mechanism, and it is the same one that produces two frequencies from a pair of coupled pendulums, or from two identical tuning forks joined by a bar.

Why that is heard as a warble #

The two split modes are close together — a few hertz — and both are excited. Two tones a few hertz apart beat at their difference, and the beat rate is what a cellist hears as the wolf’s warble.

That is the connection to this ladder’s earlier rungs and the reason this essay sits on the beating anchor rather than on a new one. The wolf is a beating phenomenon; what is unusual is where the two frequencies came from, which is one frequency split rather than two notes sounded.

A few hertz of difference is an envelope swelling a few times a second, which is what a wolf is to a listener. The useful part of drawing it at a second resonance and a weaker coupling is that the warble rate is set by the coupling strength and not by the note: a milder instrument has a slower wolf, not a smaller one.

Why it happens at that note, on that instrument #

The wolf sits where a string’s mode frequency coincides with a strong body resonance. On a cello that is usually the B1-type corpus mode, in the region of E to F sharp, and the coincidence is a coincidence — it is where that instrument’s plates happen to have put it.

Three consequences follow, and all three are things cellists report.

It is at a specific note, not a specific string. The same pitch on a different string is also affected, because what matters is the frequency and not which string produced it.

It moves if the instrument changes. Humidity, a new sound post position, a different bridge, or a different set of strings can move a wolf by a semitone or remove it, because any of those moves the body resonance.

And it is worse on a good instrument. A strongly coupled, efficiently radiating body is exactly one with a large $\alpha$ in the figure above. The wolf is the price of the coupling that makes the instrument loud, which is why the finest cellos are as likely to have one as any other and why “it has a wolf” is not a criticism.

What a wolf eliminator does, and what it costs #

The standard remedy is a mass clamped to the string between bridge and tailpiece: a brass sleeve, tuned by sliding it, so that its own resonance sits at the wolf’s frequency.

It works by adding a third oscillator. The added mass absorbs energy at the wolf frequency and damps that mode, so the splitting is suppressed and the note plays. What it also does is remove energy at that frequency from every note that has a partial there, which is why a heavy eliminator makes an instrument sound duller and why cellists tune them as light as they can get away with.

That is the design trade in its clearest form: the wolf is caused by strong coupling, and strong coupling is what makes the instrument work. Any cure that reduces the coupling reduces the instrument.

The size of the splitting, worked out #

The splitting has a closed form and it is worth writing down, because it says which quantity a maker is actually fighting.

Two oscillators of the same frequency $f$, coupled with a dimensionless strength $\alpha$, split into a pair at approximately $f(1 \pm \alpha/2)$. The difference between them is $\alpha f$, and that difference is the beat rate.

Put numbers in. A cello’s wolf near 175 Hz warbling five times a second needs $\alpha \approx 0.029$ — under three per cent. That is a small number, and it is the whole difference between a note and a stutter.

Two things follow. The beat rate is proportional to the frequency, so a wolf on a higher note warbles faster from the same coupling — which is why it is more disruptive on a cello, where the affected notes are low enough that the warble is in the range a listener counts rather than hears as roughness.

That does not explain why the wolf region is narrow, which is a separate quantity and comes from the resonance rather than from the splitting. A corpus mode has a bandwidth of its frequency over its Q, and on a violin body those are:

Under a semitone. A string mode has to land inside that to be strongly coupled at all, so the affected region is roughly one note wide — which is exactly what players report, and it is a fact about how sharply tuned a corpus mode is rather than about how far the splitting goes.

And the string impedance, which the essay names and does not compute #

The quantity said below to decide which instruments suffer is the ratio of the string’s characteristic impedance to the body’s. The string half is computable from the site’s own numbers:

Every cello string is heavier than every violin string by this measure, and the comparison that matters is between the strings the wolf actually sits on. A cello’s wolf near 175 hertz is stopped on the G string, whose Zc is 1.07; a violin’s B1− mode at 460 hertz is met on the A string, whose Zc is 0.19. A factor of 5.7, and it is the string side of the ratio alone.

That is a large enough difference to carry the qualitative claim on its own, and it is worth saying that it is only half of it. Nothing here computes a body’s mechanical impedance at its own resonance — the site has corpus frequencies and Q factors and not admittances — so what has been established is that the numerator is much larger on a cello and not that the ratio is. A cello body is also larger and more massive, which pushes the denominator the other way; whether the ratio ends up five times worse or two is not something these numbers settle.

The within-instrument gradient is the part that is fully determined, and it is steep: a cello’s C string has 2.7 times the impedance of its A string, and a violin’s G 1.8 times its E. So on either instrument the low strings are the wolf-prone ones, which is why a cello’s wolf is on the G or C and never on the A.

And the coupling needed is tiny, which is why the phenomenon is so widespread. Any bowed instrument with an efficiently radiating body has a few per cent of coupling somewhere in its range; the question is only whether the coincidence happens at a note that is played.

Why not on a violin, and why worse on a cello #

The wolf is chiefly a cello problem and to a lesser degree a viola and double bass one. Violins have them, and less often and less severely.

The reason is where the body resonances sit relative to the playing range. A violin’s strongest corpus modes are near 460 and 540 Hz, which is around A4 to C5 — well inside the range but in a register where the instrument is played with a good deal of bow and where the string’s own impedance is high. A cello’s equivalent modes sit lower relative to its range, its strings are longer and heavier, and the body is much larger relative to the frequencies involved.

The quantity that decides it is the ratio of the string’s characteristic impedance to the body’s at the resonance. A heavy string on a light body is the bad case, and a cello is closer to it than a violin.

The two wolves, side by side #

Since the word is shared, the differences are worth listing explicitly.

The last row is the point. The only thing they have in common is that both are audible as roughness and both were named by musicians who found them unpleasant, which is enough for a word and not enough for anything else. The tuning wolf is arithmetic — it is a property of the numbers 3 and 2, and it exists on an instrument that has never been built. The cello wolf is a property of one instrument on one day.

The other wolf — the tuning one — is a fixed interval a temperament has to put somewhere. This one is not fixed anywhere: it sits wherever the body’s resonance is, and moving it means rebuilding the instrument rather than retuning it. Two things called a wolf, and only one of them is a choice.

Mode splitting is everywhere once it is named #

The wolf is the most audible instance, and once the mechanism has a name it turns up across this field and the rest of the site.

A timpani with an unevenly tensioned head has its nodal-diameter modes split into pairs, because a mode oriented one way sees a different tension from one at right angles. The result is two frequencies a few hertz apart where there should be one, and a drum that will not settle — the same figure, on a membrane.

A bell that is not circular does the same, and for the same reason: two orientations, two frequencies, a warble. Founders check for it and correct it, and a bell with a badly split nominal is scrap.

A piano’s unison strings are three oscillators coupled through a bridge, deliberately mistuned by a hertz or so. That is mode splitting used on purpose: the beating between them is what gives a piano note its shimmer and its two-stage decay, and a perfectly unified unison sounds dead.

The last of those is the interesting one, because it is the same physics being exploited rather than fought. Whether a split mode is a defect depends entirely on the beat rate: at a hertz it is warmth, at five it is a wolf, and at twenty it is roughness. One mechanism, three names, and the boundaries between them are the perceptual ones this site measured in its previous phase.

What the picture cannot show #

The model is two coupled linear oscillators and a wolf is not linear. A real wolf is a limit cycle — the note jumps between two regimes of Helmholtz motion rather than sounding two frequencies at once, and what a listener hears is that switching. The two-peak picture drawn here is the right first account of why there are two frequencies; it is not the mechanism of the stutter.

The coupling strength is a parameter, not a measurement. The figure’s $\alpha$ is a dimensionless number chosen so the splitting is the size a real wolf’s is. Measuring it on an instrument means measuring the bridge admittance at the wolf frequency, which is done and is not what this figure shows.

That is the same gap the impedance table above runs into from the other side, and it is the one number this essay most wants. Everything here that is a ratio between instruments — the 5.7 between a cello’s wolf string and a violin’s, the 2.7 between a cello’s C and its A — is computed from string tensions and lengths and is solid. Everything that is an absolute — how much coupling a given instrument has, whether it will wolf at all — needs the body’s admittance, and the site carries corpus frequencies and Q factors rather than admittances. So this essay can say which instruments are more exposed and cannot say which ones actually have a wolf, which is a fair description of the state of the subject: no maker can predict a wolf from a drawing either.

And the two halves of the ratio are measured in different ways. A string’s characteristic impedance follows from three numbers a maker chooses; a body’s admittance is measured by tapping the bridge of a finished instrument. That asymmetry is why the wolf is discovered rather than designed, and why the remedies are all applied afterwards.

And the bow is not in it. Whether a wolf sounds depends strongly on bow force and speed — cellists suppress a wolf by changing the bow, not only by fitting a mass — and that is a matter of which regime the string is in rather than of where the modes are.

What a player does with one, which is most of the story #

The mechanical remedies are the ones written about, and they are not what a cellist actually relies on.

A wolf is a regime failure — the string cannot settle into stable Helmholtz motion because its termination is moving — and regime failures are sensitive to bow force and bow position, which are the two things a player has continuous control over. So the first response is a bow change: more force, closer to the bridge, which pushes the system toward the top of the force window and makes the stable regime easier to hold.

The second is to change the string’s own impedance where possible: playing the same note on a different string, in a higher position, means a shorter, more strongly tensioned segment, which changes the ratio the coupling depends on. Cellists reroute passages around a wolf as a matter of course.

The third is to change the body, briefly. Pressing the knee harder against the ribs damps the corpus mode and moves the wolf; it is a standard piece of practical technique and it is a direct intervention in the mechanism.

That last one is worth a sentence of its own. A player who suppresses a wolf by leaning on the instrument is changing the boundary conditions of a resonator with their leg. The mechanism is not remote from the playing; it is being adjusted, by hand, in performance, by people who mostly have no account of what they are adjusting.

Whose instruments, and when #

The wolf has been known for as long as bowed instruments have been made, and the coupled-oscillator explanation is twentieth-century — it appears in Raman’s work in the 1910s and was developed by Schelleng, Firth and others through the 1960s and 70s. The nonlinear limit-cycle account, which is what actually happens, is more recent still and is largely from the 1980s onward.

The claim about which note is affected is a claim about the modern cello as it has been built since roughly 1700, and it is a statistical claim rather than a rule: there is no note at which every cello has a wolf.

Where this goes #

This closes the beating ladder’s third rung, and the pattern it leaves is worth stating. Beating between two tones was arithmetic; beating used as a tuning instrument was a technique; beating produced by an object splitting its own mode is a mechanism. The same audible phenomenon, three times, from three different causes — which is exactly why a name is not an explanation.

The field’s last question is what happens to the sound after it leaves. It does not leave equally in all directions, and how directional an instrument is depends on frequency — so the spectrum a listener receives depends on where they are sitting, and a recording is a choice of seat.