The note the body will not let start

Three rungs of this ladder have described a bowed string and none of them has described an instrument. The corner travels round the string and puts a sawtooth on the bridge; the bow force has to sit between two bounds whose ratio closes as the bow approaches the bridge; and starting a note is the same inequality read at the speed the bow has after one period rather than at the speed it will settle to.

Every one of those treats the string’s ends as fixed points. The third rung said what was wrong with that, and said it in the terms a player would: the body feeds back on the string strongly enough to make some notes hard to start on a particular instrument and not on another.

Putting the body in takes one term, and this collection already computes it for a different purpose.

Which bound the bridge is in #

Schelleng’s two bounds are not two versions of one thing, and the difference is what makes this rung possible.

The maximum is about the string sticking to the bow for longer than one period. Too much force and the corner arrives while the string is still gripped, the slip does not happen on schedule, and the motion falls into a subharmonic crunch. That is a fact about friction, bow speed and where the bow is. The bridge is not in it.

The minimum is about the corner staying sharp. A Helmholtz corner triggers the slip when it arrives, and it triggers it because it is a discontinuity; anything that rounds it off makes the trigger less reliable, and the bow has to press harder to compensate. What rounds it off is loss — every pass round the string, some energy leaves — and on a bowed instrument almost all of that loss is the bridge. That is the whole point of a bridge: it is the leak through which the string’s energy becomes sound.

So the minimum bow force is proportional to how much the string sheds per cycle, and how much it sheds is the body’s admittance at the frequency being played. The body’s response is a comb of measured resonances that stays put while the note moves — the same function that turns a sawtooth into a violin’s radiated spectrum turns a bow force into a bow requirement.

The notes it makes hard #

Run the two bounds across a violin’s compass with the minimum scaled by the admittance and the answer is not diffuse. The window collapses at three frequencies and they are the three resonances: C♯4 at the air resonance, B♭4 and C♯5 at the two corpus modes.

The window at B♭4 is 1.7 — the largest usable force is less than twice the smallest — against 25 at the frequencies between resonances. A player has fifteen times less latitude on that note than on its neighbours, with the same bow, the same string and the same position.

That is a prediction about a specific instrument rather than a general observation, and it is testable in the crudest possible way: it says the difficulty sits at the body’s resonances, so an instrument whose resonances are elsewhere has its difficulty elsewhere.

The difficulty is not in the note and not in the string. It is in the box, and two instruments of the same family with different plates have their awkward notes in different places — which is what players say about individual instruments, and which no figure in this ladder could previously express.

Where it closes, and that is the wolf #

The window is a ratio, and a ratio can go below one.

Schelleng’s minimum goes as 1/β² and his maximum as 1/β, so the window itself goes as 1/β: it closes as the bow approaches the bridge, which is why sul ponticello is difficult by a power law. Multiply the minimum by an admittance that peaks at a resonance and the two effects compound.

At β = 0.04 the window at B♭4 is 0.8. That is not a hard note; it is an impossible one, and what the player gets instead is the sound the two failures make — surface noise, or a subharmonic, or the two alternating.

Two data points and an exponent are enough to turn that into a map. Schelleng’s minimum goes as 1/β² and his maximum as 1/β, so the window is exactly proportional to β — which the two figures confirm, since 1.7 at β = 0.09 predicts 0.76 at β = 0.04 and the computation returns 0.8. So the bow position at which any note closes is its window at one position divided by that position:

Seventeen millimetres against one, on a string 325 millimetres long. A note on a body resonance runs out of instrument a centimetre and a half from the bridge; a note between them runs out at a distance no bow can be placed at, because a millimetre from the bridge is inside the bridge’s own foot.

That is the useful form of the result, because it is what a player actually navigates. The window’s ratio is an abstraction; the position at which it reaches one is a place on the string, and the difference between the resonance notes and their neighbours is not that one is harder than the other but that one has a forbidden region on the fingerboard side of the bridge and the other has none. Ponticello is available on most of the instrument and unavailable on a handful of notes, and which notes those are is a property of the box.

It also says what the fifteen-to-one in the window means in the units a player has. Fifteen times less latitude in force is the same fifteen times less territory on the string, because the two are proportional — so a player who cannot find a bow force on a wolf note and a player who cannot find a bow position on it are describing one constraint from two directions, and neither has more room than the other.

Which is one of the two things a wolf is. The other wolf on this site is the same physics from the other side: a string mode landing on a body resonance so that the two exchange energy, split into a pair, and warble at the difference. That essay’s model is a coupled oscillator and this one’s is a bow force inequality, and they describe the same event. The coupled-oscillator account says what the sound does; the Schelleng account says what the player can do about it, and the answer it gives — move the bow away from the bridge, press harder, and take the note faster — is exactly the list of remedies cellists are taught.

One geometric fact sits under all of this and is easy to miss because the bow looks like a line.

So the bow is not a point on the string, and the corner does not pass a place — it passes under a width. Close to the bridge that width becomes a large fraction of the excursion, which is one more reason the near-bridge regime is the hard one, and it is a reason of geometry rather than of force.

The transient, which is where a player meets it #

The window is a statement about a note already sounding, and the third rung of this ladder found that the harder question is the start. Both bounds scale with the bow’s speed, so at the instant a note has to catch — one period in, with the bow accelerating from rest — the window sits a hundred times lower and is exactly as wide as a ratio.

Which means the body’s narrowing applies to the attack undiminished. A ratio of 1.7 is 1.7 at every bow speed, including the speed the bow has after one period of B♭4, and there is no acceleration that widens it.

The register where an instrument’s own resonances sit is therefore also the register in which its attacks are least forgiving.

The bow-position map above sharpens that too, because it says the two remedies a player has are not independent. Moving the bow away from the bridge widens the window in proportion, so a note whose window is 1.7 at the ordinary position has 3.4 at twice the distance and 5.1 at three times — which is why the standard advice for a wolf begins with play further from the bridge and why it works. But moving out also raises the absolute forces, since both bounds go as inverse powers of β, and a note taken far over the fingerboard needs a bow force the arm has to supply steadily. The remedy that widens the ratio makes the absolute target larger, which is a trade the ratio alone cannot express and which is the reason the advice is not simply “always play over the fingerboard”.

So a hard note is hard at the start and easy to hold, which is the ordinary experience and which neither Schelleng alone nor the coupling model alone predicts. Schelleng’s window is a ratio and the ratio is the same at every speed; what makes the attack hard is that the absolute force has to be in the right hundredth of a newton at the right moment, and a window fifteen times narrower is fifteen times less forgiving of getting there late.

Which computation produced the numbers #

Three ingredients and one new assumption.

Schelleng’s inequality is unchanged: minimum force proportional to Zv/(μ²β²), maximum to Zv/(μβ), in arbitrary units, since only the ratio and the slopes carry any argument. This ladder’s second rung is built on it.

The body’s admittance is bodyGain over VIOLIN_BODY, which is the four published resonance means this site has used since the spectrum ladder. Nothing about it was chosen here.

The new assumption is one sentence and it is the whole model: the minimum bow force scales with the body’s admittance at the played frequency. The physical argument is above — losses round the corner, the bridge is where the loss goes — and the scaling is taken as linear because that is the simplest relation consistent with the argument and there is no measurement here to justify a different one. It is normalised to the geometric mean of the admittance over the range being drawn, so the vertical axis is “relative to this instrument’s own average” rather than a force in newtons.

That normalisation is why no absolute claim is made about how much bow force any note needs. What is claimed is the shape: fifteen to one between the best and worst notes on one instrument, and where the worst ones are.

Whose music, and when #

The wolf is a cello problem in the repertoire and a violin problem only occasionally, and the arithmetic says why: a cello’s strongest corpus resonance falls inside its most-used register, while a violin’s fall near the top of the G string and the bottom of the E, which are less exposed.

The remedies are old and are documented as craft rather than as acoustics. A wolf eliminator is a mass clamped to the string beyond the bridge, which detunes the resonance the string is colliding with — a fix to the body’s term, which is the one this essay says is doing the work. Players’ remedies — more bow speed, less pressure, a bow position further from the bridge, a shift to a different string for the same pitch — are all moves in Schelleng’s diagram, and all of them widen the window. The design was forced in the same way a wind instrument’s bell is: the feature that lets the sound out is the feature that makes the note hard to produce.

The fourth remedy is the interesting one. The same pitch on a different string is a different note as far as this calculation is concerned, because the string’s characteristic impedance changes and the body’s does not, so the ratio between them moves. That is a claim this collection can compute and has not — and a guitar cannot be in tune for a related reason, that the same pitch on different strings is not the same physical object.

What the picture cannot show #

The scaling from admittance to minimum bow force is asserted rather than derived. Schelleng’s own treatment has a bridge resistance in it and this essay has replaced it with a proportionality. The direction is not in doubt and the exponent is: if the relation were to the square root of the admittance, the window would narrow by four to one at B♭4 rather than fifteen, and the notes would be the same notes.

Four resonances is not a body. A real violin has hundreds above about a kilohertz, and the model’s floor between its four is far smoother than a real instrument’s response. The three hard notes are real features of this body; a measured admittance curve would have more of them and they would be narrower.

The cello is a violin body scaled by one number. Corpus modes, air resonance and bridge hill do not scale together in the real family — a cello’s air resonance sits at a different ratio to its corpus modes than a violin’s does — so the cello figure demonstrates that the hard notes follow the box rather than predicting where a cello’s actually are.

And nothing here is a player. A window of 1.7 is narrow and a good player works inside narrow windows all the time; the claim is about latitude and not about possibility, except where the ratio goes below one, and even there the claim is about Helmholtz motion rather than about sound.

What a player does with a narrow window #

There is a reading of the whole picture that is about technique rather than about acoustics, and it is worth setting down because it is what the numbers are for.

A window of twenty-five is a note that can be played loudly or softly, near the bridge or over the fingerboard, with a fast bow or a slow one, and will speak in every combination. A window of 1.7 is a note with one bow. Everything a player does on such a note — the exact speed, the exact distance, the exact weight — is not expression but navigation, and the same passage played on a different instrument needs a different navigation because the box has moved the difficulty somewhere else.

That is a claim about why string players are attached to particular instruments in a way that is not sentiment. An instrument’s identity, in this model, is a list of which of its notes have latitude, and a player who has learnt one list is not carrying it over to another box.

It also explains the direction of a piece of standard advice that otherwise looks arbitrary. Told that a note is not speaking, the instruction is almost always to use more bow and not more weight — and more bow means more speed, which raises both bounds together and leaves the ratio untouched, but it also moves the absolute forces up out of the region where the player’s control is coarsest. The window does not widen; the part of it the hand can aim at does.

Where this ladder goes next #

Four rungs. The corner and the sawtooth; the window it can be sustained in; the trajectory required to reach that window from rest; and now the term all three were missing, which is the box the string is stretched over.

The rung after it is the one the fourth remedy names. Every figure in this ladder holds the string’s characteristic impedance constant, because there has only ever been one string in it — and a violin has four, of very different mass, tuned a fifth apart, so the same written pitch has four different impedances available to it. Schelleng’s bounds both carry that impedance, and the body’s admittance does not care which string produced the frequency, so the window at one pitch is a different width on each string it can be played on. That is a computation with nothing new in it, and it turns the choice of string from a matter of colour into a matter of how much force the note will tolerate.