How much bow is allowed

Helmholtz motion is what a bowed string does when it is working. It is not what a bowed string always does, and the conditions under which it happens are narrow, computable, and much narrower near the bridge than over the fingerboard.

Two straight lines with different slopes, and everything about how a bowed instrument is played sits between them.

Why there is a minimum #

The corner arriving at the bow has to break the string free. What it delivers is a change in the transverse force at that point, and its size depends on the amplitude of the motion.

If the bow’s normal force is too large relative to that, the corner arrives and the string does not slip. The cycle fails, the motion decays toward something else, and what is heard is not Helmholtz motion. The condition for the slip to happen every time gives a minimum bow force below which the regime cannot be maintained.

Schelleng’s result is that this minimum goes as one over $\beta^2$, where $\beta$ is the bow’s distance from the bridge as a fraction of the string. The reason for the square is that both the amplitude available at the bow and the force change delivered scale with $\beta$, so the requirement scales with its square.

Below the minimum, what is heard is surface sound: the bow slipping intermittently and irregularly against the string, producing a hiss with a pitch in it rather than a note. Every beginner produces it and every player uses it deliberately at the very quietest dynamics.

Why there is a maximum #

At the other end the bow’s force is large enough that once the string has stuck, the corner’s arrival is not sufficient to break it free again — so it stays stuck through the arrival, and slips only on the next one, or at some other time.

The result is a period that is no longer the string’s period. What is heard is a raucous, crunching sound, often at a subharmonic of the intended note, and it is what a bow pressed hard produces on any instrument.

The maximum goes as one over $\beta$ — linearly, not quadratically, because it depends on the force available at the bow to hold the string against a single reflection rather than on the accumulated amplitude.

The window, and what it explains #

Divide one bound by the other. The minimum goes as $\beta^{-2}$ and the maximum as $\beta^{-1}$, so the ratio between them goes as $\beta$: the usable window closes in direct proportion to the distance from the bridge.

That single line explains a set of facts about string playing that are usually taught as separate pieces of technique.

Playing near the bridge is hard, and it is hard by a specific amount. At a twentieth of the string the window is a factor of five; at a fiftieth it is a factor of two. This is what sul ponticello costs, and it is why the effect is unstable — a player operating in a factor-of-two window will fall out of it regularly, and the characteristic glassy, breaking-up quality of the effect is those failures.

Loud playing must be near the bridge. The maximum force is larger there, and loudness needs force. A player asked for a fortissimo moves the bow toward the bridge, not because of the tone but because that is where the ceiling is high enough.

Quiet playing must be away from it. The minimum force is smaller over the fingerboard, so the floor is low enough for a real pianissimo. Sul tasto and quietness go together for the same reason and in the same direction.

And the two cannot be independent. A player changing dynamic must change bow position, or leave the window. This is the coupling between two of the three parameters the previous essay described as nearly independent, and it is the sense in which they are not: speed and position are independent in what they do, and coupled in what is permitted.

What a cello has to do about it that a violin does not #

The window’s width is a ratio and is therefore dimensionless, which means it transfers unchanged from a violin to a double bass. What does not transfer is everything the window is measured in.

A double bass string is far heavier and under far more tension than a violin’s, so both bounds sit much higher: the forces involved are a matter of arm weight rather than finger pressure. A bass player at the bottom of the window is applying more force than a violinist ever applies at the top of theirs.

The bow reflects it. A bass bow is short, heavy and held so that arm weight can be delivered directly; a violin bow is long, light and balanced for speed. Those are not stylistic differences — they are two solutions to the same diagram at two very different absolute scales, and the family in between is a smooth interpolation.

The scaling also explains something about response. Establishing Helmholtz motion takes a number of round trips of the corner, and a round trip on a bass string takes several times as long as on a violin string. A bass note therefore takes several times as many milliseconds to settle, which is why bass players start notes early and why the instrument is described as slow. It is slow by a computable amount: the corner has further to travel.

The instrument-maker’s side of it #

The bounds contain the string’s characteristic impedance and the friction coefficients of the rosin, and both are things somebody chooses. Dividing one bound by the other collapses most of that, and the closed form is worth having because it decides which of the four parameters can widen the window and which only move it:

$$\text{window} \;=\; 100\,\mu\,\beta.$$

Two of the four vanish. The string’s characteristic impedance is in both bounds and cancels; so is the bow’s speed. Doubling either raises the floor and the ceiling by the same factor, so a heavier string and a faster bow both need more force and neither buys any extra room to be wrong in. That is a stronger statement than “it is not simply louder”: a player who moves to a heavier gauge is not making the control problem harder or easier at all, only moving it up the force axis.

Bow speed cancelling is the one worth pausing on, because it is the parameter a player has most direct control over and the section above listed it among the ones that are coupled. It is coupled in magnitude and not in width: a faster bow requires proportionally more force to stay inside the window and offers exactly the same fractional latitude when it gets there.

Rosin does not merely move the window; it scales it. The friction ratio μ multiplies the width linearly, so a rosin giving half the ratio halves the room a player has at every bow position. At nine per cent of the string the window is a factor of 9 at μ = 1, of 4.5 at μ = 0.5 and of 18 at μ = 2 — which is a bigger effect on playability than moving the bow from a twentieth of the string to a tenth.

That also gives the point at which the window shuts. Setting the width to one and solving, the two bounds meet at β = 1/100μ: one per cent of the string at μ = 1, two per cent at half that ratio, half a per cent at double. Inside that distance from the bridge there is no bow force that produces Helmholtz motion at all, and where the boundary falls is decided by what is on the bow hair rather than by the instrument.

And a bow’s own tension and hair count decide the force distribution across the contact width, which the model here treats as a point. A rehaired bow behaves differently for a fortnight, which players describe in terms nobody can defend and which is a real change in a parameter of this diagram.

The window is why a string instrument is hard and a keyboard is not #

There is a general observation here that this field keeps producing in different forms.

A piano’s control is a single number per note with no permitted range at all — any key velocity produces a note, and the map from velocity to sound is smooth. A bowed string’s control is two numbers that must jointly stay inside a region whose shape depends on a third, continuously, for the whole duration of every note.

That is not a difference of degree. It is the difference between an instrument where the difficulty is in choosing what to play and one where the difficulty is in producing it at all, and it is why a competent adult can make a recognisable tune on a piano in an afternoon and cannot on a violin.

One thing the steady-state window cannot say is what happens at the beginning, and it is the part a player spends most of their attention on.

A constant bow force cannot start a note. The window at the moment of attack is not the window the sustained note has, because the string is not yet moving at the speed the sustained bounds assume — so the force that will hold a note is the wrong force to begin it with, and every player learns to move through the window rather than sit in it.

A window, and the other windows this site has drawn #

The shape of this argument has appeared three times already on the site, in fields that have nothing to do with each other, and it is worth collecting because the collection is the point.

The beat has a preferred rate, and the window inside which a series of events can be felt as a pulse at all runs from about 100 ms to about 2 s with a preference near 550. Outside it, a listener re-hears the metre at a different level rather than losing the beat.

The precedence effect has a window of roughly 1 to 35 ms inside which a second copy of a sound is not heard as a second sound.

A category has a width: a major third can be about seventeen cents from just and remain the same chord.

And here a physical system has one, bounded by two failure modes that are not each other’s opposites — surface sound and crunch are different kinds of wrong. In every case the interesting content is not that a middle exists but that the boundaries are measurable and asymmetric, and that the width is a quantity a designer or a player or a listener is working inside without being able to see it.

The regimes outside the window are not noise #

It is worth resisting the idea that everything outside Helmholtz motion is failure. C. V. Raman showed in the 1910s that the bowed string has a whole family of periodic regimes — with two slips per period, or three, or with the slip occupying a different fraction of the cycle — and several of them are musically used.

Double-slipping produces a note an octave above the expected one and is one route to a bowed harmonic. Anomalous low frequencies — subharmonics produced by excessive force — are used deliberately in some contemporary writing and in Nordic folk fiddling. Surface sound is a legitimate colour at the edge of audibility.

So the diagram’s shaded region is the region in which the ordinary note happens, and the areas either side are not empty. They are where the extended techniques live, which is why those techniques are unstable: they are being produced in regions with no window at all, by players holding a system near a boundary rather than inside one.

What the picture cannot show #

The bounds are asymptotic, and the axes are logarithmic for a reason. Schelleng’s derivation gives the scaling — one over beta and one over beta squared — with constants that depend on the string, the rosin, the bow and the instrument. The figure’s absolute force values are in arbitrary units; what is real is the two slopes and the width of the gap between them.

A bow is not a point. It contacts a centimetre or so of string, and the force is distributed across it. Near the bridge, where beta times the string length is only a few centimetres, the contact width is a substantial fraction of the distance to the bridge and the model is being stretched.

Nothing here includes the left hand. Stopping the string changes its length and therefore the absolute position the bow occupies as a fraction — a player bowing at a fixed point on the instrument is bowing at a larger beta as they go up the fingerboard, and their window is widening as they do. That is a real effect on real instruments and it is a rung further up this ladder rather than in this diagram.

The closed form is the model’s, not the string’s. The width being exactly a hundred times μβ is a consequence of the constant Schelleng’s minimum carries in this implementation, and that constant is a stated one. What does not depend on it is the structure of the result — which parameters cancel and which do not — because the cancelling follows from the two bounds sharing a factor rather than from any number in front of it. So the claim that impedance and bow speed buy no width is safe, and the claim that the window shuts at one per cent of the string is only as good as the constant.

And there is no sound button on the diagram itself. The site’s synthesiser produces notes from parameters; surface sound and double-slipping are failures of a self-sustaining oscillation and are not describable as a list of partials. The essay’s audible claim — that Helmholtz motion at any position produces the same sawtooth — is on the second figure, where it can be heard.

The attack is a separate problem with a separate answer #

One thing this diagram does not describe, and it is the half a player spends most of their practice on.

Schelleng’s window is a steady-state condition: it says which forces sustain Helmholtz motion once it is established. Starting a note is a different question, and the answer is a different diagram — Guettler’s, from the 1990s, which relates bow acceleration to bow force and shows the narrow wedge inside which a note begins cleanly rather than with a scratch or a slow build.

The two are worth keeping apart, because a player can be comfortably inside the steady-state window and still be unable to start the note. Almost everything a string teacher says about bow “contact” and “catching the string” is about the second diagram, and almost everything said about “tone” is about the first.

Whose instruments, and when #

John Schelleng’s “The bowed string and the player” (1973) is the paper, and it is one of the clearest pieces of writing in the acoustics literature; the diagram is his. Raman’s classification of the regimes is from 1918. The experimental confirmation, with real bows and force sensors on real instruments, comes mostly from Knut Guettler and the Stockholm group from the 1990s onward, and it also produced the other half a player needs — the conditions for a clean attack, which is a transient problem rather than a steady-state one.

The claims about technique are claims about the Western bowed-string family as played since roughly the nineteenth century. The physics applies to any bowed string; what a tradition does with the window differs, and the sarangi and the erhu use regions of it that a violinist would consider marginal.

Where this goes #

The bowed string has been treated so far as though what it produced went straight to the listener. It does not: a string radiates almost nothing on its own, and everything a listener hears has been through the instrument’s body, which is a filter with measured resonances — the same source-and-filter arithmetic as a vowel, applied to wood.

And the body has one further consequence that belongs to this ladder. When a string mode lands on a strong body resonance the two stop being separable and exchange energy, which produces a warbling note that is not a mistuning and shares a word with one that is.