The bow makes a corner

A plucked or struck string is given energy once and then left alone; everything about its sound follows from where it was touched and how the energy leaks away. A bowed string is different in kind. It is being driven continuously, by something that is in contact with it the whole time, and what it does under that drive is not what the word “vibration” suggests.

Hermann von Helmholtz worked this out in the 1860s, using a vibration microscope — a lens attached to a tuning fork, so that a spot of white on a bowed string traced its motion against a known reference. What he saw was the corner. It is called Helmholtz motion, and it is one of the results that founded the subject.

Stick, slip, and why the corner sustains itself #

The bow’s hair is rosined, and rosin has the property that its friction falls sharply once sliding begins. So there are two regimes: the string stuck to the hair and carried along with it, and the string sliding back underneath it.

The cycle goes like this. The string is stuck to the bow and being dragged sideways at the bow’s speed. The corner, travelling round the string, arrives at the bow. Its arrival delivers a sudden change in the force at that point, which breaks the string free; the string slips back rapidly while the corner travels the other way round; the corner returns, the string re-sticks, and the cycle repeats.

The corner triggers its own release once per period. That is what makes this a self-sustaining oscillation rather than a forced one: the string is telling the bow when to let go, and the bow is supplying the energy to replace what was lost. The frequency is set by the string, not by anything the player does — which is why bow speed changes loudness and not pitch.

The two segments stay straight because a corner on an ideal string travels without changing shape. Between corners there is nothing to bend the string, so it is a straight line under tension, and the whole motion is described by the corner’s position.

Why the result is a sawtooth, and why that matters #

The force the string exerts on the bridge is proportional to the angle the string makes there. With the string in two straight segments and the corner at a moving position, that angle changes linearly with time while the corner travels one way, then linearly the other way as it returns — a ramp up and a ramp down, in the ratio $\beta : (1-\beta)$ where $\beta$ is the bow’s distance from the bridge as a fraction of the string.

For $\beta$ small — the bow near the bridge, which is where it usually is — that is very nearly a sawtooth: a slow ramp and a fast return.

And a sawtooth’s Fourier series is exactly one over $n$ with every harmonic present. Not approximately, not mostly: exactly, from a shape produced by a piece of horsehair and some tree resin.

A sawtooth’s partials fall as one over n and it contains every one of them, which is why a bowed string’s waveform looks nothing like a clarinet’s: a stopped tube’s boundary conditions delete the even partials and a corner travelling round a string deletes nothing. The corner is the spectrum. A discontinuity in a periodic waveform is a full harmonic series with a 1/n envelope, and that is a statement about the shape rather than about the instrument.

That is the reason a violin section can play a chord that locks together, and the reason the roughness calculation for strings behaves so well: a spectrum with every harmonic present and nothing else has the maximum possible number of coincidences with another such spectrum at a simple ratio, so consonance and dissonance are as sharply distinguished as they can be.

The surprise: bow position hardly changes the spectrum #

Everything in the plucked-string essay said that where a string is excited decides which partials it has. A bowed string appears to contradict that, and the contradiction is the most interesting thing here.

Moving the bow changes $\beta$, which changes the duty cycle of the sawtooth. A sawtooth with a duty cycle other than zero is not quite a pure sawtooth: it has nulls, at exactly the same places the plucked string’s are — the $n$-th partial vanishes when $\beta = k/n$.

That is the claim, and computing both spectra from this site’s own machinery refuses it. The bridge force is a triangle in time with its apex at β; a plucked string is released from a triangle in space with its apex at β; and the Fourier magnitudes of a triangle are the same function whichever axis it is a triangle on. Evaluated at the same two positions the two are identical to every figure printed:

In the ideal model a bowed string and a plucked one have the same spectrum and the same sensitivity to position, partial for partial, and the spectral centroid of both falls by exactly the same factor of 0.68 across that move. The surprise this section is named for is not in the arithmetic.

That does not make the observation wrong — every player knows bow position is a weaker tone control than plucking position — it locates it. What separates a real bowed spectrum from a plucked one is not in this model: the bow has a finite width, the corner is rounded by the string’s own damping as it travels, and the rounding fills in the notches and flattens the envelope toward the 1/n a pure sawtooth would give. Those are the mechanisms Cremer’s account of the rounded corner is about, and none of them is here.

One further correction falls out of the same computation, and it applies to the plucked figure’s own caption below. Both spectra run as 1/n, not 1/n², up to the first null — 1.00, 0.49, 0.32, 0.24, which is a sawtooth’s envelope — and only above the null at n ≈ 1/β does the 1/n² take over. At a bowing point of nine per cent that null is at the eleventh partial, so the whole of the audible part of the spectrum is in the 1/n régime for both. The steep envelope a plucked string is usually credited with is a property of the partials above the null and of how fast they decay afterwards, which is a separate mechanism this essay does not compute either.

The identity is worth one further reading before it is left, because it says which of the two ladders is doing the work. Every result this collection has about plucking position — the nulls at k/n, the null at the seventh partial that decides where a hammer lands, the tone control a guitarist has between the bridge and the fingerboard — transfers to bowing position unchanged, since the two spectra are one function. What does not transfer is everything about decay, because a plucked partial is launched once and a bowed one is re-supplied every period. So the excitation ladder and the bowed-string ladder share their spectra and share nothing about time, which is a cleaner division than either had assumed.

What bow position does change decisively is the attack. Near the bridge, the corner has a short distance to travel between reflections, the periodic slip is harder to establish, and the note takes many cycles to settle into Helmholtz motion. Far from the bridge it settles almost at once. So the difference a player hears between sul ponticello and sul tasto is largely a difference in transient behaviour rather than in steady spectrum — and the first fifty milliseconds are where an instrument’s identity lives.

Why a sawtooth is the best possible spectrum for ensemble playing #

There is a consequence for how string instruments behave together, and it follows from the partial set rather than from anything about players.

Two notes are consonant to the extent that their partials either coincide or stay far apart, and rough to the extent that they land close without coinciding. A spectrum with every harmonic present has the maximum number of partials available to coincide — at an octave every second partial of the lower note meets one of the upper, at a fifth every third, at a fourth every fourth.

Which partials two notes share at each interval is the same table for a sawtooth as for any full harmonic spectrum: the octave shares six of twelve, the fifth four, the fourth three. A bowed string supplies every partial, so it supplies every coincidence a simple ratio can produce, which is why a string section tunes and blends as precisely as it does.

This is why a string quartet can tune itself to a precision no other ensemble matches, and why string players are taught to adjust intervals by ear toward pure ratios rather than toward equal temperament. The instrument gives them the sharpest possible feedback: with a full harmonic series on both notes, a fifth two cents from pure beats audibly, and two cents is around the limit of what a listener can detect any other way. The bow’s sawtooth is what makes that feedback available.

It also explains a smaller thing, which is why a string section playing in unison sounds like more than one instrument rather than like one loud one. Each player’s sawtooth is very nearly identical in spectrum, so what separates them is small differences in fundamental and in attack timing — and those are exactly the cues the ear uses to keep sources apart.

The bow is a nonlinearity, and that is the point #

A linear system driven at one frequency responds at that frequency. Nothing linear can take a steady input — a bow moving at constant speed — and produce an oscillation at a frequency the input does not contain.

The stick-slip friction curve is the nonlinearity that makes it possible, and it is the same structural role that a reed plays in a woodwind and felt plays in a piano. Every self-sustaining instrument has one somewhere, and in each case the nonlinearity is what converts a steady supply of energy into a periodic one.

The three are worth comparing because the mechanisms are so different. A reed is an active valve, opening and closing under the pressure it controls. Felt is a passive spring whose stiffness varies. Rosin is a friction law with a falling characteristic. Three completely unrelated physical effects, doing the same job in the same place in the argument.

What the player actually controls #

Three parameters, and they are nearly independent, which is unusual.

Bow speed sets the amplitude. The string is dragged further before each slip, the sawtooth is larger, and the note is louder. It barely touches the spectrum.

Bow force must lie inside a window — too little and the corner fails to trigger a slip every time, too much and the string stays stuck for more than a period. That window is computable and it narrows toward the bridge, which is the next essay.

Bow position sets the sawtooth’s duty cycle, and the computation above says the spectral consequence is not weak in the ideal model — it is exactly the plucked string’s, which is strong. What makes it weak on a real instrument is the rounding of the corner, which is a thing the bow does rather than a thing the position does. What position changes with no such qualification is the attack and the force window.

A fourth, which belongs to the left hand rather than the right: where the string is stopped, which sets the length and therefore the pitch, and which on a fretless instrument is continuous. That continuity is the reason string players are the group in an orchestra least constrained by any particular temperament — they are not choosing from twelve positions, and the feedback that tells them which position is right is the beating the sawtooth makes available.

The near-independence of speed and position is why a violinist can play a crescendo without the tone changing character, which a pianist cannot: a piano’s one control couples loudness to brightness and a bow’s do not.

What Helmholtz had to build to see it #

The observation deserves a paragraph of its own, because the difficulty of making it explains why the result waited until the 1860s.

A violin string vibrates a few hundred times a second, with an amplitude of well under a millimetre. Nothing in the nineteenth century could photograph it. What Helmholtz did was mount a microscope objective on the prong of a tuning fork, so that the field of view swept sideways at a known frequency, and put a spot of luminous paint on the string. The spot’s motion in one axis and the fork’s in the other traced a figure — a Lissajous curve, in effect — from which the string’s displacement against time could be read off directly.

What appeared was two straight ramps meeting at a point. Not a sine, not a sum of sines that looked like anything in particular: a triangle wave in the string’s displacement, which is the signature of a single corner passing the observation point once per period.

This is worth a sentence about method. The result is a claim about a shape, and it was settled by an instrument built to make the shape visible rather than by an argument about which modes ought to be present. The modal description and the corner description are the same motion — a triangle wave’s Fourier series is one over $n^2$ in displacement, which is one over $n$ in the bridge force — and neither is more true than the other. But nobody deduced the corner from the spectrum. Somebody looked.

What the picture cannot show #

The corner is not sharp. On a real string it is rounded, by an amount that grows with each round trip because stiffness and internal damping remove the highest partials fastest. The rounding is a large part of what distinguishes a good instrument’s tone from a synthetic sawtooth, and it is the first thing a physical model has to add.

The figure draws the string, and what is heard is the bridge. The string radiates almost nothing directly. The sawtooth force at the bridge drives the body, which imposes its own resonances, and the spectrum a listener receives is the product of the two.

The motion drawn is the ideal regime. A real bowed string spends part of every note in other regimes — multiple slips per period, raucous double-slipping, or the surface sound that comes from insufficient force. Helmholtz motion is what a player is aiming for, and the aim is not always achieved.

And the sound buttons play a sawtooth, not a violin. They demonstrate what a one-over-n spectrum sounds like, which is the essay’s claim. They do not include the body, the rounding of the corner, the bow noise or the vibrato, and a listener who expects a violin will hear how much of a violin is not in the string.

Whose instruments, and when #

Helmholtz’s On the Sensations of Tone (1863) contains the observation and the analysis, and the motion carries his name. C. V. Raman worked out the full family of possible regimes in the 1910s — the Helmholtz motion is one member of a set of periodic solutions, and Raman classified the others. The modern treatment, including the force window in the next essay, is largely John Schelleng’s from 1973 and the Stockholm group’s from the 1980s onward.

The claim about spectra is a claim about bowed strings in general — a violin, a viol, an erhu and a sarangi all produce Helmholtz motion, because the mechanism depends on rosin and a string rather than on any tradition’s design. The claim about sul ponticello being an attack effect is a claim about how the instrument is used in Western art music since the nineteenth century, where it is a colour rather than a default.

Where this goes #

The next rung is the force window, which turns “bowing near the bridge is hard” into a slope on a log-log plot: the minimum force goes as one over beta squared and the maximum as one over beta, so the usable range closes in proportion to the distance from the bridge.

Beyond that the bowed string leads into the body, because a bowed instrument radiates almost nothing without one. A violin’s body is a filter with measured resonances, the same arithmetic as a vowel’s formants applied to wood — and when a string mode lands on one of those resonances the two stop being separable, which is a wolf note and is not a tuning error.