The note has to start somewhere

The bow makes a corner and there is a window of bow force within which it can keep making one. Those two rungs describe a note in progress: a steady Helmholtz motion, a periodic sawtooth at the bridge, a spectrum that barely changes with bow position.

Every quantity in them is a steady-state quantity. Schelleng’s inequality contains a bow speed, and the bow speed it contains is the one the note is being played at.

Which leaves the question a player would ask first entirely untouched. The bow starts at rest.

What the steady-state map is a map of #

The minimum force is what is needed to capture the string on each cycle; below it the string slips more than once per period and the tone is a surface noise. The maximum is what the string can escape from; above it the bow holds the string through what should have been a release, and the tone is crushed. Both scale with the bow speed, so both are proportional to how fast the bow is moving.

That last clause is where the attack problem is hiding. If both bounds are proportional to speed, and the speed at the start of a note is nearly zero, then both bounds at the start of a note are nearly zero.

The same inequality, read at the first period #

A bow accelerating from rest at a has a speed of a·t. After one nominal period of the note it is going at a/f₀ — for a bow accelerating to 0.6 metres a second over a second, playing an open G at 196 hertz, that is about three millimetres a second, which is one part in two hundred of the eventual speed.

Evaluate Schelleng’s two bounds at that speed and the picture changes completely.

Two facts come out of that and neither was in the slate for this essay.

The window is not narrower. Its width as a ratio is 9 to 1 at every acceleration, which is the same 9 to 1 the sustain diagram has at the same bow–bridge distance. The attack is not hard because the target is smaller.

The window has moved. At an acceleration of 0.6 the first-period force window runs from 0.0076 to 0.068 in the model’s units; the sustain window at the eventual speed runs from 1.48 to 13.3. That is a factor of 196 — which is exactly f₀, because the two speeds differ by f₀ times the acceleration time.

So the force that is right for the note is nearly two hundred times too large at the moment the note has to start. Whatever a player does at an attack, it is not “apply the force the note needs”.

Why the force must rise with the speed #

The wedge’s shape is the whole prescription and it is a simple one: both bounds are proportional to acceleration, so a clean start requires the force to be proportional to the bow speed at every instant of the transient, not merely to arrive at the right value eventually.

That is a statement about a trajectory. A point in the acceleration–force plane is not a description of an attack; an attack is a path, and the requirement is that the path stays inside a region that is itself opening out as the bow accelerates.

The two failure modes at the edges of the wedge are the two Schelleng named, arriving at the beginning of the note rather than during it.

Below the wedge — too little force for the speed the bow has reached — the string slips several times per nominal period. The sound is a high-pitched surface scratch that resolves into the note when the force catches up.

Above it — too much force for the speed — the bow holds the string past its release point, and the first period is longer than the ones that follow. That is the low, choked, momentarily flat attack, and it is the failure that sounds like a mistake rather than like an effect.

What this predicts about how attacks are made #

Three consequences, and each is something string players are taught without the reason.

A note started with the bow already moving is easier than one started from rest. Setting the bow in motion above the string and landing it — the gesture that produces a portato or a flying attack — starts the transient at a speed where the window is already wide, and skips the region where the wedge is at its narrowest. The instruction to “have the bow moving before it touches” is exactly this.

A pressed start requires a slow one. If the force is fixed before the note begins, the wedge says there is one acceleration at which it is correct, and it is a low one. That is the marcato or accented attack: press, then move slowly at first. The alternative — press and move fast — is above the wedge and produces the crushed onset.

And a fast attack requires a light start. The mirror image: a rapid spiccato stroke begins with the bow moving quickly and needs correspondingly little force, which is what a bouncing bow provides by its own dynamics.

The same problem on the instruments that do not have it #

It is worth noticing which instruments this does not apply to, because the contrast says what is peculiar about a bow.

That is the general division. Instruments with a continuous excitation have an attack problem and instruments with an impulsive one do not. A hammer, a plectrum and a finger all release the object and let it ring; a bow, a reed and a pair of vocal folds all have to establish a self-sustaining oscillation, and establishing one is a different problem from maintaining it.

The reed’s threshold pressure is the wind version of exactly this, and the parallel is close: a reed below its threshold does not sound at all, so a wind attack is a matter of crossing a boundary rather than of staying inside a corridor. The bow’s problem is harder because its corridor has two walls and both of them move.

How long the transient is #

The number that matters to a listener is how many periods pass before the motion is periodic, and the model gives it indirectly: the transient lasts until the bow’s speed brings the force inside the window.

For a fixed force and a fixed acceleration that is a single crossing, so the transient duration is the time for a·t to reach the speed at which the applied force equals the minimum. At 196 hertz a transient of ten periods is fifty-one milliseconds; twenty periods is a hundred and two.

The same arithmetic gives the quantity the wedge picture implies and does not state: how long a constant force is inside the window at all. Both bounds are proportional to the bow’s speed and the speed is proportional to time, so a constant force enters the window when the maximum rises to meet it and leaves when the minimum does — and both instants are the force divided by a constant times the acceleration:

A constant bow force is inside the window for between half a period and seven, and a transient needs ten to twenty. So the geometric statement — that a horizontal line crosses the wedge at one point — becomes a temporal one: a constant force cannot get a note started, not because it is the wrong value but because no value is right for more than a few periods.

The trajectory it does produce is worth spelling out, because it is the sequence a listener hears. For the first millisecond and a half the force is above the maximum, which is the crushed onset; for the next twelve it is inside, which is the note beginning properly; and from thirteen milliseconds onward the window has risen past it and it is below the minimum, which is the scratch. A constant-force attack does not fail in one way. It fails, works, and fails again, in that order, inside the first fiftieth of a second.

There is one exact relation in that table worth having. The time at which the force leaves the window divided by the time it enters is 4.0 at β = 0.04, 9.0 at β = 0.09 and 15.0 at β = 0.15 — which are the window ratios themselves. That is not a coincidence: both times are the force over a constant times the acceleration, and the two constants stand in exactly the ratio the window does. So the fraction of the transient a constant force survives is fixed by the bow–bridge distance alone, and neither the force nor the acceleration can change it. Pressing harder or moving faster shifts both instants together and buys nothing.

Which is the sharpest form of the essay’s prescription. A player over the fingerboard has fifteen times as long inside the window as they had entering it, and near the bridge four times — so the corridor at β = 0.15 is not merely wider in force, it is longer in time by the same factor, and that is the same number doing both jobs.

Both of those are inside the window in which a note’s identity is established, which is the reason the difference between a good and a bad attack is so audible. A transient is not a small imperfection at the front of a note; it occupies exactly the interval a listener uses to decide what the instrument is.

So the window a player has to enter is narrower at the top of the compass and reached sooner, because the period is shorter there. A high note gives less room and less time in the same breath, which is the transient’s version of the difficulty the sustained map already showed.

The steady state the transient is heading for #

Putting the two side by side makes the omission visible. The corner has to form, and nothing in the description of a formed corner says how one forms. A model of a periodic motion is a model of a fixed point, and a fixed point does not carry its own basin of attraction with it.

That is a general feature of steady-state acoustics and it is worth naming, because this collection is full of steady-state descriptions: the modes of a tube, the resonances of a body, the spectrum of a plucked string. Each is exact about a sound that is already happening, and each is silent about the beginning — which is the part of a note a listener uses to identify the instrument.

Three objects whose partials are not the harmonic series, drawn against one that is, make the negative case: none of them has an attack problem of this kind, because none of them has to establish a self-sustaining motion. A struck or plucked object is given all its energy at once and then decays; only a bowed or blown one has to arrive at its steady state, which is why this essay has no counterpart in the struck families.

Where this model stops, and it stops early #

Three limitations, and the first is severe enough to be stated as a warning about what has been shown.

This is not Guettler’s criterion and nothing of his is evaluated here. Knud Guettler is the person who posed the question — what has to be true for periodicity to establish at the start of a note — and his answer is a measured map of attack quality over bow acceleration and force, obtained from real bowing machines and real strings. What this essay does is much weaker: it reads Schelleng’s steady-state inequality at a time rather than at a speed, which is a derivation from a model already in use here. The wedge it produces has the shape of the region Guettler’s diagrams show, and that agreement is the reason to think the reading is not silly. It is not a reproduction of his result.

The string has no torsional motion and the bow has no width. A real bow hair band is several millimetres across and touches the string over that width; the string twists as well as bends, and torsional waves have their own speed and their own damping. Both matter most during the transient — which is to say, both matter most exactly where this essay is looking.

And there is no player in it. Every real attack is a coordinated change of force, speed and position, corrected continuously by ear within the first few tens of milliseconds. The model has an open-loop bow.

Which computation produced the numbers #

Schelleng’s maximum is proportional to bow speed over the bow–bridge fraction, and his minimum to bow speed over the square of it; the constants used here are the ones the sustain rung already uses, so the two figures are the same function called with different arguments. The window ratio comes out as a hundred times the bow–bridge fraction and is independent of speed, which is why it is 9 at 0.09 and 4 at 0.04.

The first-period speed is the acceleration divided by the frequency, which is uniform acceleration over one period and nothing more.

The factor of 196 between the two windows is the ratio of the two speeds, which for a bow reaching its final speed in one second is the frequency itself. It is not a general constant — halve the time to final speed and it halves — and it is quoted here for the specific case drawn.

Whose instruments, and when #

The bowed string is the object, and the acoustic model is general to it: violin family, viol family, erhu, sarangi, kemenche, rebab and the rest all produce Helmholtz motion and all obey the same inequality.

What differs between them is where in the wedge the tradition sits. A gut string at low tension with a light bow occupies a lower and narrower part of the diagram than a modern steel-strung instrument with a Tourte bow, and the Baroque bow’s shape and lighter head are usually explained in terms of articulation — which, read through this essay, is a statement about how easily the force can be made to track the speed at the start of a stroke.

The attack repertoire is also historically specific. The catalogue of named bow strokes that string players learn is a nineteenth-century codification, and several of the strokes in it are ways of navigating the wedge that were discovered by ear long before anybody wrote an inequality down.

What the picture cannot show #

It cannot show the sound of a bad attack. The figures are forces and regions; a scratch is a spectral event with its own character, and no point in a plane conveys it.

It cannot show correction. A player who begins badly fixes it in tens of milliseconds and the resulting note is often perfectly acceptable. The model produces a verdict per attack, and real playing is a feedback loop.

And it has no string in it above the first period. The wedge is evaluated at one instant. What happens over the next ten periods — whether the motion converges on Helmholtz, oscillates around it, or settles into a different regime — needs a simulation of the string, and nothing here is one.

The duration table above is the closest this model gets to a trajectory and it is not a substitute for one. It says when a constant force is inside the steady-state window evaluated at the speed the bow has reached, which is a statement about whether Helmholtz motion is possible at each instant rather than about whether it has had time to establish. Being inside the window for two periods is necessary for a clean start and is nowhere near sufficient, because the motion needs several periods to converge even when the conditions are right throughout. So the numbers bound the problem from one side: a force outside the window certainly cannot start a note, and a force inside it briefly may well not either.

The ladder from here #

Three rungs: the corner that travels, the window it can be sustained in, and the trajectory required to reach that window from rest.

What this ladder still lacks is the thing every string player would name first, which is the body. Every figure here treats the string in isolation, and the instrument’s own resonances feed back onto the string strongly enough to make some notes hard to start on a particular instrument and not on another. That coupling has a rung elsewhere in the collection and does not yet have one here.