What the rosin is worth

The way to find what a ladder has been assuming is to look for the parameter every one of its figures sets to the same value. This one has two, they sit side by side in the same expression, and only one of them was worth sweeping — which is a result rather than a disappointment, because the demonstration that the other cancels is what makes the first worth trusting.

How much bow is allowed is this ladder’s second rung and it introduced the object everything since has been drawn on. Schelleng’s two bounds on the bow force are

$$F_{\max} = \frac{2 Z_c v}{\mu \beta}, \qquad F_{\min} \propto \frac{Z_c^{2} v}{\mu^{2} \beta^{2}}$$

where $\beta$ is the bowing point as a fraction of the string, $Z_c$ the string’s characteristic impedance, $v$ the bow speed and $\mu$ the difference between the static and sliding friction coefficients — which is to say, the rosin.

Nine rungs have drawn the fraction — 0.02, 0.03, 0.09, 0.16, and every value in between as a swept axis. Not one has ever set the other two to anything but one.

The one that cancels #

Divide the two bounds and the impedance and the speed both go:

$$W = \frac{F_{\max}}{F_{\min}} = 100\,\mu\,\beta$$

The bow speed is not in it. Both bounds are linear in $v$, so doubling the bow speed doubles the least force that will speak and doubles the most the string will take, and leaves the ratio between them exactly where it was.

That is a null, and it is the reason the ladder has never had to name a bow speed. It is also worth stating carefully, because the thing that cancels is the window and not the playing. A fast bow needs more force, and the whole usable band moves up with it, so a player changing bow speed mid-phrase must change arm weight to stay inside — which is the coordination every teacher describes and which nothing here disputes. What the arithmetic says is that the band does not get narrower or wider while they do it, so the difficulty of a passage does not depend on how fast the bow is going.

There is one place the speed does not cancel, and this ladder has already used it. The note has to start somewhere reads Schelleng’s inequality at the speed the bow has after k periods rather than at the speed it will eventually reach, and there $v$ is the whole subject: a bow accelerating from rest is climbing through a window that is opening beneath it. That is a statement about a moving speed and not about a chosen one, and it is untouched.

The one that does not #

The friction contrast is in the window linearly. More rosin is a wider window at every bowing point, and the whole family of curves is one curve slid sideways.

A word about what the number is. mu here is a scale on the friction contrast and not a coefficient: the constant inside the minimum force absorbs the termination resistance and everything else, and one is a normally-rosined bow by definition rather than by measurement. That normalisation is why every claim below is a ratio. What is not absorbed is the pair of exponents — the maximum force carries the contrast once and the minimum carries it twice — and those are the whole content.

The two exponents are worth understanding separately, because they come from two different failures and only one of them is obviously about friction. The maximum force is the force at which the string stays stuck past the moment the corner arrives, and what has to be overcome is the difference between holding and slipping — so it carries the contrast once, in the denominator, and a stickier bow can hold the string too well. The minimum is the force below which the corner fails to trigger a slip on every pass, and the corner’s ability to trigger one depends on the contrast and on how far the resulting slip carries, which is itself set by the contrast — so it carries it twice. Two mechanisms, two powers, and the ratio between them is the one power that is left.

The published range for the two coefficients on rosined horsehair is roughly 0.8 for the static and 0.3 for the sliding, so a contrast of about 0.5. Nobody measures a rosin cake’s effect on that difference directly, but the direction is not in doubt: more rosin raises the static coefficient more than it raises the sliding one, and a bow that has not been rosined for a fortnight has a contrast approaching nothing at all, which is why it will not speak. A factor of two either side of normal is the range this essay sweeps and it is defensible; a factor of five is not.

Where the absolute forces go, which is the opposite way #

Widening the window sounds like a gift, and the way it is granted is not what the phrase suggests.

Doubling the rosin halves the largest force the string will take and quarters the smallest that will make it speak. The window widens from a factor of nine to a factor of eighteen, and it does so entirely by the floor dropping away beneath a ceiling that is also coming down.

That is a specific prediction about how a freshly rosined bow feels, and it is the one every player would recognise: an over-rosined bow grabs, and the response is to play lighter. The model says lighter by a definite amount — the whole usable band has moved down by a factor of between one and two, depending which end of it the player was near — and it says the passage that would not speak now does, because the floor has moved four times as far as the ceiling.

The reverse is the more familiar failure. Under-rosin the bow and the floor climbs faster than the ceiling: at half the contrast the least force that will speak is four times what it was, and the most the string will take is only twice. A dry bow is not a bow that grips less. It is a bow whose usable band has moved up and shrunk, which is why the note that comes out of it is thin and late rather than merely quiet.

What it does to the three limits #

The bow is not a point either found three separate limits on how near the bridge a bow can go, and the interesting question was their order. Two of the three are pure geometry — the hair ribbon’s near edge reaching the bridge, and the ribbon covering half the corner’s bridge-side excursion — and neither contains the rosin at all. The third is Schelleng’s window narrowing to a factor of three, and it is inversely proportional to it.

At ordinary rosin the eighth rung’s result comes back exactly: the violin is the one instrument on which geometry binds first, by a quarter of a millimetre, and the viola, cello and bass are limited by force alone with room to spare.

The margins are not equal, and the reason is a ratio the fifth rung already carries. The window limit is the same fraction of the string on every instrument — three per cent, because the threshold and the contrast are the same — so in millimetres it scales with the speaking length: 9.8 on a violin, 11.7 on a viola, 20.7 on a cello, 31.5 on a bass. The ribbon’s limits scale with the hair width, which barely changes across the family at all. So the gap between them widens with the instrument, and it widens fast: the cello has 8.7 millimetres of margin where the violin has minus 0.25.

Now move the rosin. The window limit slides in proportion and the two geometric ones stay put, so the ordering flips instrument by instrument, and each instrument has a friction contrast at which it flips. Those numbers are:

The first column is the eighth rung’s coincidence, restated in the variable that was not being swept when it was found. A violin at ordinary rosin sits 2.5 per cent below the contrast at which its two limits exchange places. The viola is six per cent above it, the cello is three quarters of the way to twice, and the bass would need a rosin nobody makes.

The second column is the stronger statement. At twice ordinary rosin, a violin’s force window does not narrow to three until 4.9 millimetres from the bridge — and the hair’s near edge touches the bridge at 5.0. The force limit has moved to a place the bow cannot reach. From there inward there is no bow position at which the window is the reason the note fails; the only reason left is that the ribbon has run out of room.

Which tests a prediction this ladder made and could not check #

The eighth rung’s account of extreme sul ponticello ends with a prediction it had no way to evaluate:

a heavier bow, or a stiffer stick, or more rosin — everything a player has for the force problem — should not help at all below the ribbon’s own limit, and should help above it.

The arithmetic says yes, and says where the boundary is. Above ten millimetres from a violin’s bridge, more rosin moves the force limit inward and the ribbon is not yet in the way: rosin buys bow positions. Below ten it buys nothing at all, because the limit it moves has already retreated behind one it cannot touch. The prediction is confirmed with a number attached, and the number is the same ten millimetres the eighth rung arrived at from two independent directions.

It also explains a piece of practice that otherwise looks like superstition. Players who want a reliable ponticello rosin heavily and are told it helps; players who want it at the very edge of the bridge rosin heavily and find it does not. Both are right, ten millimetres apart.

And what it is worth in milliseconds #

The last consequence is the one that leaves this ladder. The hardest place is also the latest read Schelleng’s window as an attack time: a bow accelerating from rest catches the string when the force it is applying enters the window, and a player aiming at the geometric centre of a window $W$ wide catches it in a time proportional to $1/\sqrt{W}$.

The window is proportional to the friction contrast, so the capture time goes as one over its square root.

On the G string the note begins at 24.5 milliseconds at 0.4 times the rosin, 15.5 at ordinary rosin, and 9.8 at 2.5 times. On the E string the same range runs from 18.2 to 7.3. The whole span is about fifteen milliseconds on the lowest string and eleven on the highest, and the threshold at which a listener starts to hear one instrument as displaced against another is around twenty.

That number belongs beside two others this collection has computed for the same quantity. A blown note does not start late found the spread between a wind instrument’s own partials to be between two and seven milliseconds, well inside the threshold; a note starts twice found the gap between when a bowed string catches and when a listener places it to be twenty-eight. Fifteen milliseconds of rosin sits between the two, which makes it a term worth carrying rather than a curiosity: it is larger than the effect a whole essay was written to dismiss and smaller than the one the anchor was opened to describe.

So rosining a bow is a change to when its notes arrive, and by an amount that is most of what an ensemble can detect. That is not a quantity anybody discusses. Rosin is described as affecting grip, tone and reliability; the model says it also moves the note by up to two thirds of the asynchrony threshold, in the direction that an under-rosined bow is late.

The claim needs its caveat stated as loudly. The capture model has a bow acceleration in it that this collection asserts rather than measures, and the whole family of curves scales with it. What survives that is the shape: a square-root law, the same on all four strings, and the ordering — the G string is always the latest, because it has the highest impedance and the narrowest window.

Which computation produced the numbers #

The window is the ratio of the two bounds above, which is $100\,\mu\beta$ in the normalised units this ladder has used since its second rung. The 100 is the reciprocal of the constant inside the minimum force and carries the termination resistance; it is why nothing here is in newtons.

The three limits are the eighth rung’s, unchanged: the ribbon’s floor is half the ribbon fraction, the occupancy limit is the ribbon fraction over twice the occupancy threshold, and the window limit is the threshold over a hundred times the contrast. The ribbon fractions are ten millimetres on 325, eleven on 390, twelve on 690 and fifteen on 1,050.

The attack times are speakingCensus’s own — the same computation the onset ladder uses, with the friction contrast carried into the two schelleng calls instead of left at its default.

Where the model stops #

A friction contrast is not a friction curve. Schelleng’s bounds contain the difference between two coefficients, and the modern account of what a bowed string does replaces both with a friction force that depends continuously on the sliding speed, and then on the temperature of the contact patch, which changes within a single note. Sweeping one number is sweeping the classical model honestly; it is not a model of rosin.

Nothing here measures a cake of rosin. The two-to-one range is defensible from published coefficients and from the direction the mechanism runs, and no experiment in this collection ties a number of strokes with the cake to a number on this axis.

The threshold of three is a convention. The window limit is where $W$ falls to three, and three was chosen two rungs ago because it is about where a player stops being able to hold the force. Move it to two and every crossover contrast in the table moves by a third.

And the geometric limits have their own softness. The occupancy limit assumes a flat ribbon on a straight string, and a tilted bow narrows the ribbon, which the eighth rung showed moves the geometric limit and not the force one. A player who tilts and rosins is moving both limits at once, in opposite directions, which is a combination this model can compute and no figure here draws.

What the picture cannot show #

It cannot show the sound. Everything here is about whether Helmholtz motion is available at a given force, and the timbre of a note played near the middle of the window against one played near either edge is a separate matter that nothing in this ladder has priced.

Nor the stroke’s own history. Rosin transfers from the cake to the hair and from the hair to the string, and it is scraped off again by playing. The contrast during a long passage is not the contrast at its start, and this is a static model.

And it cannot show the second thing a player changes at the same time. Rosin arrives on a bow along with a decision about how much hair is on the string, which is the tilt the eighth rung priced, and along with a choice of string, which is the fifth rung’s impedance. All three move the same window and this figure moves one of them.

Where this ladder goes next #

Ten rungs. The bow makes a corner; the corner sustains inside a force window; reaching it takes a computable number of periods; the body is in the window’s floor; the string’s impedance is in it twice; the bowing fraction is in it twice as well; the two multiplied give the instrument a worst place; the bow at that place has a width; the bow at that place has an impedance, which does nothing to the string; and now the window’s last silent parameter has been moved, which turns out to relocate every force limit and no geometric one.

Which leaves the ladder in an unusual position: its model has no parameters left that nobody has swept. What it has instead is a constant — the bow acceleration in the attack calculation, asserted at 0.6 metres per second squared since the third rung and never measured. It is the only number in this anchor that scales an answer directly rather than through a ratio, and unlike the rosin it cannot be swept honestly, because there is nothing to compare a sweep against. What it needs is a measurement: a recorded attack, a bow position, and the interval between the arm starting and the periodicity establishing. That is a corpus rather than arithmetic, this collection has none, and saying so is the honest end of a ladder that has now moved everything it can move by calculation alone.