The bow is not a point either

The hardest place on the fingerboard multiplied the last two variables this ladder had and produced a map with a worst place on it. The rung it named next — Schelleng’s window read as an attack time — has since been written from the other side, as the third rung of the onset ladder, so it is not owed.

What is owed is something none of the seven rungs noticed, and the excitation ladder found the same omission on its own object three rungs in.

A bowing point is a number: β, the bow’s distance from the bridge over the string’s speaking length. Every figure in this anchor sets it — 0.09, 0.03, 0.16 — and draws it as a place. A violin bow’s hair ribbon is about ten millimetres wide, and a violin’s speaking length is 325. So the bow is 3.1 per cent of the string, and at a normal bowing point of 30 millimetres it is a third of the way to the bridge from where it is standing.

The control first, because it comes back empty #

The hammer is not a point either gave a piano hammer a finite contact and found a second low-pass: the excitation is the mode shape integrated over the contact, which is a sinc in partial number, with its first null at two over the width. The obvious thing to do here is the same calculation on a bow.

The two spectra agree to within three decibels up to partial 29, and a bowed string’s own spectrum is thirty decibels down long before that. The ribbon’s smearing is real and it is worth nothing.

The reason is the same shape as the four-cornered accounting on the excitation ladder: the comb’s first null is at one over β, which is partial 11 at a normal bowing point, and the width’s is at two over the ribbon fraction, which is partial 65. A corner six times higher than the binding one cannot decide anything.

So the bow’s width does not change what a bowed string sounds like. That is the first half of this rung and it is a null, and stating it is what makes the second half worth trusting.

What it does change is where the bow is allowed to be #

The Helmholtz corner runs round the string: bridge, bow, nut, bow, bridge. On the bridge side of the bow it travels β of the string and comes back, which is 2β of the round trip.

The ribbon covers 3.1 per cent of the string, always. At a normal 30 millimetres from the bridge, 2β is 18.5 per cent, so the ribbon covers 17 per cent of the corner’s bridge-side excursion. At eleven millimetres it covers 45 per cent. At six millimetres it covers 83.

Below about ten millimetres the corner spends more of its bridge-side trip under the bow than clear of it. That is not a small correction to a bowing point; it is the bowing point ceasing to be one. The corner does not arrive at a bow, get released and depart — it emerges from under the ribbon, reflects, and goes back under it before the hair has finished with it.

And there is a hard floor beneath that. The ribbon’s near edge reaches the bridge at half the ribbon’s width, which on a violin is five millimetres. A figure in this ladder that draws β = 0.01 is drawing a bow inside the bridge.

Two limits, one instrument, and they arrive together #

How much bow is allowed is this ladder’s second rung and it gives the other reason ponticello is hard: Schelleng’s two bounds go as one over β and one over β squared, so the window between them closes in proportion to β. Near the bridge there is barely any force that works.

That is a statement about force and this one is a statement about geometry, and they are independent. So which stops the player first?

On a violin the force window narrows to a factor of three at 9.8 millimetres from the bridge. The ribbon takes half the excursion at 10.0. They arrive at the same place, to within a fifth of a millimetre.

That coincidence is worth naming as a coincidence. The window’s threshold of three is a convention, and the constant inside Schelleng’s minimum force is asserted rather than measured. Move either and the two numbers separate. What is not a convention is the ordering across the family, because the ribbon fraction changes by a factor of two between a violin and a bass and the window’s dependence does not.

Which is why the violin is the awkward one #

A bow’s hair ribbon is between ten and fifteen millimetres across the whole family. It is as wide as it can be made and still be controlled by a hand, and a hand is the same size on all four instruments. A speaking length is set by the pitch and runs from 325 millimetres to 1,050.

So the ribbon is 3.1 per cent of a violin’s string, 2.8 of a viola’s, 1.7 of a cello’s and 1.4 of a double bass’s — a factor of 2.2, entirely because the bows did not scale and the strings did.

Run the two limits across the family and the ordering flips exactly once. On the viola the window binds at 11.7 millimetres and the ribbon at 11.0; on the cello, 20.7 against 12.0; on the bass, 31.5 against 15.0. On the violin alone the ribbon binds first.

The violin is the only bowed instrument on which the bow’s own width has caught up with the physics of the bowing point. Everything larger is limited by force alone and has room to spare.

That is a prediction about playing and it is one players state without the arithmetic. Cello ponticello is a usable colour with a recognisable quality, asked for constantly and reasonably reliable. Violin ponticello is fragile, unstable and famously unrepeatable between instruments and players, and orchestral writing treats it as an effect rather than as a register. The model says the cello has twice the room.

What an extreme ponticello actually is #

Putting the two together says something about the sound rather than about the difficulty.

Below the bow’s own crossing there is no regime in which a sharply-defined corner arrives at a bow, triggers a slip, and departs. The corner is under the hair for most of its bridge-side life, and the force available to hold it is inside a window a factor of two or three wide. What comes out is the multiple-slipping and surface noise the second rung describes as the below-minimum-force behaviour — but arrived at by a route that has nothing to do with force.

There is a third thing the geometry does and it is the least obvious. With the corner under the hair for most of its bridge-side trip, the timing of the release stops being set by the corner’s arrival and starts being set by whatever the hair is doing when the corner is already there. A Helmholtz cycle is a clock triggered by an event, and near the bridge the event has been spread over most of the interval it is supposed to be timing.

So extreme sul ponticello has two independent causes and they produce the same sound, which is why it has never needed disentangling: nobody plays there deliberately enough to care which one they are hearing. The prediction that separates them is that 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.

And the left hand moves the ribbon fraction too #

One more consequence, which is the sixth rung’s own variable seen from here.

A player holds the bow at a fixed distance from the bridge, and the speaking length shortens as the left hand goes up the string. So β rises up the fingerboard — that is the sixth rung’s whole subject — and the ribbon fraction rises with it, because the ribbon’s width is fixed and the string is getting shorter.

At the top of the E string in fourth position the speaking length is about 130 millimetres, so a ten-millimetre ribbon is 7.7 per cent of the string. The occupancy limit that sits at ten millimetres from the bridge on an open string sits at twenty-five up there.

That is a real constraint on a real passage: a violinist playing high on the E string has less room between the bridge and the ribbon’s limit than the same player on an open G, in absolute millimetres as well as in fractions. It is also, as far as this collection can tell, not something anybody has written down.

What a player does about it, which the model half explains #

There is a standard technique for playing close to the bridge and it is to tilt the bow, rolling the stick so that only the outer edge of the hair contacts the string. Every teacher gives it, usually as a way of controlling the weight, and the arithmetic above gives it a second job: tilting narrows the ribbon.

A bow tilted to contact over half its width has a ribbon fraction of 1.5 per cent rather than 3.1, and its occupancy limit moves from ten millimetres to five — which is where the geometric floor is. So a tilted violin bow has the cello’s margin: the force window closes at 9.8 millimetres and the ribbon does not bind until five, so the player is back to fighting one limit instead of two.

That is the model’s clearest practical statement and it should be read carefully, because the same tilt does two things at once. Less hair on the string also means less force can be applied before the hair slips, which moves the upper bound of Schelleng’s window down and narrows it. So tilting buys geometry and spends force, and the model here prices only the first half.

The second half is why the advice is always given with a warning attached — tilt too far and the note stops speaking at all, which is exactly a minimum-force failure.

Which computation produced the numbers #

The ribbon fraction is the hair width over the speaking length: 10 millimetres on 325 for a violin, 11 on 390 for a viola, 12 on 690 for a cello, 15 on 1,050 for a double bass. Those are ordinary modern dimensions and none of them is measured here.

The spectrum is the mode shape integrated over the contact — sin(nπp) averaged over p from the near edge to the far edge, over 41 samples — divided by n for the bowed envelope. Averaging the amplitudes rather than the spectra is the right operation because the hair is one rigid object moving at one speed, so every point under it drives the string in phase.

The occupancy is the ribbon fraction over twice β, which is the ribbon’s width as a fraction of the corner’s bridge-side excursion. The floor is half the ribbon fraction.

The force window is schelleng’s, unchanged from the second rung: the maximum force as one over β and the minimum as one over β squared, so the ratio is proportional to β with an asserted constant in it.

Where the model stops #

The bow is flat and a bow is not. Hair is tensioned across a curved stick and a player tilts it, so the contact is narrower when the bow is tilted and full when it is flat — which is one of the first things taught and is a continuous control over the very quantity this essay is about. Tilting the bow is how a violinist plays close to the bridge, and the model has no tilt in it.

The contact is not uniform. The hairs at the edges of the ribbon carry less load than those in the middle, so the real excitation is weighted rather than flat. That makes the spectral null even more null and does not touch the geometry.

There is one corner. Real Helmholtz motion has a corner that is already rounded by the time it comes back, and near the bridge the rounding and the ribbon are comparable widths. Which of the two smears the arrival more is a calculation this ladder could do and has not.

And the floor is a rigid bow on a flat string. Five millimetres is where the hair meets the bridge on a straight instrument with a flat bridge, and a real bridge is curved and a real string sits above it. The number should be read as “about half a ribbon” rather than as a measurement.

What the picture cannot show #

It cannot show the sound of ponticello. Everything here is about whether the mechanism holds, and the characteristic glassy multiphonic sound is what happens when it does not — which is a regime this ladder’s model is defined by not being in.

Nor can it show the rosin. The friction law is a constant here and it is the thing a player changes when a passage will not speak. Whether more rosin helps below the ribbon’s limit is the prediction above and it needs a friction model this collection does not have.

It cannot show the bow’s own vibration. A stick 700 millimetres long has modes in the low hundreds of hertz, and near the bridge the forces are large and the room for error small. That is an entirely separate account of why ponticello is unstable and nothing here excludes it.

Whose instruments, and when #

The dimensions are modern: a Tourte-pattern bow with a ribbon of about ten millimetres, and a violin of the standard 325-millimetre stop, both nineteenth-century arrivals.

The baroque bow is the interesting case and it points the same way. Its ribbon is narrower — commonly seven or eight millimetres rather than ten — and its stick is lighter, and the instrument’s speaking length is the same. So a baroque violin’s ribbon is about 2.3 per cent of its string rather than 3.1, and by the arithmetic above its occupancy limit sits at seven and a half millimetres rather than ten, inside where the force window closes.

On a baroque setup the violin behaves like the modern viola: the force limit arrives first and the ribbon has room to spare. Whether that is audible in how the two instruments behave near the bridge is a question for somebody with both, and it is the cleanest prediction this rung makes.

It also puts a date on the problem. Sul ponticello appears as a written instruction in the seventeenth century and is used sparingly through the eighteenth; it becomes a standard orchestral colour in the nineteenth, which is precisely when the ribbon got wider. That is a correlation with two centuries in it and no mechanism connecting them, and it is stated here only so that the temptation to connect them is visible.

Where this ladder goes next #

Eight rungs. The bow makes a corner; the corner sustains inside a 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; and now the bow that sits at that place has a width, which does nothing to the spectrum and sets a floor under the whole travel.

What is owed after this is the other end of the hair. Everything above treats the bow as a contact and the string as the object, and the bow is a tensioned ribbon on a curved stick with modes of its own in the range the instrument plays. This collection has the machinery for a driven string terminated in something with an impedance — the bridge and the body are exactly that, at the other end — and it has never asked what the bow’s impedance does at the point of contact. A player pressing harder is loading the string with a stick, and near the bridge, where the string’s own impedance is highest, that is the place where the two are most nearly comparable.