The hair runs the wrong way

The bow is not a point either gave the bow a width and found that the width does nothing to the spectrum and everything to how near the bridge a player can go. Its last paragraph names what is still owed, and it is a change of subject rather than a refinement:

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.

The machinery is the body’s, where the string’s own characteristic impedance is set against the bridge’s admittance and the minimum bow force rises where the bridge lets the string shed most. Pointing that machinery at the other end of the string is a matter of finding one number for the bow.

There is no one number. There are three, they differ by more than a factor of twenty, and which of them matters is decided by a piece of geometry that nobody in this ladder had written down.

The geometry that decides which impedance counts #

A bow is drawn across the string. The hair ribbon therefore runs perpendicular to the string in the horizontal plane, from frog to tip, and the bow’s motion is along the ribbon’s own length. Friction acts along the direction of sliding, so the force the bow puts on the string is along that same axis, and so is the string’s response: Helmholtz motion is polarised in the bowing plane, which means the string moves back and forth along the hair.

That single sentence rearranges the whole question. A tensioned ribbon has a large transverse impedance and a much larger longitudinal one, and the string’s motion is in the second direction, not the first.

Take the ribbon as ordinary modern hair: about 175 strands of horsehair 0.18 millimetres across, 0.65 metres of it between the frog and the tip, tightened to about 60 newtons. That is 4.45 square millimetres of keratin, 5.79 grams per metre, and 3.8 grams of hair in the whole ribbon.

Across its own length the ribbon’s characteristic impedance is the root of tension times mass per length: 0.589 kilograms per second. A violin’s strings run from 0.187 on the E to 0.338 on the G. So the transverse number is between 1.75 and 3.16 times the string’s — the same kind of object, within a factor of three, which is exactly what the eighth rung’s guess was about. It is also, by the paragraph above, at right angles to the motion it would have to load.

Along its length the ribbon’s impedance is its cross-section times the root of Young’s modulus times density. Horsehair is a keratin fibre and published values for its modulus run from about 3 to 8 gigapascals; at 5, the longitudinal impedance is 11.35 kilograms per second, nineteen times the transverse one and between 34 and 61 times the string’s. Every number below that depends on the modulus does so under a square root, so a factor of two in it is a factor of 1.4 in the answer and no conclusion here turns on that.

Which makes the classical assumption right, and says by how much #

A point load of impedance $Z$ sitting on a string presents itself against the $2Z_c$ the two half-strings offer together, so an arriving wave is reflected in the ratio $Z/(2Z_c + Z)$ and transmitted in the ratio $2Z_c/(2Z_c + Z)$.

With the longitudinal number that is a reflection of 94.4 per cent on the G string and 96.8 on the E. Every rung of this ladder from the first onward has drawn the bowing point as a place where the string is held — during the stick phase the contact moves at the bow speed, which is a fixed point as far as any partial is concerned — and that assumption is now priced. It is right to within 5.6 per cent at the bottom of the instrument and 3.2 at the top, and it is right because of a stiffness nobody would have thought to compare, since the impedance anybody would have reached for is the transverse one and the transverse one is not in the path.

The order of the two matters more than either. Had the ribbon’s stiff direction been the transverse one, the bow would be reflecting only 46 per cent of the corner and the fixed-point model would have been wrong from the ladder’s first rung.

Except that the ribbon is not infinite #

A characteristic impedance is what a line presents before its own reflections come back. The hair ribbon is 0.65 metres long, clamped at the frog and the tip, and its longitudinal wave speed is 1,961 metres a second — so its own reflections come back after well under a millisecond, and what the string actually sees is the driving-point impedance of a finite object with modes.

Those modes are at 1,509 hertz and its multiples. That is between F♯6 and G6, which is squarely inside the range every one of these instruments plays, and it is the eighth partial of an open G string.

The picture is not a curve with a correction on it. Below the first hair mode the ribbon is a stiff spring, and the fixed-point assumption is better than the characteristic-impedance estimate rather than worse: 0.6 per cent of the fundamental gets past at the middle of the bow, and 0.25 per cent an eighth of the way along. Up to the sixth partial the leak stays under eight per cent.

At the eighth partial — 1,568 hertz, within four per cent of the ribbon’s first longitudinal mode — the constraint opens. At the middle of the bow 44.6 per cent of that partial goes straight through the contact. A third of the way along it is 32.9 per cent, and an eighth of the way along, where the near clamp is close enough to stiffen everything, it is 7.5.

Then it shuts again. At the ninth partial the leak at the middle is back to 12 per cent and at the tenth to 6.3, because the ribbon is off its resonance and stiff once more.

And the partial that opens is chosen by the player’s arm #

The ribbon’s modes are the modes of a clamped line, so a mode with a node at the contact point cannot be driven from there at all — it presents an effectively infinite impedance and the bow is more rigid than usual. Halfway along the hair the even modes have exactly that node, so the odd ones open and the even ones slam shut. At three tenths the pattern is different, and at an eighth different again.

On the E string the ribbon’s first mode sits at 2.3 times the fundamental, so it opens the second partial by 7.2 per cent at the middle and 5.2 at three tenths — and by the seventh partial the middle of the bow leaks 16.9 while three tenths leaks 2.1. The two positions differ by a factor of eight on one partial of one note.

That is a claim about the sound of a stroke and it is not the usual one. A player moving from the frog to the tip is normally described as changing weight, speed and the angle of the hair — and the sixth rung added the bowing fraction, which moves with the left hand rather than the right. This says that the transparency of the contact changes too, at particular partials, in a pattern set by the ribbon’s own length. Whether it is audible against everything else moving at the same time is a listening question this collection cannot settle, and the prediction that separates it from the others is that it should not depend on the bow force at all.

The damping of the ribbon is the number this collection does not have. Rosined horsehair on a wooden stick is lossy and nobody here has measured it; the figures assume a quality factor of 30. Moving that to 10 lowers the eighth partial’s leak from 44.6 per cent to 26.3 and moving it to 120 raises it to 45.0, so the notch’s depth is soft and its position is not.

What the transverse impedance is in the path of #

The comparable number — the 0.589 that is the same kind of quantity as the string’s — has not disappeared. It has changed jobs. It lies along the direction the bow force is applied in, so it is not between the arm and the string’s motion; it is between the arm and the string’s force, which is the only quantity Schelleng’s two bounds contain.

Below its own first mode at 78 hertz the ribbon is a spring, and its stiffness at a contact a fraction $x$ along is $T/(L\,x(1-x))$ — softest in the middle, firming up toward either clamp.

At half a newton — an ordinary bow force — the hair gives 1.35 millimetres at the middle of the bow, 1.09 at the balance point about 28 per cent along, and 0.38 an eighth of the way from the frog. That is a factor of three and a half within a single stroke, and it is the “give” every player feels and describes as the bow being more solid near the frog. It is not the stick’s flex and it is not the arm; it is the hair’s own tension, and it is the reason tightening the screw changes the feel before it changes anything else.

Which turns the bow into a mass on a spring #

A spring with the bow’s own mass riding on it is an oscillator, and its frequency is where the arithmetic touches something a player does deliberately.

Treating the bow as a free 60-gram mass gives 12.5 bounces a second at the middle of the hair and 23.4 an eighth of the way from the frog. Treating it as a rod pivoted at the frog, so that the effective mass at the contact is its moment of inertia over the square of the distance, gives 6.8 at the balance point and 17.3 three quarters of the way to the tip. The truth is between the two, because a hand is neither a free pivot nor absent, and this collection has no measurement that would choose.

What both models agree on is the shape of the answer. The rate is a function of where along the hair the string is standing, it is between six and thirty a second, and it goes as the square root of the hair tension. All three of those are things a string player states as technique: there is a place on the bow where it bounces naturally, moving away from that place changes how fast, and tightening the bow makes it faster and higher. None of them needs a name for the mechanism to be used, and the mechanism is a ribbon of hair 0.65 metres long under 60 newtons.

The four bows are more alike than the four instruments #

Running the same arithmetic on the viola, cello and bass bows produces the one number in this essay that surprised the calculation rather than confirming it.

The four ribbons’ transverse fundamentals are 78, 76, 79 and 80 hertz — within five per cent of one another — and their bounce rates at the middle of the bow are 12.5, 11.8, 12.2 and 10.8 a second. The strings underneath them run from a violin’s G at 196 hertz to a double bass’s E at 41, a span of more than two octaves.

So the bows were not scaled to the strings. They were scaled to a hand: a bass bow is shorter and thicker-haired and more tightly strung than a violin bow, and those three changes very nearly cancel in the one quantity that decides how it bounces. The eighth rung found the same thing from the other side, when the ribbon’s width turned out not to scale with the string either, and the excitation ladder found the same about a piano’s hammers, which are sixteen millimetres of felt on a wire whether the wire is half a metre long or two. A bow is an object built to the same human specification four times over, and the instrument it is used on is a separate matter.

And what the force window will take #

The last thing the transverse spring does is set how much the bow force is allowed to wander, which is a question the second rung can answer directly.

Aiming at the geometric centre of the window, the tolerable wander is the square root of the window’s width either way, which as a fraction of the mean force is 63 per cent over the fingerboard, 50 per cent at an ordinary bowing point, and 27 per cent at 9.8 millimetres from the bridge — which is where the window has narrowed to a factor of three and where the ribbon’s own occupancy limit arrives at the same time. Multiply that tolerance by the fingerboard map, where the window is narrowest on the body’s own resonances, and the tightest cell on a violin will take under a fifth.

A bouncing stroke is a hundred per cent modulation by construction, since the hair leaves the string, so it lies entirely outside this curve and is a series of separate notes rather than a wobble inside one. What the curve prices is the involuntary part: an arm’s tremor arrives at the string through a spring of 369 newtons per metre, and near the bridge there is no room left for any of it.

Which computation produced the numbers #

The ribbon’s constants are the two wave impedances of a line: the root of tension times mass per length across it, and cross-sectional area times the root of modulus times density along it. The mass per length is 175 strands of 0.18-millimetre horsehair at 1,300 kilograms per cubic metre. The tension is 60 newtons and the free length 0.65 metres.

The driving-point impedance is the two sides of the contact taken as lines terminated rigidly, each contributing $-jZ_c\cot(k\ell)$, added because both carry the same velocity and share the force. Below the first mode that sum collapses to the static spring, and the figures check it against that spring rather than trusting it.

The leak is the transmission of a point load: $2Z_c/(2Z_c+Z)$, with the string’s impedance the same $\sqrt{T\mu}$ used five rungs ago and the load complex.

The bounce is that spring against the bow’s mass, twice, under the two inertia assumptions named above; the timing of a stroke that begins from rest is the third rung’s and is untouched by any of this. The window and its tolerance are Schelleng’s, unchanged.

Where the model stops #

The friction contact is not a bond. Everything above treats the hair as coupled to the string, which is true during the stick phase and not during the slip: while the string is sliding the force is set by the normal load and the kinetic coefficient, and the ribbon’s impedance drops out of the coupling entirely. Helmholtz motion sticks for most of each period, so the numbers stand for most of it, but the slip is exactly the moment the corner arrives, and this account cannot say what happens in the microseconds either side of a release.

The modulus is borrowed. Nothing here measures horsehair, and the range in the literature is nearly a factor of three. Both hair impedances go as its square root and the mode frequencies do too, so the 1,509 hertz should be read as “somewhere between about 1,200 and 1,900”.

The stick is not in it. A bow’s wooden stick has bending modes of its own in the low hundreds of hertz, it is curved, and it is what the hair’s tension is reacted against. Treating the frog and the tip as rigid clamps is the assumption that the stick is stiff at audio rates, and near its own modes it is not.

And the damping is a guess. The ribbon’s quality factor sets how far the notches go and this collection has no measurement of it. Everything about where they are survives; everything about how deep does not.

What the picture cannot show #

It cannot show the two polarisations exchanging. A real bowed string does not stay in the bowing plane — the bridge and the stopping finger couple the two transverse directions — and in the out-of-plane one the hair’s transverse impedance is the load rather than the longitudinal. That is the calculation this rung has set up and not done.

Nor the string’s torsion. A bowed string rolls as well as bends, the rolling is what makes the friction contact behave the way it does, and the ribbon is a completely different object to a torsional wave than to a transverse one.

It cannot show what a listener does with any of it. A partial that passes the bow instead of reflecting from it is a partial withdrawn from the Helmholtz cycle, and whether losing the eighth partial of a G is audible against a spectrum that is already thirty decibels down there is not a question about the bow.

Where this ladder goes next #

Nine 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, which does nothing to the spectrum and sets a floor under the travel; and now the bow has an impedance, which does nothing to the string’s motion except at one partial, and does all its work on the force instead.

What the ladder owes now is the constant that has been sitting in the middle of every one of those rungs. Schelleng’s window is a hundred times the friction contrast times the bowing fraction, and this ladder has drawn the bowing fraction thirty-nine times and the friction contrast exactly once, at one. That number is the rosin — the thing a player reaches for when a passage will not speak, and the one variable in the whole model that is changed deliberately between one rehearsal and the next. Every limit this ladder has priced in millimetres moves as one over it, and none of the geometric limits move at all, so sweeping it is a test of the eighth rung’s own prediction: that everything a player has for the force problem should stop helping below the ribbon’s limit and should help above it.