The hardest place on the fingerboard

Every violinist believes there is a hardest place on the instrument. The belief is specific — a particular note, high on a particular string — and it is the kind of thing that gets passed on in lessons and never written down as a number.

This ladder has both halves of the arithmetic and has never multiplied them. The note the body will not let start put the bridge into Schelleng’s minimum bow force: a string sheds energy fastest where the body’s admittance is highest, so a note sitting on a body resonance needs more bow before it will lock into Helmholtz motion, and the window between the least and the most bow it will take is narrower there. Two dials the player turns together put the left hand into the same window from the other side: the bow stays a fixed distance from the bridge while the sounding length shortens, so the bowing fraction rises as the hand goes up, and each string has its own characteristic impedance.

One is a function of frequency and the other is a function of position. Neither figure could contain the other, so neither did.

Why that cell and not another #

The map has a minimum in one place and the two curves put it there together. Neither of them puts it there alone.

C♯ above middle C is 277 hertz. The violin’s lowest body resonance — the A0 air mode, the Helmholtz resonance of the box breathing through its f-holes — is at about 275 hertz on the instrument this collection models. So that note sits almost exactly on the resonance where the bridge is most compliant, and the minimum bow force is at its highest.

The G string is the heaviest of the four and has the largest characteristic impedance, and Schelleng’s two bounds carry the impedance to different powers — the maximum force goes as Z_c and the minimum as Z_c², so the window goes as one over Z_c. The heaviest string therefore has the narrowest window at every position, which is what the sixth rung found and is why the G string sounds the way it does.

And the sixth position on the G string is short: the sounding length is about two thirds of the open string, so a bow held three and a half centimetres from the bridge is at a bowing fraction half as far up the string again as it was at the nut. A larger bowing fraction narrows the window from the other end.

Separating the two curves says how much each is worth and how far they actually compound. Over the seventy-six playable cells:

The body is the larger term by more than a factor of two, which the phrase three effects, all mild on their own does not convey. And they compound only partially: 39 against a possible 69, which is 57 per cent, because the geometry’s worst place and the body’s worst place are not the same place. The geometry is worst at the open G — longest string, heaviest wire, smallest bowing fraction — and the body is worst at 275 hertz, six semitones above it.

So the finding is sharper than the hardest place is not where any single curve has its minimum. It is that the map’s minimum sits where the larger curve’s minimum is, displaced a little by the smaller one, and the two agree closely enough for the product to reach more than half its theoretical worst.

And there is a reason they meet on that string in particular. C♯ above middle C is 277 hertz and the open D string is 294 — so C♯ is a semitone below the D string’s lowest note and cannot be played on it at all. The body’s worst frequency is therefore available on exactly one string, and it happens to be the heaviest one, which is the string the geometry already penalises most. A body resonance twenty hertz higher would be reachable on two strings and a player would have a choice; this one has none.

That is a coincidence of this instrument’s dimensions rather than a design, and it is the kind that would move on any other violin. It is also the reason the string-choice figure below finds so little to offer at the hard cell: there is no second string to compare.

What the choice of string is actually worth #

A player who meets a difficult note has one obvious remedy, which is to play it somewhere else. The map says how much that is worth, and the answer is smaller than the reputation of the manoeuvre suggests.

For every note playable on more than one string, the window on the best string over the window on the worst never exceeds about 1.7. Moving up the fingerboard, by contrast, moves the window by a factor of thirty-nine.

The reason the two numbers are so far apart is worth saying, because it is not obvious that they should be. A string change at a fixed pitch changes the impedance and the bowing fraction and leaves the body’s admittance exactly where it was — the note is the same note, so the bridge is doing the same thing. So a string change can only move the geometry term, which spreads by 5.4 across the whole instrument and by very much less between two strings at one pitch. The larger of the two curves is the one a fingering decision cannot touch. So the string choice is a decision that changes the window by tens of per cent while the position changes it by an order of magnitude and a half.

That is not an argument that fingering does not matter. It is an argument that fingering matters for a different reason: the two strings are different colours — different impedances, different sets of body resonances excited, different bowing fractions and therefore different spectra, which is what the same note is a different width is about. What a fingering decision mostly is not is a decision about whether the note will start.

Move the body and the dark cell moves with it #

The strongest evidence that the map is a consequence of the two curves rather than a fact about C♯ is to move one of the curves and watch the minimum follow.

Scaling the whole body resonance list puts the air mode somewhere else, which is the fourth rung’s crude model of another instrument in the family. Move it up by a little over a quarter and the dark cell leaves the G string’s sixth position for F above middle C on the D string, third position — because the air mode has climbed to about 350 hertz and 349 is where F sits. Nothing else about the map changes: the general fall of the window as the hand goes up is a property of the geometry, and the dark patch riding on it is a property of the box.

Move the resonances the other way, to six tenths of where they were, and the hardest cell stays at C♯ on the G string and gets worse — a window of six rather than twelve — because the moved corpus modes have arrived where the air mode used to be. Two different bodies, two different answers, one of which is the same note for a different reason.

That is the difference between a method and a fact, and it is worth being explicit about which this is. The map is a method. Applied to this collection’s four-resonance model of a modern violin it names C♯ on the G string; applied to a different admittance curve it will name something else, and applied to a measured admittance curve for a named instrument it would name that instrument’s own worst note — which is a thing a maker could check in an afternoon and this collection cannot.

The window, and the number of periods #

There is a translation waiting here that this rung does not make and the next one should, and it is worth naming because it changes what the map is a map of.

A window is a ratio of forces, which is not a quantity anybody has direct access to. What a player feels is a delay: a note that starts late, or that speaks after a scrape. The note has to start somewhere is the rung about that delay, and what it found needs stating more carefully than the window is somewhere else during the attack. Both of Schelleng’s bounds are proportional to bow speed, so during the attack the window is at a much lower absolute force and is exactly the same width as a ratio — a factor of twelve is twelve at every speed, including the speed the bow has one period in.

That makes the map’s numbers transfer to the attack unchanged, which is the useful half, and it makes the map’s units the wrong ones, which is the half the next rung owes. Thirty-nine to one in a ratio of forces says how forgiving a cell is; what a player wants is how many periods a note takes to catch, and the conversion between them needs the bow’s acceleration, which nothing here has.

So the map above is a map of the sustain, and every cell in it has a companion the third rung drew: a wedge in the plane of bow acceleration against force, through which a bow stroke is a trajectory rather than a point. A narrow sustain window is a narrow wedge, and a bow whose force is fixed while its speed climbs crosses such a wedge quickly and leaves it.

Which computation produced the numbers #

Schelleng’s diagram is the model, and it is named because everything here rests on it: the bow force that will sustain Helmholtz motion lies between a minimum, below which the corner does not get released cleanly, and a maximum, above which the bow drags the string through more than one slip per cycle. The window is the ratio of the two.

The maximum goes as the characteristic impedance over the bowing fraction; the minimum goes as the impedance squared over the fraction squared. The string impedance is the site’s own stringImpedance, from each string’s tension and its open frequency, and the four strings are the collection’s own VIOLIN_STRINGS.

The body enters through the minimum force, scaled by the body’s admittance at the note’s frequency relative to that body’s geometric mean admittance over the range. That reference is the same one the fourth rung uses, and it exists so that the scaling says “relative to this instrument’s own average” rather than smuggling in an absolute bow force in newtons, which nothing here has.

The bow distance is three and a half centimetres from the bridge, held fixed as a player holds it, and the reach is nineteen semitones — a twelfth, which is roughly fifth position on the G string and rather less than a violinist actually uses on the E. Every position from the nut to that reach is included.

The body is VIOLIN_BODY, which is four resonances — the A0 air mode at 275 hertz, the two corpus modes at 460 and 540, and the bridge hill at 2,500 — with published centre frequencies and Qs. It is a mean over instruments rather than a measurement of one, and the desc on every figure that uses it says so.

Where the model stops #

Schelleng’s diagram is a steady-state model of a transient. Its two bounds are conditions for Helmholtz motion to be sustained, and what a player experiences as a note being hard is mostly about the attack. The note has to start somewhere computed how many periods a start takes, and joining that number to this map would say how hard a note is in milliseconds rather than in ratios.

The body model is four resonances and a real violin has dozens. Between about 600 and 2,000 hertz a measured admittance curve is a forest, and this model is smooth there. The map’s fine structure in the middle of the range should not be read.

The bow is at one distance and the player is not. A violinist meeting a narrow window moves the bow — toward the bridge for more force, away for less — and that changes the bowing fraction, which is the very variable the map holds fixed. What the map shows is the difficulty of a note at one bow position, which is the difficulty a player is solving rather than the difficulty they end up with.

And there is no vibrato. A note held on a body resonance with a vibrato of fifty cents is spending most of its time off the resonance, which is a real and possibly deliberate escape from the worst cell on the map.

What the picture cannot show #

It cannot show the wolf. The wolf note is a different phenomenon from a narrow window — it is a genuine coupling between the string and a strong body mode, with the two exchanging energy and neither settling, and the other wolf is the essay about it. A wolf lives near the strongest corpus resonance rather than the air mode, and a map of Schelleng windows cannot produce one because the model has no coupling in it.

Nor can it show the player. Every cell is a property of an instrument. Whether a note is hard is a property of an instrument and a person, and the whole of technique is the business of turning the second into a compensation for the first.

And it is one instrument. Scaling the body list moves every resonance and therefore moves the dark cell — which is the fourth rung’s own finding, that a different fiddle makes different notes hard. The map is a method rather than a fact about violins.

Whose instruments, and whose repertoire #

The four strings, the scale length and the body resonances are a modern violin’s. A baroque instrument with gut strings has lower tensions and therefore lower impedances, a shorter neck and a different bass bar, and its A0 sits lower; the map would have its dark cell somewhere else, and the whole exercise would have to be redone rather than transposed.

The repertoire claim is narrower and is about where composers put things. The region around C♯ and D above middle C on the G string is exactly the register orchestration treatises describe as the violin’s most intense — Berlioz’s “a most beautiful and passionate quality” for the G string in that reach. A narrow bow-force window is a note that will not play itself: it demands a firm, committed bow, and a firm committed bow on a heavy string is what that quality is. The hardest place on the instrument and its most characteristic sound are the same place, and the map says why they have to be.

Where this ladder goes next #

Seven 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; and now the two variables that move independently have been multiplied and the instrument has a worst place.

The rung after it is the one the steady-state limitation names, and every term for it is already here. Schelleng’s window says whether a note can be sustained; the third rung says how many periods a start takes, as a function of how far into the window the bow is. A note near the edge of a narrow window is a note started slowly, so the map above is a map of attack times under a change of units — and an attack time is a quantity a player and a listener both have direct access to, where a ratio of forces is not.