The hardest place is also the latest

The hardest place on the fingerboard multiplied two curves that had never been multiplied — the body’s admittance, which is a function of frequency, and the bowing fraction, which is a function of position — and produced a map of the violin with a worst place on it. Its last paragraph named what was missing:

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.

That is right, and the change of units is not the trivial one it sounds like.

Where the time comes from #

Both bounds of Schelleng’s window are proportional to the bow’s speed. That is the third rung’s whole finding: at the moment a note has to start, the bow is barely moving, so the window it has to be inside is nearly zero and centred nearly at zero, and it opens out as a wedge from the origin as the bow accelerates.

A player chooses one force. That force sits somewhere in the window the note will eventually occupy — say at its geometric centre, which is where a player who wants to be equally far from a scratch and a crunch would aim. The bow’s upper bound rises through that force at a definite moment, and that moment is when the note can first catch.

The upper bound at full speed is Fmax. At a fraction v of full speed it is v·Fmax. So the catch happens when

v = F / F_max = √(F_min · F_max) / F_max = 1 / √W

where W is the window’s width. If the bow reaches its playing speed in a ramp of T seconds, the note catches at T/√W.

The attack time goes as one over the square root of the window width. That is the change of units, and the exponent is the part that could not have been guessed.

What a half does to the difficulty #

The exponent is what makes this rung a rung rather than a relabelling.

Across the violin’s playable range the window varies by a factor of twenty-nine: from 13 at C♯4 on the G string, which the sixth rung named the hardest place, to 383 at C6 on the A string. That is an enormous difference and it is why the map is worth drawing.

The attack times vary by a factor of five and a third: from 22 milliseconds down to 4. The square root has compressed the difficulty by more than five to one.

So the hardest place on the fingerboard is also the latest place, necessarily — the transformation is monotone and cannot reorder anything — but it is much less late than it is hard. A player at C♯4 on the G string is fighting a window a twenty-ninth the width of the one at the top of the A string, and paying about eighteen milliseconds for it.

That is a small number in one sense and not in another. Eighteen milliseconds is well above what an ensemble notices: a note is heard after it starts puts the just-noticeable asynchrony for a listener at around twenty milliseconds and the spread across a mixed scoring at rather more than that. A violinist crossing from the top of the A string to the bottom of the G is crossing a real timing boundary, and it is one they correct for without being told.

The awkward place is one note wide #

Reading the map note by note, with the player free to choose which string to play each pitch on, gives a curve with a very sharp feature in it.

From G3 up to B3 the best available attack time sits between 13.7 and 15.2 milliseconds. At C4 it rises to 17.8 and at C♯4 to 21.9 — the peak. At D4 it drops back to 15.4 and at E4 to 11.9, and from there it falls smoothly to about 4 milliseconds at the top of the range.

The peak is one note wide because of where the strings run out. C♯4 is the highest note this model allows on the G string within a fifteen-semitone reach, and the D string has not yet started — its open note is D4. So C♯4 is the one pitch on the instrument with no alternative: it must be played high on the G string, where the sounding length is short, the bowing fraction is large, and the window is at its narrowest.

Every other note in that region has a choice, and the choice is worth several milliseconds. That is a design fact about a four-string instrument tuned in fifths and it is visible in the timing rather than in the tone.

Drawn in milliseconds the map keeps its shape: the same cell is worst. What the change of units adds is how much worse — the difference between the easiest and hardest places is a factor a player experiences as a delay rather than as a risk, which is why the near-bridge register feels sluggish as well as precarious.

What the ramp decides, and what it does not #

One number in this calculation belongs to the player rather than to the instrument: how long the bow takes to reach its playing speed. Eighty milliseconds is a normal détaché and it is asserted rather than measured.

It scales every attack time in the map by exactly the same factor. A player with a forty-millisecond ramp — a sharper attack, more of the bow’s weight arriving at once — halves every number here and changes no ordering, no ratio and no shape. The map is a map of the instrument, and the ramp is a knob the player turns that moves the whole thing.

That is a useful thing for the model to be. It means the twenty-nine-to-one window range and the five-to-one time range are properties of the violin, and the absolute milliseconds are properties of a performance — so a player who wants a fast attack in a hard place buys it by accelerating harder, and the figure says exactly how much harder: a factor of 5.3 on the ramp to make the hardest place as quick as the easiest.

Whether that is available is a different question, and it is the one the third rung already answered. Accelerating harder moves the force window up, so a player who ramps five times faster must press five times harder at the same instant or fall out of the bottom of the wedge. The trade is not free and the map does not say it is.

The same law on the other family #

The half-power law is a statement about Schelleng’s window, so it says nothing about a wind instrument — and comparing the two families is what the first rung of this ladder was for.

A wind instrument’s wait is a resonator filling up, and it goes as Q/f. A bowed string’s is a bow catching, and it goes as T/√W. Those are different functions of different quantities, and the only thing they have in common is that both come out in milliseconds.

The numbers are not close. The violin’s range is 4 to 22 milliseconds; a trumpet’s is 20 to 153 and a horn’s 54 to 161. The whole of the violin’s fingerboard is faster than the whole of a horn’s compass, and the fastest violin note is as quick as the quickest note on a trumpet.

That is a real difference and it has nothing to do with which instrument is harder to play. It is that a bowed string does not have to fill up: the amplitude of Helmholtz motion is set by the bow’s speed, and the bow has reached its speed by the time the force is right. There is no reservoir to charge.

It also explains a thing about orchestral scoring that is usually stated the other way round. A string section can articulate faster than a brass section not because strings are more agile but because the mechanism has no settling time in it at all — and the passages that expose the difference are the ones where both are asked for the same short note at the same moment.

Where the aim goes #

The other free choice is where in the window the player puts the force, and unlike the ramp it does change the shape.

Aiming at the geometric centre is the neutral choice and is what the figures use. Aiming low — a fraction a of the way up the window in log terms, with a below a half — makes the catch earlier, because a lower force is reached sooner by the rising upper bound. Aiming high makes it later.

The dependence is W^(a−1), so at a = 0 the attack time is independent of the window entirely and at a = 1 it goes as W itself. The half is the point at which the exponent is a half, and it is the centre.

So a player who wants a hard note to speak on time has a second option besides bowing harder: aim low, and accept that a low aim is close to the bottom of the window, where the corner fails to trigger on every pass and the tone is a surface noise. That is precisely the trade every teacher describes for a difficult start, and it comes out of the geometry rather than out of the pedagogy.

A number a listener has #

The reason the sixth rung wanted this change of units is that a ratio of forces is not a thing anybody perceives, and a millisecond is.

This collection has two independent accounts of what a listener does with a few tens of milliseconds at the start of a note, and they say different things. The first fifty milliseconds is about identity: the attack carries most of what tells one instrument from another, and a transient that varies by eighteen milliseconds across an instrument is a transient that varies in character. A note is heard after it starts is about placement: a listener puts a note somewhere after its physical onset, and the delay depends on how the note rises.

The second is the one that binds here. A violinist playing the same rhythm on the G string and on the E string is producing onsets four milliseconds apart at the top and eighteen at the bottom of the range, and the required leads for a mixed ensemble are of exactly that size. So the map is not a curiosity about difficulty; it is a term in the same account that decides who in an orchestra plays early.

The two quantities are one quantity. A narrow window is a slow start, exactly and by a power law, because the bow has to accelerate until its force reaches a value the window’s own width determines. The hardest place to play and the latest place to speak are not correlated — they are the same measurement in two units.

Which computation produced the numbers #

The map is the sixth rung’s fingerboardMap unchanged: every written pitch from G3 to E6, on every string that can reach it within fifteen semitones, with the bowing fraction from a fixed 35-millimetre bow-bridge distance against the sounding length, the string’s own characteristic impedance from its tension and its fundamental, and the body’s admittance from the violin resonance model, normalised to its own geometric mean over the range.

The window is Fmax/Fmin with Fmax going as the impedance and Fmin as its square, which is Schelleng’s own scaling and is what the fourth and sixth rungs used.

The time added here is T·W^(a−1) with T the ramp and a the aim, which is three lines and no new physics. The forces are in arbitrary units throughout, as they have been since the second rung, because only the ratio carries the argument.

Where the model stops #

Schelleng’s window is a steady-state criterion. It says whether Helmholtz motion can be sustained at a given force and speed, and using it at a moment during the transient is exactly what the third rung called a weaker and derivable thing than Guettler’s measured diagram of attack quality. Nothing here is Guettler’s result and nothing here contradicts it.

The window is a ratio and this is a time, so every asserted constant in the sixth rung’s map is still asserted here. In particular the body’s admittance enters Fmin and not Fmax, which is the modelling choice the fourth rung made and which this rung inherits without re-examining.

A catch is not a note. The moment the force enters the window is the moment periodic motion becomes possible, and a real attack has a few periods of irregular slipping before the corner settles into its round trip. The third rung counts those periods; this rung counts the wait before they can begin. The two add.

The body is a mean. The admittance curve is an average over instruments, and a particular violin’s wolf and its own resonances would move the map by more than the string choice does at the notes where they sit. The note the body will not let start is the rung about that, and it is a reminder that one instrument’s map is not the family’s.

And the reach is a convention. Fifteen semitones is roughly fourth position, and a player who goes higher on a lower string finds windows narrower than anything on this map. The peak at C♯4 is a peak because the model stops the G string there.

What the picture cannot show #

It cannot show what a listener does with eighteen milliseconds. A late note in an ensemble is heard as late; a late note in a solo line is heard as nothing at all, because there is nothing to be late against. Whether the map is audible depends entirely on the texture, and this collection’s account of that is on the perceptual-centre ladder rather than here.

Nor can it show the player’s correction. Every one of these numbers is what happens if the player does the same thing everywhere. No player does — the whole business of learning an instrument is learning where it is slow and starting earlier there — so the map is a description of what has to be corrected rather than of what is heard.

It cannot show the string crossing. The one-note peak at C♯4 assumes a player takes each note on the best string available, and a passage does not work that way: a phrase across the break is played on the string the phrase needs, and what a tablature keeps is the essay about how much of that a notation records. The map is a map of best cases.

And it cannot show the down-bow. Everything here treats the ramp as symmetric, and a down-bow and an up-bow have different weight distributions and different accelerations at the frog and at the tip. That is four maps rather than one, and this collection has no model of the bow’s own mass.

Whose playing, and when #

The instrument is a modern violin: 325-millimetre stop, four strings at the tensions a shop fits, a bow held 35 millimetres from the bridge. The 80-millisecond ramp is a normal orchestral détaché rather than a martelé, which is much sharper, or a sul tasto stroke, which is much gentler.

The one historical claim available here is a prediction rather than a finding. A gut-strung baroque violin has a lower characteristic impedance on every string, and the window goes as the reciprocal of the impedance, so a baroque setup has wider windows everywhere and therefore earlier catches. Players of both instruments describe the gut-strung one as speaking more readily and the modern one as more even, and the arithmetic points that way — but the body, the bow and the bridge all differ too, and this model changes only the strings.

Where this ladder goes next #

Three rungs. A wait is a Q in one unit and a Q over a frequency in another; up a brass instrument the two units disagree monotonically; and on a bowed string the wait is a capture whose length is one over the root of a window width, so the hardest place is the latest place by a factor of five rather than of twenty-nine.

What the anchor owes now is the listener. Everything in it is a duration measured at the instrument, and a duration only matters if somebody can hear it — and this collection has a threshold for exactly that, on a different ladder: a note is heard after it starts measures the delay between a note beginning and a listener placing it, from the shape of the onset rather than from its length. The two accounts are about the same event and use different inputs, and running one against the other would say whether an instrument that settles slowly is one that is heard late, or whether the two are different quantities that have been sharing a word.