Instruments and their design

Two dials the player turns together

Schelleng's window is a ratio of two bow forces, and an earlier essay found that the string's own impedance is in it twice. What it held still was the bowing point — as though a player kept β constant while moving up the fingerboard, which is the opposite of what a bow does. Let the bow stay where a bow stays and the window widens up every string, the spread between two strings at one pitch goes from 1.4 to 1.5 the other way round, and the string that is easiest changes.

Assumes: The same note is a different width · How much bow is allowed

Schelleng’s window is the ratio of the largest bow force a note tolerates to the smallest — above the maximum the string is crushed, below the minimum the corner does not sustain, and between them is Helmholtz motion and a note. Getting into that window from rest is its own rung and takes a computable number of periods. How much bow is allowed is the essay about it, and it names both variables:

F_max ∝ Z_c / β F_min ∝ Z_c² / β²

with β the bowing point as a fraction of the sounding length and Z_c the string’s characteristic impedance. The fifth rung put the four strings’ different impedances into that and found the window is a different width on each. Its last paragraph named what it had left alone:

The bowing point enters both bounds with the same asymmetry the impedance does, so β and Zc are two dials on one product — and a player moving up the G string is turning both at once, in opposite directions, because a higher position is nearer the bridge.

Fig. 1 Schelleng’s window for every written pitch on every string that can play it, with the bow 35 millimetres from the bridge and staying there. Each solid line is one string; each dashed line is what was drawn earlier, which held β at 0.09 for every note. The window widens up each string, which the fixed-β figure could not show, and the widening is enough to change which string is the easy one.

What a bow actually holds still

The variable in Schelleng’s expressions is β, the bowing point as a fraction of the sounding length. That is the right variable — a corner has to be launched at a fraction of the way along to work at all — and it is not a quantity a player controls.

What a player controls is a distance. The bow sits somewhere between the bridge and the fingerboard, at a place a player chooses for tone and then largely leaves; on a violin it is a few centimetres from the bridge, and the whole vocabulary of bow placement is distances — sul ponticello is near the bridge, sul tasto is over the fingerboard. Nobody thinks in fractions and nothing about the technique suggests the fraction is what is being held.

Meanwhile the left hand shortens the string. A note twelve semitones up the G string is played on half the string, and half a string with the bow in the same place is twice the fraction. So β doubles across each octave the left hand climbs, without the right hand moving at all.

That is the dial the fifth rung held still, and turning it is the whole of this rung.

G5, in every place it can be played. A guitar neck with the 1 places G5 can be stopped, drawn at the fret spacing a 64.8-centimetre scale actually has. The stave writes one note and the tablature writes one of these; each notation says exactly what the other leaves out. The speaking lengths run from 27.2 down to 27.2 centimetres, so a hand plucking 3.5 centimetres from the bridge meets between 13 and 13 per cent of the string. — Fig. 2 The geometry, drawn for one written pitch. The same note is available on more than one string at more than one position, and each position has its own sounding length. A fixed bow distance divided by each of those lengths is a different β — so a note on two strings is a note at two bowing fractions as well as at two impedances, and there is no way to change one without changing the other.

The window widens as the hand climbs

Both bounds carry β and they carry it to different powers, so the ratio — the window — goes as β over Z_c. Doubling β across an octave therefore doubles the window on that string.

On the G string at the twelfth semitone up, β is 0.215 and the window is 63.8. On the D string at the fifth, β is 0.144 and the window is 59.7. Those are two ways of playing the same written G, and the fixed-β model said the D string’s window was 37.3 against the G string’s 26.7 — a ratio of 1.4 in favour of the D string. With β moving, it is 63.8 against 59.7, a ratio of 1.07 in favour of the G.

The advantage does not merely shrink; it changes sides. At a written E a fourth higher the reversal is complete: the A string at the seventh semitone gives 85.5 and the E string open gives 57.7, a spread of 1.48 the other way from the one the fifth rung computed.

Of the two reversals that is the one to keep, and the limitations below say why: the written-G case is a seven per cent margin that survives only if the bow’s distance from the bridge is held almost exactly constant, while the written-E case survives almost any bowing discipline at all.

The mechanism is not subtle once it is stated. Going up a string means going up in position, which means shortening the sounding length, which means raising β, which widens the window. Going across to the next string means starting again at the bottom, in first position, with the longest sounding length and the smallest β. So a high position on a heavy string and a low position on a light one are two different bargains, and the heavier string’s position advantage can outweigh its impedance disadvantage.

The window is a different width on each string. The width of Schelleng's bow-force window across each string's own playable range, with both terms in it: the body's admittance, which was added earlier, and the string's own characteristic impedance, which every figure had held at one value. The four curves are the same shape displaced vertically, because the impedance is a constant per string — the G string's is 1.81 times the E string's, and the window is one over that. Where the ranges overlap, the same pitch sits on two or three curves at once. — Fig. 3 The impedance half of the trade on its own, which is an earlier finding: four strings of very different mass, tuned a fifth apart, with the window each gives at one fixed bowing fraction. The G string is the narrowest and the E the widest, by a factor of about 1.8 end to end. Everything in this essay is that ordering being partly undone by the other dial.

Which passages are genuinely hard

The useful form of the result is not about a note but about a passage, and the distinction the figures make available is between hard and dark.

A player asked why a phrase on the G string is difficult will often say it is because the G string is heavy, and that is the fifth rung’s answer. What the position-aware figure says is that the difficulty is concentrated at the bottom of each string, where the sounding length is longest and β is smallest — and that the same string high up is not difficult at all in this sense.

So the genuinely awkward places on a violin are the low notes of each string, and there are four of them rather than one. A phrase sitting on the open G and the first two semitones above it is in the narrowest window the instrument has. The same phrase an octave up on the same string is in a window twice as wide, and would be described as darker, not harder.

That is testable in a way this collection can state and cannot perform: the prediction is that bow-force tolerance, measured as the ratio of maximum to minimum sustaining force at a fixed dynamic, should rise monotonically up each string and drop discontinuously at each string crossing. Every player’s intuition about tone runs the other way — a note high on a lower string is prized for its colour — so the two would be separable.

How much bow force is allowed, and where. Schelleng's diagram. The lower bound is the least force that will trigger a slip on every pass of the corner and goes as one over beta squared; the upper bound is the most the string will take before it sticks for more than a period and goes as one over beta. At beta = 0.09 the usable range spans a factor of 9.0; at 0.03, near the bridge, it is 3.0, and at 0.2, over the fingerboard, 20.0. The window closes in proportion to beta, so the difficulty of playing near the bridge is a slope on this picture rather than a matter of opinion. — Fig. 4 The window itself, for one note: force against bowing fraction, with the two bounds and the region between them. Everything in this essay is a walk along the horizontal axis of this picture, and the reason the walk matters is that the two bounds are not parallel — the maximum falls as one over β and the minimum as one over β squared, so the gap between them opens as β grows.

Fig. 5 The same map with the bow twice as close to the bridge, at 20 millimetres. Every window is wider and the ordering between the strings is the same, because moving the bow scales β everywhere at once and does not change which position each note is played in. What it does change is the absolute forces, which this ratio has divided out — and that is the next section.

What sul ponticello costs, and where

The bow distance is a dial too, and it is the one a player turns deliberately.

Moving the bow toward the bridge raises β at every position at once, which by the argument above should widen the window everywhere — and every string player knows that playing near the bridge is harder, not easier. That is worth resolving, because a model whose prediction contradicts universal experience is either wrong or being read wrongly.

It is being read wrongly, and the reading is the ratio. The window is a ratio of two forces and it does widen near the bridge. What also happens near the bridge is that both bounds move down together: the minimum force falls as one over β squared and the maximum as one over β, so at large β both are small numbers and the absolute tolerance in newtons is tiny. A player near the bridge has proportionally more room and, in the units their arm works in, far less.

So the model and the experience agree once the question is asked in the right units, and the disagreement is a good illustration of a hazard that runs through this whole ladder: a dimensionless ratio and an absolute force answer different questions, and Schelleng’s window is the first.

Fig. 6 One written pitch on every string that can play it, at a fixed bowing fraction — the earlier picture. The E string wins, because it is the lightest. With the bowing fraction moving as the left hand does, the A string wins by a factor of 1.48, because the note sits at the seventh semitone on the A and on the open E. The same note, the same instrument, two models, opposite answers.

Which computation produced the numbers

The four strings are a violin’s, with published tensions of 43, 46, 54 and 71 newtons on a 328-millimetre scale. Linear density is derived rather than quoted — μ = T/(4L²f²) follows from the string equation — and the characteristic impedance is √(Tμ), which comes to 0.338, 0.241, 0.189 and 0.187 kilograms per second from G to E.

For each written pitch, each string that can reach it within thirteen semitones of its open note is taken; the sounding length is the scale length times two to the minus position over twelve; β is the bow distance over that sounding length. The maximum force uses Z_c and the minimum uses Z_c squared, which is Schelleng’s asymmetry and is where the whole ladder’s structure comes from.

Two things about the numbers are worth flagging as conventions rather than results. The window is reported as a bare ratio with the constants of proportionality left out, because those constants involve the bow’s velocity and the string’s damping and neither is being varied here; every comparison in this essay is between two windows, and the constants cancel. And thirteen semitones is a generous reach — first position to about fourth — chosen so that every written pitch in the range has at least two strings available. Shortening it to nine narrows the range over which the comparison can be made and does not change any ordering inside it.

Where the model stops

The body is not in this figure and it belongs in it. The note the body will not let start put the bridge’s admittance into the minimum force: a body resonance is where the bridge sheds most energy, so the minimum force rises there and the window narrows. That effect is a function of frequency and this one is a function of position, so the two are separable and both are real — and the fifth rung’s figure that carries the body has a fixed β, while this rung’s figure that moves β has no body. Putting them together is arithmetic and nobody has done it.

β is not the only thing the left hand changes. A shorter sounding length is a stiffer string relative to its length, so its inharmonicity rises; and a note high on a low string is being played on a string whose tension is the same but whose vibrating mass is a quarter of what it was. Schelleng’s expressions have neither of those in them.

And the bow distance is not really constant, which is the caveat that could sink the whole result and is therefore the one worth pricing rather than conceding. Players do move the bow as they move position, partly because a shorter string wants a proportionally similar contact point and partly because the geometry of the arm makes it hard not to. The two models are the ends of one dial: let the bow distance follow the sounding length as (L/L₀)^α, and α = 0 is this essay while α = 1 is the fifth rung. The question is how far up that dial each of the two reversals survives.

α	written G4: G string vs D	written E5: A string vs E
0.0 — a fixed distance	G by 1.07	A by 1.48
0.2	D by 1.01	A by 1.37
0.5	D by 1.14	A by 1.21
1.0 — a fixed fraction	D by 1.40	E by 1.01

The two examples this essay gives are not equally robust and it presents them as though they were. The written-G reversal dies at α = 0.167: let the bow follow the hand a sixth of the way and the D string is back in front. The written-E reversal survives to α = 0.973 — the bow can follow the hand almost the whole way and the A string still wins.

So the honest form of the finding rests on the second example and not the first. A player would have to hold the bow’s distance from the bridge almost perfectly constant for the G string to beat the D at a written G; nothing about technique suggests that discipline and the margin there is seven per cent, which is inside every approximation in the model. The A-string result needs no such assumption: it holds at every α up to a fixed bowing fraction, and at the fixed fraction it is a dead heat rather than a reversal, so the fifth rung’s 1.48 in the E string’s favour was never available at any bowing discipline.

It also turns “the truth is between the two and nothing here measures where” into a measurement somebody could make in an afternoon. α = 0.167 is one number and it corresponds to something visible: a bow that moves 3.8 millimetres closer to the bridge over an octave of climbing, against the 17.5 millimetres a fixed fraction would demand and the zero a fixed distance demands. Four millimetres over an octave of shifting is well inside what an arm does without being asked, which is why the written-G reversal should be treated as a possibility rather than a finding, and the written-E one as the result. Filming a player’s contact point at first and eighth position on one string decides which side of the crossing they are on, and the answer is not a matter of degree — either the written-G reversal is in their playing or it is not.

The sweep is also the check that the two models are the same computation. At α = 1 the ratio between the D string’s window and the G string’s is 1.40, which is exactly the 37.3 to 26.7 the fifth rung printed — the absolute numbers differ because a fixed distance of 35 millimetres over the open string is a fraction of 0.108 rather than the 0.09 that essay used, and the ratio divides that out.

Whose playing, and when

The claim is about a physical window and applies wherever a bow meets a string, but the use of it is a claim about a repertoire.

The technique of choosing a string for its colour rather than for convenience — writing sul G over a passage that could be played more easily elsewhere — is a nineteenth-century device and it depends on the two dials pulling apart. A composer asking for a phrase high on the G string is asking for a heavy string at a high position, which by the arithmetic above is a wide window and a dark tone. That combination is exactly what makes the effect usable: it sounds difficult and it is not, which is why it appears in solo writing where the sound is wanted and the reliability is required. The same reasoning explains why the device is almost always written on the G rather than on the D: the heavier string is where the position gain is largest.

Before that, and in the baroque repertoire generally, the string is usually chosen for position rather than for colour, and the writing keeps to the lower positions where — this figure says — the windows are at their narrowest. That is not an argument that baroque violin writing is harder; it is a note that the difficulty it has is in a different place, and that gut strings have their own impedances which this figure has not been run for.

What the picture cannot show

Whether a wider window is an easier note. The window is a ratio of two forces and a player’s experience of difficulty includes how much absolute force is needed, how fast the note speaks, how the bow behaves as it changes direction, and what the left hand is doing at the same time — including whether the note is in tune, which on a fretless fingerboard is a separate skill entirely. This figure is one term of many and is drawn as though it were the answer.

And it cannot show what the ear is listening for. A note played near the minimum force is quiet and sounds unstable; one near the maximum is loud and sounds harsh. The window says where the note exists, not where in it a player wants to be — and the answer to that is a question about the radiated spectrum rather than about Helmholtz motion.

Where this ladder goes next

Six rungs. The bow makes a corner; the corner sustains inside a window; reaching that window takes a computable number of periods; the body is in the window’s floor; the string’s impedance is in it twice; and now the bowing fraction is in it twice as well, moving under the left hand while the right hand stays still.

The rung after it is the one the body left out. The two effects in this ladder that narrow a window — a resonance in the bridge’s admittance and a small bowing fraction — are functions of different variables, frequency and position, and they meet on a real fingerboard: a note on a body resonance played low on a heavy string is the worst case the instrument has, and its location is computable from two curves this site already carries and has never multiplied together. What comes out is a map of the fingerboard with a hardest place on it, which is a thing every player believes exists and no one has drawn.