The same note is a different width

The note the body will not let start put the box into Schelleng’s window: the minimum bow force scales with how much the string sheds per cycle, a body resonance is where the bridge lets it shed most, and the window narrows fifteenfold on one. That rung ended by naming the term it had held constant. Every figure in this ladder holds the string’s own characteristic impedance at one value, because there has only ever been one string in it.

A violin has four, of very different mass, tuned a fifth apart.

A heavier string has a narrower window. That is not obvious from the diagram everybody draws, because the diagram is drawn for one string.

Why the two bounds carry it differently #

How much bow is allowed established the two bounds. Above the maximum the bow’s grip is too strong and the corner cannot trigger the slip at the right moment, so the note breaks into a crunch; below the minimum the string sheds more per cycle than the bow puts in, so the Helmholtz motion never establishes and the note is a surface noise — the corner never forms.

Both bounds are proportional to the bow speed and inversely proportional to the distance from the bridge. Where they differ is in the impedance:

F_max ∝ Zc · v / β, and F_min ∝ Zc² · v / β², the second of them multiplied by the bridge’s admittance.

The maximum is a statement about force against the string’s own resistance to being moved, so it goes as Zc once. The minimum is a statement about how much of the energy the bow puts in survives a round trip, and the round trip involves the string delivering to the bridge and getting a reflection back — which is a second factor of Zc.

So the window, which is the ratio, goes as one over Zc. A player’s tolerance on a note is inversely proportional to how heavy the string is.

It is worth noticing that this is the same asymmetry the hand in the bell turns on, one field away: a single geometric quantity entering two consequences with different powers, so that a maker cannot move one without moving the other. There the quantity is an aperture radius and the two consequences are tuning and radiation; here it is a string’s impedance and the two are the two edges of a window. An instrument’s design constraints are nearly always of this shape, and it is why they read as compromises rather than as choices.

The impedances, derived rather than quoted #

The four numbers are not looked up. A string’s linear density follows from its tension and its pitch: f = (1/2L)√(T/μ) rearranges to μ = T/(4L²f²), and a published tension with a 32.5-centimetre scale length gives it.

The A and the E have the same impedance to within one per cent. Their tensions differ by fifty per cent and their masses by a third, and the product comes out the same — which looks like either a coincidence or a design, and the section below finds it is neither in the way it first appears.

The impedance in that column is worth deriving once, because doing so removes the tension from it. Substituting T = 4L²f²μ into √(Tμ) gives Z_c = 2Lfμ: the tension cancels, and a string’s impedance is nothing but its mass per metre times its pitch times twice the scale length. Every number in the fourth column of that table is therefore a restatement of the third and the second, and the four impedances fall smoothly because the masses fall faster than the pitches rise.

The G string is the outlier. It is 1.81 times the E string’s impedance, its window is 55 per cent as wide, and it is the string violinists describe as the hardest to control.

There is a second reading of the A-and-E agreement that is worth stating because it is a claim about design rather than about acoustics, and doing the algebra rather than gesturing at it turns the claim round.

Substituting the string equation into the impedance gives Z_c = 2Lfμ — the tension cancels out entirely, and an impedance is a linear density times a pitch times a length. So equal impedances at a fixed scale need μ ∝ 1/f, not 1/f², and the G string would have to be 3.3 times the E string’s mass rather than eleven. It is 6.0 times.

So the G string is not lighter than equality wants — it is 1.81 times heavier, and the direction matters because it names a different constraint. A G string light enough to match the E string’s impedance would carry 24 newtons where the real one carries 43, which is a very slack string: it would be hard to stop cleanly, easy to pull sharp with the finger, and quiet, because the force it delivers to the bridge is what makes the sound.

So the narrow window on the G string is bought rather than conceded. What the maker is holding roughly constant across the four is not the impedance but something much closer to the tension — 43, 46, 54, 80 newtons, a spread of 1.9 against the impedance’s own 1.81 in the other direction — and a set of strings at equal tension has impedances that fall as 1/f, which is exactly the pattern in the first column.

The A-and-E agreement is then the coincidence and the G string is the rule. Two adjacent strings whose tensions and masses happen to trade off to the same product is the kind of thing that falls out of a monotone series; a whole set designed for equal windows would look nothing like this one, and would be unplayable at the bottom.

One note, two or three strings #

The consequence is not about the strings but about the notes, because most of a violin’s range is available on more than one of them.

Choosing a string is choosing how much force the note will tolerate — a fact about what a tablature keeps, since a tablature specifies the string and a stave does not, and it is a choice a player makes for reasons of colour, of position and of legato — none of which has anything to do with this.

And the derivation says which quantity a player is really choosing between. Because the impedance is 2Lfμ, and f is fixed by the note, choosing a string for a given pitch is choosing μ — the mass per metre — and nothing else. The two strings are not offering two different combinations of tension and mass; at that pitch they are offering two masses, and the window is inversely proportional to the one chosen. A fingering decision on a violin is a decision about how much wire is vibrating, stated as directly as it can be.

It also says something about what a string crossing costs. A passage that crosses from the D string to the A string in the middle of a phrase changes the window width by a factor of 1.28 at the crossing point, which means the bow force that was correct on one side of it is nearer an edge on the other. Players describe string crossings as needing “a change of weight”, and the size of the change is now a number.

And it is a number with a direction that the impedance formula fixes. Crossing to a lighter string widens the window and lowers both of its bounds, since the maximum goes as Z_c and the minimum as its square — so the same absolute bow force that sat in the middle of the window on the heavy string sits nearer its top on the light one. A crossing outward needs less weight and not merely different weight, which is what the instruction actually means and is a thing the ratio alone does not say.

The two terms are separable, and that is the useful part #

The fourth rung’s finding was that the window narrows on a body resonance, sometimes to nothing. This rung’s is that it narrows on a heavy string. The two multiply, and because one is a function of frequency and the other is a constant per string, a note that is hard on one string can be easy on another at the same pitch.

That is a prediction with teeth, because it names specific notes. The body’s worst dips sit where the violin’s main air and corpus resonances are, and a pitch that lands in a dip is available on the G string with 0.55 of the window and on the A string with 0.99 of it. The same acoustic problem, on the same instrument, with a factor of nearly two between the two ways of playing it.

Which computation produced the numbers #

Four steps and no fitting.

The linear density comes from the string equation as above. The characteristic impedance is √(Tμ), which is also μc where c is the wave speed on the string — the two forms are the same number and the second makes the interpretation clearer: it is the force the bow must supply per unit of transverse velocity.

Schelleng’s bounds are the site’s own schelleng, unchanged since the ladder’s second rung, evaluated with each string’s impedance in the place where the constant used to be. The maximum takes it once and the minimum twice, which is the whole of the new content. Everything else — the bow speed, the bowing point, the friction law — is the second rung’s and is untouched.

The body’s admittance is the fourth rung’s own bodyGain on the measured violin resonance list, normalised to its geometric mean over the instrument’s range so that the vertical scale is relative to this instrument rather than to an absolute force.

And a note is counted as available on a string if it lies between the open pitch and nineteen semitones above it, which is a generous but not absurd reach.

Where the model stops #

The four tensions are nominal. String makers publish tensions for their own sets and they differ by ten or fifteen per cent between makers and between gauges. The impedances scale as √T at fixed pitch and length, so a fifteen per cent tension spread is a seven per cent impedance spread — enough to blur the A-and-E coincidence and not enough to touch the G string’s factor of 1.81.

A wound string is not a uniform one. The G and D strings are wound — a core with metal wrapped round it — and the winding adds mass without adding stiffness, which is the whole point of the construction. The impedance calculation above uses the effective linear density and is right for the wave speed; what it misses is that a wound string’s bending stiffness is much lower than a solid one of the same mass, which affects the corner’s sharpness and therefore the maximum force rather than the minimum.

And the minimum bound has a constant in it that this collection has never pinned. Schelleng’s minimum carries a factor for the coefficient of friction’s dependence on sliding speed, which is a material property of rosin and is quoted over a range of about three to one. Every window number here inherits that, which is why the essay is written in ratios between strings rather than in newtons.

Whose music, and the string a player chooses #

The physics is about violins and applies to the whole bowed family with different numbers. The practice is about a repertoire.

String choice is notated. sul G and sul D are instructions to play a passage on a named string, and the reason given in every treatise is timbre: the lower strings are darker, thicker, more intense. That is true, and the arithmetic here adds a second consequence nobody writes down — a passage moved to the G string is a passage with 55 per cent of the force tolerance it had, which is harder to play cleanly and much more likely to break into the crunch above the maximum or the surface noise below the minimum.

That is a fair account of what sul G actually sounds like. The intensity players describe is partly a spectral fact about a heavy string near the bridge — where the exciter lands decides the spectrum — and partly a fact about operating close to the edges of a narrow window, where the tone is unstable and the effort is audible. The instruction is used at expressive climaxes, which is where the instability is wanted.

The other place it shows is in the writing for the instruments a violin’s family scales into. A cello’s C string is far heavier again, and its window is correspondingly narrower; the passages orchestrators avoid at the bottom of a cello section are the ones where a narrow window meets a body resonance, and both terms are now computable.

What the four numbers do not say #

One thing this rung deliberately does not claim. The window is a tolerance, not a quality: a wide window means a note is easy to hold and says nothing about whether it is worth holding. The E string has the widest window on the instrument and is also the string most often described as thin, and an instrument is not one timbre is the reason — a light string driving the same body puts less energy into the low corpus resonances, whatever the player does with the bow.

So the four impedances give an ordering of controllability that runs opposite to the ordering of weight, and the ordering of colour runs with it. A player choosing a string is choosing a point on both, and this ladder can now compute one of them.

What the picture cannot show #

Whether a player experiences the window as width. Everything here is a ratio of two forces, and a player experiences bow pressure through an arm with its own dynamics. A window twice as wide in newtons is not obviously twice as easy.

Nor whether the window is what limits a player at all. The note has to start somewhere found that reaching the window from rest takes a computable number of periods, and a passage of fast notes may be limited by the transient rather than by the steady state. The window is about holding a note; the attack is about getting one, and a heavy string is worse at both.

And the position on the string is not in it. Playing a note high on a low string means playing near the bridge, which is a change in β — the distance from the bridge as a fraction of the length — and β enters the maximum once and the minimum twice, exactly as the impedance does. So the position and the string are two changes to the same product, and separating them needs a fingerboard geometry this figure does not have.

Where this ladder goes next #

Five rungs. The bow makes a corner; the corner can be sustained inside a window; reaching that window from rest takes a computable number of periods; the box the string is stretched over is in the window’s floor; and now the string’s own impedance is in it too, twice, so the same written pitch has a different tolerance on each string it can be played on.

The rung after it is the one the position limitation names. 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. Whether the two cancel or compound along a real fingerboard is arithmetic this ladder now has every term for, and the answer would say which passages on which strings are genuinely hard rather than merely dark.