Acoustics & Music Theory
why it sounds like that
Concept

Bow force — where it appears

How hard a bow is pressed into a string, which must lie between a minimum that captures the string each cycle and a maximum it can escape from. Both bounds are proportional to bow speed, so the window a note is sustained in is not the window it can be started in.

Named by 7 essays across one field — each of them below, with the objects they name alongside it.

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.

How much bow is allowed

Too little force and the corner fails to trigger a slip on every pass; too much and the string sticks for more than a period. Both bounds depend on where the bow is, and they depend on it differently — one as the square of the distance from the bridge and one linearly — so the window between them closes in proportion as the bow approaches the bridge. Sul ponticello is difficult by a power law.

instruments · Bowed string A constant bow force cannot start a note. Schelleng's minimum and maximum bow force, evaluated at the speed the bow has after one period rather than at the speed the note will be sustained at. Both bounds are proportional to bow speed and the speed after one period is the acceleration divided by the frequency, so both are proportional to acceleration and the region is a wedge through the origin. Its width as a ratio is 9.0 to 1 at every acceleration — the attack is not a narrower window than the sustain, it is the same window somewhere else. A fixed force, drawn here at 0.02, is inside it at one acceleration, about 0.18, which is why the force has to rise with the speed and why beginning a note is a trajectory through this plane rather than a point in it.

The note has to start somewhere

Schelleng's diagram says how hard a bow may press at a given distance from the bridge for a steady tone to be possible. It is a map of a note that is already sounding, and it contains no information whatever about how to begin one. Read the same inequality at the speed the bow has after a single period rather than at the speed it will settle to and the admissible region turns out to be a wedge through the origin — same width as a ratio, a hundred and ninety-six times lower in force. A constant bow pressure is inside it for one instant of the attack and outside it for the rest.

instruments · Bowed string The bow's window across a compass, with the bridge in it. Schelleng's window — the ratio of the largest usable bow force to the smallest — at a bow position of 0.09 of the length, across 196 to 1568 hertz, with the minimum scaled by the body's own admittance at each note. The window is widest between resonances and narrowest on them: it falls from 25 to 1.7 at B♭4. The notes a player finds hard to start are the local minima, and they sit on the body's resonances by construction — C♯4, B♭4, C♯5 for this body. At this bow position it never closes; move the bow toward the bridge and it does.

The note the body will not let start

Every figure until now treats the string as though it ended at a rigid point, and an earlier essay admitted it: the body feeds back on the string hard enough to make some notes difficult on one instrument and easy on another. Put the body's own admittance into Schelleng's minimum bow force and the window narrows by fifteen to one at the corpus resonances — and near the bridge it closes.

instruments · Bowed string 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.

The same note is a different width

Schelleng's two bounds both carry the string's characteristic impedance and they carry it to different powers — the maximum force as Zc and the minimum as Zc squared — so the window a player has to stay inside goes as one over Zc. A violin has four strings whose impedances differ by a factor of 1.81, the same written pitch is available on two or three of them, and the tolerance is nearly twice as wide on one as on another. The two lightest strings turn out to have almost identical impedance, which nobody chose by accident.

instruments · Bowed string The fingerboard, with the hardest place on it. Every written pitch from G3 to G6 on every string that can reach it, shaded by the width of Schelleng's bow-force window there — dark is narrow, which is a note that is hard to start. Two effects are multiplied and neither earlier figure could show the other: the body's admittance, which depends on the frequency, and the bowing fraction and string impedance, which depend on where the hand is. The worst place is C♯4 on the G3 string, 6 semitones up it, at a window of 12.2 against 471 at the easiest — a factor of 39 across the instrument. It sits where the body's A0 resonance crosses the heaviest string played high, which is the compounding this figure was drawn to find: neither variable alone puts a minimum there.

The hardest place on the fingerboard

Two things narrow a bow's window and each has been drawn alone. The bridge's admittance is a function of frequency; the bowing fraction is a function of where the left hand is. They meet on a real fingerboard, and multiplying the two curves gives a map with a worst place on it — C♯ on the G string, sixth position, where the body's air resonance crosses the heaviest string played short. The window there is twelve, against three hundred and eighty at the easiest.

instruments · Bowed string Three impedances meet at the bowing point and only one is in the way. Every impedance at the contact between a violin bow and a violin's strings, on one logarithmic axis in kilograms per second. The four strings run from 0.19 on the E to 0.34 on the G. The hair ribbon's TRANSVERSE impedance, across its own length, is 0.59 — 1.7 to 3.2 times the strings', which is comparable and is the number expected earlier to matter. It lies perpendicular to the string's own motion, because the bow is drawn ACROSS the string and the hair therefore runs along the direction the string vibrates in; what it is in the path of is the bow force. The impedance that does lie along the string's motion is the ribbon's LONGITUDINAL one, 11.4, which is 19 times the transverse and 34 to 61 times the strings'. A point load of that size against the 2Zc a string offers reflects 94.4 per cent of an arriving corner, so the fixed bowing point every earlier figure has assumed is right to within 5.6 per cent on the G string.

The hair runs the wrong way

Eight earlier essays treat the bow as a contact and the string as the object, and the last of them asked what the bow's own impedance does at the point of contact. It has three, and the one that is comparable to the string's — 0.59 kilograms per second against 0.19 to 0.34 — lies at right angles to the direction the string moves in. The one that lies along it is thirty-four times the string's, which is why a fixed bowing point has been the right assumption all along; and it stops being right at one partial in the middle of the instrument's range.

instruments · Bowed string Schelleng's window at five states of the rosin. The ratio of the largest usable bow force to the smallest, against distance from the bridge on a violin, drawn at friction contrasts of 0.5, 0.75, 1, 1.5, 2 times the normally-rosined bow every other figure assumes. The window is a hundred times the friction contrast times beta, so each curve is the next one shifted bodily: the position at which it narrows to a factor of 3 moves from 19.5 millimetres at 0.5 to 4.9 at 2. The two vertical marks are the geometric limits found earlier and neither of them moves with the rosin: the ribbon covers half the corner's bridge-side excursion at 10.0 millimetres and its near edge reaches the bridge at 5.0. The friction contrast at which the force limit retreats behind the first of those is 0.98, and behind the second 1.95 — so a violin at ordinary rosin is sitting within a few per cent of the exchange, and at twice it the force window has no say at any playable bow position.

What the rosin is worth

Schelleng's window has three parameters and it has been drawn thirty-nine times with two of them held at one. The bow speed turns out to cancel exactly — it multiplies both bounds and leaves the ratio alone — and the friction contrast does not: the window is proportional to it, so every limit priced in millimetres moves as one over it and none of the geometric ones move at all. A violin at ordinary rosin is sitting 2.5 per cent from the point where the two exchange places, and at twice it the force window has no say at any playable bow position.

instruments · Bowed string

Named alongside it

The objects these essays reach for when they reach for this one.

Helmholtz motionAttack transientViolinBody resonanceImpedanceSchelleng diagramBowed stringBowing pointEnvelopeExcitation pointFrictionSlip stick

All concepts