Concept

Bowing point — where it appears

How far from the bridge a bow is drawn, expressed as a fraction of the sounding length. It decides the width of the force window a player has to hit and the sharpness of the corner they produce, and both get worse toward the bridge.

Named by 5 essays across 2 fields — each of them below, with the objects they name alongside it.

The bow's window along each string, with the bow held still. Schelleng's window — the ratio of the greatest bow force a note tolerates to the least — for every written pitch on every string that can play it, with the bow 35 mm from the bridge and staying there. The window widens up each string because β is the bow's distance over the SOUNDING length and the sounding length is shortening, and it steps down at each new string because a heavier string carries a narrower one. The dashed lines are what was drawn earlier, which held β at 0.09 for every note on every string. The widest disagreement between two strings at one pitch is 1.48 to one, at A♭5.

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.

instruments · Bowed string $The violin's attack times, string by string. Every playable cell of the earlier map, with its bow-force window turned into a time. A player aiming at the geometric centre of the window has to wait until the accelerating bow's maximum force rises to that value, which is a fraction one over the square root of the window width of the way through the bow's ramp — so a narrow window is a late note and the exponent is a half. The latest cell is C♯4 on the G3 string at 21.9 milliseconds, against 4.1 at C6 on the A4: a factor of 5.3 in time out of a factor of 29 in window width. The darkest cell is the same hardest place, unchanged — the map is the same map under a monotone change of units, and what is new is that the units are milliseconds, which a player and a listener both have access to.$

The hardest place is also the latest

A bow-force window is a ratio of forces, which nobody can hear. An attack time is milliseconds, which a player and a listener both have. The map of the violin's windows becomes a map of its attack times under a change of units, and the exponent turns out to be a half — so a window twenty-nine times narrower is only five times later.

rhythm · Onset time

The violin is the one where the bow's width catches up. Three separate limits on how near the bridge a bow can go, for the four bowed instruments. The ribbon's near edge reaches the bridge at 5, 6, 6, 8 millimetres; the ribbon covers half the corner's bridge-side excursion at 10, 11, 12, 15; and Schelleng's force window narrows to a factor of 3 at 10, 12, 21, 32. On the viola, cello and double bass the force window binds first, by a margin that widens to a factor of two on the bass. On the violin the ribbon binds first, at 10 millimetres against 9.8. A bow's hair is as wide as a hand can control and a string is as long as its pitch requires, so the ratio between them is a fact about the violin rather than about bowing.

The bow is not a point either

Seven earlier essays set the bowing point to a number between 0.02 and 0.3 and drew it as a point. A violin bow's hair is ten millimetres wide on a string of three hundred and twenty-five, so near the bridge the ribbon is wider than its own distance from it. In the spectrum that turns out to be worth nothing. In the geometry it is the reason sul ponticello has a floor, and the violin is the one instrument of the four where it arrives before the force does.

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.

Bowed string Helmholtz motion Playability Characteristic impedance Violin Attack transient Bow force Timbre Admittance Excitation point Fingerboard Friction

All concepts