Concept

Characteristic impedance — where it appears

The ratio of force to velocity a travelling wave meets in a string or a tube, from its mass per length and its tension or cross-section. It is what decides how much of a wave reflects at a junction, and on a string it appears in both bounds of the bow-force window.

Named by 3 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

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

Named alongside it

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

Bowed string Bowing point Helmholtz motion Playability Admittance Attack transient Bow force Fingerboard Impedance Quality factor Register String tension

All concepts