Acoustics & Music Theory
why it sounds like that
Concept

Playability — where it appears

Whether an instrument will do what a player asks without a fight, treated here as a computed quantity rather than an opinion. Every figure that prices it does so as a window — of force, of pressure, of embouchure — and a narrow window is a difficult note.

Named by 6 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 trumpet: what the cup does to every peak. Each impedance peak of a trumpet drawn twice: as the bare bore, and with its own catalogue mouthpiece in front of the throat. The bare heights fall monotonically, from 19.2 at the pedal to 1.96 at the top of the series, which is what a tube does. The fitted heights do not: they fall to 5.4 at E♭6 and rise again to 30.1 at E♭5, giving the instrument a maximum of support in the middle of its compass that the tube alone has nowhere. The dashed line is the mouthpiece's popping frequency, 642 hertz — the note the cup sounds when it is slapped on the palm — and the maximum sits beside it.

What a cup does to the support

The mouthpiece's job was settled four essays ago and settled in cents: it decides which harmonic each mode is. Measured instead in the currency a player buys one in — how hard the note pushes back, and how narrowly it holds its pitch — the cup does something else entirely. It multiplies the support in the written register by about five, and it puts a ceiling on the instrument that the bell had not put there.

instruments · Bore profile 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 A straight mute where its corks put it, and the annulus left over. The last 220 millimetres of a trumpet's bell, drawn to scale in radius and in length, with a straight mute in it. The mute is a cone 150 mm long running from 35 mm across at the tip to 105 at the base, and it is pushed in until its clearance is the cork's 3.0 mm — which stops it 63 mm inside the rim. The sound goes past it through the annulus between cone and wall, and the narrowest place in that annulus is 393 square millimetres, 59 mm in, which is 27 per cent of the bell's own section there. The lower curve is that annulus turned into the only thing a transmission line cares about: the radius of a round tube of the same area, which falls to 11.2 mm at the throat and recovers to 53 at the rim.

The resonator at the far end

A cup at the throat owns the top of a brass instrument and puts a ceiling on it. A straight mute is a second closure at the other end of the same air column, and the prediction written down before it was computed was that it would push that ceiling further down. It does the opposite: the ceiling goes from C6 to E♭6, and it goes there by reviving exactly the resonances the cup had killed.

instruments · Bore profile 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 stringBowing pointHelmholtz motionAttack transientBoreBrassCharacteristic impedanceImpedanceMouthpieceQuality factorResonanceTimbre

All concepts