Acoustics & Music Theory
why it sounds like that
Concept

Directivity — where it appears

How much more strongly a source sends sound in one direction than it would if it radiated equally everywhere. It grows with the size of the radiating surface against the wavelength, so an instrument's upper partials are beamed while its fundamental is not, and what a listener hears depends on where they sit.

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

Where a woodwind's A♭3 leaves it, below its corner and above it. The same fingering — 6 holes open on a 15-millimetre bore 567 millimetres long, sounding A♭3 at 207 hertz — drawn twice, with each opening's circle scaled by the share of the radiated power that leaves through it. At 400 hertz 78 per cent of it leaves through the first open hole, the bell takes 0 per cent, and the number of apertures really doing the radiating is 1.6; At 2600 hertz 6 per cent of it leaves through the first open hole, the bell takes 49 per cent, and the number of apertures really doing the radiating is 3.3. The lower frequency is below this fingering's corner and the higher one above it: below the corner the instrument is a short tube with one opening at the end of it, and above the corner it is the whole lattice at once. The power-weighted station — where a listener would say the sound is coming from — moves from 392 millimetres to 515.

Where a woodwind actually sounds from

Every number so far is read at the mouthpiece, and the corner's whole musical meaning is at the other end. Run the same solver forwards and it gives the flow leaving every hole — from which a clarinet's radiating aperture turns out to be a function of fingering and of frequency, but not the way it was predicted to: the fingering sets how far the aperture opens, almost exactly to the number of open holes, and barely moves the frequency at which it does.

instruments · Tone holes Where in the hall a raised lid makes a C3 staccato darker, and where brighter. A floor in front of a 35 cm piston radiator pointing up the page, in units of the critical distance of a source radiating everywhere, three wide and one and a half deep, in a 2-second room, divided into 160 seats. Each seat is marked by what a damped C3 does to its colour over the 400 ms after its release: darkens (42 seats), brightens but by no more than with no lid (26), or brightens more than with no lid (92). The darkening seats form a narrow wedge on the axis; everywhere else a raised lid makes the end of a note brighter, and over 58 per cent of the floor brighter than it would be with the lid closed.

A lid darkens a staccato only inside a narrowing cone

Along its axis, a raised lid makes a damped note's release darker, because the brightness it beams at the listener is the part the damper stops. Most of an audience does not sit on the axis. Off it, the same beaming does the reverse, and a little way off it does the reverse more strongly than no lid at all would. The region where the release darkens is a cone whose half-width obeys one law — the fundamental times the radiator's radius times the sine of the angle is 23.6 hertz-metres — so it narrows from 28 degrees at C3 to 8 at C5.

timbre · Envelope A cone opens its aperture as far as a cylinder does only if its holes grow with it. For every fingering of four woodwinds with twelve holes each, each tuned to its own semitones — a clarinet's cylinder with graded holes and with holes all 9 millimetres across, a conical bore of an oboe's proportions with holes 0.6 of the local bore radius, and the same cone with every hole 7 millimetres across — the largest number of openings really radiating together at any frequency above 700 hertz, as a participation number counting the bell. The dotted line is every open hole and the bell sharing equally. clarinet: a cylinder: 1 open 2.0, 2 open 3.0, 3 open 3.9, 4 open 4.8, 5 open 5.6, 6 open 6.5, 7 open 7.2, 8 open 7.8, 9 open 8.8, 10 open 9.6, 11 open 10.1, 12 open 10.5. cone, holes growing with the bore: 1 open 2.0, 2 open 2.9, 3 open 3.9, 4 open 4.9, 5 open 5.9, 6 open 6.7, 7 open 6.6, 8 open 7.2, 9 open 7.7, 10 open 7.4, 11 open 8.4, 12 open 8.9. cone, every hole 7 mm: 1 open 2.0, 2 open 2.8, 3 open 3.7, 4 open 4.5, 5 open 5.4, 6 open 5.6, 7 open 6.1, 8 open 5.6, 9 open 6.5, 10 open 6.7, 11 open 7.1, 12 open 6.8. clarinet, every hole 9 mm: 1 open 2.0, 2 open 2.9, 3 open 3.9, 4 open 4.9, 5 open 5.9, 6 open 6.8, 7 open 7.7, 8 open 8.6, 9 open 9.6, 10 open 10.1, 11 open 9.2, 12 open 10.2.

A cone radiates like a cylinder only if its holes grow

The prediction was that a conical woodwind radiates from fewer of its holes than a clarinet, because a hole near the wide end of a cone sits on a tube that is already easy to push air along, and so pulls less of the passing flow out. The mechanism is right and the conclusion is wrong for the instruments that exist. With holes that grow with the bore, as an oboe's and a saxophone's do, a cone opens its aperture as far as a cylinder up to six open holes and falls short above that by a hole or two. Only a cone whose holes are all one size tops out far below — near seven apertures with thirteen open. What a cone really changes is where and how much: its aperture opens nearly a major third lower, its bell carries twice the share, and it gets out at best under half of what reaches it, where the clarinet gets out seven tenths.

instruments · Tone holes A section whose upper partials beam keeps its pizzicato deep into a hall on its axis, and loses it faster than an unpointing one a little way off. The margin at the strike for a pizzicato section of twelve over a held clarinet, 15 m into a hall, from C4 to F♯5, with each partial's direct sound scaled as from a piston of 10 cm radius. Above zero the pluck owns the note. no beam: A♭4 to C♯5 -0.3, -1.7, -3.4, -3.5, -3.3, -1.7; on the axis: A♭4 to C♯5 5.2, 4.1, 2.8, 3.0, 3.5, 5.3; 15° off: A♭4 to C♯5 4.4, 3.2, 1.8, 1.8, 2.2, 3.9; 30° off: A♭4 to C♯5 1.9, 0.3, -1.6, -2.2, -5.1, -2.1.

A section keeps its pizzicato only inside a cone

In a hall a section's pizzicato loses its corpus notes to a held clarinet within two metres, on the assumption that a violin radiates equally in every direction. It does not, and twelve players sharing a beam cannot average it away. Directly on the players' axis, the same section keeps every note for twenty-five metres. Thirty degrees off, with the beam's upper partials thrown elsewhere, it is worse than if it had never pointed. A question about how far back a pizzicato carries turns out to be a question about which way the listener's seat faces.

timbre · Spectrum Antiphonal second violins own their worst pizzicato note almost nowhere, unless a hard wall stands close behind them. A plan of a 24 by 36 m hall with the second violins at the right of the stage, plates turned away from the audience, and a perfectly reflecting wall 2 m behind them. A seat is shaded if the section's worst note from A♭4 to C♯5 stays a pizzicato rather than a clarinet with a click. The seconds keep 30 per cent of seats, the firsts on the other side 30.

A wall puts antiphonal seconds back beside the firsts

Second violins seated opposite the firsts have their plates turned away from the audience, and in a hall a section's pizzicato depends on its beam. With nothing behind them they keep their worst note in about one seat in a hundred. A perfectly hard wall two metres behind gives them back a cone as large as the firsts' own, but a wall that loses three decibels takes most of it away again. And the seats that cone serves turn out to be, within a seat or two, the seats the same section would serve sitting beside the firsts.

timbre · Spectrum

Named alongside it

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

ReverberationBoreCritical distanceCutoffImpedancePizzicatoRadiation efficiencyTimbreTone holeBrightnessDampingDecay

All concepts