Acoustics & Music Theory
why it sounds like that
Concept

Dispersion — where it appears

The dependence of wave speed on frequency, which in a stiff string makes high partials travel faster than low ones. It is why a piano's partials are not whole multiples and why a corner that leaves the hammer does not come back a corner.

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

The partial above which the returning corner has lost its phase. Up a piano, the lowest partial whose extra travel after one round trip to the near end exceeds a quarter of a cycle — above it, the component comes home somewhere other than where a corner needs it. The curve follows the inharmonicity coefficient, which is smallest in the tenor and rises both ways, so the corner survives best at A2 — to the 24th partial — and worst at A7, where only 5 partials are still in step. The earlier coupled model assumes a clean corner at every partial it uses.

The corner does not come back a corner

Stepping a hammer forward against the string's own returning wave showed that the coupling matters most in the bass. It assumed the wave that came back was the shape that left. On a stiff string it is not: partials travel at different speeds, and by the top octave of a piano the sixth partial is the first to arrive out of place. But a periodic wave can only see phase modulo a cycle, and counted that way twenty-one of forty-eight partials are still in step — which is why the returning wave is degraded rather than destroyed.

timbre · Excitation point Which of the four terms is doing the removing, note by note. Each of three terms has a corner — a partial number above which it is the one taking the energy away — and the lowest corner at each pitch is the term that binds. The comb is flat at 8, because where the hammer lands is a fraction of the length and does not change with the note. The contact time's corner falls from 34 at A0 to 0.5 at A7. They cross at about E3, 165 hertz: below it the strike point decides the spectrum and above it the contact time does. The dispersion's corner is above both at every pitch on the instrument — 5.8 at the top against a contact corner of 0.5 — so that earlier term is real and is nowhere the binding one. That is a statement no one of them could make on its own.

Four terms, and only one of them binds

Six earlier essays each computed one term of a piano's excitation spectrum and handed it to the next, and nobody had multiplied them. Multiplied, three of the four have a corner — a partial above which they are the term doing the removing — and putting the three corners on one axis says which is in charge at each pitch. Below E3 the strike point decides; above it the contact time does; and the dispersion, which is the newest and most laborious term, is nowhere the binding one.

timbre · Excitation point Three instruments, three wires, and an order of magnitude between them. The inharmonicity coefficient of each instrument's own string design, across its own compass. At middle-of-the-keyboard C a piano's wire is 315 millimetres long and 1.02 thick and gives B = 1.5e-3; a harpsichord's at the same pitch is 355 millimetres and 0.30, and gives 8.1e-5 — a factor of 19. The harpsichord's line is nearly flat across four octaves because its scaling is Pythagorean: the length doubles for every octave down, exactly, until the case runs out. The piano's is a V, because its case runs out two octaves earlier and everything below the break is a compromise. Every column before now used the top line for all three instruments.

Three exciters and three wires

Every column drawn so far has run a plectrum and a beater on a piano's string, because a piano's was the only scaling to hand. A harpsichord's wire is an order of magnitude less stiff, for a reason that turns out to be the tensile strength of iron — and with all nine combinations computed, the crossing found before stays exactly where it was, on every wire.

instruments · Excitation point A plucking point 50 millimetres from the nut, across a harpsichord's compass. The jacks stand in a rail and the rail is one object, so a register plucks at a fixed distance from the nut while the strings shorten by a factor of ten from the bass to the treble. At 50 millimetres that is one 36th of the string at C2 and one 3.5th at C6 — a plucking fraction that changes by a factor of 10.1 without the maker moving anything. The comb corner is one over that fraction and falls with it, from 36 to 3.5; the dispersion corner has the fraction under a cube root and falls only from 106 to 21. They never meet. Setting one over p equal to the cube root of two over three times the inharmonicity and the fraction gives a plucking point at p = √(1.5B), which on this instrument's iron wire is between 3.4 and 9.9 millimetres from the nut — nearer to it than any register a harpsichord has ever carried, the lute stop included. So the maker's one free choice is the binding term at every pitch and every register position, which is exactly what the piano's is not: on a piano the contact time takes the decision away above E3.

What the second register is for

Nine earlier essays take the plucking fraction as given — an eighth, a sixth, a seventh — and a harpsichord's jacks stand in a rail, so what is given is a distance and the fraction is a consequence: 50 millimetres is one thirty-sixth of the string in the bass and one three-and-a-halfth in the treble. Engage two registers on one string and the amplitudes add exactly, the near one fills the far one's missing partials, the pair digs holes of its own that neither has, and the combination comes out louder and rounder than either — including the bright one.

instruments · Excitation point Second order and fourth order, from the same sequence of wavenumbers. Frequency against wavenumber for a string and for a bar, with each object's own wavenumbers marked on the axis. A string's equation is second order in space, so frequency goes as wavenumber and the straight line carries its modes at 1, 2, 3, 4. A bar's is fourth order, so frequency goes as the SQUARE of wavenumber and the parabola carries the same kind of sequence to 1, 2.757, 5.404, 8.933. Both sequences of wavenumbers are arithmetic — a string's are the even multiples of π/2L and a bar's the odd ones, except that the first is 3.011 rather than 3 — and squaring an arithmetic sequence is the whole of why one object has a fundamental and the other has none. Every number is derived from the boundary condition; none is measured.

A bar's partials are the odd numbers, squared

A string's wave equation is second order in space and a bar's is fourth, so a string's frequencies go as its wavenumbers and a bar's go as their squares. Both objects fit a nearly arithmetic sequence of wavenumbers between their ends; squaring one is the whole of why a marimba key has no note in it. The ratios are 1 : 2.756 : 5.404 : 8.933, which are the odd squares over nine, uniformly 12.9 cents flat, and the 12.9 cents is one root that refused to be 3π/2.

instruments · Struck bar

Named alongside it

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

InharmonicityExcitation pointPartialHammerStiffnessHarpsichordPianoRegisterSpectrumBoundary conditionBrightnessContact time

All concepts