Acoustics & Music Theory
why it sounds like that
Concept

Excitation point — where it appears

The place along a string at which it is plucked, struck or bowed, which decides how strongly each of its modes is set moving. A mode with a node at that point is not excited at all, which is one line of trigonometry and a piano design decision.

Named by 17 essays across 2 fields — each of them below, with the objects they name alongside it.

A string struck at one 7th of its length. The amplitude of each partial of an ideal string excited at 0.1429 of its length. The mode shape is a sine, so a partial with a node at the excitation point cannot be set moving at all: partials 7, 14 are silent here. The envelope over the rest is one over n, a struck string's.

Where the hammer lands

Strike a string at exactly one over n and the nth partial is silent, because the hammer has landed on that mode's node and cannot move it. A piano's hammers strike between a seventh and a ninth of the way along, which puts the seventh partial — the most dissonant member of the series — at or near a null. That is a design decision made in wood, and it takes one line of trigonometry.

instruments · Excitation point The same note, hit at a middling dynamic. The spectrum of a struck string with the hammer's own contact time applied as a low-pass. Contact lasts 1.60 ms at this force, against 2.26 ms at the softest and 0.95 ms at the loudest drawn — felt is a nonlinear spring, so a harder blow is a shorter contact and a brighter note. The spectral centroid moves from partial 1.5 to partial 2.2, which is a change of timbre and not of loudness.

A hammer is not an impulse

Contact lasts a couple of milliseconds, which low-passes the note — any partial whose half-period is shorter than the contact is barely excited. Piano felt is a spring that stiffens as it compresses, so a harder blow makes the contact shorter, the corner higher and the note brighter. A loud note is not a scaled-up quiet one, and no linear model gives that.

instruments · Excitation point Helmholtz motion, bowed at 9% of the way from the bridge. Above: the string at 5 instants of one period. It is two straight lines meeting at a corner, and the corner travels round the string rather than the string swinging. Below: the resulting force on the bridge, a sawtooth whose two segments are in the ratio 0.09 to 0.91 — the bow's own position. A sawtooth contains every harmonic at exactly one over n, so the spectrum barely changes with bow position even though the waveform plainly does.

The bow makes a corner

A bowed string is not a string swinging. It is two straight lines meeting at a single sharp corner that travels round the string once per period, triggering the slip that keeps it going. The force on the bridge is therefore a sawtooth, and a sawtooth is every harmonic at exactly one over n — which is why a bowed string is the most nearly perfect harmonic series in the orchestra.

instruments · Bowed string How much bow force is allowed, and where. Schelleng's diagram. The lower bound is the least force that will trigger a slip on every pass of the corner and goes as one over beta squared; the upper bound is the most the string will take before it sticks for more than a period and goes as one over beta. At beta = 0.09 the usable range spans a factor of 9.0; at 0.03, near the bridge, it is 3.0, and at 0.2, over the fingerboard, 20.0. The window closes in proportion to beta, so the difficulty of playing near the bridge is a slope on this picture rather than a matter of opinion.

How much bow is allowed

Too little force and the corner fails to trigger a slip on every pass; too much and the string sticks for more than a period. Both bounds depend on where the bow is, and they depend on it differently — one as the square of the distance from the bridge and one linearly — so the window between them closes in proportion as the bow approaches the bridge. Sul ponticello is difficult by a power law.

instruments · Bowed string What is struck, and where its partials land. Partial ratios for an ideal string, an ideal membrane, a kettledrum, drawn on a logarithmic axis so that a whole-number ratio is a fixed distance from the last. An ideal string: 1, 2, 3, 4, 5, 6. An ideal membrane: 1, 1.59, 2.14, 2.29, 2.65, 2.92. A kettledrum: 1, 1.50, 1.99, 2.44, 2.89. The membrane's are Bessel zeros — a derivation, and they land nowhere near the whole numbers. The kettledrum's have been pulled toward 2 : 3 : 4 : 5 by the enclosed air, on a fundamental an octave below the lowest partial present.

What a drum is doing instead

An ideal membrane's modes are zeros of Bessel functions — 1, 1.59, 2.14, 2.30 — which support no common fundamental, so a drum rings without a note. A timpani is that problem solved — the kettle's air and the radiation load drag four modes onto 1, 1.5, 2, 2.5, and the pitch a timpanist tunes is the fundamental those four imply and none of them is.

instruments · Missing fundamental C4, in every place it can be played. A guitar neck with the 4 places C4 can be stopped, drawn at the fret spacing a 64.8-centimetre scale actually has. The stave writes one note and the tablature writes one of these; each notation says exactly what the other leaves out. The speaking lengths run from 61.2 down to 27.2 centimetres, so a hand plucking 12 centimetres from the bridge meets between 20 and 44 per cent of the string.

What a tablature keeps

Middle C can be stopped in four places on a guitar. The speaking lengths run from 61 to 27 centimetres, so a hand plucking twelve centimetres from the bridge meets between a fifth and nearly a half of the string, and the comb of missing partials is different at every one: the second partial is thirteen decibels stronger in the best position than in the worst. A stave writes one note for all four. A tablature writes four different things and cannot say which note any of them is.

instruments · Notation How near the node a hammer has to land. The level of partial 7, relative to the loudest partial of the same note, against how far the hammer is from that partial's node — for a contact 2.6 per cent of the speaking length wide, which is 16 mm on a 620 mm string. At the node exactly the partial is absent whatever the width, because a mode shape is odd about its own node and integrating an odd function symmetrically gives zero. Twenty decibels of suppression needs the hammer within 2.82 mm, and thirty needs 0.89 mm. The width itself barely matters in the musical range: its sinc reaches its first zero only at partial 77.

The hammer is not a point either

The comb of missing partials is computed throughout for a contact of no width, and a piano hammer is sixteen millimetres of felt on a string of six hundred. Integrating the mode shape across the felt turns out not to fill the null in: a mode is odd about its own node, so a symmetric contact centred on it gives an exact zero however wide it is. What destroys the null is being in the wrong place, and the tolerance is 2.8 millimetres for twenty decibels of suppression and 0.89 for thirty. The design decision found at the outset is a manufacturing tolerance.

instruments · Excitation point 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 Trumpet at three dynamics, as a spectrum rather than a level. The radiated partials of a trumpet at 45, 70, 95 decibels, each normalised to its own strongest partial so that only the SHAPE is compared. A linear source would give three identical pictures. This one does not: the spectral centroid moves from partial 2.19 to 6.41, a factor of 2.92, because the excitation is nonlinear and blowing harder steepens the pressure front rather than scaling it. The tilt used is 3 decibels per octave of partial number per ten decibels of level, referred to 70 dB — a stipulated, ordinal number, not a measurement of any instrument.

A dynamic mark changes what a note is

Every spectrum until now is a shape with a level in front of it, so that playing ten decibels louder raises every partial by ten. That is true of exactly one instrument in an orchestra. Everybody else steepens their own spectrum as they lean on it, and a trumpet's centre of gravity moves from the second partial to the sixth across a dynamic range while an organ flue pipe's does not move at all.

timbre · Orchestration The fingerboard, with the hardest place on it. Every written pitch from G3 to G6 on every string that can reach it, shaded by the width of Schelleng's bow-force window there — dark is narrow, which is a note that is hard to start. Two effects are multiplied and neither earlier figure could show the other: the body's admittance, which depends on the frequency, and the bowing fraction and string impedance, which depend on where the hand is. The worst place is C♯4 on the G3 string, 6 semitones up it, at a window of 12.2 against 471 at the easiest — a factor of 39 across the instrument. It sits where the body's A0 resonance crosses the heaviest string played high, which is the compounding this figure was drawn to find: neither variable alone puts a minimum there.

The hardest place on the fingerboard

Two things narrow a bow's window and each has been drawn alone. The bridge's admittance is a function of frequency; the bowing fraction is a function of where the left hand is. They meet on a real fingerboard, and multiplying the two curves gives a map with a worst place on it — C♯ on the G string, sixth position, where the body's air resonance crosses the heaviest string played short. The window there is twelve, against three hundred and eighty at the easiest.

instruments · Bowed string 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 Where each exciter's contact corner sits against its comb. The merger's four corners, drawn for three exciters. The comb is the strike point and is a horizontal line, because it is a fraction of the string and has no pitch in it. The contact corner is the exciter's, and it falls up the compass because the contact time is a fixed number of milliseconds against a shrinking period. Where they cross, the binding term changes. piano hammer crosses at E3; harpsichord plectrum crosses nowhere in the compass; dulcimer beater crosses at A5. So the crossing found earlier is not a fact about pianos. It is a fact about compliance: a felt hammer stays on the string for about 1.6 milliseconds and a plectrum for 0.05, and a difference of that size is a corner in a different place.

The crossing belongs to the felt

On a piano the strike point decides the top of the spectrum below E3 and the hammer's contact decides it above. The merger's own bookkeeping asked whether that crossing is a fact about pianos or about hammers. A plectrum never crosses at all and a hard beater crosses two octaves higher, and the boundary is a contact time of about an eighth of a millisecond — ten times shorter than felt.

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 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 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 What each strike point on a marimba bar excites. The amplitude each of the first 4 partials receives from a strike at each point along the bar, each partial drawn to its own maximum. On a string this comb is |sin nπx| and is periodic, so one strike point silences a whole family; on a bar it is the mode shape itself, whose zeroes are not at rational fractions, so no strike point silences a family. The exception is the middle: every antisymmetric mode is zero there by symmetry, so a centre strike drives partial 1 at 100 per cent of its maximum and partial 2 at nothing at all. The cord's hole at 0.2031 is the mirror case — the fundamental is the partial a strike there cannot reach.

The four ways a marimba loses what its arch placed

The centre of a bar is a node of every even mode by symmetry, so a stroke there gives the second partial exactly nothing and a mallet's width cannot recover it; five millimetres off centre returns 14 per cent. Add the yarn mallet's 3-millisecond contact, the tube's rigid lid and a decay four times faster than the fundamental's, and four independent mechanisms take the tuned partial 80 decibels below it. On a xylophone the same four come to 24.

instruments · Struck bar Two registers 50 microseconds apart on one string. The partials of a 355-millimetre string plucked at 32 and 50 millimetres from the nut, drawn twice: once with the two quills releasing together, once with the far one releasing 0.05 milliseconds later, which is 0.026 of this string's period. A delayed release turns partial n through 2·pi·f_n·dt, so the rotation is proportional to the partial number: the fundamental is turned 9 degrees and partial 24 is turned 226. Simultaneous, the pair is missing partials 17 and 20 — holes it digs for itself where the two combs are equal and opposite. Staggered, it is missing none of them: a rotation of anything at all takes two amplitudes out of opposition. The partials neither comb can excite at all — 7, 11, 22 — are filled either way, because where one comb is zero the sum is the other one whatever its phase. The fundamental's gain over the far register alone falls from 4.36 decibels to 4.34.

The interval between two quills

Two jacks on one key are voiced separately and do not let go at the same instant. That interval turns each partial of the later pluck through an angle proportional to its number — so it leaves the fundamental alone and inverts the twentieth partial, and the holes the pair digs for itself vanish at a hundredth of a period. What a regulator can tolerate turns out to be one fixed fraction of a period at every pitch, which on a five-octave instrument is a factor of sixteen in milliseconds.

instruments · Excitation point

Named alongside it

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

PartialSpectrumInharmonicityBrightnessTimbreDispersionHammerHarpsichordRegisterStanding waveAttack transientEnvelope

All concepts