Where the hammer lands

A string vibrates in all its modes at once, and which of them it vibrates in, and how strongly, is decided almost entirely by one number: where it was touched.

The rule is exact and it is one line. A string’s displacement is a sum of modes, the nth of which is a sine with nodes at $k/n$ of the length. Setting the string in motion at a point $p$ excites each mode in proportion to that mode’s displacement at $p$, which is $\sin(n\pi p)$. When $p = k/n$ for whole $k$, that sine is zero. The mode has a node under the hammer and the hammer cannot move a node.

Why this is more than a curiosity #

Every stringed instrument in the world has an excitation point, and in almost every case it is fixed by the construction rather than chosen by the player.

A piano has hammers, mounted on a rail, striking at a position the builder decided. A harpsichord has plectra on jacks at a fixed distance from the nut. A harp and a guitar let the player move, and players use it as a control — playing near the bridge for a bright, nasal tone and over the fingerboard for a round one — but even there the useful range is a small fraction of the string.

So an instrument’s characteristic tone is set, to a substantial degree, by one length that a designer chose once. And because the rule above says which partial is nulled, that choice is legible: measure the striking point, take its reciprocal, and the answer is the partial the designer decided to do without.

The seventh partial, and why a piano avoids it #

Piano hammers strike between about one seventh and one ninth of the speaking length, depending on the register and the maker. The usual explanation is that this removes the seventh partial, and the usual explanation is right, but it is worth saying why the seventh in particular deserves removal.

The seventh partial of a harmonic series is at a frequency ratio of 7:1 to the fundamental. Reduced into an octave that is 7:4, which is 969 cents — a minor seventh 31 cents flat of the equal-tempered one and 27 cents flat of the just 16:9. It is not close to any note in the twelve-tone system.

That matters because roughness is computed from the interaction of partials and a partial that sits between two scale degrees is rough against notes on both sides of it. A piano is a chordal instrument: it plays several notes at once, constantly, in a twelve-tone temperament. A strong seventh partial in every note would be a strong out-of-tune component in every chord.

Removing it costs almost nothing, because the seventh partial is weak anyway in a struck string’s spectrum — its amplitude before the null is applied is about a seventh of the fundamental’s. What is removed is a small amount of energy in a bad place, and both halves of that are quantities.

Take a C major triad on a one-over-n spectrum, null one partial at a time, and measure what happens to the summed roughness.

One and three tenths per cent of the energy buys seven per cent of the roughness, which is the “small amount in a bad place” made a ratio: a factor of five and a half more roughness removed than energy.

The row above it is the sharper result. The sixth partial is dominated outright — nulling it removes more energy and saves less roughness than nulling the seventh. That is not a trade-off; it is a strictly worse choice, and it is exactly what the cents ruler predicts, since the sixth partial is a perfect fifth two octaves up and sits two cents from a scale degree while the seventh sits thirty-one.

Against its other neighbours the seventh is a genuine trade rather than a domination. The fifth saves half as much again and costs twice as much energy; the eighth costs three quarters as much and saves three quarters as much. So the honest statement is that the seventh is the point where the curve of savings per unit of energy turns over among the partials a designer could plausibly sacrifice — and that a maker who put the hammer at one sixth instead would be paying more to get less.

Reading a striking point off an instrument #

The rule runs backwards, which makes it a small piece of forensic work rather than only an explanation.

Measure the distance from the agraffe or nut to the point the hammer contacts, divide it by the speaking length, and take the reciprocal. On most mid-nineteenth-century and modern grands the answer in the middle of the compass is between 7 and 9. On many earlier fortepianos it is nearer 9 to 11, and on some eighteenth-century instruments further still — a smaller fraction, a null higher in the series, and a brighter tone.

The trend is worth stating because it runs against the way the instruments are usually described. Early pianos are described as clearer and less powerful and modern ones as fuller and richer, and the striking point moves in the direction that removes a lower partial as the instrument develops. Some of that fullness is a lower null, some is heavier hammers with longer contact, and some is a heavier frame carrying more tension; the point is that at least one component of a tonal history that is usually discussed in adjectives is a length that can be measured with a ruler.

The same measurement on a harpsichord gives numbers in a different range and a different logic behind them. A harpsichord’s plucking points are close to the nut — often a fifteenth to a twentieth of the string — which nulls nothing audible and excites everything. That is not carelessness: a plucked string’s spectrum falls as one over $n^2$, so a harpsichord starts with far less high-partial energy than a piano and can afford to throw none of it away. Two instruments, opposite decisions, both following from the same rule applied to different excitations.

That curve is what a maker actually has to hit, and it is the reason the traditional figure is a range rather than a point.

The null is not as clean as the arithmetic #

Three things soften it, and the essay would be dishonest without them.

A hammer is not a point. It is a felt-covered head perhaps 10 to 15 mm wide, which is a substantial fraction of a short treble string. It therefore excites a band of positions rather than one, and the null is filled in by the parts of the contact patch that are not exactly on the node.

A string is not perfectly flexible. Stiffness pushes each partial sharp by an amount that grows as the square of the partial number, so the seventh partial is not at exactly seven times the fundamental and its nodes are not at exactly the sevenths. The null lands slightly off where the ideal calculation puts it.

The striking point is not constant. A piano’s hammer line is a curve, and the fraction of the speaking length it strikes changes across the compass — nearer one eighth in the middle, further toward one ninth or beyond in the treble, and quite different in the bass where the strings are so long that a hammer at one seventh would be a metre from the agraffe.

So what a real piano has is a dip rather than a null, in roughly the right place, over roughly the right range of notes. That is still a design decision and still legible in the instrument; it is not the textbook zero.

The plucked and the struck string are not the same envelope #

The rule above decides which partials are nulled. Something else decides how fast the rest fall away, and it differs between a pluck and a strike.

A plucked string starts as a triangle — two straight segments meeting at the plucking point — and a triangle’s Fourier coefficients fall as one over $n^2$. A struck string starts with a velocity distribution concentrated near the hammer rather than a displacement, which gives an envelope falling as one over $n$: substantially brighter, with far more energy in the upper partials.

That difference is larger than anything the excitation point does. A harpsichord and a piano differ far more because one plucks and the other strikes than because their excitation points differ, and the reason a harpsichord cannot play loudly is bound up in the same fact: a plectrum releases the string from a fixed displacement, so the amplitude is set by the plectrum rather than by the player.

What a player controls, on the instruments that allow it #

On a guitar, an oud, a harp or a pizzicato violin the player chooses the point, and what they are choosing is which partials to suppress.

Playing near the bridge — say a twentieth of the length — puts the first null at the twentieth partial, so nothing in the audible part of the spectrum is removed and every low partial is excited relatively strongly. That is the bright, nasal sul ponticello pizzicato and the flamenco guitarist’s attack.

Playing over the sound hole or the twelfth fret — a quarter to a third of the length — nulls the fourth or the third partial and gives the round, flute-like tone classical guitarists use for lyrical lines.

This is a genuine instance of a performance practice that is usually taught as a matter of feel being exactly computable. A guitarist who plays “a third of the way along for a sweeter sound” is nulling the third and sixth partials, and the sweetness is their absence.

What it sounds like, and what it does not #

It is worth being careful about how large this effect is, because the arithmetic is so clean that it invites overclaiming.

Moving the excitation point from a third of the string to a twentieth changes the tone unmistakably — every guitarist and every harpist uses it, and the buttons under the figures above make it obvious. But most of that change is not the null. It is the envelope: exciting near the bridge puts more energy into the high partials across the board, and exciting near the middle puts less. The null is a notch in a spectrum whose overall tilt is doing more of the work.

The split can be taken apart, and it comes out further than “most”. The plucked spectrum’s centroid moves from 1.08 to 1.78 partials between a third of the string and a twentieth — a shift of 0.695. Replace every sine by its own rising envelope, so that the tilt is kept and the zeros are filled in, and the same move gives 0.770. The envelope accounts for a hundred and eleven per cent of the brightening, and the nulls account for minus seven: they pull the centroid very slightly back down, because at any position the lowest null is the deepest hole and it is always below the centroid.

So the nulls contribute nothing at all to why plucking near the bridge sounds brighter. They are a real feature of the spectrum, they are audible as a colour, and the direction of the tone change a player is reaching for is entirely the tilt.

The place where the null itself is decisive is exactly the piano’s case: a fixed instrument, a specific partial known to be out of tune with the temperament, and a designer able to place a notch on it. There, one partial out of the first dozen is being singled out and removed, and the reason it is that partial rather than another can be stated in cents.

Elsewhere — a guitarist’s right hand, a harpist’s — the null is real, computable and secondary. Saying “the sweetness is the absence of the third partial” is true and is not the whole truth; the rest of it is that there is less of everything above the fourth as well.

Where else this argument appears #

The rule “excitation at a node does not excite that mode” is completely general and it turns up in every part of this field.

A register vent works by the mirror image of it — placing a hole where a mode has a pressure node so that opening it disturbs that mode least and every other mode most. Here the hammer avoids a node to excite a mode; there the hole seeks one to preserve a mode.

A bell is struck at its rim, at the position of the maximum displacement of the modes a founder wants and the node of some they do not.

A timpani is struck a third of the way in from the rim rather than at the centre, and the reason is exactly this rule: the centre is a node of every mode except the axially symmetric ones, so striking there excites the modes that make a drum sound like a thud and suppresses the ones a kettledrum’s whole design exists to produce.

Three instruments, three completely different mechanisms, one line of trigonometry underneath all of them.

What the picture cannot show #

The spectrum at the moment of excitation is not the spectrum a listener hears. Partials decay at different rates — high ones faster, because losses rise with frequency — so a note’s spectrum a second after it starts is much duller than the one drawn here. What the figures show is the initial spectrum, which is what the excitation point decides.

The soundboard and the bridge are not in this model. What reaches the room is the string’s spectrum through the instrument’s radiation response, and the body of a violin or the soundboard of a piano imposes its own resonances on it. A null in the string’s spectrum is a null in what is available to be radiated, and the body can amplify what remains very unevenly.

Nothing here says how loud each partial actually is. The envelopes above — one over $n$ and one over $n^2$ — are the ideal shapes for an ideal excitation. A real hammer’s finite contact time low-passes the spectrum on top of them, and that low-pass moves with how hard the note is struck, which is where a piano’s dynamics come from.

Whose instruments, and when #

The mathematics is Daniel Bernoulli’s and d’Alembert’s, from the 1740s and 50s, and the null rule was understood before any of the instruments discussed here reached their modern form. Whether early piano builders reasoned from it or arrived at the position by ear is genuinely unclear; the striking point drifted through the eighteenth and nineteenth centuries and settled where it is by about 1850, which is the period the modern instrument settled in generally.

The claim about the seventh partial being unwanted is a claim about a repertoire and a tuning system: it is unwanted because the music is chordal and the temperament is twelve-tone. In a tradition that uses the seventh partial as a consonance — and several do, from barbershop singing to some tunings of the guqin — the same partial is an asset, and an instrument built for that music would have no reason to null it.

Where this goes #

The next rung is the hammer itself, which the model above treated as an idealised point delivering an impulse. It is neither, and the departures are where a piano’s most characteristic behaviour comes from: contact lasts a couple of milliseconds and the felt is a nonlinear spring, so a loud note is not a scaled-up quiet one but a brighter one.

Sideways, the excitation point argument has an exact counterpart in the bowed string, where the bow’s distance from the bridge does something quite different: it does not null partials at all, because the bow’s motion produces a sawtooth whose spectrum barely changes with position — and understanding why the two cases differ is a better test of the model than either case alone.