The hammer is not a point either

This ladder has two rungs and both of them are about the same event, taken apart along different axes. Where the hammer lands decides which partials have a node under it and are therefore not set moving at all. How long the hammer stays decides how high up the series the force pulse reaches before its own first null cuts it off.

Both rungs are about a piano hammer, and both were checked against what the instrument does. What neither of them varied is the one dimension a hammer obviously has and a point does not.

Where, and for how long. There is a third: how wide. A hammer is not a point and neither is a finger, a plectrum or a ribbon of bow hair, and every comb this site has drawn assumes a contact of no extent whatever.

What a width actually does #

Setting the string moving over a width rather than at a place means integrating the mode shape across the contact. The arithmetic is short: a force spread uniformly over a width w centred at p gives partial n an amplitude proportional to sin(nπp) · sin(nπw/2) / (nπw/2), which is the point-contact answer multiplied by a sinc.

Two things follow and the first is the surprise. The sinc multiplies; it does not add. The first factor — the one that produces the nulls — is untouched by the width, so a partial with a node exactly under the middle of the contact is still exactly absent, however wide the contact is.

The reason is worth having in words as well as in algebra. A mode shape is a sine, and a sine is odd about its own node: as much of it is positive on one side as is negative on the other. A symmetric contact pushes on both halves equally, the two contributions cancel exactly, and the partial gets nothing. Widening the felt makes the cancellation happen over a wider region; it does not make it stop happening.

What the width does instead #

The sinc has its own zeros, at partial 2/w and its multiples, and it rolls off before them. For the piano hammer above that first zero is at partial 77, and the response is down three decibels at about partial 34.

A piano’s audible partials in the middle register run to perhaps twenty or thirty. So the width is a low-pass filter whose corner sits at or above the top of the series that matters, and its effect on the note is small.

That is a genuinely deflating answer to the question this rung asked, and it is worth stating as one: the third variable is dominated by the second. The contact time puts its first null between the second and the sixth partial depending on how hard the note is struck. The contact width puts its first null at the seventy-seventh. The two effects are the same kind of low-pass and they differ by more than a factor of ten, so a discussion of hammer width that does not first dispose of hammer time is discussing the smaller of two things.

The treble is in that régime #

The deflating answer above is computed at one note, and it does not survive the compass. A hammer shrinks from about eighteen millimetres in the bass to about ten at the top, a factor of not quite two; the speaking length falls from 1,150 millimetres to under fifty, a factor of twenty-five. So the ratio the sinc depends on grows by a factor of thirteen up the instrument:

Above about A6 the width’s first zero falls inside the audible series, and at the top of the instrument it is at the ninth partial — which is squarely the range the contact time is usually credited with. The claim that the third variable is dominated by the second is true across the lower five octaves and false across the top two, and it is false because the string shrinks and the hammer does not.

That also changes what the top of a piano is. In the bass the hammer is a point and the strike place is everything; in the treble the hammer covers a fifth of the string, its own low-pass sits among the first ten partials, and the strike place is a much weaker term because the comb it produces has been rolled off before it can be heard.

And the tolerance shrinks with the string #

The same factor of twenty-five applies to the tolerance, because a tolerance stated as a fraction of the length becomes a distance when the length is fixed. Twenty decibels of suppression on the seventh partial needs the hammer within 5.2 millimetres in the bass and 0.24 at the top — a quarter of a millimetre, on a component struck fifty times a second by a mechanism made of wood and felt.

Which is the real content of the closing paragraph below. The strike line is a surveyed curve rather than a proportional rule not because a proportion is hard to compute but because the error budget changes by a factor of twenty-five along it, and a rule stated as a fraction gives the treble a tolerance nobody can hold.

The number that does matter #

If the width does not spoil the null, something does — because no piano has a genuinely silent seventh partial. The something is placement.

Near a node the mode shape is linear, so a partial’s surviving amplitude at an offset ε from its node is nπε times what it would be at the antinode. Inverting that gives a tolerance directly.

Under three millimetres is a manufacturing tolerance rather than a design idea. It is achievable — a piano action is built to better than that — but it is not free, and it has to hold across a compass in which the speaking length changes by a factor of twenty.

So the design decision the first rung of this ladder identified is really a specification, and one whose difficulty is not constant along the instrument. Putting the strike point between a seventh and a ninth of the length is a choice about which partials to suppress; holding it there to a millimetre is what makes the choice mean anything. And it explains a thing about the instrument that otherwise looks fussy: the strike line on a piano is a carefully surveyed curve across the plate, not a proportional rule applied per note.

It is worth seeing what the idealisation actually predicts before seeing what spoils it.

That figure is the whole of the textbook claim, and the striking thing about it is how much it deletes — eight of the first sixteen partials, not one. A point excitation at a rational fraction kills the whole family of partials with a node there, which is a far stronger prediction than “the seventh is suppressed” and is correspondingly easier to falsify.

What else spoils the null, and it is not small #

The tolerance above is the geometric part. There are two more contributions and one of them is bigger.

Stiffness moves the nodes. A real string is not perfectly flexible, so its modes are not exact sines and its partials are stretched sharp by an amount that grows as n². A mode whose frequency is not n times the fundamental does not have its node at exactly p = k/n either, and the discrepancy grows with partial number — which is exactly where the strike point is being asked to be most accurate.

And the string is struck at one point but terminated at two. The comb assumes ideal, immovable ends. A real bridge moves — that is how the sound gets out, and it is the whole of what a unison’s aftersound depends on — so the boundary is not a perfect node and every mode leaks a little into every other.

The same argument on a drum, where it is visible #

The odd-function result is not about strings and it is easiest to see where the geometry is two-dimensional.

Strike a circular membrane at its exact centre and every mode with a nodal diameter gets nothing, because the centre lies on all of them — the excitation is symmetric under every rotation and those modes are not. What survives is the family of modes with nodal circles only, which is a much smaller and much more nearly harmonic set.

The same argument is visible on a drum, where the modes of a membrane are not a harmonic series at all: an ideal membrane’s modes sit at ratios nobody would call an octave, and a kettledrum’s are pulled into something nearer one by the air inside it. A strike point that suppresses a mode there suppresses something that was never a partial of anything, which is what a drum is doing instead of having a fundamental.

The width point transfers with it. A drumstick’s tip is several millimetres across and the modes it cancels are cancelled just as exactly, because the contact is symmetric about the point it is centred on. What breaks the symmetry is being off centre, not being large, on a membrane exactly as on a string.

What it means for the ladder’s first result #

The first rung of this ladder made a claim about design: 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, and that is a decision rather than an accident.

Everything above sharpens that claim in one direction and weakens it in another.

It sharpens it because the null is robust to the thing that looked most likely to spoil it. A reader meeting the comb for the first time reasonably objects that a hammer is not a point, and the objection turns out to be answerable exactly: the width cancels out. So the design idea survives contact with the felt.

It weakens it because the null is fragile to the thing that looked least likely. Under three millimetres of placement, on a string whose partial positions are themselves moved by stiffness, is a demanding specification — and it means the suppression achieved in a real instrument is not the −120 decibels of the ideal calculation but something in the twenties or thirties. The seventh partial is 31 cents from anything a keyboard has, and it is present at thirty decibels down rather than absent.

Both halves are worth having, because the interesting property of a design argument is which of its assumptions it can afford to lose.

Which computation produced the numbers #

The width factor is the analytic integral of the mode shape over a uniform contact, which is the sinc above; nothing about it is fitted. The point-contact comb is pluckSpectrum unchanged, so a placement with no width passes through the same code it always did.

The tolerance is the linearisation of sin(nπx) about its zero, evaluated for a stated suppression in decibels — 2.82 mm at 20 dB and 0.89 at 30 for the seventh partial on a 620 mm string. It is exact in the limit and good to a per cent over the range drawn.

The hammer width of sixteen millimetres and the speaking length of 620 are ordinary figures for the middle of a medium grand. Both are stated rather than derived, and the ratio between them is the only thing the sinc sees.

Whose instrument, and when #

The strike-point convention — between a seventh and a ninth — belongs to the piano and dates from the instrument’s nineteenth-century consolidation. Harpsichords pluck much nearer the end and get a completely different spectrum for the same reason read the other way. A guitarist moving the right hand from the sound hole to the bridge is sweeping p over a range the piano fixed once and for all.

What travels is the odd-function argument, and it travels anywhere: a symmetric excitation centred on a node cannot drive that mode, whatever its extent. It is the same statement as a symmetric load not exciting an antisymmetric mode in a beam, and as a centrally struck drumhead producing no modes with a nodal diameter.

What the picture cannot show #

The force is taken as uniform across the contact. Felt is not: it is a nonlinear spring compressed most in the middle, so the real distribution is peaked rather than flat. A peaked distribution has a wider sinc and a lower first zero — it low-passes less, not more — so the finding that the width barely matters is if anything understated.

The contact does not move. A real hammer rolls slightly on the string as it compresses, and the strike point drifts by a fraction of a millimetre during a contact that lasts a couple of milliseconds. That drift is of the same order as the tolerance computed here, which is the one place where the two axes of this ladder genuinely interact and neither rung models it.

The bridge end is not a node either. A piano’s bridge moves — that is how the sound gets out — so the boundary the mode shape is measured from is itself in motion, and the node it defines is not exactly where the ideal calculation puts it. On a millimetre tolerance that matters.

Nothing here is measured on an instrument. The tolerance is a consequence of a mode shape, not a reading from a piano, and a real note’s seventh partial is suppressed by all three mechanisms at once with no way to separate their contributions from the outside.

The bow is not modelled at all. A bow’s hair ribbon is several millimetres wide and it does not set the string moving by a blow — it drives it continuously through a stick-and-slip cycle, and the width of the ribbon enters that cycle at the point where the corner passes under it rather than as an integral over an initial condition. The sinc argument here says nothing about it.

Nothing here uses the string’s own stiffness in the mode shapes. The tolerance is computed from an ideal sine, and a stiff string’s modes are not sines — they are pushed toward the ends, which moves the node the tolerance is measured from. The direction of that correction is known and its size is not computed here.

And the width and the time are treated as independent. They are not: a harder blow makes the contact both shorter and, because felt compresses, wider. The two low-passes therefore move in opposite directions with dynamic level, and the essay on dynamics computes only one of them.

Where a player uses this, and it is not a piano #

A pianist cannot move the strike point and a guitarist does nothing else. A bowed player is in between: the bowing point is a continuous control, and the region in which a note can be sustained at all is drawn against that same coordinate. Moving the right hand from over the sound hole to the bridge sweeps p from roughly a fifth of the string to a twentieth, which moves the first suppressed partial from the fifth to the twentieth and changes the timbre from round to nasal — the same one-parameter family this ladder computes, driven continuously in performance.

The width matters more here than on a piano, for a reason the sinc makes obvious: a finger pad is perhaps ten millimetres on a 650-millimetre string, which is about the same ratio as the piano hammer — but a nail is a millimetre, and a plectrum less. That factor of ten in the contact width moves the sinc’s corner by a factor of ten, from around partial 34 to around partial 340, which is why a plectrum sounds brighter than a fingertip on a string plucked at the same place. It is the one place in this essay where the width is the whole of the difference.

One more use is worth recording, because it turns the tolerance into a test. If the seventh partial’s suppression depends on placement to within three millimetres, then measuring how suppressed it is on a real instrument measures the placement — and the measurement is available from a recording rather than from a ruler. The spectrum of a struck note at a stated dynamic contains the answer, and the two mechanisms that would confuse it — the contact time and the string’s stiffness — both have their own signatures elsewhere in the same spectrum.

Where this ladder goes next #

Three variables of one event: where the exciter lands, how long it stays, and how wide it is. The third turns out to change almost nothing about the spectrum and to sharpen the first into a tolerance in millimetres.

What is left is the exciter’s own mass, which is the variable none of the three has touched. A hammer that is heavy compared with the string it strikes rebounds slowly and stays in contact through more than one reflection of the corner it makes — and a hammer that is light bounces off before the corner returns. That is a different ladder’s shape of argument, since it makes the contact time depend on the string rather than on the felt, and it is the only remaining thing about a struck string this collection has never computed.