A hammer is not an impulse

The previous essay treated a hammer as a point delivering an impulse at a position, and got a real result out of it — the one-over-n null, and the reason a piano’s hammers strike where they do. Everything else about a piano hammer is in what that idealisation threw away.

A change in which partials are present is a change in timbre, not in loudness. That is what the handle is doing, and it is the thing an idealised impulse cannot produce.

Why a finite contact is a low-pass #

An impulse — a force delivered in zero time — contains every frequency equally. That is what makes it a useful idealisation and what makes it wrong here.

A real hammer is in contact with the string for a finite time, call it $\tau$. During that time the string is not free: the hammer is holding it. A partial whose period is much longer than $\tau$ sees the contact as effectively instantaneous and is excited fully. A partial whose period is comparable to $\tau$ has time to complete a whole cycle while the hammer is still there — pushing the string one way and then the other against a mass that is not moving with it — and is excited far less.

The force pulse a hammer delivers is roughly a half-sine of duration $\tau$, whose spectrum is

$$\left|\frac{\cos(\pi f\tau)}{1-(2f\tau)^2}\right|$$

flat at low frequency, with its first null at $f\tau = 3/2$ and falling as one over $f^2$ above that. For a contact of 1.6 ms the roll-off begins well below 1 kHz, which on a note at middle C is somewhere around the third or fourth partial.

So the hammer is a filter with a corner frequency of about $1/\tau$, sitting between the string’s ideal spectrum and what is actually excited.

The part that makes it interesting: τ is not a constant #

If contact time were a property of the hammer, all of the above would be a fixed filter and a piano would have one timbre played at any dynamic — louder and quieter, and otherwise the same.

Piano hammer felt is not a linear spring. Its stiffness rises as it is compressed: pressing twice as hard compresses it less than twice as far. A harder blow therefore meets a stiffer spring, and a stiffer spring gives a shorter contact.

Measurements by Donald Hall and Anders Askenfelt in the 1980s put the exponent between about one fifth and one third — contact time falling as force to the power of roughly $-1/4$. Over the dynamic range of a piano, which is a factor of tens in hammer velocity, that is a change in $\tau$ of about a factor of two, and therefore a change of about an octave in where the roll-off begins.

This is the only place on this site where timbre is a function of how hard something is hit. Everywhere else — a vowel, a bore, a plucking point — the spectrum is set by a geometry and the dynamic scales it. Here the dynamic changes it.

Why this matters more than it sounds #

A pianist has, on the face of it, exactly one control per note: how fast the key goes down. Everything else — the escapement, the check, the damper — is on or off. There is a long-running argument about whether “touch” can therefore be anything more than velocity, and the acoustics is unambiguous that key velocity is the whole of what reaches the string.

The mechanism above is what rescues the musical intuition without contradicting the physics. Velocity is one number, but the map from velocity to sound is not a gain control: it moves the spectrum as well as the level. A pianist voicing a chord by playing the melody note louder is not only making it louder; they are making it brighter, and brightness is one of the strongest cues by which the ear separates one line out of a texture.

So a single-parameter control produces a two-dimensional effect, and the second dimension is the one players talk about.

The other half of what a hammer decides is the amplitude envelope: a struck string has an attack of a couple of milliseconds and a long decay with no sustain, and a plucked one has a shorter attack still. That half is not what this essay is about, and it is worth naming so that the spectral half is not mistaken for the whole of the hammer’s contribution.

The number, against the numbers this site already has #

A contact time of one and a half milliseconds is easy to state and hard to feel, so it is worth putting next to the other short intervals the site has established.

It is shorter than the shortest gap two clicks can have and still be heard as two — the precedence window opens at around one millisecond and closes near thirty-five — so the whole of the hammer’s contact happens inside an interval a listener cannot resolve into events.

It is an order of magnitude shorter than the asynchrony that segregates a partial from a note, which is about thirty milliseconds. Everything the hammer does arrives as one event.

It is roughly the period of the note itself at the top of a piano’s compass. The top C of a piano is 4,186 Hz, a period of 0.24 ms, so a 1 ms contact spans four cycles of the fundamental — which is the sense in which the treble is a different regime rather than merely a higher one.

And it is comparable to the fastest onset the ear can use as a timbre cue at all. The first fifty milliseconds of a note carry most of what identifies an instrument, and the hammer’s contact is the first three per cent of that: the whole of the spectral decision this essay is about is made before the attack transient has properly begun.

The contact time this essay has been treating as one number is not one number across the instrument.

Where the corner completes several trips inside the contact the hammer is still there when the string returns, so the two are a coupled system rather than a strike — and the half-sine stops being an approximation and starts being the wrong object. At the bottom of the compass this essay’s own model does not apply, which is the honest boundary on everything above it.

Voicing: the mechanism as a maker’s control #

Because everything above depends on the felt’s stiffness, the felt is what a technician adjusts.

Needling a hammer — pushing needles into the shoulders of the felt — breaks up fibres and softens it. A softer hammer has a longer contact, a lower corner and a duller note. Ironing or applying a hardener does the reverse.

Voicing is therefore not a metaphor and not a matter of taste applied after the fact: it is a direct adjustment of $\tau$, note by note, across 88 notes, and it is why two identical instruments from the same factory can sound substantially different and why the same instrument sounds different after an hour’s work with no part replaced.

It also explains a fact about old pianos that is otherwise puzzling. Hammer felt compacts over years of playing, becoming harder where the strings strike it. An old, heavily played piano is bright and hard in a way that is often described as the instrument “opening up” and is more accurately described as $\tau$ having fallen without anybody deciding it should.

Where the hammer’s width comes back in #

The previous essay noted that a hammer is not a point and softened the null accordingly. The same width has a second effect that belongs here.

A hammer contacting 10 mm of a 1,200 mm bass string is touching under one per cent of it and behaves nearly as a point. The same hammer on a 60 mm treble string is touching a sixth of the string, which is not a point by any reading. The high treble of a piano is therefore a different acoustic regime: the hammer is comparable in size to the string, contact time is comparable to the period of the fundamental, and the idealisations in these two essays are being pushed hard.

That is a large part of why the top octave and a half of a piano sounds as it does — few partials, fast decay, almost percussive — and why it is the hardest part of the instrument to voice.

What a synthesiser has to do about it, and usually does not #

There is a practical consequence that shows how large this effect is: it is the thing sampled pianos are built around.

A sampler that records one note at one dynamic and scales its amplitude produces something immediately identifiable as wrong, and the reason is exactly this essay. Scaling a mezzo-forte sample down does not give a piano; it gives a mezzo-forte piano played quietly, with all its brightness intact, which is a sound no instrument makes.

The response is velocity layering — recording the same note at many dynamics and crossfading between them — and a serious piano library records a dozen or more layers per note across 88 notes, which is a five-figure number of samples. The whole of that expense exists to reproduce a curve that comes out of a compressed piece of wool.

How many layers the felt asks for #

That count is an industry fact and the model implies one, so the two can be put beside each other. Divide the spectrum into bands of partial number, level-match two dynamics, and ask how far apart in force two samples can be before some band is audibly wrong.

Every band is relative to the note’s own total power, so the level has already been taken out and what is left is the shape. The fourth-to-seventh band moves twenty-three decibels across the range while the total stays fixed — which is the essay’s whole argument in one row, and it is why gain cannot substitute for a layer.

Now solve for the spacing. A force ratio of 1.25 puts every band within 1.5 dB of its neighbour, a ratio of 2 within about 2, a ratio of 4 within 7. Over a piano’s dynamic range — a factor of a thousand in hammer force — that gives:

• 45 layers to keep every band within 1 dB;
• 22 layers within 2 dB;
• 8 layers within 3 dB.

A dozen or more per note is exactly where those bracket. The model was told nothing about samplers, nothing about listening tests and nothing about what any library does; it was given a felt exponent measured in the 1980s and asked how fast the spectrum moves, and it asks for roughly the number of layers the industry independently settled on.

That is the strongest form of the argument this section is making. The five-figure sample count is not a matter of thoroughness; it is the felt’s exponent divided by a listener’s tolerance for a spectral error, and either number can be moved to get a different library. A designer willing to accept three decibels of band error needs eight layers and a designer chasing one needs forty-five, and there is no amount of care that removes the requirement, because the thing being sampled is a one-parameter family and a sample is a point on it.

It is worth noticing which way this cuts. The effect is large enough to have shaped an entire industry’s approach to recording an instrument, and it comes from a mechanism simple enough to write on one line. That combination — a large audible consequence from a small stated nonlinearity — is what makes it worth a rung of its own rather than a paragraph in the previous essay.

What the picture cannot show #

The half-sine pulse is a model. A real hammer’s force history is asymmetric — it rises faster than it falls — and it can even lose contact and re-establish it, because the string’s own motion pushes back. Measured force histories look like a half-sine to a first approximation and not to a second.

The string is pushing back. The model here treats the hammer as imposing a force on a passive string. In reality the two are coupled: the string’s reaction is what ends the contact, and on a bass string the reflected wave from the agraffe can return while the hammer is still there. This is the single largest omission and it is why serious piano models are numerical rather than closed-form.

Multiple strings per note. Almost every note above the bass has two or three strings, tuned very slightly apart, struck by one hammer. That is a substantial part of a piano’s tone — the slow beating between them is what gives the note its shimmer and its long compound decay — and none of it is in these figures.

The layer count is as good as its tolerance and no better. Bands of partial number are a crude stand-in for a listener’s frequency resolution, and “every band within two decibels” is a criterion nobody has measured for this task. What the calculation establishes is the shape of the requirement — that the number of layers goes as the logarithm of the dynamic range divided by the logarithm of an acceptable force ratio, and that the force ratio comes straight from the felt’s exponent. The agreement with what libraries actually record is a check on the model rather than a derivation of the practice.

And nothing here is measured on a piano. The exponent is Hall and Askenfelt’s; the contact times are their range; the spectra are computed from those numbers and from the ideal string. What the figures show is the model’s behaviour, and its value is that it produces the right qualitative fact — brightness rising with force — from a mechanism rather than from a curve fitted to the observation.

The same trick, on two other instruments #

A nonlinearity that couples loudness to spectrum is not unique to the piano, and the two other places it appears in this field are worth naming because they arrive by different routes.

A brass instrument gets brighter as it gets louder for a reason that is nothing to do with its excitation: at high amplitudes the wave steepens as it travels down the bore, because the compressions travel slightly faster than the rarefactions, and a steepening wave is one gaining upper partials. A fortissimo trombone is generating high frequencies in the air column itself that were not present at the mouthpiece. The audible result is the same kind of thing as the piano’s and the mechanism is entirely different.

Three fixed spectra — a string, a reed and a pure tone — are worth having in the ear as something to hear the moving one against. Nothing about them changes with how hard a button is pressed, because a spectrum specified as a list of amplitudes has no dynamic in it at all, which is exactly the assumption this essay is dismantling.

A bowed string does the opposite, or at least does not do the same. Its spectrum is a sawtooth set by the geometry of the Helmholtz corner and it is remarkably stable across dynamics: a violinist’s control over brightness comes largely from moving the bow toward the bridge rather than from pressing harder, and that is a change of position, which is the previous essay’s variable rather than this one’s.

Three instruments, three answers to “does it get brighter when it gets louder”, and the differences are diagnostic. A piano’s brightness comes from a material, a brass instrument’s from the air, and a violin’s is bought with a hand movement rather than with effort. Whether loudness and timbre are separable is a property of the instrument, and the ones where they are not are the ones players describe as having colour.

Whose instruments, and when #

The nonlinear felt hammer is a nineteenth-century development and is bound up with the rest of the modern piano: heavier hammers, higher tension, an iron frame. Earlier fortepianos had leather-covered hammers with quite different mechanical properties and a correspondingly smaller change of timbre with dynamic, which is one concrete component of what is meant when a fortepiano is called clearer and less “coloured” than a modern instrument.

The measurements are from a body of work in the 1970s and 80s — Hall’s papers on piano string excitation and Askenfelt’s Five Lectures on the Acoustics of the Piano (1990) — and they are measurements on particular instruments. The exponent range quoted here is a range because it varies with the hammer, the note and the state of the felt, and any single number for it would be a claim about one piano.

Where this goes #

This closes the struck string for the moment. The field turns next to a string that is not struck at all but driven continuously, where the excitation is a stick-slip cycle rather than a blow, and where the string’s shape turns out to be two straight lines meeting at a travelling corner rather than anything a sine could describe.

Further out, the same nonlinearity that makes felt interesting is what makes a reed and a bow work at all. A linear system driven at one frequency responds at that frequency and nothing else; every self-sustaining instrument in this field is nonlinear somewhere, and the hammer is the one case where the nonlinearity is a passive material rather than an active valve.