The pulse that was assumed

The hammer that is heavier than its string ended by naming the rung this makes computable and that essay declined to attempt. The contact is a coupled problem — hammer, felt, string, and a reflection arriving mid-contact — and it has a standard numerical treatment: step the hammer forward against the felt’s nonlinear force and against the string’s own returning corner, and read the force pulse rather than assuming a half-sine.

It also came with a prediction, and the prediction is the interesting part, because the computation refuses half of it.

The half-sine has nulls and the real pulse does not. That is the first finding and it is the one that matters most for every figure in this ladder, because a null in the excitation is a partial that is not there.

What the integration is #

Three equations, none of them new.

The felt is a nonlinear spring: the force is K times the compression raised to a power p, and p is 2.5, which is the middle of the published range for piano felt. That nonlinearity is what a hammer is not an impulse is about — it is why a harder blow makes contact shorter and therefore brighter, and it is the only place in this collection where timbre is a function of how hard something is hit.

The hammer is a mass: its acceleration is minus the force over its mass, and nothing else acts on it.

The string is a characteristic impedance plus its own history. A force applied at a point on an infinite string produces a velocity of F/2Z; a real string is finite, so the wave launched at the strike point runs to each end, reflects inverted from the fixed termination, and comes back — and when it does it subtracts from the drive-point velocity. The string pushes back harder once its own corner has come home.

v_s(t) = (1/2Z)·[ F(t) − F(t−T₁) − F(t−T₂) + F(t−T₁−T₂) − … ]

T₁ and T₂ are the round trips to the near and far ends, which for a strike at one eighth of the length differ by a factor of seven — and where the hammer lands is the rung about why it is one eighth.

The felt stiffness K is not published and is not fitted to anything musical. It is solved for, at every note, so that the same hammer against a rigid wall gives exactly the contact time this site has published since the ladder’s second rung. That makes the comparison a controlled one: the hammer’s own behaviour is held fixed and only the string is allowed to differ, so everything that separates the two curves is the string.

Why a null is the thing to fix #

Before the prediction, the reason this rung is worth an essay rather than a footnote.

A half-sine of duration τ has zeros in its spectrum at every multiple of 1/τ above the first, and those zeros are exact. So every excitation figure in this ladder has been drawing a small number of partials at literally nothing — not attenuated, absent — and those partials are wherever the arithmetic happens to put them.

The real pulse has no zeros at all. It is not a half-sine, it is not symmetric, and its shoulders are set by a string that is still moving; a function like that has minima but no nulls. So the ladder’s figures have been asserting the strongest possible statement about certain partials on the strength of a shape nobody measured.

That is the same class of defect as the excitation nulls a plucking point produces, and it is worth keeping the two apart. A plucking null is real: excite a string at one over n of its length and the n-th partial has a node under the finger and receives nothing, exactly. A contact null is an artefact of assuming a sine.

The prediction, and the half of it that fails #

The previous rung’s prediction was stated precisely, which is what makes it possible to say it is wrong. In the bass, where the corner never returns during contact, it should differ hardly at all, and in the treble, where it returns forty-five times, it should differ completely.

The bass is where the two contact times differ most, and the reason is the previous rung’s own finding. A bass hammer weighs more than the string it strikes; the string yields under it and moves away; and a hammer chasing a retreating string stays in contact far longer than one meeting a wall. The round trips have nothing to do with it — at C2 the corner comes home three times and at C7 forty-nine, and the contact-time error is worst where the trips are fewest.

That last clause is the part to hold loosely, and the limitations below say why: the comparison is against a rigid-wall contact held at 1.60 milliseconds at every note, which is a calibration and not a measurement. Re-running with a rigid contact that falls across the compass the way real hammers do makes the discrepancy U-shaped rather than falling — largest at both ends and smallest in the middle. What is robust is that a bass hammer chasing a light string is one mechanism that lengthens contact; what is not robust is that it is the only one showing up in the duration.

So the prediction is half right and the half that is right is about the spectrum rather than the duration.

Two mechanisms, and the previous rung conflated them. The mass ratio governs how long contact lasts; the round-trip count governs how far the spectrum departs from a smooth pulse. They run in opposite directions across the instrument, which is why one prediction covering both was going to be wrong somewhere.

The calibration test below sharpens that into its final form and takes one word out of it. Both mechanisms lengthen the contact, at opposite ends of the compass, and only one of them scatters the spectrum. So the division is not duration versus spectrum; it is that the round trips do two things and the mass ratio does one, and the instrument is arranged so that each mechanism owns an end. The middle of the piano is where neither is in charge, and it is the only place where a half-sine was ever going to be a fair approximation.

Which computation produced the numbers #

The integration is explicit Euler at a two-hundred-nanosecond step, which is about eight thousand steps per contact and is small enough that halving it moves the contact time by under a tenth of a per cent.

The string’s mass, length, wave speed and impedance at each note come from pianoString, which is the scaling this site has used since the perception phase. The hammer’s mass comes from hammerMassRatio, which is the previous rung’s own function.

The felt stiffness is found by bisection on the rigid-wall contact time, as above. The felt exponent is 2.5 and is stated rather than derived; at 2.0 the pulses are broader and at 3.0 sharper, and the comparison between computed and assumed moves by a few decibels and keeps its shape.

Each spectrum is the magnitude of the force pulse’s Fourier transform evaluated at each partial’s own frequency, normalised to the first — which is what an excitation spectrum is, and is exactly what hammerRolloff computes analytically for the half-sine.

Where the model stops #

The string is a pure impedance and it is not. A real piano string is stiff, so its high partials travel faster and its corner does not come home as a corner. The piano is tuned wrong on purpose is the essay about that dispersion, and it means the returning wave is smeared over a time comparable with the contact in the treble — which would round off exactly the serrations this model draws most sharply.

The hammer is a point and it is not. The hammer is not a point either established that the contact has a width of a centimetre or so, which spreads the excitation over a region of the string and low-passes it a second time. That effect is in this ladder already and is not in this integration, so the computed spectra above are too bright at the top.

And the treble contact times are too long against measurement. Real piano contact times fall from a few milliseconds in the bass to well under one in the treble, and this model gives three milliseconds at C7. The reason is in the calibration: the felt stiffness is solved for so that the rigid-wall contact is 1.6 milliseconds at every note, and a real treble hammer is not merely lighter but very much harder.

Fixing that properly needs a published stiffness scaling across the compass, which this collection does not have. What it can do is ask how much the finding depends on the calibration, by re-running the integration with a rigid-wall contact that falls the way real hammers do — 2.5 milliseconds at C2 to 0.5 at C7, which is asserted rather than measured and is the point of the exercise:

The computed contact as a multiple of the assumed one.

The headline result does not survive. Under the essay’s own calibration the discrepancy is largest in the bass and that is the whole of “the prediction is wrong in the direction nobody expected”. Under a scaled one it is U-shaped: 2.55 in the bass, a minimum of 1.35 around C5, and 2.70 at the top. The bass is no longer where the two models differ most.

What survives — and is a better version of the same finding — is the two-mechanism reading. Both mechanisms lengthen the contact, and they act at opposite ends: the mass ratio in the bass, where a heavy hammer chases a light string, and the returning corner in the treble, where the string answers back before the hammer has let go. The essay’s calibration flattened the second by handing treble hammers a rigid contact three times longer than they have, which is exactly the regime where the round trips do their work. The minimum in the middle is where neither mechanism is in charge, and the middle of a piano is where the half-sine was always going to be least wrong.

The trip counts move with it. C7 falls from 49 round trips inside the contact to 23, because a shorter contact contains fewer of them — so the forty-nine quoted below is a calibration artefact too. The crossover the last section names, at about ten trips, sits between C5 and C6 either way.

Whose music, and what a voicer is doing #

The physics is about pianos. The practice it explains is one that is usually described as a mystery.

A piano technician “voices” an instrument by needling the hammer felt — pushing needles into it to soften it, or applying lacquer to harden it — and the effect is described entirely in adjectives. What the model says is that voicing moves K, that K sets the rigid contact time, and that the contact time sets the corner frequency of the excitation. A softer hammer is a longer pulse is a darker note — the same chain the mark that is not a level follows from the other end, where the player rather than the technician moves it, and the whole of the technician’s vocabulary maps onto one number.

What the model adds is that the mapping is not the same at both ends of the instrument. In the treble the contact is ended largely by the string’s returning corner, so softening the felt changes the pulse’s shape more than its length; in the bass the contact is ended by the hammer running out of momentum against a string that is lighter than it is, so softening the felt lengthens it directly. That is a fair description of why voicing the top two octaves is a different job from voicing the bottom one, which every technician says and no acoustics text derives.

The claim about the repertoire is narrow and worth stating anyway. Everything here is about the modern piano, which is to say an instrument of roughly the last hundred and fifty years with heavy felt hammers on a metal frame. A fortepiano’s leather-covered hammers are lighter and harder, its strings are lighter still, and both terms move — so the balance between the two mechanisms above is different, and there is no reason to expect the two instruments to be voiced by the same reasoning.

What the picture cannot show #

Whether any of the difference is audible. The two spectra differ by tens of decibels at partials that are already forty decibels down, and the ear’s own masking spread may be wider than the differences. What can be said is that the assumed pulse’s nulls are audible in the sense that matters — a null puts a partial at nothing, and nothing is a very different amount from something.

Nor what the string does with it afterwards. The force pulse is the input; what a listener hears is that input through the string’s own modes and then through the soundboard, and both have far more structure than the excitation does. A twenty-decibel difference in excitation at the eighteenth partial is a twenty-decibel difference in a partial that the board may radiate at a tenth of the efficiency of its neighbours.

And nothing here is measured. The integration is a standard treatment evaluated at this site’s own parameters, and no piano was recorded. The one measurement that would settle it is a force sensor between hammer and string, which has been done and whose published pulses are the shape this integration gives rather than the shape it replaces.

The number worth keeping #

If one thing survives from this rung into the rest of the collection it should be the trip count, because it is a single number that says which regime a note is in.

The corner’s round trip is 2x/c for the near end, where x is the strike distance and c the wave speed. Divide the contact time by it and the answer runs from three at the bottom of the piano to forty-nine at the top. Below about ten the string is effectively infinite during the blow and the hammer is talking to an impedance; above it the string is a resonator that answers back before the hammer has let go.

The crossover sits around the fifth octave, which is where piano makers change several things at once — the strings stop being wound, the strike point moves, the hammers get abruptly lighter. Whether those changes are responses to this crossing or to any of the half-dozen others that happen near it is not something arithmetic can settle, and it is a strikingly crowded part of the instrument.

Where this ladder goes next #

Five rungs. Where the hammer lands and which partials it silences; that contact takes time and low-passes the excitation; that the contact has a width as well as a duration; that the hammer’s own mass is compared with the string’s and the ratio inverts across the compass; and now the coupled problem all four were approximations of, which agrees with them at the bottom of the spectrum and does not at the top.

The rung after it is the one the dispersion limitation names. The returning corner is what makes the treble pulse serrated, and a real corner does not return as a corner because a stiff string’s partials travel at different speeds. So the sharpness of the serrations is set by the inharmonicity coefficient — a quantity this collection has computed for every note on the piano since the harmonic-series ladder’s second rung — and putting the two together would say how much of this rung’s own treble result survives the instrument it is about.