The corner does not come back a corner

The fifth rung of this ladder stopped assuming the hammer’s force pulse and computed it, stepping the hammer’s motion forward against the felt’s nonlinear spring and against the string’s own reaction — which is its characteristic impedance plus every wave that has already been launched and come back. That impedance is the same quantity a bow feels, met from the other side. The returning wave is the interesting term: it is the corner the hammer itself made, arriving home after a round trip to the near end, and it subtracts from the drive-point velocity because a fixed end reflects inverted.

That rung’s own last paragraph named what it had taken for granted:

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.

Why partials on a stiff string do not keep step #

An ideal string has no bending stiffness, so every frequency travels at the same speed and any shape put on it travels without changing. That is why a corner stays a corner and why the whole Helmholtz picture works.

A real string is a beam as well as a string, and a beam is dispersive: the higher the frequency, the more the bending stiffness contributes and the faster the wave goes. This site has carried the consequence since the second rung of the harmonic-series ladder — the n-th partial of a stiff string is not n times the fundamental but

with B the inharmonicity coefficient. That expression is normally read as a statement about pitch: the partials are sharp, a tuner stretches the octaves, the Railsback curve exists. Read as a statement about speed it says something else, and the something else is this essay.

A round trip to the near end takes x/L periods of the fundamental, with x the striking distance. In that time the n-th partial accumulates a phase of 2π·n·(x/L)·√(1+Bn²) instead of 2π·n·(x/L), so it arrives with an extra phase of

Cubic in the partial number. That is the whole finding compressed: B is a small number and n³ is not, so nothing happens for a long while and then the top of the spectrum scatters all at once.

Which end of the instrument this changes #

The interesting part is where the effect lands, because it is not where the previous rung’s own result landed.

The fifth rung found that the coupled contact differs most from the assumed half-sine in the bass, by a factor of 3.8 in contact time, because a bass hammer is heavier than its string and the returning corner therefore does most of the work of ending the contact. That was itself a refutation of a prediction the fourth rung had made in the other direction, and it is recorded as such.

Dispersion runs the other way. B on a real piano is U-shaped across the compass — high at the very bottom, where the strings cannot be made long enough and are wound to get the mass, lowest around A2, and rising steeply through the treble as the strings shorten. So the corner survives best in the tenor and worst at both ends, and worst of all at the top.

So the two effects are largest in different places, and the fifth rung’s most robust result is in the region where this one bites hardest. Its bass finding — the factor of 3.8 — is on strings whose B is around 1.2 × 10⁻³ at A1 and 5 × 10⁻³ at A0, where the corner is still intact to the sixteenth and tenth partials respectively. Its treble finding is on strings where the sixth partial is the first to arrive out of place — which, as the next section shows, is not the same claim as nothing above the sixth arriving in step, and the difference is most of what survives up there.

That is the prediction a returning corner has to be measured against, and it is a strong one — a whole family of partials at zero rather than merely quiet. Everything below is about how much of it survives once the corner comes back rounded rather than sharp.

How much of the treble result survives #

The honest answer is: the ordering does, and the size does not.

The fifth rung’s treble claim is that the coupled model’s contact time differs from the assumed half-sine — that the hammer is light relative to its string up there, that the corner comes home dozens of times inside the contact, and that each return stiffens the drive. Every part of that except the last is untouched by dispersion, because they are statements about mass ratios and travel times rather than about wave shape.

What dispersion changes is the last term. A returning corner delivers a sharp velocity step to the drive point; a returning ripple delivers a smeared one spread over a substantial fraction of a period. The impulse is the same — dispersion moves energy about in time and does not destroy it — so the average reaction over a contact is unchanged, and the contact time, which is an integral, is nearly unchanged with it.

The first partial out of step is not the count of partials out of step #

There is a control this essay did not run on its own headline number, and running it changes what the number means.

blurredFrom is the lowest n whose extra phase exceeds a quarter of a cycle, and the phase it tests is the raw accumulated one — 2π·n·(x/L)·(√(1+Bn²) − 1), which grows without bound. But the wave being drawn is periodic. It is a sum of sines over one cycle, so a component whose extra phase is 2π arrives exactly one period late and is indistinguishable from one that arrives on time. The picture can only see phase modulo a cycle, and the number reported beside it is computed from a phase that is not.

Counting the partials whose wrapped phase is within a quarter cycle of zero gives a different instrument entirely:

The first-failure number falls by a factor of four across the compass and the count of partials in step falls by a factor of one and a half. In the top octave, where the essay’s own caption says “there is no corner”, partials 9, 10, 12, 15, 18, 20 and 21 are all back within a quarter cycle of where they left, and twenty-one of the forty-eight are. What has gone is any run of consecutive partials in step, which is what a discontinuity needs; what remains is a scattered set of them, which is what a ripple is made of.

That is exactly what the slope column already said and nobody read it as such. The maximum slope in the top octave is 0.22 of the undispersed corner’s, not zero — a fifth of the original steepness survives, and it survives because a fifth of the spectrum is coincidentally back in phase. The bass end reads 0.21, which is the same number, so by the measure this essay chose, the top of the piano and the bottom are damaged equally and only the first-failure statistic makes the treble look unique.

Two of these measures are right about different things and the essay had been using one for both. The unwrapped phase is the correct quantity for a single returning transient: on the first trip the sixth partial really is a quarter cycle behind in absolute time, and a hammer in contact for a few milliseconds is meeting a transient rather than a steady state. The wrapped phase is the correct quantity for the periodic wave the figures draw and for anything held long enough to become one. The hero figure draws the second and labels it with the first.

What is destroyed is the serration. The fifth rung reported that the treble force pulse is serrated by the returning corner and that the serrations are what put its spectrum where the assumed half-sine’s is not. Those serrations are the corner arriving, and above the sixth partial there is no corner to arrive. So the mechanism is right, the contact time is safe, and the spectral feature that mechanism was invoked to explain is at least partly an artefact of an ideal string.

That is a smaller correction than a refutation and a larger one than a caveat, and it is the kind this ladder keeps producing: a model that is right about the quantity it computes and wrong about the picture it draws to justify it.

The width of that tolerance is the practical form of the whole argument. A finite contact turns a null into a dip, and the dip is shallow enough that a maker aiming at the node has a range rather than a point — which is why the strike ratio in a piano is quoted as a band and not as a number.

Which computation produced the numbers #

The corner is built as a sawtooth: partials at amplitude one over n, which is the bridge force of an ideal Helmholtz corner, summed to the forty-eighth. Each partial is given the extra phase above and the sum is re-evaluated.

Two measures are read off the result and the choice between them was not free.

The obvious one is a rise time — the ten-to-ninety fall across the steepest descent — and it is useless here. Once the dispersion is large the wave has several descents of nearly equal steepness, the search picks whichever is marginally steepest, and the answer jumps between 0.4 per cent and 50 per cent of a period from one note to the next with nothing physical moving. A measure that reports a smooth physical trend as noise is not a measure.

What is used instead is the maximum slope anywhere in the cycle, as a fraction of the undispersed corner’s. It is one number over the whole period, it falls monotonically as the components scatter, and it is 1 by construction at B = 0. Beside it is the first blurred partial: the lowest n whose extra phase exceeds a quarter of a cycle, which is monotone in B and is the number the hero figure draws.

The inharmonicity coefficients come from this site’s own piano scaling — a length that follows the ideal until the case limits it and a diameter that tapers with pitch — which is the same scaling every essay in the string ladder uses. The striking distance is an eighth of the speaking length, which is where a piano hammer lands and is why the seventh partial is at a null — a null this essay’s smearing does not move, because a null is a property of where the hammer landed rather than of what came back.

Where the model stops #

The sawtooth is an idealisation of an idealisation. A corner on a struck string is not a Helmholtz corner — that is a bowed-string object — and the wave a hammer launches is a pair of steps travelling in opposite directions rather than a sawtooth. What the sawtooth has in common with the real thing is a discontinuity and a one-over-n spectrum, which is what the dispersion argument needs, and nothing else. The struck string’s actual initial shape is the hammer’s own width and duration convolved with a step, and it has its own comb. The partial at which coherence is lost is a property of the phases and is right; the waveform drawn is illustrative.

The wrapped count is not a claim that the treble is fine. Twenty-one partials in step is twenty-one out of forty-eight, and they are scattered rather than consecutive, so the wave they sum to is a ripple with a steepest point rather than a corner. The distinction the census draws is between degraded and absent, and the fifth rung’s serration needs the corner rather than the steepest point.

The dispersion relation is the small-B one. √(1+Bn²) is a two-term expansion of the stiff-string dispersion and it is being evaluated at n = 48 and B = 0.028, where Bn² is 64 and the expansion is well past where anybody would defend it. The direction and the ordering are safe — the phase error is monotone in both n and B however the relation is written — and the specific partial numbers in the top octave should be read as “about six” rather than as six.

And nothing here is fed back into the coupled model. The right thing to do with this result is to run the fifth rung’s step-forward computation with a dispersive delay line instead of a pure one, and read the contact times and spectra off the result. That is a rewrite of that model rather than a figure, it is the obvious next thing, and this essay has established only that it would change the answer and roughly where.

Whose instruments, and when #

Dispersion is a property of steel and applies to every string on every instrument. What varies is how much of it a design accepts.

The piano is the extreme case and it is extreme by choice. Its top strings are short because the instrument has to fit in a room, and short and thick is exactly the recipe for a large B; a wound string is the trick that keeps the bass under control and there is no equivalent trick for the treble. A harpsichord, whose strings are thinner and under much less tension, has a far smaller B through most of its range — which is one of the reasons its treble sounds clearer and less percussive, and it is a reason that has nothing to do with the plectrum. The same scaling argument decides how long a bass string has to be before it stops being makeable at all.

The clavichord goes further still, and a modern concert grand’s scaling is a nineteenth-century compromise: iron frame, high tension, short treble, and a great deal of inharmonicity accepted in exchange for power. Every essay in this ladder is a consequence of that trade, and this one says the trade also costs the treble its returning corner.

What the picture cannot show #

Whether any of it is audible. The serration this essay dissolves is a feature of a force pulse, and what reaches a listener is that pulse filtered by the string’s own response, the bridge, the soundboard and the room. Two force pulses with the same integral and different fine structure may well produce indistinguishable notes, and nothing here says otherwise. What is known is that the ear is remarkably insensitive to the phases of a steady spectrum and rather sensitive to them during an attack, which puts this effect in exactly the region where the question is open.

And it cannot show the first round trip against the tenth. The figures are drawn after one trip because that is when the corner first comes home and is what the coupled model needs. On a treble note the corner returns dozens of times inside a single contact, and the phase errors accumulate — but they accumulate modulo a full cycle, so partials drift in and out of step and the measure stops being monotone in the number of trips. That non-monotonicity is real and it is not drawn, because a figure of it would show a physical fact and a numerical artefact in the same wiggle and there is no way to tell them apart at this resolution.

Where this ladder goes next #

Six rungs. Where the hammer lands and which partials it silences; that contact takes time; that it has a width as well as a duration; that the hammer’s mass against the string’s inverts across the compass; the coupled problem all four approximated; and now the assumption inside the coupled problem, which is that the string sends back what it was given.

What is owed is the merger. Every rung of this ladder has computed one term and handed it to the next, and there are now four of them — the comb, the low-pass, the mass ratio and the dispersion — with no single computation that carries all four at once. The obstacle is not conceptual; it is that a dispersive delay line inside a step-forward hammer model costs a Fourier transform per time step, and the result would be a spectrum for every note on the instrument that could be compared against a recording. That is the first thing in this ladder that would need a measurement to check rather than an argument.