Four terms, and only one of them binds

The corner does not come back a corner ended with a stocktake and a complaint. Six rungs, four terms — the comb, the low pass, the mass ratio and the dispersion — and no single computation that carries all four at once. Every rung had computed one thing and handed it to the next, which is how a ladder is supposed to work and is not the same as having a model.

The complaint had an obstacle attached: a dispersive delay line inside a step-forward hammer model costs a Fourier transform per time step. That is true of the full merger and it is not necessary for the question the merger is actually for, which is not “what is the waveform” but which of these four terms decides the answer.

The four terms, and what each of them is a corner of #

The comb. Where the hammer lands is the ladder’s first rung: a hammer at a fraction x of the length silences every partial with a node there, so partial n is scaled by the absolute value of sin(nπx). At the usual x of one eighth, partial eight is the first casualty. That number is a fraction of the length and therefore identical at every pitch — a horizontal line.

The width. The hammer is not a point either is the third rung: the contact has an extent as well as a place, which multiplies the comb by a sinc factor. At a contact two per cent of the string’s length the first null of that factor is at partial one hundred, which is far above everything else here. The width is a real term and it never binds.

The contact time. A hammer is not an impulse is the second rung and the hammer that is heavier than its string the fourth: contact lasts a finite time, which low-passes the excitation, and the time is set by whichever of two mechanisms ends the contact first — the felt’s own recoil, or the string carrying the hammer’s momentum away. The corner is the first null of the force pulse’s spectrum, at fτ = 3/2, which in partials is 3/(2f₀τ). It falls steeply with pitch, because the frequency is in the denominator.

The dispersion. The corner does not come back a corner is the sixth: a stiff string’s partial n travels at a speed that depends on n, so the corner returning to the hammer has been smeared, and the re-excitation of partial n is degraded by the phase error it has accumulated — πBn³x for a round trip. Its corner falls as the cube root of one over B, and B rises steeply toward the treble, so this corner falls too, but only as a cube root.

The crossing, and the term that never binds #

Three corners, three different behaviours with pitch, and the answer is a crossing and an absence.

The crossing is at about E3, 165 hertz. Below it the contact-time corner is above eight and the comb is the binding term: what decides a bass note’s spectrum is where the hammer lands. Above it the contact-time corner has fallen below eight and the comb is irrelevant, because the partials it would have silenced are already gone.

That is a statement about piano design that this collection could not previously make. A piano’s striking point is chosen — famously, and with a great deal of argument about sevenths and ninths — and the choice can only matter where the comb is the binding term, which is the bottom two octaves and a bit. Everywhere above E3 the striking point is a decision about a term that is not deciding anything.

And the dispersion is nowhere the binding term. At the top of the instrument its corner is 5.8 partials, which is the lowest it ever gets, and the contact-time corner there is 0.5. Whatever the returning corner has smeared, the low pass had already removed.

That is not a refutation of the sixth rung. Its arithmetic is unaffected and its finding — that the smearing is cubic in the partial number, so it is nothing in the bass and everything in the treble — is exactly right about the term it computed. What the merger says is that the term arrives somewhere the spectrum has already been emptied, and no rung looking at one term could have known that.

What the crossing says about a design choice #

There is a piece of piano lore that this figure speaks to directly, and it is worth being careful about what it does and does not say.

The received account of the striking point is that a hammer at one seventh to one ninth of the speaking length suppresses the seventh partial, which is the flattest and most dissonant member of the series against a tempered scale, and that this is why the fraction is what it is. The account is old, it is in every technician’s manual, and it is a claim about the comb.

The corners say the comb is the binding term over the bottom two and a half octaves and nowhere else. Above E3 the seventh partial has already been removed by the contact time, at a level forty decibels down, and moving the hammer would change nothing a listener could hear. So the striking point is a decision about the bass, and the fact that it is expressed as a fraction of the length — and therefore applied to the whole instrument — makes it a bass decision imposed on the treble for reasons of manufacture rather than of sound.

That is a hypothesis rather than a finding, and the difference matters. What is computed here is which term binds where; whether a maker choosing the fraction was hearing the bass is a historical question. What can be said is that the usual justification is stated as though it applied across the compass, and under this collection’s own model it does not.

Measuring the sensitivity rather than reading the corner #

A corner is a proxy, and the quantity the argument is actually about is how much the spectrum moves when the strike point moves. That is one sweep of the merged model: hold everything else, take the strike point across the whole range a maker would use — one ninth to one seventh — and read the spectral centroid.

The strike point’s whole effect on the spectrum falls by a factor of two hundred from the bottom of the instrument to the middle, and it does so smoothly rather than at a switch. That is the corner argument confirmed on the full model and corrected in one respect: E3 is where the binding term changes hands, and the sensitivity has already fallen to a fifth of its bass value by the time it gets there and goes on falling. There is no pitch above which the choice does nothing; there is a decade over which it stops mattering, and the middle of that decade is where the corners cross.

The second column is the received account priced directly. The seventh partial at the usual strike point is eighteen decibels below the fundamental at A0 and fifty-one at A4 — which is to say it is a real component of a bass note’s sound and is not a component of a treble note’s at all. A maker choosing a fraction to suppress it was choosing it for the register in which it exists, and the fraction then applied itself to the rest of the instrument because a fraction is what a stringing scale can hold.

So the hypothesis survives the measurement, and what the measurement adds is that it is not a claim about a boundary. The bass and the treble are not two regimes with a line between them; they are two ends of a two-hundred-fold decline, and the received account is right at one end of it.

Why a corner is the right way to compare them #

The four terms are multiplied, and a product of four curves is not a thing anybody can read. Comparing them by their effect on a single summary — a centroid, say — hides the structure, because a term that removes a great deal of a spectrum that has almost no energy left in it moves the centroid by nothing and is not thereby unimportant.

A corner is the comparison that survives that. Each of the three terms has a frequency above which it is doing the removing, and those three numbers are commensurable in a way the terms themselves are not: they are all partial numbers, and the smallest of them is by definition the one that gets there first.

It also makes the pitch dependence legible. One corner is flat, one falls as 1/f₀τ with τ itself falling, and one falls as a cube root — three different exponents, which on a log-log axis are three different slopes, and the crossings are where the regime changes.

The check the merger had to pass #

Multiplying four things together is the kind of change that can be right in every part and wrong as a whole, so the merger has to reproduce each rung’s own figure when the other three terms are switched off. It does, because switching them off is what the argument list does: passing only the comb gives the first rung’s spectrum exactly, and passing the comb and the contact time gives the second rung’s.

That is not a coincidence of implementation; it is the reason the terms are a list rather than four functions. A comparison between one model at five settings is a different kind of evidence from a comparison between five models, and this collection has been caught by the difference before — the key-finding ladder spent an essay establishing that two of its readings differed because of an argument nobody had passed rather than because of anything about the music.

The second check is the width. Its corner is at partial one hundred for a contact two per cent of the length, which is so far above the others that it should never bind — and it never does, at any pitch, for any contact width a hammer could have. A term that is real, correctly computed and never decisive is a useful thing to have identified, and it is the same verdict the dispersion gets for a different reason.

Which computation produced the numbers #

Every term is the ladder’s own, unchanged, and the merger multiplies them:

a(n) = (1/n) · |sin(nπx)| · |sinc(nπw/2)| · R(n, f₀, τ) · cos(πBn³x / 2)

where R is the half-sine force pulse’s spectrum from the second rung, B is the inharmonicity coefficient computed from pianoString’s own diameter and length at each note, and τ is the shorter of the felt’s recoil and the string’s momentum time, which is the fourth rung’s finding rather than the second’s fixed number.

The dispersion factor is the new piece of arithmetic and it is one line. The phase error of partial n after a round trip of x/L periods is πBn³x; a returning component that far out of phase re-excites the mode by the cosine of half of it, and the factor is clamped at zero rather than allowed to go negative, since a component a quarter-cycle out contributes nothing rather than subtracting.

The corners are: 1/x for the comb, 3/(2f₀τ) for the contact, the cube root of 2/(3Bx) for the dispersion — where the coherence has fallen to a half — and 2/w for the width.

Where the model stops #

This is a spectrum and not a waveform. The obstacle the sixth rung named is real and is not solved here: a genuine dispersive delay line inside a step-forward hammer model would produce a time signal, would show the second and third contacts a real hammer sometimes makes, and would cost a transform per step. What is done instead is to multiply the four transfer functions, which assumes they act independently and in sequence.

They do not act independently, and one place is knowable. The contact time depends on how much the string pushes back, which depends on what the returning corner looks like, which is the dispersion term. So the fourth term feeds the third and the merger treats them as separate. The error is small — the dispersion changes the corner’s shape rather than its arrival time — and it is an error.

And τ comes from a model with an asserted reference. The felt’s 1.6 milliseconds at the reference blow is the second rung’s number and everything above it inherits it. The crossing at E3 moves if that number moves; the existence of a crossing does not, because one corner is flat and the other falls monotonically.

What the picture cannot show #

It cannot show what any of this sounds like. A corner is where a term starts removing energy, not how much of the sound it takes. The sound buttons on the merged figures play twelve partials at their computed levels, which is the honest thing to do and is not a piano.

Nor can it show the sustained tone. Every term here is about the excitation — what the hammer puts into the string at the strike. What a listener hears afterwards is that excitation filtered by the string’s own decay, which is a different set of rates per partial, and by the soundboard’s own filter and which changes the balance again over the note’s life.

And it cannot show where a maker put the strike point and why. The received account is that one eighth avoids the seventh partial, which is dissonant against the fundamental. This figure says the choice only bites in the bottom two octaves, which is a different argument from the received one and is not evidence against it — a maker who could hear the difference in the bass and not in the treble would have arrived at the same fraction.

Whose instruments, and when #

The striking point of one seventh to one ninth of the speaking length is a fact about pianos from roughly Cristofori onward, and it is the number every piano technician quotes. The compass over which it binds — up to about E3 — is the bottom two and a half octaves of a modern instrument, which is nearly the whole compass of a Viennese fortepiano’s bass, and it is the register that instrument’s repertoire lives in less than the modern one does.

The dispersion term belongs to the modern instrument in a different way. Inharmonicity is a consequence of steel wire under enormous tension in a short frame, so B on an eighteenth-century instrument with thin brass and iron wire is very much smaller, and its corner correspondingly higher. The sixth rung’s term is a term the modern piano brought with it, and this rung says the modern piano’s contact time got there first.

Where this ladder goes next #

Seven rungs, and the four terms are one calculation.

The rung after it is the one the merger’s own bookkeeping names. Every corner here is a property of the hammer and the string, and one term of the four — the strike point — is chosen by a maker while the other three are consequences of materials. So the figure above is a design diagram: it says over what part of the compass a maker’s one free choice can do anything at all, and the obvious next question is what the answer looks like for the other struck instruments this collection carries. A harpsichord plucks rather than strikes, so its excitation has no contact-time term worth the name and the comb binds everywhere; a dulcimer’s hammer is hard and its contact very short. The same three corners drawn for those two would say whether the piano’s crossing is a fact about pianos or a fact about hammers, and both instruments are already in this collection.