The crossing belongs to the felt

Four terms, and only one of them binds put the whole excitation ladder into one calculation and found a crossing. Below E3 the piano’s spectrum is cut off by where the hammer lands; above E3 it is cut off by how long the hammer stays. Its last paragraph asked the obvious question and named the two instruments that answer it:

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.

The answer is neither. It is a fact about felt.

Two of the four corners are the string’s #

Before the exciters can be compared it is worth separating what belongs to each.

The comb is the strike point. Where the hammer lands is the cleanest design fact in the subject: striking at a fraction 1/n of the length silences the nth partial exactly, because the exciter lands on that mode’s node. It is a property of where and not of what, so it is the same horizontal line for a hammer, a plectrum or a finger at the same fraction.

The dispersion corner is the string’s stiffness. The corner does not come back a corner shows that a stiff string’s partials travel at different speeds, so the sharp corner an exciter makes spreads out on its journey and the highest partials arrive smeared. That depends on the string’s inharmonicity and on the strike fraction and not at all on the exciter.

The width corner is how wide the contact patch is — a hammer is not a point, and a contact spread over a fraction of the string is a low-pass of its own. It is the exciter’s, but it is a large number for all three of these and never binds.

The contact corner is the exciter’s own duration, and it is where they differ. A force applied over a time τ cannot excite anything much above about 1.5/τ, so the corner in partial number is 1.5/(f₀τ).

Two of the four corners are properties of the string. Only the contact corner really distinguishes a hammer from a plectrum, and it distinguishes them by an order of magnitude.

Three contact times, an order of magnitude apart #

A piano’s felt hammer stays on the string for about a millisecond and a half at middle C, and longer in the bass, because the hammer that is heavier than its string sets the contact by a mass ratio and a bass hammer is much heavier than the string it hits. That number is the one the ninth rung computed and it is the one used here, deliberately: using the sixth rung’s felt law instead would move the crossing from E3 to E5 and make this a comparison of two models rather than of three exciters.

A harpsichord’s plectrum releases in something like a twentieth of a millisecond. A quill or a delrin plectrum plucks by displacing the string and then slipping off its own tip, and the slip is a fast event governed by the plectrum’s stiffness rather than by the string’s mass.

A dulcimer’s beater is on and off in about three tenths. A hard wooden head with almost no compliance against a light string is much closer to an impulse than felt is, and much less close than a plectrum.

The two non-piano figures are asserted rather than measured, and that is exactly why they are swept rather than quoted below.

What the three exciters do #

Running the corners over the compass gives three quite different pictures.

The piano crosses at E3, which is the seventh rung’s result reproduced. Below it the comb binds at partial 8 and the contact corner sits above; above it the contact corner has fallen through 8 and the hammer’s own duration decides the top of the spectrum.

The harpsichord never crosses. Its contact corner runs from partial 728 in the bass down to partial 17 at the top of the compass, and its comb sits at 6. The strike point binds at every pitch on the instrument, by a wide margin everywhere.

The dulcimer crosses at A5, near the top of its range. Its contact corner starts at 121 and falls through its comb of 7 only in the last octave.

So the three instruments have three different answers, and the ordering is the ordering of their contact times. The crossing is not a property of struck strings, of hammers, or of pianos: it is a property of an exciter that stays on the string long enough for the string to have started moving underneath it.

The boundary, which is the point of the rung #

Two of the three contact times above are asserted, so a comparison of three verdicts is not worth much. Sweeping the contact time and asking where a crossing appears at all converts three assertions into one boundary, and that is a number worth having.

A crossing enters the compass at a contact time of about 0.12 to 0.15 milliseconds. Below that the comb binds everywhere; above it, a crossing exists and moves down the compass as the contact lengthens.

The three exciters sit at 1.49, 0.30 and 0.05 milliseconds. The plectrum is a factor of three below the boundary and the dulcimer’s beater is twice above it; the piano’s felt is ten times above it.

So the conclusion does not depend on the two asserted numbers being right. A plectrum would have to be three times slower than assumed to acquire a crossing at all; a dulcimer beater would have to be twice as fast to lose the one it has. Those are the sensitivities, and the piano’s position is not in question at any plausible value.

Why felt is the odd one out #

The reason a piano hammer is so slow is not an accident of materials, and it is worth saying because it makes the finding a design fact rather than a curiosity.

Felt is compliant, and its compliance is what lets a piano have a dynamic range. A hammer is not an impulse shows that a harder blow compresses the felt further, which stiffens it, which shortens the contact and brightens the note. That is the mechanism behind a dynamic mark changing what a note is: the piano’s timbre moves with its loudness because its exciter is soft enough to be squashed.

A plectrum cannot do that. It plucks with a displacement set by the key’s travel, and pressing harder moves the key faster without changing where the string is released from — which is why a harpsichord has almost no dynamic range from the keyboard and why its registration is done with stops instead.

So the long contact and the crossing come from the same property as the dynamic range, and an instrument that has one has the other. The harpsichord’s spectrum is decided by the plucking point at every pitch and at every touch, and that is the same statement as a harpsichord sounds the same however hard it is played.

What the three spectra actually look like #

Corners are a summary and the spectra they summarise are worth seeing, because the summary hides how gently the terms bite.

Every one of these roll-offs is a smooth curve rather than a cliff. The comb is a set of nulls at multiples of 1/p with maxima between them; the contact low-pass is a sinc-shaped envelope; the dispersion smear is a gradual loss of the highest partials. What a listener gets is the product, and the “binding” term is the one whose curve is lowest first rather than the only one doing anything.

That matters for the harpsichord case in particular. Saying that the comb binds everywhere on a harpsichord does not mean the plectrum’s release contributes nothing; it means the plectrum’s contribution begins at partial 17 at the very top of the instrument and higher everywhere else, by which point the comb has already taken out the eighth partial and the fourteenth.

The strike point is the same fraction on all three, and that is a coincidence #

The three combs in the figure sit at 6, 7 and 8 — a plucking point a sixth of the way along for the harpsichord, a beater at a seventh for the dulcimer, a hammer at an eighth for the piano. Those are close enough that the figure’s horizontal lines nearly coincide, and it is worth saying that this is not a shared principle.

The piano’s eighth is chosen to put the seventh partial near a null, which is the cleanest design fact in the subject and is about avoiding the flat seventh. A harpsichord’s plucking point is chosen for tone and varies enormously between registers on one instrument — the lute stop plucks very close to the nut, which is a comb at a much higher partial and a nasal sound. A dulcimer’s striking point is chosen by where the player’s hands can reach across a fixed bridge layout, which is a constraint of a completely different kind.

So three instruments that agree to within a partial in the position of one corner agree for three unrelated reasons, and the agreement is a fact about the figure rather than about instrument design.

What a maker chooses #

One of the four corners is a maker’s free choice and three are consequences, which is the observation the seventh rung ended on.

The strike point is chosen. A piano maker picks a fraction between a seventh and a ninth; a harpsichord maker picks a plucking point, and typically a different one for each register, which is a large part of why two stops on one instrument sound different.

The contact time is a material. It follows from the hammer’s mass, the felt’s compliance and the string’s mass.

The dispersion is the string. It follows from the wire’s diameter, its length and its tension, all of which are decided by the scaling.

And the contact width is the hammer’s geometry, which never binds.

Reading the crossing as a design diagram therefore says something different for each instrument. On a piano, the maker’s one free choice controls the spectrum only in the bottom two octaves and is overruled by the felt above them. On a harpsichord, the maker’s choice controls the spectrum everywhere. That is a very large difference in how much of an instrument’s voice a maker has direct access to, and it comes out of one number.

Which computation produced the numbers #

The four corners are the merger’s, expressed as partial numbers so that they can be compared on one axis: the comb at 1/p, the contact at 1.5/(f₀τ), the dispersion at the cube root of 2/(3Bp), and the width at 2/w.

The string at each pitch is the piano scaling — length from a stated law with a case limit, diameter from a stated taper, inharmonicity from both — and it is used for all three instruments, which is an approximation discussed below.

The piano’s τ is the mass-ratio calculation from the ninth rung: the shorter of the two candidate contact times, given the hammer’s mass against the string’s at that pitch. The other two τ values are stated constants, and the sweep is what prices them.

Where the model stops #

All three exciters are hitting a piano’s strings. That is the largest simplification here. A harpsichord’s string is longer, thinner and at much lower tension than a piano’s at the same pitch, so its inharmonicity is smaller and its dispersion corner higher; a dulcimer’s is shorter and lighter. Fixing the string is what makes the three columns comparable and it is not what any of the three instruments is.

Two contact times are asserted. The sweep is the answer to that, and it is a partial answer: it prices how wrong they would have to be to change the verdict, and it does not measure them.

A plectrum is not a hammer with a short contact. It displaces the string and releases it, which is a different initial condition from a force applied and removed — a triangle in displacement rather than an impulse in force. The pulse that was assumed is the rung about how much that matters, and the corner arithmetic here treats the two the same way.

And a corner is not a spectrum. Every one of these numbers is the partial at which a term begins to bite, and what a listener hears is the product of four roll-offs rather than the position of the first one. The merger draws the product; this rung compares the corners.

What the picture cannot show #

It cannot show the harpsichord’s registration. A harpsichord has two or three sets of strings plucked at different points, and the instrument’s voice is the combination. One comb per register is what the figure would need, and the interesting question — how much the two combs together fill in each other’s nulls — is not asked here.

Nor can it show the dulcimer’s beaters. Dulcimer players carry several pairs, hard and soft, and choose between them as a piano’s felt cannot be chosen. Softer beaters would move that instrument’s crossing down and hard ones up, so the dulcimer is really a family of exciters rather than one.

It cannot show the soundboard. Every one of these instruments radiates through a bridge and a board with resonances of its own, and the body is the filter is the essay about how much that shapes what leaves. A corner at partial 8 is a statement about what the string carries and not about what the room hears.

Nor can it show the decay. The note that gets duller as it dies shows that the top of a struck string’s spectrum drains faster than the bottom, so the corner an excitation puts in place is not where the corner is a second later. The excitation ladder is entirely about the first instant.

And it cannot show what any of this sounds like. A corner at partial 8 and a corner at partial 17 are both far above where the ear’s resolution ends, and whether a listener can tell which term is binding is a question this collection has no way to ask.

Whose instruments, and when #

The piano here is the modern one, with felt hammers and a cast-iron frame — an instrument of the second half of the nineteenth century onwards. Its predecessors are the interesting case for this argument: a fortepiano’s hammer is leather-covered and much lighter, so its contact time is shorter and its crossing is higher up the compass. That predicts the fortepiano’s strike point matters over more of its range than a modern piano’s does, which is consistent with the care makers took over it and is not evidence about anything.

The harpsichord is the eighteenth-century instrument, with quill plectra. Modern instruments often use delrin, which is stiffer and if anything faster, which moves the plectrum further below the boundary.

The hammered dulcimer stands here for a large family — santur, cimbalom, yangqin — whose beaters differ enormously and which have never been treated together in this collection. The one used here is a hard wooden head, which is the santur’s rather than the cimbalom’s; a cimbalom’s cotton-wound beaters are much softer and would sit nearer the piano.

Where this ladder goes next #

Eight rungs. The strike point silences a partial; the hammer is not an impulse; nor a point; nor lighter than its string; the pulse’s shape was assumed; the corner does not come back a corner; four terms and one crossing; and now the crossing belongs to the exciter’s compliance rather than to the instrument.

What the ladder owes now is the string these exciters are actually hitting. Every column above uses a piano’s scaling, and a harpsichord’s string at the same pitch is longer, thinner and slacker — which moves its inharmonicity down by an order of magnitude and its dispersion corner up with it. This collection has the string-scaling machinery for all three instruments and has never run the excitation model on anything but a piano’s, so the honest next rung is three instruments with three sets of strings, and the question it settles is whether the plucked instrument’s comb still binds when its own wire is under it.