Three exciters and three wires

The crossing belongs to the felt ran three exciters — a piano’s felt hammer, a harpsichord’s quill and a dulcimer’s hard beater — through the four-cornered excitation model, and found that only the hammer produces a crossing: below E3 the strike point’s comb decides what the spectrum looks like, above it the hammer’s contact time does, and the plectrum and the beater never hand over at all.

Its last paragraph names the thing it could not do:

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.

That is not a caveat about precision. One of the four corners — the one where stiffness disperses the travelling corner and smears the comb — is computed entirely from the wire, and it was being computed from the wrong wire for two of the three instruments. If the corners had come out differently the previous rung’s verdict would have gone with them.

Three scalings, stated #

None of these is fitted to anything. They are stated designs in the same spirit as the piano scaling this collection has used since the piano is tuned wrong on purpose, which says it is a piano rather than the piano.

The piano is that same one: a speaking length falling as f⁻⁰·⁹³ until the case runs out at 1.15 metres, steel wire tapering from 1.3 millimetres in the bass to 0.89 at the top.

The harpsichord is Pythagorean: the speaking length doubles for every octave down, exactly, from about 355 millimetres at c″, until the case runs out at 1.8 metres and the bottom two octaves are foreshortened. Iron wire, 0.30 millimetres at c″ thickening to about 0.5 in the bass.

The dulcimer is a trapezium’s compromise: strings falling more slowly than the pitch, because the bass end of a portable instrument cannot be four times the treble end, held at near-constant tension around 55 newtons. Steel wire of about a third of a millimetre throughout.

At the C above middle C a piano’s string is 315 millimetres of 1.02-millimetre steel under 725 newtons. A harpsichord’s, at the same pitch, is 355 millimetres of 0.30-millimetre iron under 98. Longer, a third of the diameter, and an eighth of the tension — and the inharmonicity coefficient that comes out is 8.1 × 10⁻⁵ against 1.5 × 10⁻³, a factor of nineteen.

Why a harpsichord’s wire is so slack, and it is not a musical reason #

The factor of nineteen has a cause and the cause is not taste.

The maximum product of frequency and speaking length a wire can survive is half the square root of its tensile strength over its density — a quantity this collection has had since it priced what a raised pitch standard would have cost the strings — and it depends on the material and on nothing else. Divide each instrument’s own f·L by that maximum and the numbers are:

A Pythagorean scaling is a scaling at constant f·L. That is what “the length doubles every octave” means, and a constant f·L is a constant fraction of the wire’s breaking point. The harpsichord’s whole treble, from A2 upward, sits at 97 per cent of what cold-drawn iron will take.

So the harpsichord’s string design is set by the tensile strength of iron wire and by nothing else. The maker made every string as long as it could possibly be, because that is what gives the best tone from a plucked string, and stopped where the wire stopped. A long slack string is a string with very little stiffness in it, and very little stiffness is a very low B.

The piano cannot do that. It has to survive being hit, its case is a metre and a bit rather than two and a half, and its strings run at two-thirds of breaking at most. That last constraint is the one the pitch-standard ladder priced when it asked what raising the pitch would have cost a piano’s frame, and it is the same margin read from the other end. Both facts push the same way: shorter and thicker, which is stiffer.

That the two instruments differ by an order of magnitude in inharmonicity is therefore a consequence of one being plucked and the other struck — but not through anything about plucking. It is through what a maker is allowed to do to the wire.

The nine instruments #

Now the question the previous rung left open, which is whose the crossing is.

There are two candidate answers and one wire could not tell them apart. Either the crossing belongs to the exciter — the felt’s compliance gives it a contact time that shortens toward the treble, and nothing else in the model does — or it belongs to the string, and the piano happens to be the instrument whose wire is stiff enough for dispersion to matter.

Three exciters against three scalings is nine instruments, six of which nobody has built, and the six are what decide it.

The felt hammer crosses from comb to contact at B2 on a piano’s wire, at B2 on a harpsichord’s, and at B2 on a dulcimer’s. The plectrum crosses on none of the three. The beater crosses on none of the three.

The verdict reads across and not down. The crossing is the exciter’s, exactly as the previous rung said, and now it is said against the alternative rather than in the absence of one.

Why it could not have been otherwise, once looked at #

The reason is embarrassingly simple and it is worth stating, because it is the kind of thing a model says only when it is made to do the work.

Two of the four corners have no string in them at all. The comb corner is one over the strike fraction — where the exciter lands, and nothing else. The contact corner is 1.5 over the frequency times the contact time — the exciter’s compliance, and nothing else. The crossing on a piano is between those two, so the wire could not have moved it if it had tried.

The string enters only the dispersion corner. And the dispersion corner never binds on any of the nine combinations. Its closest approach is on a piano’s own wire at the bottom of the compass, where it comes within a factor of 1.67 of taking over and does not.

So the answer to a debt that looked as though it might overturn a rung is that the correction is real, is large, and changes no verdict — because the quantity it corrects was never in a position to decide anything.

That is worth separating from the more comfortable kind of null. This is not a parameter that turned out not to matter; it is a parameter that could not have mattered, and the model says why in one line. The four-cornered accounting had that line in it from the day it was written and nobody read it that way, because the four corners were introduced as four candidate mechanisms of equal standing and two of them are not functions of the same object as the other two. Sorting them by what they depend on rather than by what they do would have answered the previous rung’s question before it was asked.

The same question can be put to the model from the other direction, and it gives the same answer with a number attached. Hold the plectrum’s strike fraction and its wire and ask how long it would have to stay on the string before a crossing appeared inside the compass at all. The answer is a contact time far longer than any plectrum has, and it does not move when the wire under it changes.

What the wire does change #

Not nothing, and the quantity is worth having.

A plectrum on its own iron beats the next corner by a factor of 10.9. The same plectrum on a piano’s steel beats it by 4.1, and on a dulcimer’s by 5.8. Nothing has changed hands; the comb still wins; it wins by two and a half times as much.

That factor is audible in a way the verdict is not. The dispersion corner is the partial above which the strike point’s comb stops being visible in the spectrum, because the travelling corner has spread out before it comes back. On a harpsichord’s own wire that partial is 28 to 81 across the compass. On a piano’s it is 8 to 27 — which is to say, right where the comb is.

A harpsichord has a clean comb over four or five octaves of its own spectrum and a piano’s comb is smeared before it starts. That is the difference between an instrument whose plucking point is an audible design decision, argued about for three centuries, and one where the strike point is chosen for a reason about the seventh partial and is otherwise inaudible.

The previous rung could not have said that, because on a piano’s wire the harpsichord’s margin is 4.1 and a factor of four does not sound like a difference in kind.

The dulcimer is the one that surprises #

The beater’s numbers are the least expected of the three, and they come out of the shape of the instrument rather than out of anything acoustic.

A hammered dulcimer is a trapezium. Its bass strings cannot be four times its treble strings, because the case would have to be a metre and a half wide and be carried. So its scaling is much flatter than Pythagorean, its bass strings are short for their pitch, and their B rises accordingly: at E3 the dulcimer’s wire is about three times stiffer than the harpsichord’s at the same note, and at the top of its compass twelve times.

That gives the instrument a bass with a good deal of stiffness in it and a treble with very little, which is the reverse of a piano and is exactly what a dulcimer sounds like — a bright, slightly clangy bottom and a pure top. The clang is inharmonicity and the inharmonicity is a carrying handle.

None of that changes which corner binds, because the dispersion corner on a dulcimer runs 18 to 47 and its comb corner is 7.

Which computation produced the numbers #

The four corners are the seventh rung’s, unchanged: the comb at one over the strike fraction, the contact roll-off at 1.5 over the frequency times the contact time, the dispersion corner at the cube root of two over three times B times the strike fraction, and the width corner at two over the contact width in string lengths.

The contact times are the eighth rung’s, unchanged: the felt hammer’s from the mass-ratio calculation, so it lengthens toward the bass; the plectrum’s a flat 0.05 milliseconds; the beater’s 0.30. The strike fractions are an eighth, a sixth and a seventh.

The only new thing is the string, and it enters through B = π²E d² / (64 ρ L⁴ f₀²), which is the same expression this collection has used since its second essay about a stiff string. Young’s modulus and density are the material’s: iron and modern music wire at 2.0 × 10¹¹ pascals and 7,850 kilograms per cubic metre, brass where a design calls for it.

The compass filter is off for the nine-cell grid on purpose. Six of the cells are instruments that do not exist and asking where a plectrum on a piano’s bass wire crosses over is exactly the question, so restricting each column to a real instrument’s range would have thrown away the experiment.

Where the model stops #

Three stated designs are not three instruments. Harpsichord scalings vary enormously between builders and traditions — Italian, Flemish and French practice differ in the treble length by more than a semitone’s worth — and the Pythagorean idealisation here is a description of what makers were aiming at rather than a measurement of any surviving instrument.

The bass of every one of them is a fudge. A piano’s bottom two octaves are overwound, which adds mass without stiffness and is the entire purpose of the winding; the model reports the core diameter and refuses to pretend it knows the rest. A harpsichord’s bottom is foreshortened and often brass rather than iron, which changes both constants. Neither of those is modelled and both are where the B curves in the figure bend.

The breaking fraction is the softest number here. Historical iron wire’s tensile strength varies by a factor of two between analyses of surviving samples, so “97 per cent of breaking” should be read as “at the limit, within the accuracy of knowing what the limit was”. The constancy across the treble is the robust part, because it follows from constant f·L and not from the strength at all.

And a corner is not a spectrum. All four of these are the partial number at which some mechanism starts to matter, which is a summary of a filter rather than the filter. The pulse that was assumed is this ladder’s essay on what that summary discards.

What the picture cannot show #

It cannot show the soundboard. Every difference in tone between these three instruments that a listener would name first is a difference in the body, and this model ends at the bridge. A harpsichord’s clean comb reaches a soundboard that is thin, light and heavily resonant, and what comes out is not what went in.

Nor can it show the corner’s own history. The dispersion corner is derived from how far a travelling corner spreads before it returns, which the sixth rung computed and which is a statement about one round trip. A harpsichord note lasts for hundreds of them.

Nor can it show the plucking mechanism. A harpsichord jack releases the string in a way that depends on how the plectrum is voiced, and voicing is the single largest thing a harpsichord technician does. The 0.05-millisecond release used throughout is one number for a whole craft.

It cannot show the courses. Every note on a dulcimer is three or four strings, on a piano two or three, on a harpsichord one or two per register — and what three strings do to a decay is a large effect that has nothing to do with anything here.

And it cannot show the six instruments that do not exist. A harpsichord strung with piano wire is a column of numbers, and the reason it is a column of numbers is that a quill would not pluck it and the case would collapse. The grid is a way of separating two explanations, not a set of predictions.

Whose instruments, and when #

The piano is a modern small upright. The harpsichord is an idealised eighteenth-century single-manual, five octaves, of the kind whose scalings are recorded in surviving instruments and in the makers’ own marked-out soundboards. The dulcimer is a modern hammered one; a Persian santur is smaller and a Hungarian cimbalom much larger, and neither is this.

The historical claim available here is the one about the wire and it is narrow. The reason harpsichord scalings are Pythagorean in the treble and foreshortened in the bass is a documented builders’ practice with a documented rationale — the treble is at the wire’s limit and the bass is at the case’s — and this calculation reproduces both boundaries from the material constants without being told them. It says nothing about why the practice was arrived at, which was by breaking strings.

Where this ladder goes next #

Nine 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; the crossing belongs to the exciter’s compliance; and now three exciters on the three wires they actually hit, which leaves that verdict standing and multiplies the margin behind it by two and a half.

What the ladder owes now is where the exciter is allowed to be. Every rung above takes the strike or pluck fraction as given — an eighth, a sixth, a seventh — and those are three numbers copied off three instruments. A harpsichord has two or three registers plucking the same string at different fractions, which is the one place in this collection where a maker’s choice of excitation point is a control the player operates rather than a number fixed at the factory. The comb corner is one over that fraction and the dispersion corner has it under a cube root, so moving it moves two of the four terms at once and in different proportions — and this ladder has never asked what the second register is for in the units it has spent nine rungs building.