What the second register is for

Three exciters and three wires gave each exciter the string its own instrument actually strings and found that the eighth rung’s verdict survived it. Its last paragraph names what is still owed:

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.

Two things came out of doing it and neither was on the list. The first is that a harpsichord’s plucking fraction is not a number a maker chooses at all. The second is what happens when two of them are engaged at once, which is not what adding a brighter register to a rounder one sounds as though it should be.

A plucking point is a distance #

The jacks stand in a rail. The rail is one piece of wood across the whole compass, and the strings under it shorten by a factor of ten from the bass to the treble.

So a register plucks at a fixed distance from the nut, and the fraction — the quantity every rung of this ladder has argued about — is what that distance becomes on each string in turn.

At fifty millimetres — an ordinary distance for the further of two unison registers — that is one thirty-sixth of the string at C2 and one three-and-a-halfth at C6. A factor of ten in the fraction, with the maker having moved nothing.

This is the same shape as three things already in this collection: an end correction, which is a fixed length against a shrinking tube; a bow held at a fixed distance from the bridge while the left hand shortens the string; and a hand in a bell. In each case a constant in one unit is a variable in the unit the argument is actually about.

It is also, in this case, half of why a harpsichord’s treble sounds the way it does. A string plucked at a third of its length loses the third partial and everything above it falls off as one over $n$ squared from a comb that has already thrown away most of the top. Makers know it and angle the rail to compensate, which reduces the factor of ten without abolishing it, because the strings shorten faster than any rail can be angled.

And on this instrument that is the only term that decides anything #

Four terms, and only one of them binds put the excitation model’s four terms — the comb, the exciter’s contact time, its width, and the dispersion the string’s own stiffness causes — on one axis as corner partials, and found that on a piano the strike point decides below E3 and the contact time above it.

On a harpsichord the comb wins everywhere. Across the whole compass its corner runs from 36 at C2 to 3.5 at C6; the quill’s contact time gives a corner from 459 down to 29, its width gives 500, and the dispersion runs from 106 down to 21. The plucking point’s corner is the lowest line at every pitch on the picture and it is not close.

There is a way to check that this is a fact about plectra and not about the particular contact time assumed for one. The crossing belongs to the felt swept the exciter’s contact time until a crossing appeared, and asked how compliant an exciter has to be before its own contact starts deciding anything.

A quill would have to hold the string for something like the time a felt hammer does before its contact reached down to where the comb is, and a quill is the stiffest exciter in the collection — it is a wedge of bird quill or delrin that snaps past the string in a twentieth of a millisecond. The comb’s supremacy on this instrument is not an artefact of the number 0.05; it survives moving that number by a factor of twenty.

The debt asked specifically about the dispersion corner, because it moves when the plucking point moves and it moves differently: the comb corner is one over the fraction and the dispersion corner has the fraction under a cube root, so a plucking point that changes by a factor of ten changes one by ten and the other by 2.2. That is a real convergence and it is nowhere near enough. Setting them equal gives a plucking point at the square root of one and a half times the inharmonicity coefficient, which on this instrument’s iron wire is between 3.4 and 9.9 millimetres from the nut — nearer to it than any register a harpsichord has ever carried, the lute stop included, and near enough that the string is not free there at all.

So the answer to the ninth rung’s question is no: no choice of plucking point can make dispersion the binding term on a harpsichord. The maker’s one free choice is the term in charge at every pitch and every register position, which is exactly the opposite of the piano, where the same choice stops mattering above E3.

That is a difference in kind and it explains why the two instruments are built the way they are. A piano gets its variety from how hard the key is struck, because the contact time is what a dynamic changes and the contact time is the binding corner over most of the compass. A harpsichord has no dynamic at all — the plectrum releases at the same displacement however the key is pressed — so its variety has to come from the only term it has, and it comes as registers.

Two registers on one string #

Which raises the question this rung was written for. What happens when two of them pluck at once?

The answer is easier than it looks, for a reason worth stating. A plectrum is voiced to a force and not to a displacement: the quill’s stiffness and the depth it engages set how hard it pulls before it slips off, and a maker adjusts that until the two registers feel and sound balanced. A string under tension is linear, so two static deflections superpose exactly. The modal amplitudes therefore add, term by term:

$$a_n \propto \frac{\sin(n\pi p_1) + \sin(n\pi p_2)}{n^{2}}$$

Three things happen at once and only the first is expected.

The fundamental gains 4.4 decibels. Two plucks of equal force at nearby points add almost exactly in phase at the bottom of the spectrum, so the pair is close to twice as loud there. Not quite six, which is what two identical sources would give, because the two combs are not identical at the fundamental either — a pluck closer to the nut moves the string less for the same force, and the deficit is what the shortfall from six measures.

Every hole either register leaves is filled. The far register at fifty millimetres kills partial 7; the near one at thirty-two kills 11 and 22. In the pair all three are present, because where one comb is zero the other is not, and the sum is simply the other one’s value.

And the pair loses two partials that neither register loses. At 17 and 20 the two combs are equal and opposite, and the sum is zero. Those are holes the combination digs for itself.

The trap is the arrangement a plan would suggest #

Where the two registers sit relative to each other decides where those new holes fall, and one arrangement is much worse than the rest.

Put the near register at exactly half the far one’s distance and its comb has nulls at 14 and 28, which are two of the far register’s nulls at 7, 14, 21 and 28. Where one comb is zero the other is zero too, so the pair keeps every hole the near register had, and gains one at 19 into the bargain.

That is the arrangement a draughtsman would reach for. Twenty-five and fifty are the numbers a rail gets drawn with, and they are the one ratio in the available range that guarantees a shared null. The instruments do not do it, and this says why they should not.

The rule generalises and it is short. Two combs share a null whenever one plucking distance divides the other, because a comb’s nulls are the multiples of one over its own fraction. So any small whole-number ratio between the two distances puts shared nulls low in the spectrum where they can be heard — two to one at partial 14, three to one at 21, three to two at 14 again — and the only safe arrangements are the ones a ruler does not suggest. That is a design rule stated in the units the ladder’s first rung built, and it appears nowhere in the instructions a maker is given.

The opposite extreme is the lute stop, which plucks almost on the nut.

At fourteen millimetres the comb’s first null is at partial 25, which is past everything audible in a plucked spectrum, so the lute stop on its own is the one register with no missing partial. Its centre of gravity is at partial 4.57 against the far register’s 2.67 — genuinely and measurably the brightest thing on the instrument, which is what it is used for. Combined, it gains only 2.2 decibels and puts holes at 10 and 11.

The design diagram, which has no optimum #

Sweeping the near register along the rail with the far one fixed makes the trade explicit.

Nine of the thirty-two positions — 24, 27, 37, 40, 41, 42, 43, 44 and 46 millimetres — leave the pair with no missing partial anywhere in the first 24. Every other position digs at least one hole. And the level runs the other way: the fundamental gains 2.3 decibels with the near register up against the nut and 5.7 with it close beside the far one.

The two do not have a common optimum, and neither is what a maker is actually choosing. A register that sits almost on top of its neighbour is loud, clean and adds nothing a player can hear as a change of colour; one that sits far from it is a distinct voice with a comb of its own and a compromised combination. What a harpsichord is built to give is a contrast between registrations, and the contrast is exactly the thing that costs the pair its completeness.

The result that reverses the expectation #

The last measurement is the one that came out backwards.

Engaging the brighter register alongside the darker one does not brighten the combination. In every pair above, the pair’s centre of gravity is below the average of the two registers’ own, and in nearly all of them it is below the darker register as well: 3.15 and 2.67 giving 2.44; 3.58 and 2.67 giving 2.48; 4.57 and 2.67 giving 2.71.

The mechanism is the same one that fills the nulls, seen from the other end. Two plucks a couple of centimetres apart are in phase at the bottom of the spectrum, where a partial’s wavelength is far longer than the gap between them, and increasingly out of phase toward the top, where it is not. So the pair adds at the fundamental and partially cancels in the treble, and the ratio between those is what a spectral centroid measures.

The size of the effect deserves a number rather than a direction. Going from the far register alone to both engaged moves the centre of gravity from partial 2.67 to 2.44, which is a shift of nine per cent downward, at the same time as the level goes up by 4.4 decibels. Louder and darker together is an unusual combination for an instrument to be able to make, and it is the only dynamic gesture a harpsichord has.

A full registration is louder and rounder than either of its halves. That is what harpsichordists say about eight-foot-plus-eight-foot against a single eight — fuller, broader, less penetrating — and it is usually attributed to the second string’s slight mistuning and the beating that follows. This says that most of it is arithmetic, and that it would happen on two perfectly tuned strings, and that it would happen on one string plucked twice.

Which computation produced the numbers #

Each register’s spectrum is the same $\sin(n\pi p)/n^{2}$ the ladder’s first rung drew, in absolute units so that two of them can be added rather than each normalised to its own peak. The pair’s spectrum is their sum, and the figures check it partial by partial rather than trusting the sum.

A partial is called silent when its amplitude falls more than 24 decibels below where the one-over-$n$-squared envelope would put it, which is the same order as the two per cent test the single-comb figure has used since that rung.

The string lengths and the inharmonicity are harpsichordString’s, introduced by the ninth rung: Pythagorean scaling foreshortened below about C3 by the case, and gauges from 0.26 to 0.55 millimetres of iron.

The corners are the seventh rung’s four, with the quill’s contact time at 0.05 milliseconds and its width at four thousandths of the length.

Where the model stops #

The two plectra do not release at the same instant. Two jacks rise on one key, and their quills slip off at slightly different moments because they are voiced differently and sit at different depths. The stagger is a fraction of a millisecond and the arithmetic here assumes it is zero. A stagger of a tenth of a period would rotate the relative phase of the two combs at every partial and move every null this essay computes.

The two registers usually pluck two different strings. A single-manual harpsichord’s two eight-foot registers each have their own choir, tuned in unison, and the model above puts both plectra on one string. That is the right calculation for the lute stop, which shares a choir with the front eight-foot, and it is an idealisation for a pair of unisons. Two strings that beat are not the same object as one string plucked twice, and where the two accounts differ is exactly the effect this essay says is smaller than people think.

Voicing is not equal force. A maker balances the two registers by ear, and what “balanced” means is that they sound alike alone rather than that they pull with the same force. Under any other normalisation the two combs are weighted rather than added, and every null moves.

And the rail is straight here and angled in reality. The compass sweep uses one distance for every note, which exaggerates the treble’s fraction. A real jack rail is angled and sometimes curved, which flattens the trend without reversing it.

What the picture cannot show #

It cannot show the pluck’s own decay. A comb is an initial condition and a harpsichord note is mostly decay, and the upper partials die much faster than the low ones. A partial that is present at the attack and gone after a hundred milliseconds is not the same thing as one that is present throughout, and nothing here distinguishes them.

Nor the jack’s noise. A quill releasing makes a sound of its own, and it is a substantial part of what a harpsichord’s attack is. It is broadband and it has no comb in it.

And it cannot show what the extra string does to the decay. Two choirs sounding together are two coupled oscillators on one soundboard, which is the object the wolf is made of, and coupled strings exchange energy and decay in two stages rather than one. Everything above is about the first millisecond of the note, where superposition is exact and coupling has not had time to act.

Where this ladder goes next #

Ten 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; three exciters on their own three wires; and now the one instrument where the excitation point is a control, on which that control is the only term that ever binds and two of them at once is neither the sum nor the average of what each does alone.

What the ladder owes now is the stagger. Every figure above adds two combs as though the two quills let go at the same instant, and they do not: two jacks on one key are voiced separately, sit at different depths, and release a fraction of a millisecond apart. That interval is not a detail of the model — it rotates the relative phase of the two combs, by an amount proportional to the partial number, so a stagger of a twentieth of a period leaves the fundamental alone and inverts the twentieth partial. Every null this rung computed would move, the ones the pair digs for itself would move furthest, and the combination that is complete at one stagger would not be at another. This collection has the arithmetic for it and no measurement of what the interval actually is, which makes it the rare debt that is answerable by one afternoon with a microphone rather than by a corpus.