The interval between two quills

What the second register is for put two plectra on one string and added their combs, and its own list of limitations named the assumption that made the addition possible:

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.

It is a rare kind of caveat, because the correction is arithmetic rather than a measurement, and because it is not obviously small. This rung runs it, and two of the previous rung’s three findings do not survive.

The reason it matters more here than on any other instrument in this collection is the ninth rung’s result restated: a harpsichord has no dynamic. A plectrum releases at the same displacement however the key is pressed, so the plucking comb is the only term that ever binds — four terms, and only one of them binds puts the contact time and the quill’s width and the string’s dispersion three orders of magnitude away from it — and the instrument’s whole expressive range is which registers are engaged. A correction to what two engaged registers do is therefore a correction to the only gesture the instrument has.

What a delay is, to a partial #

A string is linear, which is why the two combs could be added at all: two static deflections superpose exactly, so the pair’s spectrum is the sum of the two registers’.

Delaying one of them does not break that. It rotates it. Partial $n$ of the second release arrives with a phase $2\pi f_n \Delta t$ behind the first, so the pair’s amplitude is

$$\left| A_1 + A_2\,\mathrm{e}^{-\mathrm{i}\omega_n \Delta t} \right|$$

rather than $|A_1 + A_2|$, and the phase in the exponent is proportional to the partial number, because $f_n$ is.

That one fact does all the work below. A stagger of a twentieth of a period turns the fundamental through eighteen degrees, which is almost nothing, and turns the twentieth partial through a full turn; halfway between, it inverts the tenth. So a single number that is far too small to hear as a delay is a large change to the top of the spectrum and no change at all to the bottom, and where the crossover falls is set by the interval alone.

The pair’s own holes are not real #

The previous rung’s most striking finding was that the combination acquires nulls neither register has.

At 32 and 50 millimetres on a 355-millimetre string, the far register cannot excite partial 7 and the near one cannot excite 11 or 22; the pair sounds all three, because where one comb is zero the sum is simply the other one. But at partials 17 and 20 the two combs are equal and opposite, and the sum is zero. Those were holes the combination digs for itself, and the essay drew a design diagram showing which register positions avoid them.

They do not survive. An interval of ten microseconds — a two-hundredth of this string’s period, far shorter than anything two hand-voiced quills would be matched to — removes both of them, and every interval above it removes them too.

The mechanism is exactly the phase proportionality. A self-dug null is a cancellation, which needs two amplitudes to be exactly opposed; the partials where that happens are necessarily high-numbered, because the two combs have to have run far enough apart to be antiphase; and a high-numbered partial is the one a small interval rotates most. Partial 17 of this string is at nearly nine kilohertz, and ten microseconds turns it through thirty-two degrees — which is enough, because taking two exactly opposed amplitudes sixteen degrees apart leaves half of each of them standing.

The two kinds of missing partial behave in opposite ways and the figure separates them. A null one register simply has — a node under its own quill — survives any interval whatever, because $|0 + A_2 \mathrm{e}^{\mathrm{i}\phi}| = |A_2|$ for every $\phi$. A null the pair makes survives none.

So the previous rung’s design diagram is a diagram of an idealisation. Its nine clean register positions and its twenty-three dirty ones are all clean on any real instrument, and the rule it derived — that a small whole-number ratio between the two distances puts shared nulls low in the spectrum — is untouched, because a shared null is the first kind. The half of that rung about which holes are filled stands, and the half about which holes are dug does not.

And the level does not survive either #

The second finding was that the pair gains 4.4 decibels at the fundamental, which is most of six and is the loudness a full registration buys.

That one is a fundamental, so a small interval barely touches it — and the whole question is what “small” means. Sweep the interval and the answer is a cosine: the gain falls from 4.4 decibels to −9.2 at half a period, where the two plucks are in opposition, and returns to 4.4 at a whole period. There is nothing in the curve but the string’s own period.

One decibel of the gain is spent at 0.153 of a period. That is the number this rung exists to produce, and the reason is that it is the same number at every pitch. The rotation is proportional to frequency and the period is inversely proportional to it, so the fraction cancels: 0.154 at the bottom of the compass, 0.152 at the top, and 0.153 or 0.154 at every note between.

Which is the rail’s own shape, arriving from the other side #

This is where the rung stops being a correction and starts being an argument, because a constant fraction of a period is a variable number of milliseconds, and a regulator does not work in periods.

The previous rung opened by observing that a jack rail is one piece of wood across the whole compass, so a register plucks at a fixed distance and the plucking fraction — the quantity the whole ladder argues about — is what that distance becomes on each string in turn. Fifty millimetres is one thirty-sixth of the string at C2 and one three-and-a-halfth at C6.

A stagger is the identical shape in time. A regulator sets a difference in jack height and quill stiffness, which is a difference in milliseconds, and the argument is about a fraction of a period. A harpsichord’s strings shorten by a factor of ten from the bass to the treble and its periods by a factor of sixteen, so one setting is sixteen different quantities.

That was not put in. The plucking distance and the release interval are two independent dimensions of the same instrument, chosen by different craftsmen for different reasons, and both turn out to be constants in a unit the physics does not use. It is the third instance of the shape in this collection — an end correction is a fixed length against a shrinking tube, and a bow held at a fixed distance from the bridge is a fixed length against a shortening string — and it is the first one that is a time rather than a length. The first two are geometry and could in principle be designed away by angling the rail or moving the hand; this one cannot, because a quill’s release time is set by its stiffness and the depth it engages, and nothing about either scales with the string.

Take three-tenths of a millisecond, which is a modest difference between two separately voiced quills. In the bass it is a fiftieth of a period and costs two hundredths of a decibel. At C6 it is 0.31 of a period, and the pair’s gain has gone from 4.6 decibels to −0.3: two registers together, at the top of a harpsichord, quieter than the rounder of them alone.

That is a claim about regulation rather than about design, and it is one a technician would recognise. A harpsichord’s treble is the part that has to be voiced most carefully and the part that goes wrong first, and the usual account of why is that the strings are short and thin and the quills are small. The arithmetic says something more specific: the tolerance in the only unit anybody can set is sixteen times tighter at the top than at the bottom, on a mechanism whose adjustments are made with a knife.

Whether it is still louder and rounder #

The previous rung’s third finding was the one that came out backwards, and it is the one worth testing hardest.

Engaging the brighter register alongside the darker one made the pair darker than either — the two plucks add at the bottom of the spectrum, where a partial’s wavelength is long compared with the gap between the quills, and partially cancel at the top, where it is not. That was offered as the arithmetic behind what harpsichordists say about a full eight-plus-eight against a single eight: fuller, broader, less penetrating.

Two things come out of running it across the compass and neither was on the list.

The first has nothing to do with the stagger and is a correction to the finding as stated. Released together, the pair is darker than the rounder register only in the top two octaves. In the bottom two it is brighter — the ratio runs 1.04, 1.04, 1.07, 1.07, 1.01, 0.96, 0.91, 0.89, 0.90 from C2 to C6 — and it crosses at about C4. The previous rung computed that finding on one 355-millimetre string, which is C5, and the number it got there is right; what does not hold is the generalisation to the instrument. In the bass, fifty millimetres and thirty-two are both a small fraction of a long string, so the two combs are nearly the same shape and nothing much cancels anywhere.

The second is that a real interval reverses the trend rather than shifting it. At three-tenths of a millisecond the ratio runs 0.83, 0.74, 0.83, 0.93, 0.95, 1.03, 1.05, 1.14, 1.20 — rising with pitch where the simultaneous curve falls, and crossing one at about G4 going the other way.

The mechanism is one thing seen at both ends, and it is the phase proportionality again. Where the two combs reinforce each other in the treble — which is the bass end of the instrument, because there the two combs are alike — an interval takes that reinforcement away, and it takes most away at the top of the spectrum where the rotation is largest. Where the two combs cancel in the treble, which is the top of the instrument, an interval takes the cancellation away instead. Both effects are the same rotation and they point in opposite directions because the thing being rotated is opposed in one case and aligned in the other.

So the account of a full registration as louder and rounder holds in one octave of the instrument and something different is true in each of the others. At C6 the pair is a fifth brighter than a single register and no louder at all; at C2 it is a fifth darker and 4.3 decibels up; in the middle it is close to neither. That is a claim about how the second eight-foot is used, and it is used most at the top, where a harpsichord is thinnest.

One honest caution about how far to push it. The reversal depends on the interval being a real fraction of a period, and three-tenths of a millisecond is a plausible interval rather than a measured one. At a tenth of a millisecond the staggered curve is much flatter and the crossing moves toward the top of the instrument. The direction of the effect is arithmetic; its position on the keyboard is a measurement nobody has made.

Which computation produced the numbers #

Each register’s comb 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 — which is the previous rung’s arrangement, unchanged.

What is new is one line: the second register’s contribution is multiplied by $\mathrm{e}^{-\mathrm{i}\omega_n\Delta t}$ before the sum, and the sum is taken as a complex magnitude. The partial frequencies are the stiff string’s, $n f_0\sqrt{1+Bn^2}$, rather than $n f_0$ — the dispersion the sixth rung computes stretches the upper partials, so a given interval rotates them by slightly more than proportionally. On a harpsichord’s iron wire that correction is under one per cent at the twenty-fourth partial and leaving it out would have put this figure’s ladder at odds with the ladder every other figure in the anchor draws.

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 previous rung’s threshold and the same order as the two per cent test the single-comb figures use.

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

Where the model stops #

The two releases are treated as two independent initial conditions. In reality the string is not free between them: after the first quill lets go the string vibrates while the second still holds it at a point, which is a constrained problem rather than a superposition. The approximation is exact in the limit of a small interval, and the intervals that matter here are between a hundredth and a third of a period, so the treble end of the argument is where it is weakest — which is the end the interesting result is at.

Voicing is not equal force. A maker balances two registers by ear, and “balanced” means they sound alike alone rather than that they pull equally. Under any other normalisation the two combs are weighted rather than added and every number moves, though not the phase argument, which does not care about amplitudes.

And the interval itself has never been measured. Everything above is parameterised on a quantity this collection does not have. The previous rung called that the rare debt that is answerable by one afternoon with a microphone, and it still is: two accelerometers or one microphone and a fast enough sample rate would give it in a morning, per jack, per instrument.

What the picture cannot show #

It cannot show the decay. A comb is an initial condition and a harpsichord note is mostly decay, and the upper partials — which are the ones this whole rung is about — die fastest. A rotation that changes partial 17 changes something that has left the sound in a hundred milliseconds.

Nor the two choirs. Two eight-foot registers usually pluck two different strings, tuned in unison and therefore beating; this model puts both plectra on one string, which is exactly right for the lute stop and an idealisation for a pair of unisons. Two strings a couple of cents apart have their own slowly rotating phase, and it sweeps through everything computed here several times a second.

And it cannot show whether a listener could hear any of it. A five per cent shift in a spectral centroid and a decibel of level are both near the edge of what is detectable on a plucked note that lasts a second, and nothing in this collection has put either in front of a listener.

Where this ladder goes next #

Eleven 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; two registers on one string; and now the interval between them, which fills the holes the pair was said to dig and takes back the level it was said to gain.

What the ladder owes now is the second string. Every figure above puts both plectra on one wire, and a single-manual harpsichord’s two eight-foot registers each have their own choir tuned in unison — so the real object is two strings a couple of cents apart, each plucked once, coupled through a soundboard. That changes the argument twice over: the relative phase of the two combs is no longer a constant set at the attack but a quantity that rotates through a full turn every second or two as the two strings beat, which means the pair sweeps through every interval this rung computes, in order, during one note. And the two strings are coupled oscillators on one bridge, which is the object a wolf note is made of, so the decay is two-staged rather than exponential. The first half is arithmetic and this collection has all of it: take the pair’s spectrum as a function of interval, which is the figure above, and drive the interval linearly in time at the beat rate. What comes out is a prediction about what a harpsichord’s tone does over the length of a note rather than at its attack, which is the half of the instrument every rung of this ladder has left out.