Two strings undo what two quills were regulated for

A harpsichord has no dynamic. Its quill lets go of the string at the same displacement however the key is pressed, so a player cannot play louder or softer, and the instrument’s whole range of colour is which registers are engaged. What the second register is for put two plectra on one string, at 32 and 50 millimetres from the nut of a 355-millimetre string, and added their combs of partials: the pair was 4.4 decibels louder at the fundamental than the far register alone, and it dug two holes of its own, at partials 17 and 20, where the two combs were equal and opposite.

The interval between two quills then let the two plucks happen at different instants, because two jacks rising on one key never release exactly together. A delay of dt turns partial n of the later pluck through 2π·n·f0·dt, so it leaves the fundamental alone and turns the twentieth partial twenty times as far. The pair’s own holes vanished at a hundredth of a period; the gain at the fundamental fell by a decibel at 0.15 of a period, which at C5 is 0.29 milliseconds. That essay read the second number as the tolerance a regulator has to work to.

Both essays put both plectra on one wire. The registers of a real harpsichord do not share a string. Each eight-foot register has its own choir: the front jacks pluck one string and the back jacks pluck another, tuned to the same pitch. And no two strings are ever tuned to exactly the same pitch.

A detuning is a stagger that grows #

Suppose the second string is c cents above the first. Its partial n runs at n·f0·2^(c/1200), and its phase against the first string’s partial n advances at n·Δf turns a second, where Δf is the difference between the two fundamentals. That is exactly the turn the regulation interval gave, with the interval growing with time: at t seconds the pair is the one-string pair staggered by t·Δf/f0. At two cents, Δf/f0 is 0.00116, so after one second the effective stagger is 1.16 milliseconds — more than half a period at C5.

So during a single held note the two-string pair passes through every interval the previous essay swept, from aligned to half a period apart and back, the fundamental doing so once every 1/Δf seconds and partial n doing so n times as often. Whatever the regulator achieves at the pluck, the tuner’s unison has taken over within a fraction of a second.

The figure follows the pair’s brightness, the spectral centroid in partial numbers, through the note. At the pluck it is the aligned pair’s 2.44. Two cents apart, it climbs to 3.67 within 0.8 seconds, falls back to 2.31 at about 1.6, and climbs again, a breathing that repeats every 1.65 seconds — the period of the fundamentals’ beat. The brightness swings by half its value. Half a cent apart the same cycle takes 6.6 seconds and most of a harpsichord note is over before it completes; four cents apart it takes 0.83 seconds and the pair breathes three times in a note’s audible life.

The regulation’s 1-decibel tolerance is passed at 254 milliseconds. Before that the regulator’s work is what the listener hears; after it, the tuner’s.

Every partial at its own rate #

The centroid is an average, and what it averages is a set of partials beating at different speeds. The fundamental falls 13.5 decibels below its aligned level at the bottom of its 1.65-second cycle; the second partial falls 14.8 decibels twice as often; the fifth falls 19.6 five times as often. The thirteenth partial, whose two combs are nearly equal and opposite in sign at these plucking points, plunges forty decibels thirteen times in 1.65 seconds — about eight times a second, a flicker rather than a breath.

The seventeenth partial runs the other way. It is one of the aligned pair’s own holes: on one string the two plucks nearly cancel it. With two strings it starts in its hole and immediately climbs out, spending almost the whole cycle up to 32 decibels above where the aligned pair had it, dropping back into the hole only for an instant each time the two strings’ seventeenth partials realign. Every 97 milliseconds that instant comes round.

That is the arithmetic of beats are arithmetic applied to a harmonic spectrum: each partial beats at its number times the fundamentals’ difference. The unison is the coarsest thing in the room explains why a cent or two survives any tuning: a mistuned unison beats more slowly than any other interval mistuned by the same amount, so it is the hardest mistuning to hear while tuning and the easiest to leave in. On a harpsichord the leftover does something more than beat, because the two strings are excited at different points and carry different combs. A partial whose two combs are unequal swings little; a partial whose combs are nearly equal swings enormously, and the excitation points — not the tuning — decide which partials those are.

How soon the regulator stops mattering #

The time at which the detuning carries the pair past a regulation tolerance has a closed form. A tolerance of a fraction τ of a period is reached when t·Δf = τ, so t = τ / (f0·(2^(c/1200) − 1)). It is inversely proportional to the pitch, so it halves with every octave, and inversely proportional to the detuning.

At C5 and two cents apart, the decibel tolerance goes at 254 milliseconds and the hundredth of a period at 16.5. At C2 the same two cents leave the regulation intact for two seconds, and half a cent leaves it intact for eight — longer than a bass note sounds. So the regulator’s work is heard through the whole of a bass note and only through the attack of a treble one. The previous essay found that the regulation tolerance, as a fraction of a period, is the same at every pitch, which made it a factor of sixteen in milliseconds across five octaves; the detuning reverses the consequence. The bass’s demanding tolerance is the one that lasts, and the treble’s generous one is undone almost at once.

The hundredth of a period, at which the pair’s own holes vanish, is gone within a few tens of milliseconds at any pitch and any plausible detuning. The holes the one-string arithmetic produced were never a feature of a real two-choir instrument’s sustained sound.

What the second register is worth, averaged #

Once the strings have drifted apart, the pair’s spectrum at any instant is just a point on the cycle, and what a listener hears over a note is closer to the average. Two strings beating against each other add in power on average, not in amplitude, so each partial’s averaged level is the root of the sum of its two combs’ squares, whatever the signs.

At the fundamental that costs the pair most of its gain. Aligned on one string, the near and far plucks reinforce each other, and the pair is 4.36 decibels louder than the far register alone; averaged over the beating, it is 1.54 decibels louder. Partials 2 to 6, where the two combs also have the same sign, lose about three decibels each in the same way.

Above the sixth partial the trade reverses. Where the two combs have opposite signs, the aligned pair cancels and the averaged pair does not: partial 9 comes back by 14 decibels, partial 16 by 13, and partials 17 to 20, the aligned pair’s deep cuts, by 25 to 29. The result is a spectrum with less weight at the bottom and none of its upper holes, and its centroid — 2.93 against the aligned pair’s 2.44 — is a fifth higher.

That reverses the second-register essay’s conclusion in two places. It found the second register rounding the sound and adding weight at the root; on two strings, averaged over a note, the second register adds less weight at the root than that essay computed and brightens the sound rather than rounding it. Its holes were the most interesting finding of that essay, and they are not part of the sustained sound at all.

The holes are momentary #

The pair still has holes, but they are places the spectrum passes through rather than places it stays. Three partials — 13, 17 and 20 — pass through the missing-partial threshold once in each of their cycles, and they spend 4, 1 and 3 per cent of the cycle there. At two cents apart that is a few milliseconds every hundred. The partials that swing furthest are the same ones: 42 decibels for the thirteenth, around 30 for the seventeenth to twentieth, against under a decibel for the seventh and eleventh, where one comb is nearly zero and the other does all the work.

The figure is a map of where a harpsichord’s two-register tone is unstable, and it is drawn by the plucking points. Moving a register along the rail moves which partials have equal combs, and so which partials flicker. The second-register essay presented the register spacing as a choice that decides which partials are missing; on a two-choir instrument it is a choice that decides which partials shimmer. A builder who set the near register a few millimetres further along the rail would not be likely to remove the shimmer — among twenty-four partials some pair of combs is almost always nearly equal — but would move it to other partial numbers, and so to a different register of the sound, higher or lower in the spectrum where the ear is more or less ready to notice it.

The breath across the compass #

The same two cents give very different notes at different pitches, because the beat period is the reciprocal of the fundamentals’ difference and that difference scales with the pitch. At C6 the fundamentals beat every 0.83 seconds, and the pair’s brightness completes a breath before the note has faded far. At C5 it takes 1.65 seconds, at C4 3.3, at C3 6.6 and at C2 more than thirteen. A harpsichord’s bass notes ring longer than its treble notes, but not thirteen seconds’ worth of audible life: at C2 the pair spends the whole note on the first part of its first cycle, where the fundamentals are still nearly aligned and the one-string arithmetic still describes it.

So the compass divides. In the bass, the regulation tolerance lasts most of a note and the pair sounds roughly as the one-string essays computed — the second register’s full gain at the fundamental, the holes in place for the first tenth of a second or so. In the treble, the regulation is gone in a few hundredths of a second, the fundamental breathes within the note, and the averaged spectrum of the last figure is what is heard. Somewhere in the tenor, at a cent or two, a note lasts about one beat of its fundamentals, and the pair’s brightness makes exactly one swing during it — rising from the aligned value through the bright half of the cycle and back.

That is a different account of why a harpsichord’s two eight-foot registers together sound unlike either alone. The usual account is the one the second-register essay computed: more weight at the root, a rounder spectrum. The arithmetic here suggests that above the middle of the compass the more noticeable difference is motion — a sound whose colour shifts through each note — and that the motion is supplied by the tuner’s leftover cent rather than by anything in the action.

Why this was invisible to the one-string arithmetic #

The previous two essays were not wrong about one string. Two plucks on one string add coherently forever, so every finding about their fixed relative phase stands for that configuration — which is the configuration of a single register with two jacks, or of a lute stop plucking the same string as another register, both of which exist. What changed is the object. A harpsichord’s unison choirs are two strings, and two strings are the one object in all these essays on the plucking point whose relative phase is not fixed at the moment of excitation.

That is the surprise worth carrying out of the arithmetic. Four terms, and only one of them binds found that on a harpsichord the plucking comb is the only term that ever binds — contact time, quill width and dispersion are orders of magnitude away. The detuning between the choirs is a fifth term, and on a held note it binds harder than the comb does: the comb sets the two amplitudes, but the tuning decides how they combine.

The model, stated #

Two strings, each plucked by one register at its own point, in the ideal-string comb these essays have used since where the hammer lands: a pluck at a fraction p of the length gives partial n an amplitude proportional to sin(nπp)/n².

The second string is detuned by c cents, and its partials stay harmonic, so partial n’s phase against the first string’s grows as 2π·n·Δf·t. Inharmonicity would make the beat rate of partial n slightly larger than n·Δf on both strings alike; it does not change the picture.

Both strings decay alike, so every level is relative to the pair’s own decaying envelope. A real unison’s two strings are coupled through the bridge and decay in two stages — a fast decay while they move together and a slow one after — which is left out entirely.

The regulation tolerances are the previous essay’s: 0.153 of a period for a decibel of the pair’s fundamental, and a hundredth of a period for its holes.

What two strings on paper cannot show #

What is heard of a breathing brightness. The centroid swings by half its value over a second and a half, and the thirteenth partial flickers eight times a second. Whether that is heard as life in the tone, as a slow wah, or not at all depends on the masking of the upper partials by the lower and on a listener’s sensitivity to slow spectral change, and the arithmetic says nothing about either.

What the bridge does. Two strings on one bridge are coupled oscillators. They pull each other’s frequencies, they exchange energy, and when nearly in tune they can lock or decay together in ways a pair of independent strings never does. Every figure here treats the two strings as independent.

How a harpsichord is actually tuned. The figures sweep half a cent to four. A careful tuner aims for a beatless unison, though a tuner can have reasons to do otherwise — on a piano, the octave is counted best over a unison left wide — and a harpsichord’s light strings drift with the room’s temperature and humidity faster than a piano’s. What a real instrument holds through an evening is a matter of measurement, and the answer decides, pitch by pitch, which side of a second the regulation’s influence lasts. A unison held within half a cent across the treble would keep the regulator in charge of C5 for a whole second; one that has drifted to four cents would hand it over in an eighth.

Still open: two strings that share a bridge #

The model treats the two strings as independent, and a harpsichord’s are not: both cross one bridge, and through it each drives the other. A string that decays twice is counted early worked through the same object on a piano’s unison: two strings driving one bridge split into a mode that moves together and one that moves apart, decaying at different rates, so the sound has a prompt part and an aftersound. The computation is this pair of combs carried by two coupled strings rather than two independent ones, and it would say whether the bridge’s coupling pulls a slightly detuned unison into step — leaving the regulator’s interval in charge for the whole note — or whether the in-step mode drains away first, leaving the out-of-step mode, with the pair’s holes in it, as the sound that lasts.