A doubled pizzicato gives its note away early

The blend arrives before the note does found that two players starting one note together are a balance turned by the clock. Their envelopes rise at different rates, so the level of one against the other moves during the attack, and it moves by exactly twenty times the log of their two attack times — six decibels for a clarinet with a violin — and then stops. One pair in ten changed owner inside that travel. Nine in ten did not, and the attack was a mis-balance that corrected itself in about twenty milliseconds.

It ended by naming what an attack is not: a release. A bowed note stops when the bow does and a held note stops when the player stops, but a plucked note does not stop at all. Every partial decays, the upper ones faster, so a pizzicato doubled by a held instrument is a pair whose balance falls for the whole length of the note, and the travel of the dial is not bounded by anything. The prediction written down was that ownership of such a note would be a handover with a duration, a hundred times longer than the attack’s, and that there was no reason to expect the answer to be small.

A pluck is a balance that keeps falling #

The pair is built from pieces already in hand. The plucked player is a violin string through the violin’s measured body, which is what a pizzicato radiates; the held player is a flue pipe, standing for a flute or an organ stop, with no filter moving under it. They add in power partial by partial, because two players are independent sources, and the composite is judged by the measure two players on one note used for a doubling held still: its log-spectral distance from each player, the note belonging to whichever player it is nearer.

What is new is that one player is dying. Each of the pluck’s partials loses a number of decibels proportional to time, to one over the pluck’s decay time, and to its own partial number — the loss law the note that gets duller as it dies built for a struck string, under which a pluck’s colour drains toward its fundamental. The decay time is set at one second, which is a round number for a pizzicato in the middle of a violin’s range and not a measurement of one, and the pluck starts twelve decibels above the held note, strongest partial against strongest partial — nearly ten decibels in total power — which is how a pizzicato’s onset stands out from a sustained part at the same written dynamic. Both are swept below.

The note changes hands at 70 milliseconds #

At the instant of the pluck the composite is the pluck’s — 1.8 decibels from its colour and 17.8 from the flute’s — and at 70 milliseconds it is the flute’s. By then the pluck has lost only four and a half decibels of total power, and it is still 5.2 decibels louder than the held note it has just surrendered the note to.

The dashed curve says how unexpected that is. A pluck whose partials all faded together would lose total power at almost exactly the same rate — the two curves are nearly one line — and keep its colour, and it would keep the note for 543 milliseconds. Two plucks with the same balance at every instant hand the note over eight times apart. The steady-state doubling of the same two instruments makes the point from the other side: held still, a violin keeps ownership of a doubled note with a flute until it is more than twenty decibels quieter than the flute. A real pluck gives the note away twenty-five decibels before that level is reached.

So the handover is not a balance crossing a threshold. The balance at which a steady doubling changes owner, which every steady doubling measured so far has used, does not predict when a decaying one does, and it misses by twenty-five decibels.

What hands the note over is the pluck’s upper partials #

The reason is visible in the two distances the ownership is judged on.

The distance from the composite to the flute falls steadily as the pluck loses level, which is the part a balance describes. The distance to the pluck rises, and a balance does not describe that. A pluck losing its upper partials fastest is changing shape: its own spectrum is collapsing toward its fundamental, while the composite still carries the flute’s upper partials at full strength. The composite is therefore not merely a quieter version of the pluck plus a flute. It is a spectrum whose top belongs to the flute already, and whose bottom is the only part of the pluck left.

A pizzicato is identified by its upper partials, and they are exactly what a pluck loses first. The note changes hands when the top of the composite stops being the pluck’s, and that happens long before the bottom does.

Judging the pluck against its spectrum as it was struck rather than as it currently is moves the handover from 70 milliseconds to 63. The difference is small because both readings turn on the same thing — the composite’s upper partials no longer matching a violin’s — and the drawing uses the pluck as it is now, which is the fairer comparison of the two.

Every held partner takes it within a twentieth of the pluck’s life #

The flute is one partner. The same computation for every held instrument whose radiated spectrum is in hand gives the same shape of answer and, in two cases, a stronger one.

A clarinet takes the note at 44 milliseconds, an oboe at 50, a flue pipe at 70. Under a uniform fade those would be 258, 472 and 543. A voice singing the vowel of “hod” and a trumpet do not need to wait: their steady-state crossings require the violin to be twenty-one and fifteen decibels louder than they are before the violin owns anything, so a pluck with a twelve-decibel head start never owns the note at all. Its onset is heard as an attack on the sung or brass note, not as a separate colour.

A section has a loudest member found that ownership among instruments is never cyclic: the same instruments always win. The release keeps that order and adds a clock to it. The instruments whose steady spectra dominate a violin’s — the voice and the trumpet — take a doubled pizzicato at once, and the ones that a violin can dominate held still — the clarinet, the oboe and the flute — take it within a twentieth of a second.

The loss law decides, and it is the measurement nobody has made #

The whole difference between 70 milliseconds and 543 is the exponent relating a partial’s loss to its number, which the drawings above set at one.

At an exponent of nought the flute takes the note at 543 milliseconds; at a half, 187; at one, 70; at one and a half, 28; at two, 11. The handover spans a factor of fifty across the range of plausible loss laws, and it is steepest exactly in the region where measured strings are said to sit.

The collapse belongs to the bass recorded that exponent as a measurement nobody has made, and counted in the decay, or not at all found the same unmeasured number deciding whether a hard strike costs a piano tuner the beat. It now decides, on a different instrument and in a different question, whether a doubled pizzicato’s colour lasts a twentieth of a second or half a second. Three unrelated questions — how a note’s colour drains across a keyboard, how a tuner should strike an octave, and how long a pizzicato is heard as itself inside a doubling — wait on one measurement of how fast a string’s upper partials die relative to its lower ones.

A fixed fraction of the decay time #

A one-second decay was a choice, and the dependence on it is exact.

Every loss in the model is a number of decibels proportional to time divided by the decay time, so a pluck that dies twice as slowly passes through every state twice as late. The handover is therefore a fixed fraction of the pluck’s life: seven per cent of it for a flute, five for an oboe, four and a half for a clarinet. A short, dry pizzicato of a quarter of a second loses its note in 11 to 18 milliseconds, inside the time the attack takes to settle. A string that rings for four seconds — nearer a harp’s or a guitar’s decay than a violin’s pizzicato — keeps it for 180 to 280 milliseconds, which is a clearly heard stretch of colour at the front of a crotchet and still gone before the crotchet is.

That turns an unmeasured decay time into a unit. The claim that needs no instrument measured is that a plucked note doubled by a sustained one is the plucked instrument’s for its first twentieth, and the held instrument’s afterwards.

A louder pluck buys milliseconds, not the note #

The head start was twelve decibels, and a player can make it larger.

Every extra decibel of pluck buys a few milliseconds of ownership. From level to thirty decibels up, the flute’s handover grows from 47 to 142 milliseconds, and thirty decibels is a pizzicato that would dominate the texture as a sound in its own right. The clarinet, whose colour is furthest from a violin’s in the upper partials, gains most — from 9 milliseconds to 148 — because at level the clarinet has nearly taken the note before the pluck has begun to decay.

No head start makes the note the pluck’s for more than about a seventh of its decay time. A composer who wants the pluck’s colour to survive in a doubling cannot get it from the dynamic; the upper partials go at the same rate however loud they began.

Up the range, the handover stays short #

The spectra are filtered by fixed resonances, so the handover could have depended strongly on the note.

It moves between 21 and 101 milliseconds across three octaves and has no trend, dipping at C5 for all three partners and rising again above it. An instrument is not one timbre found the violin’s radiated spectrum different at every note because its body’s resonances stay put while the note moves, and the blend table has a row for every note found the doubling rankings turning over across the range for the same reason. The handover inherits that irregularity and stays inside a tenth of a second everywhere.

The release is not the attack run backwards #

The attack’s dial was bounded and settled in about twenty milliseconds, and it changed a doubled note’s owner in one case in ten. The release’s dial is unbounded — and it changes the owner in every case where the pluck starts as the owner, within 21 to 101 milliseconds for a one-second pluck, which is the same order of time as the attack.

The reason the unbounded travel does not make the handover slow is that the travel is not what does the handing over. The attack moved a balance and kept both colours intact; the release destroys one colour from the top down, and the composite’s resemblance to a colour is decided by its upper partials long before its level is. The recorded prediction was right that there is no bound and wrong that the answer would be large, and it was wrong for a reason it could not have seen from the attack: an attack does not change a spectrum’s shape, and a decay does nothing else.

The arithmetic, and the choices in it #

The spectra are the same radiators every doubling here has used: the string’s one-over-n source through the measured violin body, the organ list for the flue pipe, the odd-dominant clarinet list cut off above its tone-hole cutoff, the reed list for the oboe with its own cutoff, a glottal pulse through the vowel of “hod”, and a sawtooth through a trumpet bell’s high pass. Two players add in power. Resemblance is log-spectral distance over the first sixteen partials with each spectrum normalised to its own power, which is the measure every steady doubling has been judged by. The pluck’s partial n loses sixty decibels over the decay time divided by n.

Three things there are choices rather than measurements, and all three are swept: the decay time, the head start and the loss exponent. The ownership rule — nearer in colour — is a convention, and the first fifty milliseconds is a reminder that what a listener uses to name an instrument is not necessarily the steady spectrum’s resemblance at all.

What a decaying spectrum leaves out #

The pluck’s own transient. A pizzicato begins with the noise of a finger leaving a string and a burst of partials that are not in any steady spectrum, and those are among the strongest identity cues a note has. The model starts the pluck as its steady spectrum at full level, so the handover here is the handover of the colour, and the attack’s identity cues end on a clock of their own.

The body’s own ringing. A violin body is a set of resonances that ring after excitation, and the pluck’s energy passes through them. The model treats the body as a fixed filter on a decaying source, which is right for the steady part and wrong for the first few milliseconds after the pluck.

The room. A pizzicato in a hall is followed by reverberation, which sustains every partial for longer than the string does and sustains the low ones longest. What that does to the handover is not obvious in either direction.

And a listener. Resemblance in log-spectral decibels licenses a comparison of colours, not a report of what anybody hears as the owner of a note. The drawing says when the composite’s spectrum stops being nearer the pluck’s, and a listener attending to the pluck’s onset may go on hearing a pizzicato after that.

Whose orchestration #

The practice is orchestral doubling of a sustained line by a plucked one — pizzicato strings under a woodwind melody, a harp doubling a clarinet or a flute — in the repertoire from the nineteenth century on, where it is taught as a way of giving a sustained line a defined start. The arithmetic supports that description exactly and more strongly than its wording: the pluck is the held instrument’s attack, owning the note for its first twentieth, and the held instrument owns the rest. A doubling of a pizzicato with a trumpet or a voice does not even give the pluck that twentieth. An orchestrator doubles a line, not a chord found a doubling to be a small decision made once; a plucked doubling is a smaller one still, since after a few tens of milliseconds it has been made for the player on the held note.

Still open: whether a room gives the note back #

The decay the pluck undergoes here is the string’s alone, and a pizzicato is heard in a room whose reverberation keeps every partial alive after the string has let it go. A room decays unevenly — its high frequencies are absorbed faster than its low ones — which is a tilt in the same direction as the string’s, and a slower one. So in a room the composite a listener receives is the direct pluck, the held note, and a reverberant tail of the pluck whose upper partials last longer than the string’s did. Whether that tail returns the note’s colour to the pluck after the direct sound has handed it over, and for how long, is the same distance arithmetic with a room’s band-by-band decay put under the pluck.