A bow holds the number a blow hides

Five essays here describe a note that nobody is holding. Something delivers energy once — a hammer, a plectrum, a mallet — and everything afterwards is the string spending it. The loss law decides how the spending goes, the spectrum collapses toward the fundamental, and the note ends.

About half the instruments in an orchestra do not work that way. A bow keeps pulling, a reed keeps opening and closing, a flue keeps splitting the jet, and a singer keeps pushing air past the folds. The energy is not deposited and then spent; it is supplied at the rate it is lost, for as long as the player wants the note, and the decay that the account here has treated as the whole of a note is postponed until the moment the supply stops.

That is one term in the model and it changes which part of the note carries the instrument’s identity. It also reverses the account’s own direction of travel, because the quantity the fourth essay had to dig into the middle of the note to find turns out to be sitting on the surface of a held one.

The arithmetic of being held #

Each partial of a string is an oscillator with its own resonant frequency and its own loss. Struck, it is given an amplitude and then loses it:

$$A_n(t) = F_n \, e^{-\gamma_n t}, \qquad \gamma_n = \gamma_1 n^{\,p}$$

The source spectrum $F_n$ decides the strike; the exponent $p$ decides everything after it. Driven at its own resonance, the same oscillator reaches a steady state where the energy arriving each cycle equals the energy leaving, and that balance has a one-line answer:

$$A_n = \frac{F_n}{\gamma_n} = \frac{F_n}{\gamma_1 n^{\,p}}$$

The loss exponent adds to the source’s roll-off. A drive falling as $1/n$ against a loss rising as $n$ gives a held spectrum falling as $1/n^2$; against a loss rising as $n^2$ it gives $1/n^3$. In decibels the statement is simpler still: a source rolling off at six decibels per doubling of partial number, under a loss exponent $p$, holds a spectrum rolling off at $6(1+p)$ decibels per doubling.

This is why the two kinds of note put the same number in different places. A strike is the source spectrum with no time elapsed, so it contains no loss at all — which is exactly what the essay that found both endpoints identical when it discovered that two notes with different loss laws are identical at the attack and identical in the silence, and that the whole difference between them lives in the part of the note neither endpoint samples. A sustain is the source spectrum divided by the loss, so it contains nothing else.

What a held note looks like when the loss changes #

Taking the eight-partial string model on C3 at eighty decibels, three loss exponents give three strikes that are the same drawing and three sustains that are not.

The strike has a centroid of 231.9 hertz whatever the exponent is. The sustains sit at 167.0, 144.5 and 133.3 hertz for exponents of a half, one and two — 5.7, 8.2 and 9.6 semitones below the note’s own attack, and 3.9 semitones apart from each other across the plausible range of the exponent. Nothing has been added to the model except the requirement that the energy be replaced as fast as it leaves.

The second partial is the clearest single cell. Struck, it sits 6.0 decibels under the fundamental in all three cases, because the source spectrum is $1/n$. Held, it sits 9.0, 12.0 and 18.1 decibels under, which is the same six decibels plus three, six and twelve more. Every higher partial repeats the pattern with a larger multiplier, so by the eighth partial the three sustains are 27.4, 36.5 and 54.5 decibels down — a spread of twenty-seven decibels on one partial of one note, produced entirely by a number that a blow cannot express.

The drive being a different shape changes the picture and not the finding. A reed opening and closing against a mouthpiece supplies a spectrum with almost nothing on the even partials, and the loss divides that spectrum the same way it divides a bowed string’s. The offset between the rows is a property of the loss and not of the source, which is why it is the same number on both.

That independence is worth stating as a limit rather than as a strength. It means the slope of a held note is the sum of two unknowns, and reading the loss law off it requires knowing the drive. A listener comparing two instruments has neither in isolation.

The corner that supplies the sawtooth #

For a bowed string the drive is not a free parameter. The bow makes a corner that travels round the string once per period, and the force it delivers to the bridge is a sawtooth — every harmonic present, at one over $n$, with a notch wherever the bowing point falls on a node. That is a strikingly clean source: a spectrum with an exact analytic shape, set by geometry rather than by a material.

So a bowed string is the case where the division actually can be inverted. The numerator is known to be $1/n$ from the geometry of the motion; anything else in the held spectrum is either the loss or the radiation. That is not true of a struck string, where the source spectrum depends on where the hammer lands, on how heavy it is and on how long it stays in contact, none of which a listener knows.

The uncomfortable half of that is that a bowed string’s losses are not the string’s alone either. Most of what leaves a bowed string leaves through the bridge, which is the same path three strings on one note found dominating a piano’s decay, and the bridge’s admittance is a function of frequency with resonances in it. An exponent is a smooth curve and a body is not, so $n^{\,p}$ is a description of the trend and not of the object.

The attack is the least informative moment of a held note #

The first two essays here found that an instrument’s identity is in its attack, and the finding held twice over: cut the first fifty milliseconds and listeners stop being able to name the sound, while the spectrum they are still hearing is unchanged. On a driven note the loss law reverses that ordering completely.

At the instant the drive begins, nothing has been lost yet, so the spectrum present is the drive’s and the loss is not in it. As the note establishes, each partial climbs toward its own balance, and the time it takes is the reciprocal of its own loss rate: partial $n$ is within a factor of $e$ of where it will settle after $1/\gamma_n$ seconds. The partials that make the sustain dark are the ones that get there first.

On the string above, at an exponent of one, the eighth partial has settled after 0.109 seconds and the fundamental after 0.869 — a factor of eight, which is the exponent applied to the partial number and nothing else. At an exponent of two the eighth partial settles in 0.014 seconds against the fundamental’s 0.869, a factor of sixty-four. So the held spectrum’s shape is correct long before its level is: the tilt a listener would read the loss law off is in place within a tenth of a second, while the note is still getting louder.

That has a consequence worth separating from the arithmetic that produced it. A bowed note that is interrupted before it reaches full loudness has nevertheless delivered the whole of its steady cue, because the cue is a ratio between partials and the ratio arrives before the sum does. A struck note interrupted at the same moment has delivered a fraction of its transient cue, because that cue is a difference that has not yet opened. The two cues are not merely different sizes; they are differently robust to the note being cut off, and in opposite directions.

It also says something about why a bowed attack has the character it has. During the first tenth of a second the upper partials are at their steady values and the fundamental is not yet at its own, so the sound is momentarily brighter than the sustain it is heading for. That is a prediction of the same equation and not an observation; whether it survives the bow’s own starting transient, which is a nonlinear regime the equation does not cover, is not something this model can say.

Two cues, and what each costs #

The fourth essay’s measurement can now be made twice. Take two notes whose loss exponents are a half and two, and ask how far apart their audible spectra are at each instant.

Struck, the answer is a curve pinned to zero at both ends: nothing at the strike, 41.3 decibels apart after 0.398 seconds, nothing again in the silence. Held, the answer is a horizontal line at 18.7 decibels, from the first moment of the sustain until the bow lifts.

The two are not ranked by that comparison and the shapes of the two offers are what matter. A struck note first matches the sustain’s standing offer after seventy-six milliseconds, exceeds it for 1.99 seconds of a note lasting 5.68, and then falls back below it and eventually to nothing. A held note is at 18.7 decibels from the instant the sustain establishes and stays there.

Those are different kinds of evidence and they fail in different circumstances. The struck note’s cue is a rate: it requires that the note be allowed to run, that the listener attend across several hundred milliseconds, and that nothing else change during the window. The held note’s cue is a slope: it is present in a single short sample of the sustain, it survives being interrupted, and it does not require the listener to have heard the beginning. A phrase in which every note is stopped after a tenth of a second destroys the first and leaves the second untouched.

It also explains why the earlier essay’ listening result — that cutting the first fifty milliseconds off a recording costs a listener the instrument’s name — was demonstrated on struck and plucked sounds. On a bowed note the same excision removes the attack transient and leaves a sustain that still carries a slope. Whether that is enough is a listening question the essays here cannot settle, but the arithmetic says the information is not gone.

The release has almost nothing left to take #

There is one part of a driven note that does decay, and it is the part after the player stops. It might be expected to carry the same colour drain the third essay built its identity cue out of, compressed into a shorter time. It does not, and the reason is the equilibrium above.

A struck note on this string starts at the drive spectrum and drains 9.9 semitones of centroid before it dies. A held note starts at the sustained spectrum, which is already 8.2 semitones duller at an exponent of one — so what is left for the release to take is 1.7 semitones, a sixth of the struck note’s. At an exponent of two the sustain is duller still and the release has 0.33 semitones left to take, which is a twelfth of a tone.

The steeper the loss, the less colour the sustain holds and the faster the remainder goes — half of it in 0.49 seconds at an exponent of a half, 0.20 at one and 0.088 at two. Over an eighty-millisecond release those two effects nearly cancel: the shares delivered are eleven, twenty-six and fifty per cent of totals that themselves fall by a factor of thirteen. In semitones the deliveries are 0.48, 0.44 and 0.16, which is not a set of numbers anybody would sort an instrument by.

So a sustaining instrument gives up the account’s transient cue and gets a steady one in exchange. That is a real trade rather than a strict improvement: the steady cue is less than half the size, it is confounded with the drive spectrum, and it says nothing at all unless the note is held long enough to establish. What it buys is that the note is informative for the whole of its length, where a struck note has spent its information by the end of the first second.

The exchange also sorts the two families by which of their parameters a player controls. A struck note’s loss law is fixed by the instrument: nothing a pianist does changes how fast the sixteenth partial leaves, so the cue goes out whether it is wanted or not. A held note’s is fixed by the instrument too, but the drive that gets divided by it is not — a bow’s speed, force and distance from the bridge all change the spectrum being supplied, and a player changes all three within a phrase. The steady cue is therefore the one that a performer can move, and the transient cue is the one that identifies the instrument regardless of the performer. That is the opposite of how the two are usually described, and it follows from one division.

Which computation produced the numbers #

The drive spectrum is this collection’s eight-partial string model, normalised so that its total power sets the stated playing level at the fundamental. The loss is $\gamma_n = \gamma_1 n^{\,p}$ with $\gamma_1$ fixed by the fundamental’s stated sixty-decibel time, which is six seconds throughout. The held amplitude is $F_n/\gamma_n$, normalised the same way, so the two rows of every figure are at the same overall level and differ only in shape. A partial is present when its level exceeds the threshold of hearing as the essays here have computed it at its own frequency.

The separation between two spectra is the root-mean-square difference in decibels over the partials present in either, with an absent partial scored at its own threshold — the same convention the fourth essay used, kept so that the two numbers on the availability figure are the same measurement.

The centroid is power-weighted and restricted to audible partials, which is what makes it a statement about a listener rather than about the arithmetic. The eight-partial model is why the drain here is 9.9 semitones where the register figure quotes 21.3 at the same pitch: that one builds every partial under the audio ceiling and this one builds the eight the account has always drawn. The comparison between struck and held is internally consistent, and the absolute drains are not comparable across the two conventions.

Where the model stops #

A drive is not a fixed force. Writing $A_n = F_n/\gamma_n$ treats the bow as supplying a spectrum that does not depend on what the string is doing, and a bow does not. The stick-slip regime is set by the interaction — the force required to maintain Helmholtz motion at all has a floor and a ceiling that depend on where the bow sits and how fast it moves — so the drive and the loss are coupled, and the clean division above is a first approximation to a nonlinear problem.

And the flue and the reed are worse. A reed is a valve whose opening is controlled by the pressure inside the bore, which is the field the reed is driving, so the source and the resonator are one system rather than two. The steady spectrum of a clarinet is not its reed’s spectrum divided by anything; it is the fixed point of a loop. The clarinet figure above should be read as showing what the addition would look like on an odd-harmonic drive, not as a model of a clarinet.

The exponent is a fitted trend. A real string’s losses come from air, from internal friction and from the bridge, and only the first two are anything like smooth in frequency. The bridge’s admittance has peaks, so the true $\gamma_n$ is a curve with structure in it, and a single exponent is what remains after that structure is averaged away.

What the picture cannot show #

It cannot show the attack of a bowed note, which is a distinct object with its own literature and its own failures — a bow can start a note in a single period or take a dozen, and an attack time is not a single number on any instrument. Everything here begins after the sustain is established.

Nor the vibrato. A held note on a string or a voice is almost never held still; the frequency wobbles by tens of cents several times a second, and every partial’s frequency wobbles with it. Whether that smears the slope a listener is supposed to read, or helps by sweeping each partial across the body’s resonances and averaging them, is a question with an answer that nothing here computes.

It cannot show radiation. The spectrum that leaves the instrument is not the spectrum at the bridge: the body has its own frequency response, and the top of a series falls off with an efficiency that depends on the radiator. That response multiplies both the struck and the held spectrum, so it does not change the difference between two loss laws — but it decides absolutely whether either is above the threshold, and no figure here has a body in it.

And it cannot show that a listener uses any of this. The 18.7 decibels is a difference between two computed spectra, not a discriminability. Turning it into a prediction needs an account of how the auditory system compares spectral slopes, which is a much harder object than the root-mean-square difference used here.

Still open: which of the two cues a real sustained instrument can afford #

The comparison above holds the loss law as the only difference between two instruments, which is the fourth essay’s design and is what makes the two numbers comparable. It is not how instruments differ. A violin and a viola differ in their strings, their bodies, their bridges and their sizes all at once, and every one of those changes the drive as well as the loss.

What would settle the size of the steady cue is the one measurement the account here has already recorded as owed, taken on a sustaining instrument instead of a struck one: the held spectrum of a bowed note at a stated bow speed, force and position, at thirty or forty pitches across the compass. The drive is known analytically for that case, so dividing it out gives the loss as a function of frequency directly — and that curve, rather than an exponent, is what would say whether the trend the account here has fitted describes anything on an instrument with a body.