A bowed bass line has an onset at every step, in its upper partials

A bowed bass line is placed by its entrance; a violin line by every note asked whether the second note of a legato line has an onset of its own, for a listener who takes a sound’s direction from its onset and whose slightly asymmetric ears displace quiet onsets more than loud ones. It read the onset in the auditory filter at the second note’s fundamental. In the bass that filter is wide in semitones, so a step of a tone or less left it nearly as full of the first note as before: no onset, and the note kept its predecessor’s position — the bowed entrance’s 6.2 detectable steps, carried through the line’s steps.

The essay said where the result was weakest. A bowed note is not a fundamental. Its upper partials sit in filters at higher frequencies, which are far narrower in semitones, and a step of a tone moves a fourth or fifth partial right out of the filter it was in. If the rule that places a sound by its onset fires on the first filter to see one, every step in the bass might be an onset after all.

It is. Read in every filter its partials reach, every step of a bowed bass line has an onset, first seen in the third, fourth or fifth partial. The line is re-placed at every note, and it is placed nearer the centre than the fundamental’s reading said, because the latency difference the asymmetric ears make is smaller at higher frequencies. And the same reading, applied to the entrance itself, displaces a bowed bass entrance a quarter less than its fundamental did.

Every partial its own filter #

The model is the earlier essay’s, applied eight times. Each of the second note’s first eight partials, falling in level as one over its number as a bowed string’s do, is given the auditory filter centred on it. Before the change that filter holds every partial of the first note, each attenuated by the filter’s level-dependent rounded-exponential skirt at its distance — the shallow lower skirt that makes a loud chord a smaller chord. Over the change the first note fades as the second rises, both over a bowed rise of a tenth of a second, and each filter’s level moves from what the first note left in it to the second note’s own partial. A filter sees an onset when its level rises a decibel above what it held — or, if what it held was below the threshold of hearing, when it crosses the threshold.

The note is placed by the first filter to see an onset: the earliest one. From that moment its position is read exactly as the earlier essays read a position — the interaural offset the brainstem latency law gives the asymmetric ears at that filter’s level, weighted over the few milliseconds after the onset — with one change that turns out to matter. The latency law depends on frequency as well as level, falling as the frequency rises, so a partial read at 300 or 400 hertz is displaced less than the same level read at 98.

Why the third to fifth partial #

The pattern follows from one proportion. A step moves every partial by the same number of cents, which is a number of hertz proportional to the partial’s number: a tone up from G2 moves the fundamental 12 hertz, the fourth partial 48 and the eighth 97. The auditory filters widen with frequency too, but more slowly — about 35 hertz wide at 98 hertz, 67 at 392 and 110 at 784 — so the higher the partial, the further out of its own filter the step takes it. By the third to fifth partial a tone is most of a filter’s width, and the first note’s partial at the old pitch leaks into the new filter far less than the fundamental did. Above that the partials grow too weak to matter first. So for a small step the onset appears in the third, fourth or fifth partial, 62 to 90 milliseconds into the change, when the rising partial has climbed a decibel above what the old one left.

Leaps behave differently. A leap down of a fourth or more leaves every one of the new note’s partials in filters the old note never reached, because the old note’s partials were all above; its fundamental crosses threshold within milliseconds, and the note is placed by its fundamental, as an entrance from silence would be — the first wavefront wins, and the first wavefront here is low. A leap up leaves the old note’s upper partials in the new note’s lower filters, so the first onset is usually in the second or third partial.

The two lines again #

The double-bass line of the earlier essay, placed by partials, has no note without an onset. It enters on G2 at 4.49 steps, and its steps up are placed at 2.3 to 3.0 — C3, which kept B2’s 3.6 before, is now placed at 2.3 on its own onset in the fifth partial. Its leap down a fifth to G2 is placed at 5.0, as before, because a leap down is read in the fundamental. E2, which kept D2’s 4.9 before, is now placed at 3.3.

The violin line changes too, less dramatically. Its fundamental already had an onset at almost every note, but its partials’ filters see onsets sooner and at frequencies where the latency difference is smaller, so its steps are placed at 1.35 to 1.86 steps rather than 1.9 to 2.1. F4, the one violin note that kept its predecessor’s position, now has an onset in its fifth partial.

So the earlier essay’s register contrast — a bass line placed in stretches, a violin line note by note — does not survive. Both lines are placed note by note. What survives is the leap: in both registers a leap down is read from its fundamental, as a near-entrance, and displaced much further than the steps around it.

Why a partial is placed nearer the centre #

The displacement comes out smaller when the onset is read in an upper partial for a reason that is in the latency law itself. The brainstem latency difference that the asymmetric ears produce has a level term and a frequency term, and the frequency term falls as the frequency to the power 0.413. Between 98 hertz and 392 hertz, two octaves, that term falls by a factor of 1.77, so the same level read at the fourth partial gives a displacement a little over half the size. The fourth partial is twelve decibels weaker than the fundamental, which on its own would make it more displaced, not less, since the level term grows as the level falls. At these levels the frequency term wins, and the fourth partial’s onset is placed nearer the centre than the fundamental’s would be.

Averaged over the double-bass line of the figure above, the two effects move every note. Placed by its fundamental, the line averages 4.40 detectable steps off centre; placed by its partials, 3.53, a fifth nearer. The violin line averages 2.70 by its fundamental and 2.25 by its partials, a sixth nearer. Neither line loses its displacement. Both lose the part of it that came from reading every note at its lowest and most asymmetrically delayed frequency.

The entrance itself #

The same reading changes the result the earlier essays were built on. A bowed note is placed as if it were quieter placed a bowed G2 entering from silence at 60 decibels 6.2 detectable steps off centre, reading the onset at the fundamental. Placed by the first of its partials to cross the threshold, it is 4.49 steps, by its fourth partial, a quarter less. At 90 decibels it is 3.61 rather than 4.36, by its second partial.

The reason is the threshold of hearing, which falls steeply from 98 hertz toward a few hundred. A bowed G2 at 60 decibels has its fourth partial twelve decibels lower, at 392 hertz — but the threshold there is about twenty decibels lower than at 98, so the fourth partial crosses the threshold first, on the rising bow stroke, and is read at 392 hertz where the ears’ latency difference is smaller. The violin’s A4 entrance is read at its fundamental whichever way it is computed, because at 440 hertz the fundamental is already where the threshold is low.

So the earlier result survives in direction and shrinks in size. A bowed bass entrance is still displaced further than a steady tone at the fundamental would be — 4.49 against 3.99 at 60 decibels — and further than the steps that follow it. But the asymmetric listener’s bass section moves toward the better ear by about a quarter less than the fundamental’s reading said, and the deformation of the orchestra that a quiet bass is further off-centre than a loud violin found is correspondingly gentler at the bottom.

A leap down is displaced further than the entrance #

One consequence runs the other way, and it is the largest displacement anywhere in the figures. A second note that drops an octave is placed by its fundamental: its partials’ filters above the fundamental are all occupied by the first note’s partials — the first note’s second partial is the new note’s fourth, exactly — so none of them sees a rise, and the only filter that starts from nothing is the fundamental’s. It is read there, at 98 hertz, from silence, and placed 6.19 steps off centre.

That is further than the line’s own entrance, which placed by its partials was 4.49. So on this reading a bowed bass line’s most displaced note is not its first. It is any note reached by a leap down of a fourth or more, which starts from silence in exactly the filter where the asymmetric ears disagree most. A descending octave in the bass — the commonest large leap a bass line makes, at cadences and in walking lines alike — is heard by such a listener as a jump sideways as well as down.

The bow’s spectrum #

How much the upper partials take over depends on how strong they are, and a bowed string’s spectrum is set by the player: bowing near the bridge brightens it, bowing over the fingerboard dulls it. With partials falling as one over their number, the brightest drawn, the steps are placed at 2.6 to 3.1 steps. With partials falling as one over the cube of their number, a very dull stroke, the steps are placed at 4.1 to 4.6, and the entrance goes back to its fundamental at 6.2.

Every step keeps an onset at every slope drawn, in the fourth to seventh partial. So the reversal does not depend on the bow: a legato bass line is re-placed at every note however it is bowed. What the bow decides is where each note is placed — and, curiously, a brighter stroke puts the whole line nearer the centre for an asymmetric listener, because its onsets are read from partials higher up, where the ears’ latency difference is smaller. Sul ponticello is heard, on this arithmetic, less displaced than sul tasto.

What counts as an onset #

The decibel of rise that counts as an onset is the one number the argument chooses. It is about the smallest change in level a listener detects in a steady sound. At half a decibel or one, every step of a semitone or a tone has an onset in some partial. At two decibels a semitone down loses its onset; at three both semitones do; the tones keep theirs to three and lose them only at six. So the reversal is robust to the criterion for steps of a tone, and holds for semitones as long as an onset is a rise of a decibel or two.

An attack time is not an attack found that the moment a note is heard to begin depends on the shape of its envelope and on what a listener counts as its start; here the question has become which envelope. A bowed note has one envelope per filter, and the one that starts first need not be the loudest.

What this changes, and what it does not #

The earlier essay’s central claim was that in the bass a legato line is placed in stretches, each stretch fixed at the displacement of its first note. On this reading that claim is withdrawn. Its secondary claim — that a leap down is re-placed almost as far as an entrance, while steps are placed near the steady position — survives and sharpens: the steps are placed nearer the centre than the fundamental’s steady tone, and only the leaps down are read from the fundamental at all.

The rest of the earlier essays’ structure stands. Direction is still taken from onsets. An asymmetric listener still displaces quiet onsets more than loud ones. A bowed entrance is still displaced more than a plucked one, since a pluck reaches its level within a few milliseconds in every filter it excites; two ears, and the whole of the difference is 655 microseconds is still the scale. What changes is that every one of those statements has to be read in the filter where the onset first appears, and for a low bowed note that is not the filter at its pitch.

What the picture cannot show #

Which filter the listener uses. The note is placed by the first filter to see an onset. A listener might instead weigh every filter that sees one, or favour the band below about 1.5 kilohertz where interaural timing is most reliable — in a room, where the two ears stop agreeing is lower still — or weight filters by their level. Each rule would give different numbers; the first-onset rule is the simplest, and it agrees with the precedence principle that the earliest arrival wins.

A revised position. The note is placed once, by its first onset. A listener might instead place it by the first onset and then drift toward what the later onsets in other filters say, over the tens of milliseconds in which they arrive. That would put each note between the partials’ reading and the fundamental’s, nearer the partials’ for steps, whose later onsets are in neighbouring partials, and unchanged for leaps down, which have only the one.

The spectrum of a real bowed note. The partials fall smoothly as one over a power of their number. A real double-bass note has a body filter, a formant-like set of resonances that lifts some partials and lowers others, and which partial sees the first onset depends on which ones the body lifts.

Forward masking. The first note’s tail raises the threshold in its own filters for tens of milliseconds, which a sound hides what came before it described in the other direction as well. That would delay the onsets in the filters the first note filled and favour the ones it did not, which is the direction this essay’s result already goes.

Still open: a filter the body lifts #

A double bass’s body has resonances of its own, like the violin’s, and they sit low enough to lift particular partials of particular notes. If the partial that first sees an onset is decided by which partials the body makes strongest, then a bass line’s placement depends on the instrument, note by note, much as one sound hides another depends on which partials stand clear. The computation puts a published double-bass body response under the partials here and asks whether, on some notes, the body’s peak moves the first onset into a filter low enough to put the displacement back near the fundamental’s — which would make an individual bass’s legato line displaced unevenly, note by note, for an asymmetric listener.