A bowed note is placed as if it were quieter

A quiet bass is further off-centre than a loud violin gave a listener ears that differ slightly in sensitivity and asked where they hear an orchestra. The worse ear is later to report every sound, by the difference the brainstem latency law gives between the two ears’ effective levels, and that difference halves every 43 decibels and grows toward the bass. With a seventh of it reaching the comparator that reads direction — the share the ear’s own thresholds allow — a listener whose ears differ by two decibels hears a pianissimo bass section moved 5.5 detectable steps toward the better ear and a fortissimo first violin 1.2. The displacement is a deformation, not a shift.

That essay treated every note as a steady tone at its dynamic, and it said in its caveats why that might be wrong. A latency is a property of an onset. Once a tone is sounding, the two ears’ neural responses are locked to its waveform, and the comparator reads the phase difference between them, which an onset delay need not survive into. And the first wavefront wins: a listener takes a source’s direction from its first arrival and holds it. So the position a note is given is the position its onset had.

The earlier essay predicted what that would do. A struck or plucked note would carry its onset displacement through the whole note and be displaced more than the steady curves said; a bowed note, beginning softly, would be displaced less. The arithmetic says the reverse, and by a large margin. A plucked bass at 60 decibels is displaced 4.7 steps against the steady tone’s 4.0. A bowed one is displaced 6.2, as far as a steady tone 27 decibels quieter would be.

Giving each note an attack #

A note here rises linearly in amplitude from nothing to its dynamic over its rise time: two milliseconds for a struck note, five for a plucked one, twenty-five for a tongued wind attack, a hundred for an ordinary bowed stroke, and a quarter of a second for a soft bowed entrance. Those rise times are stated, not measured, and an attack time is not an attack is the reminder that a published attack time fixes very little about a curve’s shape.

The note becomes audible when its level crosses the threshold of hearing at its frequency — 27 decibels at 98 hertz, 4 at 500 — and from that instant the latency difference follows the ramp, since at every moment the worse ear is hearing a note that is quieter than the better ear hears it by the asymmetry. The position the listener assigns is that difference averaged with an onset weight, which is largest at the audible onset and decays with a three-millisecond time constant. The steady state that follows carries no offset at all, so a note whose onset is ignored entirely would be heard straight ahead.

Every other quantity is the earlier essay’s: the latency law of Neely, Norton, Gorga and Jesteadt, a two-decibel asymmetry, a seventh of the difference reaching the comparator, and a detectable step of fifteen microseconds.

Why the soft beginning is the displaced part #

The reason the prediction came out backwards is in the first thirty milliseconds of each note.

Every note becomes audible at the same level, the threshold, whatever its eventual dynamic. At that level the latency law’s difference is at its largest, about 101 microseconds for this bass and this asymmetry. What differs between attacks is how fast the level then leaves the threshold behind.

A plucked note is at its full dynamic five milliseconds after it becomes audible, so its onset weight sees mostly the full dynamic’s 60 microseconds and a little of the threshold’s 101. It is heard at 70. A bowed note takes a hundred milliseconds to reach its dynamic, and for the whole of the onset window it is only a few decibels above the threshold. It is heard at 92, nearly the threshold’s value.

The earlier essay’s prediction reasoned from the displacement of a loud onset surviving into the note. But the onset of any note is quiet, because every note starts from silence, and what a listener’s onset weighting reads is how long the note stays quiet. A soft beginning is not a small displacement; it is the note spending its onset near the threshold, where the latency difference is largest.

How much quieter each attack is heard #

A convenient way to state the result is as a level: the dynamic at which a steady tone would be displaced as far as the note is.

At 60 decibels a struck note is placed like a steady tone 4.8 decibels quieter, a plucked one 9.5, a tongued one 20, a bowed one 27 and a soft bowed entrance 30. At 90 decibels the bowed stroke is placed like a tone 35 decibels quieter and the soft entrance like one 42 quieter. At 40 decibels every curve runs into a ceiling of 13 decibels, the distance from the dynamic down to the threshold. No attack can be heard as quieter than the level its onset starts from.

That ceiling is the one place where the prediction’s direction survives. A note already close to the threshold has little room for its attack to make it quieter, so at pianissimo the articulation matters least: 5.7 steps struck against 6.8 soft at 40 decibels. At fortissimo it matters most: 2.7 against 4.9.

The threshold is lower in the treble, so the treble has further to fall #

The threshold of hearing is not the same at every pitch, and it enters the arithmetic twice. It is where every onset starts, and the distance from it to the note’s dynamic is how much quieter a slow attack can be heard.

At 98 hertz the threshold is 27 decibels, so a bowed note at 60 has 33 decibels of room below it, and it is heard 27 of them quieter. At 262 hertz the threshold is 11 decibels and the same bowed note is heard 33 decibels quieter; at 523, with a threshold of 4, 34.5. Measured against the steady tone, a bowed bass is displaced 1.54 times as far, a bowed viola 1.71 times and a bowed first violin 1.74 times.

This runs against the earlier essay’s register effect, and the two partly cancel. The latency law makes a low note’s offset larger at any given level, which is why the basses moved furthest there. The threshold makes a high note’s slow attack fall further below its dynamic, which is why the violins gain proportionally more from bowing here. The net is still a stage stretched toward the bass, since the latency law’s register effect is the larger of the two, but the stretch is less in bowed passages than in plucked ones, measured as a ratio: the basses are displaced 1.96 times as far as the first violins when both are plucked and 1.76 times when both are bowed.

The same cancellation shows why the struck and plucked notes barely move from the steady curve at any register. Five milliseconds is short enough that even at 98 hertz the note is at its dynamic before most of the onset weight has been spent, and the onset reads it at a level ten decibels or so below its dynamic whatever the threshold. Those notes are heard 1.17 to 1.19 times as far round as a steady tone at every register drawn. The register-independence of that ratio is the clearest sign that for fast attacks the steady-tone arithmetic was nearly right.

An orchestra deformed by its bowing #

The earlier essay’s stage put five string sections at their seats and moved each by its register. Articulation is a second axis, and it is not a small one.

At 60 decibels, plucking moves the double basses 4.7 steps and the first violins 2.4, a spread of 2.3 steps from register alone. Bowing moves them 6.2 and 3.5. The bowing adds between 1.1 and 1.5 steps to every section, which is half as large as the whole register spread. Bowed first violins, at 3.5 steps, are displaced further than plucked violas an octave lower, at 3.2. A passage that goes from pizzicato to arco would, for this listener, move every section of the strings a step toward the better ear, and move the basses furthest.

This is the deformation the earlier essay found for dynamics, reappearing in a new variable, and it is measured in the same units: the hall through a head counted a stage in detectable steps rather than degrees, and a step here is a step there. A rigid displacement moves every source equally and leaves a listener nothing inside the sound to compare against. A displacement that depends on how each note begins changes the relations between sources whenever the articulation changes, and an orchestra changes articulation constantly. The soloist’s entrance marked niente and the pizzicato accompaniment under it are, for a listener with a two-decibel asymmetry, two steps apart that are not two steps apart on the stage.

The onset window, swept #

The one parameter in this essay without a published value is the onset weight’s time constant. The literature on onset dominance says the first few milliseconds of a sound carry most of its lateralisation, and “a few” ranges from under one to a few tens depending on the stimulus. So it is swept.

A shorter window reads every note closer to its threshold, so every attack is displaced further. A plucked note goes from 4.1 steps at a thirty-millisecond window to 5.6 at half a millisecond. A longer window gives each note more time to reach its dynamic and brings every attack toward the steady tone. But the gap between plucked and bowed survives: it is largest, 1.5 steps, at windows of three to five milliseconds, and it is still 0.9 at thirty and 1.0 at half a millisecond. The window changes how large the articulation effect is, not whether it exists.

Two things would remove it. A window long enough that even a bowed note reaches its dynamic inside it, a hundred milliseconds or more, would read every note at its dynamic; that is the earlier essay’s steady tone, and the onset-dominance literature is clear that the window is much shorter than that. Alternatively, the comparator might read the latency difference only once the note’s level has settled, which would make the whole of this essay moot and is the question the earlier essay left open. It is also the one assumption that separates this listener from the one in the error that moves straight ahead, whose offset was a constant and so had no onset to be read at.

The crescendo still gives it away #

The earlier essay’s most useful finding was a calibration signal. A still source getting louder should stay where it is, and for a listener whose offset falls with level it drifts back toward the centre — by more than a detectable step between 40 and 90 decibels for any asymmetry over about a decibel and a third. Where a wrong head gives itself away had found that a head turn betrays almost nothing of a constant offset, so a crescendo that betrays a level-dependent one mattered. Onset-reading could have removed it, since a slow attack is read near its threshold at any dynamic.

It weakens the signal in the bass and strengthens it in the middle, and removes it nowhere. At 98 hertz a soft bowed entrance drifts 1.88 steps across the crescendo against the steady tone’s 3.05, because at both dynamics its onset is read close to the same threshold. At 500 hertz the threshold is so low that even a slow attack spends its onset far below its dynamic at 90 decibels and not far below it at 40, and the drift grows to 2.11. The crescendo remains a test an asymmetric listener carries around, whatever the orchestra’s articulation.

What the onset model assumes #

Onset dominance, full strength. Every note’s position is its onset’s. Real lateralisation gives the ongoing part of a sound some weight, and the right model is a mixture. Because the steady state here carries no offset at all, a mixture scales every number in this essay down by the onset’s share, and the ordering of attacks survives any share above zero.

A linear ramp from silence. Real attacks have shapes, and a bowed string’s onset in particular is a sequence of slips before the Helmholtz motion establishes itself — the capture the hardest place is also the latest priced. A ramp that rises quickly and then creeps would read like a faster attack here; one that dwells near silence, like a slower one.

The latency law at every instant. The brainstem law was fitted to tone bursts at steady levels. Applying it moment by moment during a rise assumes that the latency follows the instantaneous level, which is the simplest assumption and is not measured.

A seventh of it. As in the earlier essay, the share of the latency difference that reaches the comparator is chosen to match the ratio of the ear’s own two thresholds, not measured. Every displacement in this essay scales with it; the ratio between attacks does not.

Whose articulation #

The contrast this is about is the one string players manage every bar: pizzicato against arco, a bow set on the string before the note against one dropped onto it, the soft entrance marked from nothing. A note starts twice found that a bowed note is both the latest instrument to settle and the last to be heard. Here the same slowness puts it further round for a listener with an asymmetry: a bowed note is heard later in time, as that essay found, and, for this listener, further toward the better ear. The two effects come from the same fact, that the note spends a long time quiet before it is loud.

For struck and plucked instruments, the piano, the harp and the guitar, the steady-tone arithmetic of the earlier essay is nearly right, and this essay adds less than a step. For bowed strings, voices entering softly and winds tongued gently, it adds one to two steps, and more at a fortissimo than at a pianissimo. The stage heard by a listener whose ears differ slightly is therefore not one stage but several, one for each way of beginning a note, laid over each other.

The voice sits among the slow attacks, and a choir is the case where this would be noticed if anywhere. A soft choral entrance is exactly the quarter-second rise drawn here, and a choir is spread across a stage in sections by register, like the strings. On this arithmetic a listener with a two-decibel asymmetry hears a choir’s basses round toward the better ear and its sopranos less so, as for the strings, and hears the whole choir swing a further step toward that ear whenever it enters softly rather than on a consonant, whose plosive gives each voice a struck onset. Diction, on this account, would be a localisation cue for such a listener as well as an articulation.

Still open: the second note #

Everything here is one note starting from silence. Most notes in music do not: they follow another note, often at the same dynamic, often without a gap. A legato line’s second note begins while the first is still sounding, and whether its onset is heard as an onset at all — whether the precedence rule fires on a change of pitch within a sound, or only on a sound starting from silence — decides whether a legato passage keeps the displacement of its first note throughout. The computation is the same ramp and window applied to a note that starts from its predecessor’s level instead of from threshold. It would say whether a bowed melody is displaced at its entrance and then held there, or displaced again with every bow change.