A bowed bass line is placed by its entrance; a violin line by every note

A bowed note is placed as if it were quieter gave the listener of a quiet bass is further off-centre than a loud violin, whose ears differ by two decibels in sensitivity, and asked where it hears each note, on the understanding that the first wavefront wins: a listener takes a sound’s direction from its onset. The worse ear is later to report a quiet sound than a loud one, by the brainstem latency law, and a delay between the ears is one of the two numbers direction is computed from, so a note’s position is displaced toward the better ear by an amount that depends on how loud it is while its onset is being read. A plucked note reaches its dynamic before that reading is over and is placed almost where a steady tone would be. A bowed note is still near the threshold of hearing throughout it, and a bowed G on the double bass at 60 decibels is displaced 6.2 detectable steps, as far as a steady tone 27 decibels quieter would be.

Every note in that essay started from silence. It ended by pointing out that most notes in music do not. A legato line’s second note begins while the first is still sounding, often at the same dynamic, and whether it is heard to begin at all — whether the rule that places a sound by its onset fires on a change of pitch within a sound, or only on a sound starting from nothing — decides whether a bowed melody keeps its entrance’s displacement or is displaced again at every bow change.

The answer turns on a quantity these essays have already computed for a different purpose, and it divides the orchestra by register. In the bass, a bowed step of about a tone or less has no onset of its own, and the note keeps the position its predecessor had. In the treble nearly every note has an onset and is placed near its steady position, except where the line leaps down.

An onset is a rise in the note’s own filter #

The ear does not hear a note’s onset in the pressure wave as a whole. It hears it in the auditory filters that the note’s energy falls in, and a new note at a new pitch is a new event in the filter centred on that pitch. What the filter held before the change is the only baseline it has.

So the question is how much of the first note is already in the second note’s filter. Before the change, that filter carries the first note attenuated by the filter’s skirt at the distance between the two pitches. Over the change the first note fades as the second rises, each over the articulation’s rise time — a tenth of a second for a bow, one of the stated rises that an attack time is not an attack showed is a curve rather than a number — and the filter’s level moves from the first note’s leakage to the second note’s full level. The second note has an onset of its own if that movement is a rise of more than about a decibel, the size of level change a listener can detect. If the leakage is below the threshold of hearing, the rise starts from the threshold, and the second note is exactly the earlier essay’s entrance from silence.

If there is an onset, the note is placed as the earlier essay placed one: the interaural offset the latency law gives at the filter’s level, weighted over the few milliseconds after the onset. If there is not, a listener that takes direction from onsets has nothing new to take it from, and the note stays where the line already was.

The first note’s leakage, above and below #

The leakage has two features, and each does its own work.

The first is register. An auditory filter is roughly a constant number of hertz wide at low frequencies and a constant fraction of its centre at high ones, so measured in semitones it is very wide in the bass and narrow in the treble. At G2, 98 hertz, a note a tone away is within a decibel or two of full strength in the filter; at A4 it is three to seven decibels down; at A6 further again. So a small step in the bass barely changes what the second note’s filter hears, and the same step on the violin changes it by several decibels.

The second is direction. At a moderate or loud level the filter’s lower skirt is shallower than its upper — the arithmetic behind one sound hides another, and it hides upward, and the reason a loud chord is a smaller chord — so a note below leaks into a filter more than a note the same distance above. At G2 an octave below is only 9 decibels down in the second note’s filter; an octave above is 37 down. That asymmetry, which these essays have always met as masking, reappears here as a statement about onsets: a line that steps up leaves more of itself in the next note’s filter than a line that steps down, so ascending notes are less of an event than descending ones.

The first figure shows both. On G2 the intervals with no onset run from a semitone down to a tone up — lopsided toward ascending. Beyond them the curve rises toward the entrance’s displacement much faster on the descending side, because a descending leap leaves the new filter nearly empty: an octave down on the double bass is placed 6.2 steps off, a full entrance, while an octave up is 4.5. On the violin every interval has an onset, and even so the pattern is the same: an octave down is 3.7 steps, the entrance, and an octave up 3.1.

Why a tone is small in the bass and large in the treble #

The width that matters can be stated in the units a player uses. An auditory filter’s equivalent rectangular bandwidth is about 35 hertz at G2 and 72 at A4, and 215 at A6. In hertz the bass filter is the narrowest of the three. In semitones it is by far the widest: 35 hertz around 98 spans more than six semitones, 72 around 440 under three, and 215 around 1,760 two. A step of a tone moves a bass note a third of the way across its own filter and a violin’s A4 more than two-thirds of the way.

That is why the pattern is a pattern of register rather than of instrument. A cello playing in its tenor register behaves like the violin here, and a violin’s lowest string behaves more like the viola’s than like its own top. What decides whether a step is an event is how many hertz it moves the note against how many hertz wide the ear’s filter is at that pitch, and below a few hundred hertz the filter’s width stops shrinking as the pitch falls while the step’s width in hertz keeps shrinking.

The consequence for a line is simple to state and a little surprising to hear stated. The lower a legato line lies, the fewer onsets it has, and the fewer chances a listener who places sounds by their onsets has to correct where it hears it.

Two lines, note by note #

Put into lines, the pattern becomes something a listener could notice. The double-bass line enters on G2 at 6.2 steps — the bowed entrance of the earlier essay. Its step up to A2 is an onset, from a filter already holding most of the G, so A2 is placed at 3.8, close to the steady tone. B2 is placed at 3.6. The semitone step to C3 has no onset at all, and C3 keeps B2’s 3.6. D3 is 3.4. Then the line leaps down a fifth to G2, which is an entrance in all but name — the D’s energy is far up the new filter’s steep upper skirt — and it is placed at 5.0. D2 below it at 4.9. The tone up to E2 has no onset and E2 keeps D2’s position. The final G2 is placed at 4.0.

The violin line enters at 3.7 on A4, and its steps are placed between 1.9 and 2.1, near the violin’s steady-tone position of 2.1. Its drop of a fifth to A4 goes back to 3.6, nearly the entrance. Then it falls a fourth to E4, placed at 3.3, and steps a semitone up to F4 — the one violin note in either line with no onset, because at 349 hertz a semitone is still inside the filter’s shallow lower skirt — so F4 keeps E4’s 3.3, and the line returns toward 2 only at G4.

So a bowed bass line and a bowed violin line are heard differently in the one respect the earlier essay was about. The violin’s line is placed almost note by note, and a listener with asymmetric ears hears it close to where a steady tone would be, apart from the notes that follow a downward leap. The bass line is placed in stretches, and each stretch is placed by its first note — its entrance or the note after its last leap down. Where that first note was displaced a long way, the stretch after it inherits the displacement for as long as it moves by small steps.

Separate bows put every note back at the entrance #

The comparison that makes the inheritance visible is the same line played with separate strokes and a small gap between them. Then every note begins from silence in its own filter, and every note is an entrance: the double-bass line’s notes would be placed between 5.5 and 6.5 steps, averaging 6.0, and the violin line’s between 3.2 and 4.2, averaging 3.7. Played legato, the bass line averages 4.4 steps over its nine notes and the violin line 2.7.

So legato playing, for this listener, pulls a line toward its steady position in both registers, and in the treble it does so at every note. In the bass it does so only at the notes that are onsets, and a note that is not an onset carries whatever the note before it carried — which, straight after the entrance or a leap down, is most of an entrance’s displacement. The same bowing is a correction in the treble and, over the bass’s short steps, a way of preserving an error.

A forte makes the ascending line one stretch #

The lower skirt’s shallowness grows with level, so the dynamic moves the boundary. At 40 and 50 decibels even a tone up in the bass has an onset of its own, placed at 5.7 and 4.8 steps. From 60 decibels it has none. At 80 and 90 decibels, a fifth up has none either: the first note fills the fifth above it within a decibel. Descending intervals keep their onsets at every dynamic, because the upper skirt does not shallow.

So at a forte an ascending bowed bass line is, as far as its onsets go, one note. A rising line from the bottom of the double bass to a fifth above it gives a listener who places sounds by their onsets exactly one onset to place it by, the first, and every note after it is heard where the first was heard. A falling line at the same dynamic gives an onset at every note. A listener with slightly asymmetric ears would hear a rising bass line at forte as fixed at the displacement of its entrance, and a falling one as drifting toward the steady position note by note.

What counts as an onset #

The one number in the argument that is chosen rather than computed is the rise that counts as an onset. A decibel is about the smallest change in level a listener detects in a steady sound, and it has been used throughout. It is not known that the mechanism that places sounds by their onsets uses the same criterion.

The criterion moves the boundary but not the pattern. With half a decibel counting as an onset, the double bass has no onset only for a semitone up or a quarter-tone down. With three decibels it has none up to four and a half semitones up and two down. On the violin, even three decibels leaves only a tone and a half up and a semitone down without an onset, and at A6 a tone. Every value of the criterion leaves the same two statements true: the bass has a band of intervals without onsets several times as wide as the treble’s, and that band is wider upward than downward.

A slower bow matters only for a leap down #

The earlier essay found that the slower a note rises the further it is displaced, because a slow note is quiet for longer while its onset is read. For the second note of a legato pair that holds only when the new note’s filter was nearly empty. An octave down on the double bass is displaced 5.5 steps tongued, 6.2 bowed and 6.5 with a soft bowed entrance, each within a fraction of its own entrance from silence. A fifth up is displaced 4.2, 4.3 and 4.3: the filter already held most of a note, and the rise starts from there, so how slowly it rises barely changes the level at which the onset is read.

That makes the articulation matter less inside a line than at its start. Whether a line is tongued, bowed or bowed softly changes its entrance a great deal and its steps almost not at all; only its downward leaps take on the articulation’s character, because only they begin from something like silence.

Whose ears, whose lines #

The listener is the one the earlier essays built: ears that differ by two decibels in threshold, which is well within what audiometry calls normal, and a seventh of the brainstem latency difference reaching the comparator that reads direction. The claim is about that listener and bowed string lines played legato in an orchestra, where the double basses and cellos routinely play long stepwise lines under a melody. It is not a claim about notes that are re-articulated — a bow change with a clear stop between the notes, or a line played détaché, begins every note from something closer to silence and is placed note by note in every register.

The arithmetic makes a prediction that could be tested with such a listener. A rising legato bass line at a forte should be heard at the position of its first note for as long as it rises, and a falling one should move toward its steady position note by note; played détaché, both should behave like the falling one. A listener with symmetric ears hears no displacement to inherit, so the difference is invisible to exactly the listeners most often asked — the same blind spot the error that moves straight ahead found in a listener whose head is the wrong size.

What the picture cannot show #

Whether direction is re-read on a change within a sound. The model re-reads position only at an onset, and gives a note with no onset the position it inherits. A listener could also track a sound’s direction continuously, at a slower rate, in which case a note with no onset would drift toward its steady position over the course of its length rather than keeping its predecessor’s. The inherited displacements drawn here are an upper bound on how long a displacement persists.

Partials. Each note is treated as its fundamental in one filter. A bowed note has many partials, and a step changes every one of them; whether the note has an onset may be decided in the filters where its upper partials land, which are narrower in semitones than the fundamental’s. The bass’s wide band of intervals without onsets is the fundamental’s band and would narrow if the upper partials dominate.

The crossfade’s shape. The first note fades and the second rises over the same time and in straight lines of amplitude. A slurred string crossing, a finger change and a bow change each have their own shape, and a bow change in particular has a dip between the notes, which could itself be read as an onset.

Still open: the upper partials’ onsets #

The second of those caveats is the one that could reverse the result. A bowed bass note’s fundamental sits in a wide filter, but its fourth and eighth partials sit in filters a quarter and an eighth as wide in semitones, where a step of a tone moves a partial right out of its filter. If the rule that places a sound by its onset listens to all the filters a note excites and fires on the first that sees a rise, then every step in the bass is an onset in its upper partials, and the bass line is re-placed note by note after all — though from filters at higher frequencies, where the latency difference is smaller and, above about a kilohertz in a room, where the two ears stop agreeing. Forward masking would pull the other way: a sound hides what came before it, and hides what follows it for longer, so the first note’s tail may bury part of the second note’s rise in exactly the filters where it would be an onset. The computation is the same leakage, summed over each note’s partials and filter by filter, with the note’s onset taken as the earliest rise in any filter above threshold, and it would say whether a legato bass line’s inherited displacement survives its own harmonics.