A blown note parts by its register, not its family

Two partials of a note are heard as parts of one thing partly because they start together. The cue that settles it showed how much that cue can decide, on spectra whose partials do start together, and left wind instruments as the exception to look at. A blown note does not start late, it starts slowly looked. Each partial of a wind note rides its own resonance of the bore and builds at that resonance’s own rate, so a note’s partials do not begin at different moments; they begin at one moment and approach their steady levels at different speeds. Whether that counts as a spread depends on how far into its rise a partial has to be before it is said to have started, and that fraction has never been measured.

Read at any fixed fraction, though, the essay’s figures made a ranking of five instruments. At nine tenths, a saxophone’s partials are 151 milliseconds apart, a clarinet’s 121, a trombone’s 92, a trumpet’s 53 and a horn’s 49. Those were offered as what they are: a census at one note each. Its last section said what the census was missing. It should be run at every note each instrument plays, to see whether the ordering of the families survives being asked about a whole range. The impedance sweeps needed were already computed and unused.

Run at every note, the ordering does not survive, and the reason it looked solid is a small embarrassment. The one note read on each instrument was its written middle, and those notes ran from a clarinet’s chalumeau D at 126 hertz to a trumpet’s A at 428. The list was sorted largely by a variable nobody had meant to include.

The same calculation at any note #

The calculation is the essay’s with one change. Eight partials are taken at their own frequencies, odd partials only on the clarinet, and each builds at the time constant of the bore’s resonance nearest it, $\tau = Q/\pi f$. A note played by fingering has its resonances somewhere other than the open tube’s, so the time constant is read between the bore’s peaks on logarithmic axes, as the nearest available statement of what a resonance at that frequency would do. At each instrument’s written middle this returns the earlier figures to within a few tenths of a millisecond, on all five, so the two calculations agree where they overlap. The spread at a criterion $p$ is $-\ln(1-p)$ times the largest time constant less the smallest, and as before the partials are said to part where that spread reaches twenty milliseconds.

What the extra notes show first is a very plain fact. Up every compass, the spread falls. At half of the steady amplitude, a saxophone’s note at 221 hertz has its partials 45 milliseconds apart and one at 556 hertz has them 13 apart. A trumpet goes from 26 at 165 hertz to 10 at 660. The horn goes from 36 to 9 over its range. Since a resonance’s time constant is $Q/\pi f$, and a higher note’s partials all sit at higher frequencies, all of its time constants are shorter, and the difference between them shrinks with them.

The higher note speaks sooner and takes longer has already shown each brass resonance’s settling time falling by a factor of seven. Here the same fall arrives as a property of a note’s own coherence: a low wind note comes apart at the start and a high one does not.

The ranking belongs to a pitch #

The one thing the old list did get right is the saxophone, and the bore says why. A settling rate goes as $Q/f$, and on a cone the $Q$ falls as the frequency rises, from 51 at its lowest peak to 35 at its third, so both terms shorten the time constant together: 69 milliseconds at the bottom and 15 two peaks up, a difference of 54. On the clarinet the $Q$ climbs against the frequency, from 26 to 51, so the same span runs from 66 milliseconds to 25. A wider range of rates inside one note is what parting means, and at low pitches the saxophone has the widest.

Now set the five bores side by side at the same pitches, which the earlier census could not do. The question is simply which parts first at a given fundamental.

There is no single order of the families. The saxophone parts earliest at the bottom of this range and the clarinet earliest at 660 hertz, while the horn, which never parts earliest, stays at the end of the list from 330 hertz upward. The list changes twice over two octaves. So the earlier sentence that one instrument’s partials part sooner than another’s has a pitch hidden inside it, and the clearest way to see the cost of hiding it is to compare how far the criterion moves in each direction.

Moving up one instrument shifts the point of parting three times as far as changing instrument at one pitch: 54 points against 18. Trumpet 42 to 90 per cent, horn 32 to 96, clarinet 34 to 100, saxophone 26 to 86. At any one pitch the five bores sit within a band: 27 to 42 per cent at 165 hertz, 39 to 54 at 330, 74 to 96 at 660. And by 466 hertz the band has moved clear of where it began. If a listener were told only the pitch of a wind note, they could say a great deal about when its partials draw together; told only the family, rather less.

That reverses the claim that opened the question. The cue that settles it proposed a statement about orchestration: which families of instrument the ear fuses easily and which it has to work at, made entirely from onset physics. Some of that remains, since the bands have widths. But a statement about families, without a register attached, is mostly a statement about the register each family was read in.

Why a high note runs out of partials to be late with #

There is a second reason the spread collapses at the top, and it is a different kind of reason.

A bore’s resonances stop at some frequency. Above a brass instrument’s bell cutoff or a woodwind’s tone-hole cutoff the peaks’ Qs collapse, and the sound is radiated rather than accumulated. So a note’s upper partials have no resonance of their own to rise on. In this calculation they have been left out, since they follow the reed or lips rather than a resonance, and the reed or lips follow the resonance the note sits on.

The count drops quickly. By 416 hertz a trumpet has six partials on resonances, and by 660 three. A clarinet, with odd partials only, has five at 233 hertz, three at 370, two at 588, and from 832 hertz just one, at which point there is no spread to speak of, because there is only one rate in the note. Every criterion gives the same answer there. A high clarinet note starts as one object, not because its partials agree but because only one of them has any say.

Here the bores do differ, and this is where the family has a real claim. A cylinder’s odd partials reach the top of its resonances twice as fast as the others do, so the clarinet’s count falls first. But a clarinet’s register is also why the criterion reads 100 per cent on its range bar: the extreme of its range is a fact about how few resonances it has left, while the extremes on the other bars are about time constants still shrinking.

Where the note comes together #

If the pitch does most of the sorting, a single statement should be available in pitch.

Read at a half, the partials of all five bores come together within one interval of a fifth, between about 290 and 440 hertz: horn 293, trumpet and clarinet 311, trombone 368, saxophone 441. That is roughly the D above middle C to the A above that. Read at a quarter, everything is together from the bottom of its compass, because $-\ln(0.75) = 0.29$ multiplies a difference of time constants that is at most 65 milliseconds. Read at three quarters, the band moves up by most of an octave and the trombone does not reach it.

So the unmeasured criterion still moves everything, as read at two different heights insisted that it must. What the compass adds is that the criterion’s effect is mainly to slide a boundary along the keyboard, with the five families sliding in rough formation. Somewhere in the compass of each wind instrument, there is a range below which its attacks come apart and above which they do not, and if the criterion is single-valued, that range is nearly the same for all five.

That gives a prediction with a sharper form than the family ranking did. If the grouping mechanism uses any single criterion, then a wind note’s onset coherence should change over a narrow range of pitch, in about the same place on every instrument. A listening experiment could look for that without needing to know the criterion beforehand, by asking where in each compass a partial stops being heard out in the first moments of a note, and whether that place is the same on a horn as on a saxophone. If the crossings of the five bores line up, the argument here is right about what is doing the work, and the pitch at which they line up measures the criterion. It would be the first measurement of it.

A connection the census could not have made #

The first fifty milliseconds holds that an instrument’s identity is largely in the sound’s first moments: cut them off and listeners lose the ability to name it. Taken with the figures here, that suggests something about register. If the first fifty milliseconds identify an instrument partly because the partials arrive in a family-specific pattern, then the pattern has most to say low in the compass, where the partials are far apart in rate, and little to say high up, where they are close together or few.

That is at least consistent with a common report in identification studies, that listeners are worse at naming instruments on very high notes. Many other things change at the top of a range: few partials are resolved, the formants are undersampled, and a musical context is usually missing. So nothing here claims to explain that finding. What it adds is one further reason for it, computed from the bores alone, and the direction is right.

There is also an unexpected link to the run of quick notes. In a fast run is a crescendo nobody wrote, the fall of the same time constants up the compass puts a slope in a fast scale’s level; here it puts a slope in the coherence of each note’s start. One property of the bore, that its low resonances are slower than its high ones, means that a wind instrument at speed plays its low notes weaker and less coherent than its high ones, and both are unmarked in the score.

It also completes a symmetry that the partials that do not die together began. That essay put time into a note’s ending: a struck string’s partials start as one and fall apart as they decay, at rates set by how each loses energy. A wind note, on this account, runs the other way round: its partials are at their most separate at the start, in the low register at least, and they are held together for as long as the player keeps blowing. So the two families’ coherence has opposite shapes in time. Whether a listener’s grouping treats a note that comes together over its first moments differently from one that comes apart over its last has not been computed anywhere here, though the calculation for each half now exists.

What is assumed here #

The model. Eight partials, each rising as one minus an exponential at a time constant from the lossy impedance sweep of a modelled bore, read between peaks at notes that are not on one. Partials beyond the last resonance whose Q exceeds ten are left out. Real fingering changes where the resonances are, and a real bore’s peak heights change the rate, neither of which is here.

The rises are passive. As in every essay on this subject, a resonance’s rate stands in for what is really a regeneration between the reed or lips and the bore. Real partials of a wind note are generated together by a nonlinear source, and how far each is free to follow its own resonance is an open question. If they were not free at all there would be no spread on any note, and the entire question would vanish. The calculation describes the largest spread the resonances could produce, not a measurement of one.

Twenty milliseconds is one value from experiments on other sounds. Onset asynchrony thresholds for hearing a partial out come from synthetic tones with abrupt onsets, and a gradually rising partial may need a different figure. The crossings in the last figure would move if it did, though they would still move together.

The compasses are nominal. Each is a stated range, with notes near the extremes that would rarely be played fortissimo or at any speed, and a brass instrument’s notes in particular are not all equally available at every fingering.

What a coherence figure cannot show #

It does not show a timbre. A note that comes apart in the first fifty milliseconds is a note whose brightness is changing over that time, and a blown note does not start late computed that sweep. Whether a listener hears it as separation, as colour or as nothing at all is not in any of these figures.

It does not show a performance. A player’s attack, the use of the tongue and the dynamic will all change these rates in ways that have been only partly worked out. A loud note, especially on brass, brightens as it grows, and that is exactly a change in the ratio between partials over time.

It says nothing about ensembles. A partial is heard out of a note only if nothing else in the texture covers it, and in an orchestral texture, the partials of one note are not the only partials starting then. Who plays what and how loud is one question deals with that, and with the masking that decides what is left. This essay only considers a single note in silence.

Whose instruments #

Five modern Western wind instruments, modelled as simple bores: a B♭ trumpet, an F horn, a tenor trombone, a B♭ clarinet and an alto saxophone. Nothing here describes double reeds, flutes or period instruments, although the argument would apply to anything whose partials ride resonances. A flute’s jet drive differs in kind, and a harmonic-rich reed organ pipe or a shawm would have very different Qs. The figures say what the resonances of a bore imply for these five, played in Western conservatory practice. The numbers are modelled and no recording was used.

Still open: the double reeds and the flute #

The five bores here are the five the earlier essays happened to compute, and the families they leave out are the interesting ones. An oboe’s narrow conical bore and a bassoon’s long one have very different Qs, and a flute is driven by a jet with no reed to set a common rate at all. Adding them requires only their geometries, since the calculation is the same, and it would say whether the band of crossings near the D above middle C is a property of wind instruments in general or of these five, and whether a flute’s partials, with their different source, part where everything else does.