A fast run is a crescendo nobody wrote

Up to now every number about when a wind note speaks has been a number about a note that is allowed to finish speaking. A note takes a number of periods to speak gives a trumpet at its written middle 62 milliseconds to reach nine tenths of its steady amplitude, a tenor trombone 110, a clarinet in the chalumeau 148 and an alto saxophone at the bottom of its tube 159. Those figures have an obvious next question in them, and six essays on speaking time have left it unasked. At a semiquaver of crotchet 240, sixteen notes a second, a note lasts 62.5 milliseconds. So the note ends at about the moment the trumpet’s speaks and well before the others’. What happens to a note that has not finished arriving when its time is up?

Stated that way it sounds like a limit, a tempo beyond which a wind instrument has no notes. It is not, and the first job is to say why, because the way the settling time is defined makes it read as more of a catastrophe than it is. The real cost is smaller and more interesting: it is not that fast notes fail, but that they fail unequally, in an order set by the bore.

Being cut off at nine tenths is almost nothing #

A resonance driven from rest rises as $1 - e^{-t/\tau}$, with $\tau = Q/\pi f$. If the note ends at time $d$, the amplitude it has reached is that expression at $d$, and in decibels it is $20\log_{10}(1 - e^{-d/\tau})$. That is all the physics the argument needs, and it contains a simplification worth noticing at once. Divide $d$ by the settling time $T = \tau\ln 10$, and $\tau$ disappears: the shortfall is $20\log_{10}(1 - 10^{-d/T})$, with nothing left in it about the instrument.

Every wind note there is falls on a single curve, and the curve is much kinder than the phrase “has not yet spoken”. A note cut off at its settling time has lost 0.9 of a decibel, because nine tenths of an amplitude is 0.915 decibels. Loudness differences near one decibel are about what a careful listener can just tell apart. Half a settling time costs 3.3. A note has to last less than about a third of its settling time before the loss passes five or six decibels.

So the settling time overstates the urgency, and that is exactly what read at two different heights warned about. The nine tenths is a convention with nothing measured behind it, and it invites the reading that a note is not there until then. Asked in decibels, the question has no convention in it: the curve does not care which height anyone named. It is the same note, now read at every height at once.

Where the loss falls #

The curve takes the argument out of the unit but not out of the instrument, because a settling time is different at every pitch. The higher note speaks sooner and takes longer found the settling time falling by a factor of seven up a brass instrument’s compass. So a fixed note length is a large fraction of a settling time at the top and a small fraction at the bottom, and the same run sits on different parts of the curve as it climbs.

At sixteen a second, the trumpet at A♭3 is 1.9 decibels short, the horn at C3 3.4, the trombone at A2 4.3, the clarinet at B2 4.1 and the saxophone at its lowest peak 4.5. All of them are inside a decibel from about 560 hertz upward, and by the top of their compasses none of them has lost anything measurable. Those are real gaps, but not the kind a player would describe as a note that failed to arrive. Four decibels is a note audibly weaker than its neighbours, and still plainly a note.

The settling time does give one honest speed limit, once it is read through the curve. A note that lasts its own settling time is under a decibel short, so the reciprocal of the settling time at the bottom of a compass is the rate below which nothing in a run loses more than that: 11.4 notes a second on the trumpet’s A♭3, 7.8 on the horn’s C3, 6.8 on the clarinet’s B2, 6.5 on the trombone’s A2 and 6.3 on the saxophone’s lowest peak. None of these is a rate at which anything stops working. Each is the rate at which the loss begins to be worth writing down, and that is well below what these instruments are routinely asked to do.

The saxophone’s curve is the steepest for a reason the bore supplies. A settling time goes as $Q/f$, and the five bores differ in what $Q$ does as $f$ rises. On the clarinet, $Q$ climbs from 26 at the lowest peak to 51 at the third, which slows the fall in milliseconds; on the saxophone’s cone it drops from 51 to 35 over the same three peaks, so both the numerator and the denominator shorten the wait. Its settling time falls from 159 milliseconds to 35 in that span, against the clarinet’s 148 to 59.

The other families do not have this at all, which is why the comparison belongs here. A note starts twice sorted the instruments by mechanism, and the mechanism decides this too. A violin’s bow captures the string in 12 to 16 milliseconds in this model, and a note a violin has captured is at full amplitude from then on. A piano hammer and a harpsichord plectrum deliver an impulse with no build at all. So a fast passage in octaves between a clarinet and the violins is balanced, in the bottom of the clarinet’s range, against an instrument that loses nothing. At sixteen notes a second the clarinet’s D3 sits 3.7 decibels further under a violin doubling it than the same two notes would when held, and by the top of the two octaves that has shrunk to 0.9. A composer doubling a low wind line with strings in a fast tutti is writing a balance that shifts with speed, and it shifts on one side only.

The crescendo no part marks #

Now put the gradient into a run, since a run is a walk along that compass. A rising scale of two octaves, with each note reached by a change of fingering, starts each note from nothing. The bottom note ends lower on the curve than the second, the second lower than the third, and so on to the top.

At sixteen notes a second a rising two-octave scale contains a crescendo of between 1.7 and 4.5 decibels that nobody wrote and the player’s breath did not supply. The trumpet has the least of it, at 1.7, the horn 2.1, the clarinet 2.8, the trombone 3.0 and the saxophone 4.5. At slower speeds it thins: the rise reaches one decibel at about 12.3 notes a second on the trumpet, 9.8 on the horn, 7.8 on the clarinet, 7.3 on the trombone and 6.5 on the saxophone. So on the trombone, an ordinary semiquaver at crotchet 110 is already there.

A falling scale has the same rise in reverse, a diminuendo as the run descends, so a run up and back down carries an arch that follows its own contour. That arch is the same shape a musician shapes on purpose, since a rising line and a louder one are often played together, and that is the surprise in the arithmetic. The most familiar expressive habit in melodic playing, getting louder as the line climbs, is partly done for a fast wind player by the instrument, and a wind player asked to keep a fast run flat is being asked to compensate for a slope they cannot hear as separate from their own playing. Whether the reported difficulty of evenness in wind technique has anything to do with this is a question the model raises rather than answers; there is no measured run here.

A listener’s own short-note penalty does not cancel this. A staccato is a dynamic mark found that the ear’s running loudness gives a short note less than a held one, by as much as 8.1 phons at a fast tempo. That loss applies alike to every note of a run, since all of them are the same length, and it cannot flatten a slope. If anything, it says that the fast scale’s overall level is lower than the steady level by more than the numbers here; the slope is left as it was, and the bore’s loss is a real addition rather than the ear’s loss seen twice.

Nor does the slope move the note in time, at least not for the listener’s clock as these essays have computed it. A note is heard after it starts places a note by the shape of its attack, and every note in the scale here has an attack shaped by its own time constant, which is longer at the bottom. So a scale’s low notes should be placed later against its high notes, and a fast rising run could seem to hurry as it climbs. That is a prediction from two models neither of which has been run together here, and it is offered as a place to look.

A note that keeps its resonance #

The picture so far assumes each note starts from nothing, and that was the honest assumption for a run because a new fingering makes a new air column. A repeated note does not. Double-tonguing the same D, the bore stays as it is, and the tongue stopping the reed for a few milliseconds lets the resonance decay only a little. When the reed starts again, the resonance is mostly still there.

A repeated note, at the same rate and the same pitch, loses a quarter of what it loses in a scale. With a ten-millisecond stop at sixteen notes a second it is 0.9 decibels short, against 3.7. At twenty a second it is 1.3, against 4.9. The settled pattern follows from the periodic solution, $(1 - e^{-d/\tau})/(1 - e^{-(d+g)/\tau})$, with $g$ the length of the stop: what the stop leaves is what the next note builds on. The note that begins a repeated figure is not protected: it starts from nothing and reads like a note in a scale, so a repeated figure should swell slightly into its pattern over its first few notes.

That reverses an intuition about what is hard. A repeated note is hard for the tongue. For the resonance it is the easiest thing a player can ask. Conversely, a fast scale, which takes no special tongue at all when slurred, is the case in which the air column is asked to start again on every note. Another way of putting it: speed in a slurred run costs the player nothing in the embouchure and costs the air column everything it has.

The same reasoning says why the articulation that was priced in what the tongue actually removes should matter more here than on a long note. That essay found that a tongue is worth about half the length of the ramp it removes, roughly the same number of milliseconds on every wind instrument. On a held note those milliseconds disappear into the length of the note. In a run they come out of a budget of sixty.

At eight notes a second a soft attack costs under two decibels, and at sixteen it costs up to seven. An 80-millisecond rise cannot finish inside a 62.5-millisecond note, and a clarinet’s D3 under it stops at 10.5 decibels short. Even a 40-millisecond rise moves that note from 3.7 short to 6.0. That is the difference between a run that loses a little evenness and a run whose notes no longer have a clear level. It supplies a reason, in the terms of the resonance, for the ordinary instruction to keep a fast passage’s articulation clean, which is usually given as a matter of clarity rather than level. Both may be true; the arithmetic concerns only the second.

What is being claimed about real instruments #

Everything here is the passive resonance filling up. The model. Each bore’s impedance peaks come from the lossy transmission-line calculation, each peak’s Q gives a time constant, and notes between peaks read their time constant from a logarithmic interpolation. A note’s level is taken as that one resonance’s amplitude at the note’s end, relative to its steady amplitude, and every partial of the note is taken to follow it.

A reed or a pair of lips is not passive. They add energy in proportion to the oscillation already present, so the real build-up is a growth process with a rate of its own, and the resonance’s Q says how fast the air column can respond rather than how fast the note must arrive. That could shorten or lengthen any of the figures above. It was accepted in a note takes a number of periods to speak for the same reason it is accepted here: the argument depends on the ratio of time constants across the compass, and that ratio is what a bore determines.

A change of fingering does not always start from nothing. Some of the air column’s motion carries over from note to note, particularly on a small interval, and the amount surely depends on the note and the tonehole. A scale here is the worst case; a slurred step could carry a good deal more, which would lessen the rise and move it towards the repeated note’s numbers.

And the player is not holding still. A trained wind player’s breath and embouchure change with the note, and part of what is called evenness is exactly that. These figures give the slope that playing has to correct for if the player does nothing. A measured run, at known speed with a microphone and level-calibrated recordings, would show what remains of it. That has not been done here, and the sign of the difference would be worth knowing.

The dynamic is also absent. Nothing here depends on how loud the note is, but a loud brass note brightens as it grows, and a blown note does not start late, it starts slowly has already said that its partials differ in rate. A short note must therefore differ in colour as well as level. No figure above shows that, and a real fast run on a trombone is surely thinner in tone as well as softer; this essay measures only the second.

Whose runs #

Fast scalar writing for winds, at twelve notes a second or more, is a feature of nineteenth- and twentieth-century orchestral and solo practice and of later jazz, and it is that repertoire these numbers belong to. In an eighteenth-century trumpet part, which sits in the upper partials on a long tube, the notes in a run are mostly high in the compass, where almost nothing is lost. The natural-trumpet clarino writing that looks most impressive on paper happens to sit where a run costs least. That is a coincidence of range, and there is no evidence it had anything to do with the writing; the tessitura has other explanations. Scales up from the bottom of a saxophone, or arpeggios up a trombone, are the cases where this effect is largest, and these are the passages to look at first.

Still open: whether anything carries from one note to the next #

Everything above sets the question of carry-over to one of two extremes: a change of fingering starts from nothing, and a repeated note keeps nearly everything. Real runs lie between. Adjacent peaks of a bore overlap, and a note a tone away shares a good part of its resonance’s motion with the note before it, more when the two are close and less for a leap. The computation is the air column’s response to a sudden change of length with the old motion as its starting condition, and it would say how much of the slope survives for a stepwise scale against an arpeggio, and whether a run in thirds should in fact be less even than a run in tones at the same speed.