The higher note speaks sooner and takes longer

The first rung of this ladder took one note from each instrument in the collection and found that the two units for measuring an onset — periods of the note, and milliseconds — put twelve instruments in different orders. That could have been an artefact of comparing a saxophone with a harpsichord.

It is not. Hold the instrument fixed, take a brass player up their own compass, and the two units still disagree, monotonically and over almost the whole range.

Why the two curves have to differ #

The arithmetic is short. A resonance of quality factor Q at frequency f settles with a time constant of Q/πf, so the wait in periods is proportional to Q alone and the wait in milliseconds is proportional to Q/f.

Going up a brass instrument, both of those change. The frequency doubles between the second peak and the fourth and doubles again by the ninth. The Q rises too, because the bell is a high-pass filter and a mode well below the bell’s cutoff is well contained, while a mode near the cutoff radiates and is therefore lossy.

The Q’s rise is slow — from 15 at the pedal to a maximum of 40 near the sixth peak — and the frequency’s rise is not slow at all. Their ratio therefore falls, and it falls fast: 153 milliseconds at the pedal, 62 at the fourth peak, 20 at the tenth.

So the milliseconds fall because the frequency wins. The periods rise because the Q rises and nothing else enters. There is no way for both readings to say the same thing, and neither is wrong.

The turn near the sixth peak #

The periods curve is not monotone, and where it turns is the interesting part.

It rises from 11 at the pedal to 30 at the sixth peak, and then falls back to 22 by the tenth. That turn is the bell’s cutoff arriving. Below the cutoff the bore contains its modes and the Q grows as the mode gets further from the reflection boundary at the bottom; above it the modes begin to leak, the Q falls, and by the tenth peak the bore is on its way to having no resonances at all.

The cutoff a maker can actually measure put that boundary at around 1,590 hertz for this bore, measured the way a maker measures it. The tenth peak here is at 1,112, so the top of this ladder is already inside the shoulder rather than past it, and the Q has begun to fall while the peaks are still comfortably countable.

That gives the instrument a most-transient note, in the sense that matters for colour: the note whose onset occupies the largest fraction of its own cycle, which sits near the middle of the useful compass rather than at either end. Whether every brass instrument has one is a question the next section puts to the horn, and the answer is no.

The crossing lines are the whole argument. The number of periods a resonance needs and the number of milliseconds it takes are different orderings, because a period is a different length at every pitch — so “which instrument speaks slowest” has two answers and the essay’s title is a statement about which unit is being used.

A longer instrument is a slower one, in one unit only #

The horn is nearly two and a half times the trumpet’s length and its ladder is correspondingly denser. Its Qs are almost the same as the trumpet’s peak for peak — 18 to 37 against 15 to 40 — because they are the same family of shapes and the same losses, which is the finding the family’s impedance ladders makes explicit.

Same Qs, lower frequencies. So the horn’s milliseconds run from 161 at its lowest peak to 54 at its tenth, against the trumpet’s 153 to 20 over the same peak numbers. In milliseconds the horn is slower everywhere above its second peak, by a factor approaching three at the top.

In periods it is not slower anywhere. Its periods run 13 to 27 and the trumpet’s 11 to 30, which is the same range — and, it turns out, not the same curve, for a reason the section on the turn already supplies.

That is a fairly complete account of a thing brass players say and no figure in this collection had drawn: a horn is a slow instrument to speak. It is slow in the unit an ensemble uses and not in the unit its own tone is made of, and the reason is not that its resonances are worse but that its notes are lower.

The horn has no turn #

The trumpet’s periods curve rises to 29.6 at the sixth peak and falls to 22.5 by the tenth, and the fall is the bell’s cutoff arriving. Reading the horn’s ten peaks the same way, the curve rises to 26.8 at the ninth and stays there — its Q reaches its maximum at the top of the ladder rather than in the middle, and there is no fall.

The reason is the one the trumpet’s turn was explained by, applied to a longer tube. Ten peaks on a trumpet reach 1,112 hertz, which is inside the shoulder of a cutoff at about 1,590. Ten peaks on a horn reach 493 hertz, which is nowhere near it. The horn’s ladder is denser because the instrument is longer, so the same count of peaks covers a third of the frequency range and never gets to the boundary where modes begin to leak.

So the most-transient note is a property of the trumpet and not of brass instruments. A horn’s onset occupies a larger fraction of its own cycle at every step up its useful compass, with no interior maximum, and a player going higher is always going toward a more transient note rather than through one. That is a difference the two curves’ ranges hide completely, and it is the reason the sentence above says the same range rather than the same curve.

Where the three quantities are least bad at once #

The claim that the middle of the compass is the compromise is worth making by ranking rather than by eye. Rank every peak on height, on Q and on speed, and ask which peak has the best worst rank of the three:

Both instruments put their compromise at the sixth peak, and on the trumpet the runners-up are the fifth and the seventh — which narrows the essay’s “fourth to the seventh” to a band of three with the sixth at its centre. On a B♭ trumpet that is roughly the written F♯ to A at the top of the staff.

The horn’s answer is the same peak and it is arrived at differently, and the difference is worth one sentence because it is the turn again. On the trumpet the sixth peak wins because the Q curve has its maximum there; on the horn the Q is still climbing, so the sixth peak wins only as the point where a monotone rise crosses a monotone fall. A minimax over three monotone curves always lands in the middle by construction, and a minimax over one curve with an interior maximum lands somewhere the instrument chose.

What the hand costs, which is nothing #

The tenth rung of the air-column ladder took the hand all the way into the bell and found the note crossing from a semitone flat to a semitone sharp; the eleventh added what the same travel does to the strength of each resonance, and found a window in which the horn loses about a fifth of the support the player pushes against.

The same sweep carries the Q of every peak at every hand position, so what the hand costs in speaking time is a change of units on a table that already exists — and the answer is that it costs almost nothing.

The mean settling time over the first eight peaks goes from 114 milliseconds with the bell open to a worst of 125 at 95 per cent closed and back to 107 with the bell shut. That is a swing of about ten per cent, against a semitone of pitch and a fifth of the support over exactly the same travel.

This is the first thing in that ladder the hand does not cost, and the reason is worth stating because it is not obvious. Closing the bell does two things at once: it makes the resonances sharper, because less energy escapes, which lengthens the settling time; and it lowers the frequencies, which also lengthens it. Both should slow the instrument down. What holds the total nearly still is that the hand also renumbers the modes — the note jumps — so the peak a player is on is not the peak they were on, and the mean is taken over a ladder whose rungs have shuffled.

In milliseconds the wait falls steadily up the compass, which is the title’s first half. In periods it does not, which is the second — and both are properties of the same nine peaks, read with the same arithmetic and divided by different things.

The register the instrument is loudest in is not the one it is quickest in #

Three quantities are now available for every rung of the same ladder, from the same sweep: the peak’s frequency, its height, and its Q. They do not agree about which note is the instrument’s best.

The height — how hard the bore pushes back on the lips, which is what a player means by a note speaking easily — falls monotonically from the pedal upwards. The lowest peak is by a long way the strongest and every one above it is weaker.

The Q — how tightly the bore holds the pitch, which is what a player means by a note being centred — rises to a maximum near the sixth peak and falls away.

The settling time in milliseconds falls monotonically from the pedal upwards.

So the strongest note is the slowest, the most centred note is the most transient, and the quickest note is the weakest. There is no peak at which all three are best, and a player’s account of where an instrument is comfortable is a compromise between them rather than a reading of any one.

That is a fairly general shape for a designed object and it is the reason the ladder of frequencies is not the instrument. A brass instrument’s written compass is chosen by makers and players who have all three of these in their hands and none of them in numbers, and the middle of the useful range — the fourth to the seventh peaks here — is where the three curves are least bad at once.

What a player would notice #

Two of these results are things brass players report and one is not.

That the low register speaks slowly is reported everywhere, and the figure gives it a number: the pedal register’s settling time is two and a half times the middle register’s in milliseconds. A pedal note asked for on the beat has to be started well before it, and the amount is in the tens of milliseconds rather than the units — comfortably inside what the perceptual-centre ladder treats as a lead a player rehearses.

That the top speaks quickly is also reported, and the figure gives it a smaller number than the register’s reputation suggests: 20 milliseconds at the tenth peak. High brass notes are hard for reasons that are about the lips rather than about the bore, and the partial the lips cannot reach is the essay about that. The bore’s own contribution to the difficulty of a high note is that it stops helping, not that it is slow.

What is not reported is the turn, because it is a claim about tone rather than about timing. The bore’s onset occupies the largest fraction of its own cycle around the fifth and sixth peaks, which on a B♭ trumpet is roughly the written G to C at the top of the staff — the part of the range players and writers describe as the instrument’s own voice. That is a coincidence until somebody listens for it, and it is stated here as a prediction rather than as a finding.

Which computation produced the numbers #

The sweep is the transmission-line solver from the fourth rung of the bore ladder: the bore in short cylindrical sections, propagated from a radiation load at the mouth back to the throat, with Benade’s visco-thermal wall losses giving each peak a width. The peaks are local maxima with their half-power points found by walking out to either side, which is the definition a maker uses and which needs the losses to exist at all.

Everything else is the identity in the first rung: periods are 0.733 Q, milliseconds are that over the frequency, and the 0.733 is ln(10)/π for a rise to nine tenths.

The hand is a short constriction at 94 per cent of the bore rather than a narrowing of everything past it, which is the geometry the tenth rung used, so that this rung’s widths are commensurable with that rung’s centres.

Where the model stops #

The Qs are sensitive to the sweep’s resolution, at about five per cent between a sixteen-hundred-point sweep and a six-thousand-point one, because a half-power point has to be found by walking a sampled curve. Every shape in this essay survives that and no third digit does.

A mouthpiece is not in these bores. What the mouthpiece is actually for shows that a cup and a throat are real geometry rather than an equivalent length, and adding one moves the upper peaks down and changes their heights. The Qs would move with them.

The losses are a smooth-tube approximation. A real instrument has valve ports, tuning slides and a bore that is not a mathematical surface, and every one of those adds loss. Real measured Qs for brass instruments run lower than these, which would shorten every settling time here in proportion and leave every ratio alone.

And a player is not a switch. The whole calculation is a resonator driven from rest by a source that appears at full strength. A brass attack is a tongue release into lips that are already vibrating in the air, which is a different initial condition and a gentler one.

What the picture cannot show #

It cannot show the spectrum settling. Each partial of a played note has its own Q and therefore its own settling time, so a note does not arrive all at once — it arrives from the bottom up, or the top down, depending on which peaks are sharpest. That is a much better model of what a listener hears as a brass attack than a single exponential, and it needs the played spectrum rather than the bore’s impedance.

Nor can it show the ensemble. A player’s lead is a decision made against everybody else’s, and the p-centre ladder’s map of required leads has instruments rather than bores in it. Joining the two would mean asking whether a horn player’s lead tracks the note they are playing, which is a question about a performance.

And it cannot say the hand is free. What the hand does not cost is settling time. It costs pitch, it costs support, and it costs the player a hand — and a horn player stopping a note is doing several things at once of which speed is the only one this figure prices.

Whose instruments, and when #

These are modern orchestral bores. The trumpet is the valve instrument of the mid-nineteenth century onwards, a cylinder over two thirds of its length with a Bessel flare over the last third; the horn is the same idea longer and more gradual, at the F length that has been standard since the double horn settled the question of which crook.

Hand-stopping is older than both, and it is worth noting where this figure sits in that history. On the natural horn of the eighteenth century the hand was the only way to get the notes between the partials, so the settling time it costs was a cost every chromatic passage paid. On the modern valve horn it is a colour, used deliberately for the sound it makes, and a player choosing it is choosing the timbre rather than paying for a pitch. The arithmetic is the same and the reason for doing it is not.

Where this ladder goes next #

Two rungs. The wait is a Q in one unit and a Q over a frequency in the other; and up a brass instrument those two run in opposite directions, monotonically, over a range where the milliseconds fall sevenfold and the periods rise threefold.

What is owed next is the other family. Everything above is a resonator filling up, and the bowed string does not fill up — its onset is a capture rather than a build, and the map of where it is slow is already drawn in a different currency. Turning that map into milliseconds is a change of units this collection can make, and the exponent that relates the two is the part nobody has computed.