What the tongue actually removes

A note starts twice set the instrument’s own settling time against the listener’s placing delay and ended by naming what the anchor still owed:

Every disagreement on this page has the same shape: the physical model computes a free response and a musician does not produce one. A tongued brass attack, a martelé bow stroke and a struck key all deliver an impulse before the steady drive begins, and this collection has a model of exactly that on the excitation ladder — a force pulse whose spectrum is set by its own duration.

Both halves of that are worth doing and only the first half is right. The impulse is real, it is computable from machinery this collection already has, and it is worth about a fifth of one per cent. What a tongue does is not add an impulse to the front of the note. It is the removal of something, and what it removes has been sitting in every figure this anchor has drawn, unnamed, as an assumption about the shape of the drive.

The step this anchor has been assuming #

A note takes a number of periods to speak computes a settling time as the response of a resonance to a drive switched on at $t = 0$. The amplitude climbs as $1 - e^{-t/\tau}$ with $\tau = Q/\pi f$, and the wait to reach nine tenths of the steady amplitude is $\tau \ln 10$.

A drive switched on at $t = 0$ is a step. A player who does not tongue does not produce one: the pressure in the mouth rises over some tens of milliseconds and the resonance follows it up. So the calculation this anchor has been doing all along is the calculation of a tongued attack, and what has a price is its absence.

A trumpet’s written middle sits on an impedance peak of Q 36.5 at 428 hertz, so its time constant is 27.1 milliseconds and the step reaches nine tenths in 62.5. Under a ramp of 80 milliseconds the same peak takes 111.7.

The excess is 49.2 milliseconds, and that is 62 per cent of the ramp’s own length. Across the range a player actually works in the fraction barely moves: 0.51 at a five-millisecond rise, 0.53 at twenty, 0.62 at eighty, 0.73 at two hundred. To within the precision anything here deserves, a ramp costs half its own length, and that one sentence is the whole model of articulation.

The half has a reason and it is worth stating because it explains why the number is so nearly constant. A linear ramp is a step delayed by the average of the times at which its own drive arrives, and the average of a uniform ramp is its midpoint — so a resonance short compared with the ramp is simply waiting for the middle of it. The drift upward, from a half toward three quarters at very long ramps, is the other end of the same argument: once the ramp is long compared with the resonance, the resonance is tracking the drive rather than integrating it, and what it is waiting for is the moment the drive itself reaches the criterion, which on a linear ramp is nine tenths of the way along. Half and nine tenths are the two limits and everything a player does lies between them.

Which is why the saving is the same on every instrument #

The consequence is not obvious until it is drawn. The excess depends on the ramp, which belongs to the player, and only weakly on the time constant, which belongs to the instrument. So the tongue is worth nearly the same number of milliseconds everywhere and a completely different fraction on each instrument.

The five wind instruments save 44.0, 43.0, 40.4, 39.0 and 38.8 milliseconds — a spread of thirteen per cent — while the wait they are saved from runs from 112 to 203 and the share runs from 39 per cent on a trumpet down to 19 on an alto saxophone.

That inverts the intuition attached to the technique. The instruments where the tongue is described as most necessary are the slow-speaking ones, and on those it buys proportionally least: the saxophone’s resonance is going to take 159 milliseconds whatever the player does with their tongue, and the tongue removes 39 of the 203 a breath attack costs. On a trumpet it removes 44 of 112. A tongue is worth an absolute amount of time and instruments differ in how much of the wait that is.

Set against the twenty milliseconds at which a listener begins to hear one player as displaced from another, the absolute number is the one that matters, and it is about two thresholds’ worth on every instrument here. Two wind players entering together, one tongued and one on the breath, are not slightly apart; they are apart by twice the interval at which the difference becomes audible, on any instrument in the section. That is why the instruction is given as an ensemble instruction rather than as a matter of individual taste.

There is a corollary about anything that raises Q. The resonator at the far end found that a straight mute raises a trumpet’s ceiling by reviving high-Q peaks that the open instrument had lost. A higher Q is a longer time constant, so it is a longer settling time — and the ramp’s cost is unchanged, because the ramp is still the player’s. So the tongue’s share falls exactly as far as the Q rises. A mute makes the instrument slower and makes the tongue matter less, which is the reverse of the advice usually given about muted articulation.

And the other route, priced #

The impulse the debt asked for is real and it can be computed exactly, with the same half-sine pulse spectrum the hammer’s contact is low-passed by on the excitation ladder.

A tongue release of duration $T$ delivers a pulse. What it puts into the mode, as a fraction of the steady amplitude that drive would eventually reach, is

$$a_0 = 2\,\frac{T}{\tau}\,\left|H(fT)\right|$$

and the two factors fight. A longer pulse carries more momentum, and less of that momentum sits at the note’s own frequency, because a pulse of duration $T$ has almost nothing above $1/T$.

The product peaks at about half a period of the note. On a trumpet’s B♭ that is 1.67 milliseconds, which no tongue achieves, and even there the saving is 2.08 milliseconds — 3.3 per cent of the wait. At the ten milliseconds a tongue actually takes it is 0.28 per cent, or 0.17 milliseconds out of 62.5.

The ceiling has a form worth writing down. Put the optimum back into the expression and the pulse’s best contribution comes out at about $\pi/Q$, so the saving is about $1.13/Q$ of the settling time. On a trumpet that is 3.1 per cent against the 3.3 measured; on a clarinet, 4.4 against 4.8.

The route is capped by the resonance’s own Q, which is the very thing making the note slow. A high-Q instrument takes a long time to speak because it accepts energy only in a narrow band, and a short impulse is broadband by definition, so almost all of what the tongue delivers is at frequencies the resonance will not take. The two properties are the same property, and that is why no amount of tonguing technique reaches this route.

The one instrument where the impulse is not negligible #

The exception is at the bottom of the compass and it is the clarinet’s.

A clarinet’s chalumeau D is at 126 hertz, so half a period is four milliseconds — inside what a tongue can do — and its Q is 25.5, the lowest of the five. Both factors point the same way, and the impulse route buys 2.6 milliseconds at a ten-millisecond release and 7.2 at its optimum of 5.5.

Seven milliseconds is not nothing. It is a third of the twenty-millisecond threshold at which a listener starts to hear one instrument as displaced against another. But it is still under five per cent of a 148-millisecond wait, and the ramp route on the same instrument is worth 39. The ordering never reverses anywhere in the collection: the ramp is worth between fifteen and two hundred and sixty times what the impulse is, and the ratio is largest on the instruments where articulation is talked about most.

Which is worth stating the other way round, because it is the more useful form. The impulse ceiling is a fraction of the wait rather than a number of milliseconds, so it shrinks in exactly the same proportion as the thing it is trying to shorten. An instrument with twice the Q has twice the wait and the same 1.13 over Q of relief available against it, and the two cancel: the absolute saving is very nearly constant while the wait it is set against is not. That is why the exception is at the bottom of a clarinet and not at the top of a trumpet. A low clarinet note is the one place in the collection where a slow resonance and a lossy one coincide, and it is lossy for a reason the bore ladder has already priced — an open hole near the top of a long tube radiates freely, so the peak that would otherwise be sharpest is the one the instrument deliberately spoils.

The bowed string’s version, which is a different ramp #

A bowed string has no resonance to build. The hardest place is also the latest computes its onset as a capture: the bow accelerates from rest and the note begins when the force it is applying enters Schelleng’s window, so the duration is the time the force takes to climb into a band.

That is a ramp too — a force ramp rather than a pressure ramp — and it is the constant BOW_RAMP that every figure in this anchor has carried at eighty milliseconds without ever varying it. A martelé stroke is exactly the removal of it: the bow is set into the string, the force is established while nothing is moving, and only then does the arm go.

The figure draws that row and it should be read as a parameter rather than as a result. Nothing in this collection measures how long a martelé takes to set its force; the twelve milliseconds drawn is a choice, and the eighty it is set against is the choice the anchor has been making silently since its third rung. What the model does say, and says without any measurement, is the form: the capture time is proportional to the force ramp, so an articulation that removes nine tenths of the ramp removes nine tenths of the capture time. It is a linear saving where the wind instruments’ is a saturating one.

And a struck or plucked string has no ramp at all. It is at full amplitude the instant the exciter lets go, so there is nothing for an articulation to remove, and nothing a pianist does with the key changes when the note begins — only how loud it is and how bright, which are the excitation ladder’s subject and not this one.

Which puts the anchor’s own census in a different light #

The three mechanisms this anchor is built on can now be sorted by whether a player can do anything about them.

A resonance that builds has a ramp in front of it and the ramp is worth up to half of itself. A bow that captures has a ramp in front of it and the ramp is worth all of itself, proportionally. An exciter that lets go has no ramp and articulation buys nothing.

So the census’s three clusters are also three degrees of player control over onset, and they run in the opposite order to the clusters themselves: the fastest mechanism is the one with no control, and the slowest is the one where the control is a diminishing return.

That is worth setting beside what a blown note does not start late established about a wind instrument’s partials — that they differ in rate rather than in onset time, with a spread of 2.2 to 6.9 milliseconds against a twenty-millisecond threshold. Articulation moves the whole note by 39 to 44. The articulation is an order of magnitude larger than the internal spread it would have to be compared against, which means an ensemble hearing a wind section arrive together or apart is hearing tongues rather than resonances.

Across a trumpet’s own peaks the ramp is an additive constant, near enough. It does not change which register speaks soonest, and it does add tens of milliseconds to all of them together, which is the part an ensemble hears.

Which computation produced the numbers #

The step response is the anchor’s own: $-\ln(1-p)\,Q/(\pi f)$ to reach a fraction $p$.

The ramp response is that resonance driven by a drive rising linearly to one over $T_r$ and held: $a(t) = [t - \tau(1-e^{-t/\tau})]/T_r$ while it is rising, and a plain exponential approach from whatever amplitude it reached at $T_r$ afterwards. The time to reach the criterion is found by bisection, and the figure checks the zero-ramp case against the closed form rather than trusting the solver.

The impulse route is the momentum a half-sine pulse of duration $T$ carries at frequency $f$, divided by the momentum needed to reach the steady amplitude — which is the same hammerRolloff the excitation ladder low-passes a hammer’s spectrum with, evaluated at the note’s fundamental rather than at a partial number.

The instruments’ Qs are speakingLadder’s: the impedance peaks of each bore, from the eleventh rung of the air-column ladder, at the peak nearest each instrument’s written middle.

Where the model stops #

The ramp lengths are chosen and not measured. Eighty milliseconds for a breath attack and ten for a tongue are ordinary figures and they are not from any experiment in this collection. Everything above scales with them: the fraction of the ramp that is lost is robust, and the number of milliseconds is only as good as the ramp.

A tongue release is not a switch. The tongue withdrawing from the reed or the lips is a continuous opening of an aperture, so the real drive is a fast ramp rather than a step, and the ten-millisecond row is that ramp. What is not modelled is that the aperture’s opening changes the impedance the mouth presents to the instrument, which is a term this collection does not have at all.

The drive is treated as linear. A brass instrument’s lips and a reed are both nonlinear valves whose behaviour depends on how much pressure is behind them, and a resonance driven by such a valve does not simply follow the mouth pressure. The whole of this anchor makes that assumption; this rung makes it once more.

And the martelé number is an input. It is drawn so that the mechanism is visible and it should not be quoted. What can be quoted is the proportionality, and the fact that the bowed row responds linearly where the wind rows saturate.

What the picture cannot show #

It cannot show when the listener hears any of it. Everything here is measured at the instrument, and turning a settling time into a moment of arrival needs the perceptual account the fourth rung paired it with — which reads the envelope at a different height, and that difference turns out to matter more than anything on this page.

Nor what a tongue costs elsewhere. A hard tongue produces a broadband transient that is audible as a consonant, and much of what a player is choosing when they choose an articulation is the sound of that transient rather than the timing of the note behind it. Nothing here is about the noise.

Where this ladder goes next #

Five rungs. A wait is a Q in one unit and a Q over a frequency in another; up a brass instrument the two units disagree monotonically; on a bowed string the wait is a capture and the hardest place is the latest; the instrument’s number against the listener’s; and now the articulation, which is the removal of a ramp rather than the addition of an impulse, worth about half the ramp’s own length and very nearly the same number of milliseconds on every wind instrument in the collection.

What the anchor owes now is the number underneath all five. Every figure this anchor has drawn asks when a resonance reaches nine tenths of its steady amplitude, and nine tenths is a convention with nothing measured behind it — the first rung says so in as many words, and adds that the ratios reported here are independent of it. The second half of that claim is true only inside the build mechanism: a bow’s capture and an exciter’s contact contain no criterion at all, so moving it rescales one of the anchor’s three clusters against two that do not move. This collection has three named criteria on its perceptual side, at fifteen decibels below peak, six decibels below peak, and ninety per cent, and the settling side has only ever used the last of them. Whether the fourth rung’s headline survives being read at either of the others is arithmetic that costs one multiplication, and nobody has done it.