A note takes a number of periods to speak

Two essays in this collection end by naming the same object, and neither could reach it from where it stood.

A resonance has a strength as well as a frequency computed every impedance peak of a brass instrument with its height and its Q, and closed by pointing out that a Q is a settling time — so the machinery for saying how long a note takes to start was suddenly all present, and no figure in that ladder had drawn it. The hardest place on the fingerboard multiplied two curves into a map of bow-force windows, and closed by pointing out that a narrow window is a slowly started note, so the map was a map of attack times under a change of units.

Two ladders, two instrument families, one phase apart, both stopped in front of the same quantity. That quantity is what this anchor is about, and the first thing to say about it is that it is not one quantity at all.

The three ways a note can start #

A note begins when its mechanism reaches the state that sustains it, and there are three quite different mechanisms in this collection, which have no reason to produce comparable numbers and are always reported in the same units anyway.

A wind instrument accumulates. The air column is a resonator being driven from rest, and its amplitude rises toward the steady value as one minus a decaying exponential. Nothing about the reed or the lips decides how long that takes; the resonator decides, and its time constant is the same quantity that sets the width of its resonance peak.

A bowed string captures. The moment Helmholtz motion begins the string is already at its full amplitude — the bow makes a corner and the corner is either going round or it is not. What takes time is getting the bow into a state where the corner can survive: the force has to lie inside Schelleng’s window, and at the start of a note the bow is barely moving, so both bounds of that window are nearly zero.

A plucked or struck string is simply released. Where the hammer lands decides the spectrum and nothing builds at all: the string is at maximum displacement at the instant of release, and everything afterwards is decay. The only duration in the event is the exciter’s own — the time a felt hammer stays in contact, or the time a released corner takes to reach the bridge.

Only the first of the three is a settling time in the ordinary sense, and only the first has a Q in it.

The wait is the Q, and it has no pitch in it #

For the wind instruments the arithmetic is short enough to state completely. A resonator of quality factor Q at frequency f has a time constant of Q/πf, so reaching nine tenths of the steady amplitude takes ln(10) of those, which is

t = 2.303 · Q / (π f)

Multiply by f and the frequency cancels. In periods of the note being played, the settling time is 0.733 times the Q and nothing else — the same for a pedal note and for a high one, for a trombone and for a piccolo trumpet, provided the peaks have the same Q. The instrument’s pitch does not appear.

In milliseconds it appears immediately, because milliseconds are periods divided by the frequency. So the same table read in the two units is two different tables, and it is not obvious in advance which one a musician means.

What the two units disagree about #

Nine of the twelve instruments in the census change place between the two orderings, and the disagreement is not a rounding effect at the margins — it reaches the top of both lists.

It also has a boundary, and the boundary is the more useful half of the result. Ranking the twelve both ways, the two orderings correlate at 0.846, and every one of the nine changes is inside a family:

The three families occupy the same three blocks in both units, with no overlap on either axis, and the reordering happens entirely within them. The winds run 18.7 to 37.5 periods and 62 to 159 milliseconds; the bowed strings 3.0 to 7.6 and 11.5 to 15.5; the struck and plucked 0.0 to 0.1 and 0.2 to 0.6. The gaps between the blocks are a factor of two and a half on the slower axis and a factor of thirty on the faster.

So the choice of unit never changes which kind of instrument is slower and always changes the order within a kind. A musician asking whether a wind instrument speaks more slowly than a plucked one gets the same answer in either unit, by a wide margin; a musician asking whether a clarinet speaks more slowly than a trombone gets opposite answers, and the question is genuinely ambiguous rather than merely unresolved.

That also confirms the identity the section above states. Dividing each wind instrument’s settling time in periods by its own Q gives 0.7329 in every case, to four figures — the same number for a trumpet, a horn, a trombone, a clarinet and a saxophone, because the frequency has cancelled and nothing else is left. It is the only exact relation in the census, and it is exact because it is a definition rather than a measurement.

In milliseconds the slowest thing here is a saxophone, at 159 milliseconds, with a clarinet’s low E just behind at 148. In periods the slowest is also the saxophone, at 37 — but the clarinet, second in milliseconds, is last of the five winds in periods, at 19. Its low E has a Q of 26 against the saxophone’s 51, and it is only slow in milliseconds because its note is low.

The four violin strings reverse completely. The G string is the slowest in milliseconds, at 15.5, and the fastest in periods, at 3.0; the E string is the fastest in milliseconds, at 11.5, and the slowest in periods, at 7.6. Nothing about the strings changes direction between the two readings — the frequency does.

Which unit a musician means #

The disagreement matters because both readings are used, by different people, for different things, and they are almost never labelled.

Milliseconds are what ensembles are made of. A note is heard after it starts is the perceptual-centre ladder’s first rung and its whole subject is a delay in milliseconds between a note beginning and a listener placing it. The players who have to be early turns that into an instruction: an instrument that speaks slowly is played early, and how early is a number of milliseconds a player rehearses into their hands. Nothing in that account cares what the pitch is except through the delay it produces.

Periods are what timbre is made of. A transient is heard as part of the note’s colour rather than as a separate event, and how much of the note it occupies is a proportion rather than a duration. The first fifty milliseconds shows that identity lives in the attack; a transient of twenty-seven periods at the top of a trumpet’s range and one of eleven at the bottom are two different fractions of the note’s own cycle, and a spectrum that takes twenty-seven cycles to settle sounds different from one that takes eleven whatever the wall clock says.

So the question “which instrument speaks slowest” has two correct answers, and choosing between them means choosing whether the answer is about playing together or about tone.

There is a case where the number can be moved by the player, and it is the one every horn player knows as a complaint.

Hand-stopping makes the note slower to speak, and by an amount that is not small beside the differences between instruments. So the response a player feels is not a fixed property of the horn — it is a property of the horn and of where the hand is, which is a third axis on a control that was already trading pitch against loudness.

The bowed string is not on this scale at all #

The capture times are an order of magnitude shorter than the winds’, and that is a fact about mechanism rather than about difficulty.

The figure the third rung of the bowed-string ladder drew is a wedge: both bounds of Schelleng’s window are proportional to the bow’s speed, the bow’s speed at the start of a note is nearly zero, so the whole window is nearly zero and opens out as the bow accelerates. A player aiming at the middle of the window the note will eventually sit in has to wait until the rising upper bound reaches that force, which happens a fraction of the way through the bow’s ramp.

That fraction is one over the square root of the window’s width, which is a number between about a seventh and a fifteenth on a violin. At a bow ramp of eighty milliseconds — the time a normal détaché takes to reach speed — the capture happens between four and twenty-two milliseconds in.

The bowed case has a wedge rather than a threshold. Both of Schelleng’s bounds go as the bow’s speed, so the admissible region opens out from the origin instead of sitting still and being aimed at — a note is caught by moving into the wedge rather than by hitting a value, and the time that takes is the bowed family’s version of the number this essay is about.

The reason this is so much faster than a wind instrument is that nothing has to fill up. A string under a bow that has reached the right force is already vibrating at the amplitude the bow will sustain, because the amplitude of Helmholtz motion is set by the bow’s speed and the bow has reached it. There is no reservoir.

And the struck string has no onset at all #

The third cluster sits three orders of magnitude below the winds in periods, and its numbers are not settling times in any sense — they are the durations of the excitation.

A hammer is not an impulse computes how long a piano hammer stays on the string: about four tenths of a millisecond at middle C, longer in the bass because a heavier hammer meets a heavier string. That contact does not delay the note; it shapes it, by low-passing the excitation. The string is moving from the first instant of contact.

A plucked string is the same case with a shorter number. The plectrum releases the string with a corner in it, and the corner reaches the bridge in the plucking fraction divided by twice the frequency — a quarter of a millisecond at middle C for a pluck an eighth of the way along. Again nothing is delayed.

A struck string has only one duration at its onset — the felt hammer’s contact time — and it is a shaping term rather than a delay: the string begins to move at the first instant of contact. So the struck family is the one where this essay’s quantity is nearly zero, which is worth having as the end of the range rather than as an exception to it.

Which computation produced the numbers #

Three separate calculations, each already in this collection before this essay.

The wind rows come from the transmission-line solver the bore ladder built: a sweep of the input impedance, the peaks found with their half-power points, and the Q of the peak the instrument’s written middle register sits on. The peak is chosen by frequency rather than by index, because a peak-finder that gains or loses a pedal at the bottom of a sweep renumbers everything above it.

The bowed rows use Schelleng’s window with each string’s own characteristic impedance, exactly as the string’s own impedance set it up, and place the player’s force at the geometric centre of the window that string gives at a bow-bridge fraction of 0.09.

The impulsive rows are the contact-time model of the sixth rung of the excitation ladder and the corner-transit time of a plucked triangle.

Two quantities are asserted rather than computed. The fraction of the steady amplitude counted as speaking is nine tenths, which is the usual convention for a rise time and has no measurement behind it — it enters as one multiplicative constant, so every ratio in the census is independent of it. And the bow’s ramp is eighty milliseconds, which is a plausible détaché and moves every capture time in proportion.

Where the model stops #

A flute plays at an impedance minimum, not a maximum. Every wind row here is a pressure-controlled valve — a reed or a pair of lips — which plays where the bore’s input impedance peaks. A flute is a flow-controlled instrument driven at a pressure node, so its Qs would have to be read off the minima of the same sweep, and it is left out rather than reported with the wrong sign.

The peaks’ Qs are sensitive to how finely the sweep is sampled, at about five per cent between a sixteen-hundred-point sweep and a six-thousand-point one. The orderings in the census are robust to that and the third digit of any Q is not.

Nothing here is a measurement of an instrument. These are bores of stated shapes, solved. A real trumpet has a mouthpiece whose cup and throat are geometry rather than an added length, valve tubing with corners in it, and a player whose lips are the fourth term. The instrument is not the model is the standing caution and it applies with force here, because a Q is exactly the quantity that losses decide and the model’s losses are Benade’s approximation for a smooth tube.

And a real onset is not one exponential. A note that has settled to nine tenths of its steady amplitude is still changing, and the spectrum settles at a different rate from the amplitude, because each partial has its own Q. Whether a listener hears a note as having arrived is a question about that spectrum and not about the total.

What the picture cannot show #

It cannot show the player. Everything above assumes the driver is switched on abruptly at full strength, and no player does that. A wind player’s tongue release, a brass player’s attack and a string player’s bow ramp are all shapes rather than steps, and a gentle onset is slower than the resonator by as much as the player chooses.

Nor can it show what a listener does with the difference. A dissonance has to last prices how long a sensory quantity must persist before it registers, and something similar must be true of a transient — but the perceptual account of onset in this collection is about when a note is placed rather than about how much of its beginning is heard as beginning.

And it cannot say that these times are audible as times. A twenty-millisecond difference between two instruments is comfortably above what an ensemble has to correct for. A difference of eight periods against twenty-seven is a claim about colour that has no listening evidence here at all.

Whose instruments, and when #

The bores are modern orchestral shapes, solved at modern dimensions. The trumpet is a cylinder with a Bessel flare over the last third, which is what a trumpet has been since the valve instrument settled in the middle of the nineteenth century; the horn is a longer and more gradual version of the same idea; the clarinet and the saxophone are a cylinder and a cone of the usual lengths.

The violin numbers are for a modern setup at a 325-millimetre stop with steel-wound strings at the tensions a shop fits. A gut-strung baroque violin has a lower characteristic impedance on every string and therefore a wider window and a faster capture, which is a prediction this collection cannot check and which fits what players of both say about how the two instruments start.

The piano hammer is a modern felt one. A fortepiano’s leather hammer is lighter and harder and would sit between the modern piano and the dulcimer, which is the direction the excitation ladder is already pointing.

Where this ladder goes next #

One rung. There is a time between a note being asked for and a note arriving; it is a settling time for a wind instrument, a capture time for a bowed string and no time at all for a struck one; and the number of periods and the number of milliseconds are two different orderings of the same twelve instruments.

What the ladder owes next is the compass. Every row above is one note of each instrument, chosen as its written middle, and both the Q and the frequency change as a player goes up — so the settling time has a shape over the instrument’s range that a single row cannot show. The impedance sweep has every peak already, so the shape is available for nothing, and the interesting question is whether the two units keep disagreeing when the instrument is held fixed and only the note moves.