An attack time is not an attack

There is a way of finding what a ladder has been assuming, and it is the closure criterion run backwards: look for the parameter every figure on it sets to the same value. On this anchor the answer is not a parameter anybody chose. It is one that was never offered.

Every heard moment on these nine rungs is a level crossing on a rising envelope. The level is discussed at length — three criteria, with three sources, drawn together on every figure so a reader can see how much of any claim survives the choice. The envelope is discussed nowhere. All nine families in the source table record the same curve; the map’s own arithmetic does not take a shape argument at all; and the word appears in exactly one figure call in the collection, where it names the default.

The assumption is that an instrument’s amplitude approaches its peak the way a resonator driven by a step does. That is a real physical model and it is right for some instruments. It is not what a published attack time says.

What a measured attack time actually fixes #

An attack time is a ten-to-ninety per cent rise: the interval between the instant the envelope reaches a tenth of its peak and the instant it reaches nine tenths. That is the convention the measurements are made under and it is the right one, because the two ends are robust to noise and to where a recording is judged to start.

It fixes one duration. It says nothing whatever about the shape of the curve joining those two points, and a great many curves join them.

Three are drawn above and they are not arbitrary. A resonator driven by a step approaches its peak as one minus a decaying exponential, which is what a pipe does when the air is turned on; that is the shape the whole ladder has used. A straight ramp is what an amplitude does when the excitation itself grows at a constant rate. A raised cosine is that ramp seen through a resonator, so it starts slowly, climbs, and eases in — which is what a driven system does when the drive is still increasing.

At the ten and ninety per cent marks the three are identical by construction. Between them they are nothing like each other, and this ladder does all of its work between them.

Which shape is right, and this collection already said #

The interesting part is that the answer is not open. It is written down on a neighbouring ladder, in that ladder’s own words, about the very instrument whose lead is the largest in the map.

Read at two different heights is explaining why the two accounts of an onset disagree about the order of the instruments, and its explanation is this:

A bowed string reaches Helmholtz motion early and then keeps growing in level, because the amplitude is set by the bow speed and the bow is still accelerating.

An amplitude set by a quantity that is still increasing is not a resonator settling toward a fixed value. It is a ramp. The same is true of a sung note, where the amplitude follows the breath, and it is why both of those families sit at the slow end of the attack table in the first place — a bowed or sung attack is long because the excitation takes time to arrive, not because a resonance takes time to build.

A blown pipe is the other case and the exponential is right for it: the reed or the lip edge reaches its regime quickly and the standing wave then builds under a drive that is not changing. A struck or plucked string is at its peak before the question arises.

So the honest reading is mixed, and it is the reading the collection’s own accounts of these instruments imply. The families this ladder gives the longest attacks to are exactly the families whose shape the ladder has wrong.

There is a way to falsify that inference and it does not need an instrument at all. A resonator driven by a step has a rise time set by its own decay time, so an exponential attack should scale with the sharpness of the resonance and with nothing else — which is the identity the settling ladder opens with, where the wait in periods is the quality factor and no more. A ramped attack has no such relation: its duration is set by the player, so it varies with the dynamic, the articulation and the individual, and it does not track the instrument’s resonance. The published ranges say which is happening. A bowed violin’s attack runs from forty milliseconds to a hundred and eighty across ordinary playing — a factor of four and a half on an instrument whose resonances do not move — and a marimba’s runs from two to six. A quantity that varies four-fold with how it is played is a property of the excitation, and an excitation with a duration of its own is a ramp rather than a step.

The ordering survives and nothing else does #

Before any map, it is worth being clear about what does not change, because it is the claim the anchor has actually leaned on.

Under every shape, the lag is a fixed multiple of the attack time. That is not obvious and it is checked here rather than assumed: it holds because each shape is a single curve rescaled in time, so reading it at a fixed fraction of its peak returns a fixed fraction of its duration.

The consequence is that an ordering cannot move. If a violin is heard after a trumpet under one shape it is heard after it under every shape, because both lags have been multiplied by the same number. Every claim on this anchor of the form this instrument is heard later than that one is safe, and so is every claim about which part of a scoring needs the largest lead.

What is not safe is any number in milliseconds, and this anchor is made of numbers in milliseconds. The multiple at six decibels below peak is 0.32 under the exponential and 0.63 under a ramp, so a violin’s lag is 28.5 milliseconds or 56.4 depending on a curve nobody recorded.

There is a second thing on that figure worth more than it looks. The three shapes converge near the top of the envelope and cross at 92 per cent of the peak — above the highest of the three criteria this ladder reads at, and essentially at the criterion the settling ladder uses. So the disagreement between the two ladders that the sixth rung of the onset anchor traced to their different reading heights has a second face: the height that ladder reads at is nearly blind to the shape, and the height this one reads at doubles with it. The ladder whose criterion was called arbitrary picked the robust one.

The map, redrawn four ways #

A single shape applied to everything multiplies the whole map by a constant: 31.5 milliseconds becomes 62.3 under a ramp and 84.4 under a cosine, and every part keeps its place. The mixed reading is not a constant, and under it the map changes its shape as well as its size.

The quiet low piano is the latest part under the exponential, at a lead of 31.5 milliseconds — not because a piano is slow but because its E1 has a hundred-millisecond floor put under it by its own period. Under the mixed reading the violin overtakes it, at 66.8 against 31.5, and the headline of that rung — the pitch floor can beat the instrument — is no longer what the four parts show.

That is a real change to a published result and it is worth being exact about its status. The mixed reading is an inference from what excites each family, not a measurement of any envelope. What the figure establishes is not that the violin is latest; it is that which part is latest is decided by a curve nobody has looked at, and that the collection’s own physics points the wrong way for the shape the ladder chose.

What it does to the constructed passage #

The seventh rung is where the assumption is actually spent, and it is spent on a comparison with a threshold.

That rung built a passage to distinguish two kinds of ensemble: a scoring changes mid-phrase, and an ensemble that has learnt the new map applies it while an ensemble that is listening has to find it. The whole verdict was that the two sound the same, because the maps differ by less than the twenty milliseconds a listener needs to notice an asynchrony. The measurement is easy and the perception is impossible, it said, and called that an unusual and useful place for a question to sit.

The verdict was a property of the curve. Under three of the four readings the constructed passage is audible, and under the one the collection’s own account of a bow implies it is audible by a factor of three.

That changes what the experiment is. The seventh rung asked for a close microphone on every part and a millisecond-accurate onset extraction, because eleven milliseconds is invisible to a listener and easy for a machine. If the honest figure is sixty, then the difference between an ensemble that has learnt the map and one that is finding it is something a person in the hall can hear for three or four beats — and the experiment needs a listener rather than a laboratory.

It also means the two hypotheses are not the quiet distinction that rung described. A feedforward ensemble handed a scoring it has not learnt would be audibly ragged for the first bar of it, which is a thing musicians describe constantly and which nothing in that rung’s arithmetic could account for.

A struck note is safe, and that is the useful half #

The size of the exposure is not the same everywhere, and where it is small it is very small.

That is worth stating as a rule, because it says which of the anchor’s results are exposed and which are not. The shape enters as a multiple of the attack time, so the milliseconds at stake are proportional to it. On a marimba at three milliseconds the three shapes disagree by under two milliseconds; on a sung vowel at a hundred and ten they disagree by fifty-nine. The same rule sorts the anchor’s own rungs: the ensemble spread of a gamelan, whose parts are all struck, is a number that cannot move, and the spread of a choir with an organ is a number that can double.

Every claim this ladder makes about struck and plucked instruments therefore survives intact, and those include the whole of the argument that a percussionist is the natural timekeeper of an ensemble. Every claim about bowed strings, voices and organ pipes carries a factor of two on it — and those are the claims the map is mostly made of, because a lead is a difference and the differences are dominated by the slow families.

Which computation produced the numbers #

Three envelope shapes, each defined by one curve and each normalised so that its ten-to-ninety per cent rise takes exactly the attack time the table reports. That normalisation is the load-bearing step and it is checked on every figure by reading the ten and ninety per cent crossings back off the drawn curve.

The heard moment is the first instant the envelope reaches the criterion, which for the exponential is a logarithm, for the ramp a proportion, and for the cosine an arc cosine. The lag as a multiple of the attack time is that quantity computed at a rise of one.

The map is the fifth rung’s own arithmetic — the larger of the family’s attack and the floor the note’s period imposes, shortened by the dynamic — with the final level crossing read off the stated shape rather than off the exponential. The mixed reading assigns the exponential to the struck and blown families and the raised cosine to the bowed and sung one.

The map change is the seventh rung’s quantity: two scorings, each map referred to its own earliest part, and the largest disagreement between corresponding parts.

Where the model stops #

The three shapes are a choice and not a census. An envelope could be anything monotone between the ten and ninety per cent marks, and these are three plausible ones rather than the range of what an instrument does.

No envelope was measured. Every figure here is arithmetic on published attack times. The shapes themselves are inferences from what drives each family, and the whole rung is a request for one measurement rather than a result.

A real attack is not monotone. A bowed onset has slipping and a chiff, a blown one has a transient before the regime settles, and a note starts twice on this site’s own account of what a bowed string does. None of these curves has a bump in it.

The mixed reading assigns one shape per family. Within the bowed family a martelé stroke and a dolce one are different mechanisms, and the second is a ramp where the first is very nearly a step.

And the criterion is still one of three. Everything above is computed at six decibels below peak, which is the middle of the band. At the detection criterion the shapes disagree by a factor of five and at the perceptual attack criterion by a third, so the exposure is a function of both constants at once.

What the picture cannot show #

It cannot show a level. Both constants multiply an attack time, so nothing here says anything about how loud a note is when it is first heard — which is what an absolute criterion is about and what the third rung turns on.

Nor can it show a listener. The shapes are physical curves; whether a listener places a note at a fixed fraction of any envelope is the standing assumption of the whole anchor and no figure here tests it.

It cannot show an ensemble. The map change above is a difference between two computed maps and not a spread of recorded onsets.

Nor can it show which instrument was measured. A published attack time is a mean over notes, players and dynamics, and a mean of curves of different shapes is not a curve of any shape.

It cannot show the room. A reverberant hall adds its own rise to every note, and a tail arriving during an attack lengthens it in a way that is neither of these shapes.

And it cannot show what the shapes do to a section. Fourteen players summed cross their own criterion at a moment that depends on the individual curve as well as on the scatter, and that arithmetic has been done for one shape only.

Whose instruments, and when #

The attack times are twentieth-century laboratory measurements of orchestral playing, reported as ten-to-ninety per cent rise times because that is the convention for a rise time in every field that has one.

The convention is the point. It was chosen so that a duration could be reported robustly, by people measuring rise times of amplifiers and filters, where the shape is known from the circuit and only the duration is in question. Carried into instrument acoustics it keeps the robustness and loses the guarantee: nobody knows the circuit, and a single number that was designed to be independent of the curve turns out to be all anybody wrote down.

That is not a criticism of the measurements. It is a statement about what a reader of them is entitled to, and this anchor has been claiming more than it was given for nine rungs.

The historical shape of the gap is ordinary and it recurs. The perceptual attack literature and the instrument-acoustics literature grew up separately, and a quantity that is a summary in one is a premise in the other: an attack time is a convenient descriptor of a note to somebody cataloguing timbres, and it is the entire input to a model of when a note is heard. The same thing happened to the criterion, which a neighbouring ladder found carrying an argument after ten placements had left it at one value. Two constants, both inherited, both quietly deciding a result.

Where this ladder goes next #

Nine rungs. A note is heard after it starts; an ensemble that mixes attack families carries a spread; the dynamic moves each attack; the pitch puts a floor under all of it; the three added for a scoring, as a map; the map found by an ensemble that was never told it; the passage that would separate a learnt map from a found one; the map scored for sections; and now the curve every one of them read the map off.

What is owed after this is one measurement, and it is smaller than anything the ladder has asked for before. The seventh rung wanted a constructed passage sight-read once by a chamber group with a microphone on every part. This wants the envelopes of ordinary orchestral notes — one bowed, one blown, one struck, at a stated dynamic — plotted rather than reduced to a number. Every study that reported an attack time had those curves on a screen and published the ten-to-ninety time from them. The recordings needed already exist in quantity, the analysis is an amplitude envelope and a curve fit, and the answer decides whether this anchor’s arithmetic is in milliseconds or in factors of two.