A note starts twice

The hardest place is also the latest finished this anchor’s third rung by naming what the whole of it had left out:

Everything in it is a duration measured at the instrument, and a duration only matters if somebody can hear it — and this collection has a threshold for exactly that, on a different ladder.

There are two accounts of when a note starts here and they were built four phases apart for different purposes. This anchor’s is physical: a wind instrument’s resonance takes Q over π f seconds to establish, a bow takes a computable time to reach a force inside its window, a struck string has no build at all. The perceptual-centre ladder’s is perceptual: a note is heard after it starts, by an amount read off the shape of its measured amplitude envelope against a level criterion.

Both come out in milliseconds. Neither has ever been given the other’s numbers.

Nothing is on the diagonal #

The pairing is by hand, because the two tables were written for different purposes and name their rows differently — there is no mechanical way to know that “a violin’s D4 string” and “bowed violin” are the same object, or that an alto saxophone has no partner at all. Five instruments appear in both.

The ratio of the perceptual number to the physical one runs:

A factor of sixty-nine, and it sorts perfectly by mechanism. Around 6.5 for an exciter that lets go; 2.2 for a bow that captures; around 0.12 for a resonance that builds.

That is not scatter. The three groups are the three mechanisms the first rung of this anchor separated, and each one has its own constant relating the two accounts. So the question the debt asked — are these the same quantity or two quantities sharing a word — has an answer with structure in it: they are one quantity for one mechanism and two for the others, and which is which is decidable.

The winds: the physics is slower than the measurement, and it should be #

Take the blown instruments first, where the ratio is about an eighth.

The impedance-peak calculation says a trumpet’s written middle takes 62 milliseconds to reach nine-tenths of its steady amplitude, and a clarinet’s chalumeau 148. The measured 10-to-90 per cent attack times for the same instruments are 30 and 45.

For an exponential build those two are almost the same quantity — a 10-to-90 rise is 2.197 time constants and a 90 per cent settling is 2.303 — so this is not a criterion problem. The model really does predict attacks two to three times longer than the ones anybody has measured.

There is a reason and it is not a defect in either. The measured attack times are of notes played by players, and a player articulates. A tongued attack delivers a pressure transient into the bore rather than a step from rest, which is exactly the thing that gets a resonance up faster than its own time constant — the same principle as a hard-driven filter. The physical model computes the free response of a bore to being switched on, which is what an organ pipe with an electric valve does and not what a trumpeter does.

That is testable and the test exists: an organ flue pipe is the one blown instrument in the perceptual table that has no player between the wind and the pipe, and it is the slowest of the blown family at 75 milliseconds — half again the flute’s, twice the trumpet’s, and the closest of the four to what the physics predicts.

The struck: the measurement is slower, and that is the room #

The other end of the table is the reverse. A plucked string’s physical onset is the time the released corner takes to reach the bridge — 0.2 milliseconds — and the measured envelope takes five to rise.

The string is not what is being measured. A piano’s amplitude envelope is the radiated one, and between the string and the microphone are the bridge, the soundboard and a few metres of room. A soundboard is a mechanical resonator with its own settling time, three strings are a coupled system with a beat in it, and the room’s first reflections arrive in the first few milliseconds.

So the physical model is measuring the string and the perceptual one is measuring the instrument. Both are right about their own object, and the factor of six between them is the soundboard and the room — which is a number this collection has never had and now has, though not to two figures.

The violin, which is the one that matters #

Everything above is a factor. The bowed string is a difference in kind.

The capture model computes when the bow first reaches a force inside Schelleng’s window, which is when periodic Helmholtz motion becomes possible. On a violin’s D string with an ordinary détaché that is thirteen milliseconds.

At that instant the string is periodic and nearly silent. The amplitude of Helmholtz motion is set by the bow’s speed, and the bow is still accelerating — it is a fifth of the way to its playing speed when the note catches. The measured envelope takes ninety milliseconds to rise, because the bow takes about that long to get there, and the listener places the note twenty-nine milliseconds into that.

So a bowed note has two starts and they are separated by a factor of three. On a wind instrument they are the same moment, because a resonance that has become periodic is a resonance that has amplitude — the two are the same exponential. On a struck string they are the same moment because both are essentially zero. On a bowed string the mechanism decouples them, and the two ladders have been measuring one each without either noticing.

Which is why the violin changes places #

Order the five by how long they take to settle and the violin is third, between the piano and the trumpet. Order them by when they are heard and it is last, behind both winds.

Three of the five change place between the two orderings and the violin moves furthest. It is the instrument on which the two accounts disagree not about a constant but about which end of the list it belongs at.

That has a consequence for the thing this anchor is ultimately about, which is ensembles. Which notes have to be played early computes required leads from the perceptual numbers, and gives the bowed violin the longest lead in a mixed scoring. If a conductor or a player reasoned instead from the physics — from how long the instrument takes to speak — they would put the violin in the middle of the pack and the winds at the back.

The perceptual account is the right one for that purpose, and this rung’s contribution is to say why: a lead is about when a note is heard, and the physical settling time on a bowed instrument is not a statement about that at all.

The reverse holds for the other question this anchor asks. When a player asks how fast a passage can be articulated, the quantity is when the note becomes periodic — a repeated note whose Helmholtz motion has been established is a note that can be stopped and started again, whatever its amplitude is doing. So the physical number is the right one for agility and the perceptual one for placement, and the violin is the instrument where choosing wrongly costs an order of magnitude. That is exactly what string players are describing when they say the instrument responds instantly and speaks slowly, which sounds like a contradiction and is two numbers.

Neither of those two orderings is the one a table of measured attack times gives, which is the third list and the one an engineer would reach for first. It is worth seeing beside them, because the disagreement between the three is the whole of this essay’s claim.

The ranges are the part worth carrying out of this essay. A bar chart of nine instruments invites the reading that the order is a property of the instruments, and for the two rows a player has most control over it is not: the violin’s range overlaps most of the table, which is the same fact the section above reached from the physics and is here reached from the measurements.

Three quantities, and the collection now has all of them #

It is worth counting what this anchor has ended up holding, because it is more than it set out to build.

A note’s onset can now be given three numbers here. In periods of its own pitch, which is a resonance’s Q and has no pitch in it. In milliseconds at the instrument, which is that count divided by the frequency. And in milliseconds at the listener, which is a level crossing on a measured envelope. The first two are related by one division and the first rung drew the crossings that produces. The third is related to the second by a constant that depends on the mechanism, and this rung is that.

None of the three orderings is the same as either of the others. A clarinet is the slowest instrument in milliseconds at the bore and the second fastest in periods; a violin is nearly the fastest at the instrument and the slowest at the listener. There is no such thing as a slow instrument, and there are three well-defined ways of being one.

That is the sort of result that looks like a caveat and is not. Every claim in this collection that some instrument is slow to speak now has to say which of the three it means, and the three ladders that make such claims — this one, the perceptual-centre one and the orchestration one — mean different ones.

A single row per instrument therefore conceals two different things at once — the range a player controls, which the figure above draws, and the register floor, which nobody controls.

What the pairing costs, and it is a real cost #

The five rows are five hand-made identifications and every one of them can be argued with.

The physical trumpet is a transmission-line solve of a trumpet-shaped bore at its written middle, and the perceptual trumpet is a spread of measured attacks from 15 to 60 milliseconds across dynamics and articulations. Those bands overlap the disagreement: at the fast end of the measured range the ratio is 0.24 and at the slow end 0.96, which is to say a hard-tongued trumpet note is three times faster than the physics and a gentle one agrees with it.

The bars on the figure are those spreads and they are wider than most of the differences between families. A claim resting on one instrument being a factor of two from the diagonal would be a claim this pairing cannot support. What it supports is the pattern: three mechanisms, three constants, one of them the wrong side of unity.

Which computation produced the numbers #

The physical side is the first rung’s census, unchanged. A wind instrument’s settling is Q over π f, with Q from an impedance-peak sweep of that instrument’s own bore, and the peak chosen by frequency rather than by index. A bowed string’s is the bow reaching the geometric centre of Schelleng’s window during an 80-millisecond ramp. An impulsively excited string’s is its exciter’s own contact time.

The perceptual side is the fifth rung of the perceptual-centre ladder, unchanged. Each instrument’s measured 10-to-90 amplitude rise, taken as an exponential approach, crossed against a criterion at six decibels below the peak — Vos and Rasch’s, which is the middle of the three the ladder carries.

The ratio is one divided by the other and nothing else. Both are in milliseconds and neither has been rescaled to fit.

The bowed-note figure superimposes the two: the bow’s speed as a linear ramp over 80 milliseconds, the note’s amplitude as an exponential of 90-millisecond rise beginning at the capture, and the placing lag measured from the capture rather than from the bow’s arrival.

Where the model stops #

The pairing is five rows, by hand. Two of them — the plucked string and the piano — are the same mechanism, so the effective sample is three mechanisms and two of them have one instrument each.

The physical settling model is a free response. It has no articulation in it and no player, which is the whole of the wind disagreement above. Putting a tongued transient into a transmission-line solve is a calculation this collection could do and would need a model of the tongue.

The perceptual criterion is one of three. At the detection threshold, fifteen decibels below the peak, every lag here shrinks by about a third and every ratio with it; at Gordon’s ninety-per-cent criterion they roughly double. The ordering survives all three, because the criterion multiplies rather than reorders.

And the exponential is an assumption on both sides. A real onset is not one exponential — a bowed attack has a period or two of irregular slipping, a wind attack has a chiff, and the measured 10-to-90 time is a summary of a shape rather than a parameter of one.

What the picture cannot show #

It cannot show a listener hearing two instruments. Every number here is one note in isolation, and the quantity a conductor cares about is a difference between two, which is the fifth rung’s map rather than this one.

Nor can it show what a player does. A violinist starting a note late by design and a violinist whose instrument is slow are indistinguishable in a recording, and the whole of this anchor is about the second while every measurement of ensemble timing is about the sum.

It cannot show the register. Both accounts vary up the compass — the physical one monotonically and in opposite directions in its two units, the perceptual one through the pitch term the fourth rung of that ladder added — and this figure takes one note from each instrument.

It cannot show the dynamic. Both accounts have a level in them and neither varies it here: the perceptual lag moves with loudness under an absolute criterion, and a wind instrument blown harder reaches its resonance sooner. The two move the same way, which is the one place on this page where they might be expected to agree, and nobody has checked.

And it cannot show the room. The factor of six on the struck instruments is attributed above to the soundboard and the room, on the grounds that nothing else is there. That is an argument from elimination and not a measurement of either.

Whose instruments, and when #

The physical bores are the modern ones this collection has used since the shape ladder: a B♭ trumpet, an F horn, a tenor trombone, a clarinet and an alto saxophone. The perceptual attack times are twentieth-century laboratory measurements of ordinary orchestral playing, and their ranges are what different articulations produce rather than measurement error.

The organ pipe is the historically interesting row, and it is the one that supports the reading above. An organ has no articulation: the pallet opens and the pipe speaks in its own time, which is why organ builders voice the speech of a pipe rather than leaving it to the player, and why organ attack times are the longest in the table. Everything a trumpeter does with the tongue, an organ builder has to do with a languid and a nick, once, in a workshop.

Where this ladder goes next #

Four 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; and now the instrument’s number set against the listener’s, which agree about the ordering of everything except the family this anchor was opened to describe.

What the anchor owes now is the articulation. 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. Running one into the front of the settling calculation would say how much of a note’s speaking time a player can buy with the tongue, which is the quantity every wind teacher is describing and nobody here has priced.