Every essay — page 18
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Timbre and acoustics
What makes a clarinet a clarinet, and what a room does to it before it reaches an ear.
The room is the slower of the two
A reverberant field is the source convolved with the room, so a partial's tail falls at the slower of the two rates rather than at their sum — and the room is slower for exactly the partials the string is losing fastest. Half a note's colour is gone in 0.163 seconds in no room at all, 0.313 in a concert hall and 1.441 in a stone church. The destination is identical in all three, because a room cannot hold a partial up above the fundamental it is also holding. What a hall takes away is the rate, and the rate was the whole of the identity cue.
A damper changes the clock, not the colour
A damper is an extra loss on the string rather than a second decay, so it adds the same number of nepers a second to every partial — and adding a constant to every rate leaves every difference between rates exactly where it was. The damped spectrum at any instant is the ringing spectrum at that instant shifted bodily down, to machine precision. The colour goes on draining at its own rate; the note simply runs out of seconds, and how many it gets is written on the page as a note value and a tempo.
One note in the compass loses its pizzicato
Dry, how long a pluck keeps the composite spectrum barely depends on which note it plays: three-hundredths of a second at the worst pitch and seven at the best, a spread of three. In a concert hall the same eight notes spread by a factor of forty-three, and in a stone church one of them never gets the note at all. The room does not scale the dry answer by a constant — it multiplies it by between four and nine times depending on the pitch, and at the one note where the two instruments' spectra nearly coincide it makes the pluck's position worse instead of better.
A damper cannot reach into the room
The essay before this one proposed the arithmetic for a damped note in a hall: take the slower of the string's rate and the room's, then add the damper's to whichever won. The composition is wrong, and it is wrong in the one place that decides the answer. A damper is a loss on the string, so it belongs inside the minimum where a room can overrule it — and past about three seconds of reverberation it is overruled on every partial, so the damper removes no audible seconds of note at all.
A seventh chord cannot be spaced to last like a triad
A struck triad keeps the partial coincidences of all its intervals for at most 1.26 seconds, whichever way it is spaced. Run the same census over every inversion and spacing of five kinds of seventh chord and none gets near: the dominant, minor and half-diminished sevenths top out at about two thirds of a second, the major seventh at 0.42. The diminished seventh, which theory calls the least stable of them, lasts longest at 0.86 — because it is the only one with no tone or semitone among its pitch-class distances, and the worst distance a chord contains sets a ceiling no spacing can lift.
Only the player hears a staccato end
A damper stops a string in a seventh of a second, and in a hall the room goes on for two. A listener hears both, mixed in proportion to how close they sit, and the question was at what distance the short part stops mattering. The answer is closer than any seat. A damped note's twenty-decibel fall has doubled in length by a seventh of a hall's critical distance — 77 centimetres in a two-second concert hall — and by a quarter of it in a jazz club. The end of a staccato is something the pianist hears and the front row does not.
The seventh has two places to be
Tune a dominant seventh's seventh as the seventh partial and its coincidences last 0.87 seconds instead of 0.66 — exactly the diminished seventh's figure, which is the highest any four-note chord in this census reaches. It gets there by levelling rather than by improving: the root's pair with the seventh gains a third of a second and the fifth's loses one, and all three pairs that contain the seventh arrive at the same number. The route is a valley. A seventh tuned six cents flat has lost the ninth-partial coincidence and not yet found the seventh, and lasts 0.46 seconds, which is worse than leaving it alone. And the same retuning halves the minor and half-diminished sevenths.
A doubling's worst note belongs to one instrument, not to the pair
The essay before this one found one pitch at which a violin and a flue pipe nearly share a spectrum and asked which pitch that is for every pair. Run over six radiators, thirty ordered pairs and forty-nine semitones, the answer is that seventeen pairs choose a worst note and thirteen simply run out of compass — and that the seventeen do not choose thirty different notes. They cluster in fours and threes, and each cluster is one instrument's own filter: the violin's worst note is B flat 4, its strongest body resonance; the clarinet's is F sharp 6, its cutoff; the voice's is C sharp 6, where its second partial meets its second formant. A doubling fails at a note that belongs to one of its players.
Half the worst notes of a doubling are notes nobody can play
A map of where two instruments' spectra come closest found seventeen pairs with a worst note of their own, gathered into four clusters, each belonging to one instrument's filter. It drew every instrument over the same five octaves. Read inside the compass each pair can actually play, eight of the seventeen worst notes are out of reach — the voice's C♯6 lies a semitone above a soprano's top C, and its E3 below her lowest note — and two pairs that seemed only to run downhill turn out to have a worst note after all. Of the four clusters, only the violin's B♭4 is advice every pair in it can use.
A pointing source's staccato ends dark, not bright
A damped note's release is heard as the player plays it only close to the instrument: a little way out, the room's copy takes over and the staccato is gone. A source that points, as a grand piano with its lid raised does, beams its upper partials along its axis, so their direct sound reaches further, and the prediction was that the release of a note's colour would survive into the hall where the release of the note does not — heard as a brief brightening before a room-length tail. Beaming does carry the second to fifth partials' articulation beyond the fundamental's. The highest never catch up, the whole note gains a fifth of its reach, and the release is heard not as a brightening but as a darkening.
Four notes cannot get below the seventh partial
Tune a dominant seventh as 4 : 5 : 6 : 7 — every note moved, not only the seventh — and its coincidences last 0.868 seconds, which is the one-note retuning's figure to the millisecond. Tuning the third and fifth buys nothing at the top and costs seven voicings a fifth of their life. Every other chord of four notes a harmonic series holds does worse, and the reason is not temperament: no four pitch classes avoid a semitone, a tone and a tritone together, and none of those three can be tuned to coincide below the seventh partial. The half-diminished seventh, which the one-note retuning ruined, reaches the same ceiling read down the series.
A lid darkens a staccato only inside a narrowing cone
Along its axis, a raised lid makes a damped note's release darker, because the brightness it beams at the listener is the part the damper stops. Most of an audience does not sit on the axis. Off it, the same beaming does the reverse, and a little way off it does the reverse more strongly than no lid at all would. The region where the release darkens is a cone whose half-width obeys one law — the fundamental times the radiator's radius times the sine of the angle is 23.6 hertz-metres — so it narrows from 28 degrees at C3 to 8 at C5.
A violin's worst doubling note is wherever its own body resonates
A pizzicato violin doubled by a held clarinet, oboe or trumpet has one note in the shared compass at which its pluck comes nearest to never owning the note, and for the modelled violin all three partners choose B♭4. That is where the violin's lower corpus mode sits, and real violins place that mode a semitone or two either way. Move it and the worst note moves with it, a semitone for a semitone, at 36 of 39 settings — so every violin has its own worst doubling note, and it can be found by knowing where the instrument resonates. Held rather than plucked, the violin's body decides nothing.
A section of violins fills in the note one violin gives away
A pizzicato violin doubled by a held clarinet has one note — the note whose fundamental sits on its own lower corpus resonance — at which the pluck never owns the sound, and real violins put that resonance a semitone or two either way. A section plucks with a dozen different bodies at once. Summed, their resonances smear into a broad shelf, and the section's worst note over the clarinet rises from 2.2 decibels below zero to about zero: the note one violin gives away, twelve nearly keep. Over an oboe the section owns every note. And if every member shares the same air resonance, that shared resonance becomes the section's worst note instead — a section is weakest where its instruments agree, not where any one of them is extreme.
A tuned chord's flutter binds it only if its notes start together
A close seventh chord tuned 4 : 5 : 6 : 7 low on the keyboard fluctuates at one rate in every auditory filter, and one rate everywhere looked like the ear's evidence that a sound is one thing. It is only half of that evidence. The filters also have to rise and fall at the same moments, and whether they do depends on when each note starts — which tuning does not decide. Spread the notes' onsets by two milliseconds and the tuned chord's filters move together no more than a tempered chord's.
In a hall a section of violins loses the notes it saved
Dry, twelve violins with different bodies fill in the note one violin's pizzicato gives away to a held clarinet, leaving a shallow shelf a few tenths of a decibel above zero. In a hall that shelf goes under all at once. The room takes the same few decibels from each note whether it comes from one violin or twelve, so a section's smoothing survives only to the front of the hall — about two metres over a clarinet, six over an oboe — and from ten metres back a section loses exactly as many notes as one violin does.
A tuned chord stays in step for half a cent-second
A close seventh chord tuned 4 : 5 : 6 : 7 low on the keyboard fluctuates in every auditory filter at once, but only if its notes start within a millisecond of each other — and a tuned chord, it was said, keeps its phases for as long as it sounds. It keeps them only as long as it stays exactly tuned. Mistune its notes by a fraction of a cent and they slide against each other, and the chord drifts out of step in a time inversely proportional to the mistuning: about half a cent-second. A tenth of a cent holds it for four seconds, a cent for half a second. A chord that starts out of step never drifts into it.
A shared vibrato cannot hold a tuned chord together
A tuned close seventh chord binds its auditory filters together only while its notes stay within a third of a millisecond of each other, and mistuned by a fraction of a cent it drifts out of step in half a cent-second. A vibrato shared by every voice looked like a way to hold it: if all four notes swing together, they slide together. They do not stop sliding. A perfectly locked vibrato leaves the drift exactly as fast as before, because it changes every note's frequency by the same factor and leaves their differences alone; a vibrato locked ninety-eight per cent takes an exactly tuned chord out of step in six seconds, and one locked ninety per cent in one.
A chord comes back into step too briefly to bind
A vibrato that is shared imperfectly does not carry a tuned chord's notes ever further apart: each note's time is warped back and forth, and the chord passes back through step twice in every swing. Locked ninety-seven or ninety-eight per cent, it does so about ten times a second. But each return lasts on average less than one cycle of the chord's own flutter, which is the rhythm the ear's filters would have to share to show the binding — fewer than one return a second lasts that long. The chord is in step often and never for long enough.
Two octaves up, a sung chord can bind at every swing
A vibrato's swing between two notes is the same number of milliseconds in every register; the tolerance for being in step halves each octave, and so does the period of the chord's own flutter. The first two make a scaling law — an octave higher is the same chord with its vibratos disagreeing half as much — and the third breaks the tie. Matched for time in step, the higher chord's returns carry twice as many flutter cycles: over C2 fewer than one return a second is long enough to bind, over C4 with the vibratos locked ninety-nine and a half per cent nearly every return is.
A narrow vibrato needs no lock
A tuned close seventh binds and releases at the vibrato's swings only where its notes' timing swings by about the width of its onset tolerance, and the swing is the vibrato's width times how far the singers' vibratos disagree. Down to a ninety-five per cent lock the two trade one for one: a 71-cent vibrato locked ninety-nine and a half per cent is the same chord as a 7-cent one locked ninety-five. Below that the trade fails, in the singers' favour. With no lock at all the swing stops growing with the disagreement, and a vibrato of three cents over middle C — twelve over the bass — binds the chord at most of its swings with each singer's vibrato its own.
A straight tone drifts out of step and stays there
A tuned close seventh binds — its filters flutter together — only while its notes keep their timing within a fraction of a millisecond, and a narrow vibrato keeps them there because a sinusoid's timing offset swings inside a fixed range. A straight-tone singer's pitch is never quite straight: it wanders slowly, a cent or less, at random. That wander's timing offset is a random walk. Over C4 a wander of half a cent takes the chord out of step within a fifth of a second and leaves it in step four per cent of the time, where a three-cent vibrato holds it nearly half. What matters is the wander's slow part, and adding a vibrato on top never helps.
A section keeps its pizzicato only inside a cone
In a hall a section's pizzicato loses its corpus notes to a held clarinet within two metres, on the assumption that a violin radiates equally in every direction. It does not, and twelve players sharing a beam cannot average it away. Directly on the players' axis, the same section keeps every note for twenty-five metres. Thirty degrees off, with the beam's upper partials thrown elsewhere, it is worse than if it had never pointed. A question about how far back a pizzicato carries turns out to be a question about which way the listener's seat faces.
A wall puts antiphonal seconds back beside the firsts
Second violins seated opposite the firsts have their plates turned away from the audience, and in a hall a section's pizzicato depends on its beam. With nothing behind them they keep their worst note in about one seat in a hundred. A perfectly hard wall two metres behind gives them back a cone as large as the firsts' own, but a wall that loses three decibels takes most of it away again. And the seats that cone serves turn out to be, within a seat or two, the seats the same section would serve sitting beside the firsts.