A pointing source's staccato ends dark, not bright

Only the player hears a staccato end gave a listener in a hall the direct sound of a damped note as well as the room’s copy of it, mixed by the textbook rule: equal at the critical distance, the direct sound stronger nearer and the room stronger further away. The damper stops the direct sound almost at once. The room’s copy goes on at the room’s own rate. So the time a damped note takes to fall twenty decibels after its release — the articulation a listener is given — has doubled within a fifth of the critical distance, and in the audience the release belongs to the hall.

That essay ended on the one assumption it had made about the source: that it radiates equally in every direction. A grand piano with its lid raised does not. It sends more of its sound toward the audience than toward the ceiling, and the critical distance along its axis grows with the square root of how much more. An instrument points had already computed that directivity band by band, and it is strongest in exactly the upper partials that carry the brightness of a note. The prediction was that the articulation of a note’s upper partials would survive to two or three times the distance its fundamental’s does, so that a staccato in the hall would be heard as a brief brightening followed by the room’s tail.

Both halves of the prediction are about the right quantity, and both turn out the other way round. Beaming carries some partials’ release further and leaves the highest behind, and the release it delivers is a darkening.

How much a radiator points #

The directivity of a radiating surface is a matter of its size against the wavelength. A source small against the wavelength radiates evenly; one comparable to it beams, and the crossover is where the circumference of the radiating area equals a wavelength. The law used here is the one the instrument essays drew for a circular piston: along its axis the direct sound is stronger by a factor of one plus half the square of that circumference-to-wavelength ratio.

For a radiator of thirty-five centimetres’ radius — a rough stand-in for a grand piano’s soundboard and lid — the fundamental of C3 is beamed by a factor of only 1.35. The fourth partial, at 523 hertz, is beamed by 6.63 and the eighth, at a kilohertz, by 23.5. Along the axis, then, the eighth partial’s critical distance is about four times the fundamental’s, and a listener at the fundamental’s critical distance is still well inside the eighth partial’s.

Each partial’s reach, pointing and not #

The earlier essay’s measure applies partial by partial: for each partial, the distance at which its own twenty-decibel fall after the release has taken twice as long as it does at the source. That is where a partial’s staccato stops being the player’s and starts being the room’s.

Radiating everywhere, the partials’ reaches fall steeply with partial number. The first three keep their release to about 0.14 of the critical distance; the fourth to 0.087; the eighth only to 0.013 — its release is the room’s almost at the keyboard. The reason is not the room but the string. The upper partials of a struck string die faster than the lower ones while the note is held, so by the time the damper comes down their direct sound has already fallen further than the room’s copy of them, which the room has been holding up. An upper partial arrives at its release with the room already comparatively loud.

Beamed at thirty-five centimetres, every partial reaches further, and each by roughly the square root of its directivity factor — the fourth by two and a half times, the eighth by five. But the upper partials start so far behind that multiplying their reach does not bring them past the fundamental. Partials two to five overtake it: the third reaches 0.283 of the critical distance against the fundamental’s 0.164. The sixth, seventh and eighth, beamed by factors of thirteen to twenty-four, still lose their articulation nearer the source than the fundamental does. No partial reaches the prediction’s factor of two or three over the fundamental: the third, the furthest, reaches 1.7 times as far.

The dial shows how the pattern depends on size. With a ten-centimetre radiator, closer to a violin’s than a piano’s, only the second and third partials pass the fundamental. At seventy centimetres, five do. The highest partials never catch up at any size drawn, because their deficit grows with partial number as fast as their beaming does.

The whole note #

A listener does not hear partials one at a time, and for the whole note the answer is dominated by the partial with the most power.

The whole note’s release reaches from 0.139 of the critical distance to 0.168 with a thirty-five-centimetre radiator, and to 0.208 at sixty — a gain of a fifth, and of a half at the largest size drawn. The fourth partial’s reach quadruples over the same range and the eighth’s grows eightfold, but neither carries enough of the note’s power to move the whole. On a string whose partials fall with number, the fundamental and the partials just above it are where the note’s energy is, and they are the ones a radiator barely beams.

So a raised lid buys the audience very little of the staccato itself. The note’s articulation still belongs to the room from a fifth of the critical distance outward, and along the axis the critical distance of the fundamental has grown by a sixth.

Two octaves up, the lid matters #

Everything so far is a note on C3, whose fundamental is too low for a thirty-five-centimetre radiator to beam. The beaming is a function of frequency, not of partial number, so a note two octaves higher starts with its fundamental where C3’s fourth partial was.

Radiating everywhere, a C5 keeps its articulation exactly as far out as a C3 does, partial by partial, because in this model a partial’s decay rate depends on its number and not its frequency, and so does the balance between its direct sound and the room’s at the release. Beamed, everything changes. The fundamental of C5 is beamed by a factor of 6.6 and reaches 0.362 of the critical distance, two and a half times its reach radiating everywhere; the whole note’s release doubles only at 0.378, against 0.139. The third partial, at 1,570 hertz, keeps its staccato out to 0.995 of the critical distance, almost to where the room and the direct sound are equal.

So the lid’s effect on a staccato is a treble effect. In the bass it gives the audience a sixth more of the critical distance; two octaves up, where the fundamental itself is beamed, it gives nearly three times as much, and in the upper partials of a treble note a listener on the axis at the critical distance still hears the damper. The darkening at the release is larger in the treble too: at three tenths of the critical distance the beamed C5’s centroid falls from 1.136 to 1.037 partials, nearly twice C3’s fall.

What is heard instead #

What the lid does change is the colour of the release, and in the opposite direction to the prediction.

From a source that radiates everywhere, a damped note gets brighter as it ends: at three tenths of the critical distance its centroid rises from 1.143 to 1.217 partials. The direct sound, whose upper partials have been dying fast on the string, stops; what remains is the room’s copy, in which the room has held the upper partials up; the tail is brighter than the note was. From a source that points, the same note gets darker as it ends, from 1.137 to 1.079. Beaming has put more of the upper partials into the direct sound, the direct sound is what the damper stops, and what remains in the room is the part of each partial that was not sent down the axis — relatively more of the fundamental.

The prediction said a raised lid would let the release of the colour reach the audience as a brief brightening. It does reach the audience, and the colour that reaches it is a dimming: the brightness is the part of the note that was travelling toward the listener, and it is the part that stops.

One partial, direct and reflected #

The fourth partial shows how the two fields add after the release.

At a fifth of the critical distance, radiating everywhere, the fourth partial’s direct sound is 96 per cent of its steady power. After the release that direct sound collapses within tens of milliseconds and the partial settles onto the room’s copy eleven or twelve decibels down. Beamed, the direct sound is 99 per cent of the partial’s power at the same seat, the room’s copy is correspondingly further below it, and the partial falls nearly eighteen decibels in fifty milliseconds before the room takes over. That is the extra reach the hero figure reports, and it is real: at this seat the fourth partial’s staccato is intact where, radiating everywhere, it would already be blurred.

The two effects, separated #

The essay’s two findings pull apart two things that the prediction treated as one. The reach of each partial’s articulation is a question of its direct share at a seat, and beaming raises every partial’s share. The colour of the release is a question of which partials lose most when the direct sound stops, and beaming makes the upper partials lose most. A lid that sharpens the upper partials’ release for a listener in its beam also makes the release, taken as a whole, a subtraction of brightness.

The surprising connection is with a note’s own decay. The note that gets duller as it dies found that a struck note darkens as it rings, because its upper partials decay fastest; a damper cannot reach into the room found that a room holds up the partials the damper would otherwise take, which is why an undirected note brightens at its release. A pointing source reverses the second effect and restores the first: its release sounds like a very quick version of the note dying on its own.

This also puts the prediction’s other half in its place. The brightening the prediction expected is what an undirected source does: a room keeps a pizzicato from giving its note away and its neighbours found the room holding up the partials a decay or a damper would take, and holding up upper partials is brightening. A pointing source adds a direct field that is brighter than the room and stops sooner, and the sum of the two is a release that dims.

The model and its one new term #

The note is the earlier essay’s: C3 on a string with a spectrum falling as one over the partial number, decay rates rising in proportion to partial number, held 0.4 seconds and damped by a felt with a sixty-decibel time of 0.15 seconds, in a room of two seconds’ reverberation. The listener’s power is the direct field, decaying at the string’s own rates while held and at string plus damper after, plus the reverberant field, decaying at the slower of the string’s and the room’s rates, mixed by the steady-state ratio at the seat. The one new term is the directivity: each partial’s direct share along the axis is computed with its own critical distance, the omnidirectional one times the square root of the piston’s directivity factor at that partial’s frequency. At a radius of nought the figures reproduce the earlier essay’s exactly. The centroid is power-weighted over the eight partials at each instant.

What a piston cannot be #

A piano is not a piston. A soundboard radiates from many modes at once and a raised lid reflects part of that toward the audience; the piston’s smooth rise of directivity with frequency is the simplest law that points the right way, and the numbers depend on the radius standing in for a real instrument.

The axis is one seat. Off the axis the upper partials are weaker, not stronger, and a listener to one side of a piano should hear the reverse of what is drawn here: a note whose direct sound is darker than an undirected one’s and whose release brightens even more.

The room is diffuse and even. The reverberant field is taken to be the same everywhere and the same for every direction of emission. A lid that sends the upper partials toward an absorbent audience rather than a reflective ceiling also changes what the room returns, and that is not modelled.

The damper is one felt. A piano’s dampers do not stop every partial at the same rate: a felt pressed on a string damps the low partials more slowly than the high ones, which would shift the colour of the release again, in the same direction as the beaming.

Brightness is a centroid. Whether a change in centroid of five hundredths of a partial is heard at the end of a staccato is a discrimination question the arithmetic cannot answer.

Whose staccato #

The case in point is the grand piano in a concert hall with its lid on the long stick, the instrument and room for which the question of what the audience hears of an articulation is most often asked — and, among other instruments, the brass, whose bells beam their upper partials far more sharply than a soundboard does. The room is the slower of the two is the rule under all of it: whatever the source does, a listener beyond the critical distance hears the slower of the note’s decay and the room’s. A lid moves the critical distance along the axis for the upper partials, not for the note.

The practice of playing with the lid on its short stick, or closed, for accompaniment is usually explained by balance: less sound toward the audience lets a singer or a string player through. The arithmetic here adds a second consequence. A lowered lid is a smaller effective radiator, and on the figures above it both shortens the reach of the treble’s articulation and softens the darkening at each release, so a pianist accompanying with the lid down is heard with blurrier staccatos and a brighter tail than with it up. Brass instruments sit at the far end of the same scale: the bell decides what gets out — a trumpet’s bell radiates what lies above its cutoff and beams it, so on this arithmetic a brass staccato heard from in front should keep its articulation further into a hall than a piano’s does, and end darker. A violin, whose radiator is its body and whose upper partials are shaped by the body that is the filter, is closer to the ten-centimetre end of the dial, where only a couple of partials outrun the fundamental.

Still open: the listener off the axis #

Everything above is drawn along the axis, where a directional source is at its strongest. An audience is not a line: most of it sits to either side of the lid’s axis, where the upper partials are weaker than from an undirected source and the release, on the arithmetic here, should brighten more than it would without the lid. The same beaming that darkens the release for a listener in front brightens it for a listener beside.

The computation is the one here with the piston’s off-axis response in place of its directivity factor — the function the instrument essays already evaluate at every angle — run across a row of seats. It would say whether the colour of a piano’s staccato is a property of where one sits, and how wide the region is in which a raised lid makes the articulation of a note darker rather than brighter.