A lid darkens a staccato only inside a narrowing cone

A pointing source’s staccato ends dark, not bright put a listener on the axis of a raised piano lid and found the colour of a damped note’s release reversed. From a source that radiates evenly, a note brightens as it ends: the damper stops the direct sound, and what remains is the room’s copy, in which the upper partials have been held up — a damper cannot reach into the room. From a source that beams its upper partials at the listener, the note darkens as it ends, because the brightness was travelling toward the listener in the direct sound and the direct sound is what the damper stops.

That essay’s last section pointed at the obvious hole. An audience is not a line. A grand piano’s lid on the long stick sends its upper partials toward one part of the hall, and most of the seats are somewhere else. Off the axis a piston’s upper partials are weaker than an even source’s, and the same arithmetic that darkened the release in front should brighten it more than usual to the side.

It does, and the useful result is where the boundary lies. Around the axis there is a cone inside which the release darkens. Its half-width is not a matter of taste or of the room: across three radiator sizes and seven notes, it obeys

$$f_0 \cdot a \cdot \sin\theta = 23.6\ \text{hertz-metres}$$

to within ten per cent, wherever the fundamental $f_0$ times the radiator’s radius $a$ is at least 45 hertz-metres. For a 35-centimetre radiator that is 28 degrees at C3, 15 at C4 and 8 at C5. Outside the cone the release brightens, and a few degrees further out it brightens more than it would with the lid closed. On the floor plan below, that is 58 per cent of the seats.

The same note, the same room, only the direction changed #

The model is the earlier essay’s, with one term added. A note on a string whose spectrum falls as one over partial number, and whose partials decay at rates rising with their number, is held for 0.4 seconds and damped by a felt with a sixty-decibel time of 0.15 seconds, in a room of two seconds’ reverberation. A listener receives each partial’s direct sound and its reverberant copy, mixed by how far they sit from the source against the critical distance, where the two are equal — the mixture only the player hears a staccato end introduced. The brightness is the power-weighted centroid of the first eight partials, in partial numbers.

The directional source is a circular piston of 35 centimetres’ radius, a rough stand-in for a soundboard and raised lid. Along its axis each partial’s direct sound is stronger by the piston’s directivity factor, $1 + (ka)^2/2$, where $ka$ is the circumference of the piston in wavelengths at that partial’s frequency. The new term is the angle. Off the axis the direct sound is further multiplied by the square of the piston’s own pattern, $2J_1(x)/x$ with $x = ka \sin\theta$ — the function an instrument points drew as polar plots. The reverberant field does not move with the seat: it is the room’s, and every listener gets the same share of it. At zero degrees the calculation reproduces the earlier essay’s axis exactly.

The first figure is the result at three tenths of the critical distance. On the axis, C3 darkens by 0.058 partials over the 400 milliseconds after the release; the same note from an even source brightens by 0.075. Moving off the axis, the darkening shrinks slowly and then collapses. It is still 0.040 at twenty degrees, crosses zero at 27.6 degrees, and by 31.1 degrees the note brightens more than it would without the lid. At forty degrees it brightens by 0.342 partials, four and a half times as much as with no lid, and at ninety by 0.55.

C5 does the same thing in a much narrower range. It darkens by 0.099 on the axis, crosses at 7.6 degrees, and passes the no-lid brightening almost at once. Its curve then rises past a full partial between fifteen and thirty degrees before falling back. That rise is where the piston’s nulls come in, and the section on the nulls below explains why its size should not be trusted.

Which partials lose their direct sound first #

The mechanism shows up one partial at a time. At the seat drawn, three tenths of the critical distance, a source radiating evenly gives every partial 92 per cent of its steady power as direct sound. The piston gives each partial a different share, and the share depends on the angle.

The fundamental of C3, at 131 hertz, is hardly beamed at all. Its direct share is 94 per cent on the axis and 93 at ninety degrees, within two points of the even source’s. The eighth partial, at a kilohertz, is beamed hard: 100 per cent direct on the axis, but on the piston’s pattern it falls through a null at 35 degrees, where its direct share is nothing at all. It recovers into the side lobe and falls again to 17 per cent at ninety. The fourth partial holds on longer and slides to half by ninety degrees.

So off the axis, the upper partials reach the listener mostly through the room while the fundamental still arrives mostly direct. This is the opposite of the axis, and it gives the opposite result. The damper stops the direct sound, which here is the dark part of the note — the part a string keeps longest, as the note that gets duller as it dies found; what remains is the room’s copy, which carries the upper partials; and the room’s copy is brighter than the note the listener was hearing a moment before. The lid does not add brightness to the release. It takes brightness out of the direct sound at the side, so that the room’s copy, which is unchanged, is brighter by comparison.

One constant sets the cone’s width #

The boundary would be a curiosity if it depended on everything, and it depends on almost nothing. Running the same crossing at every note from G2 to G5 and for radiators of 20, 35 and 50 centimetres gives a set of angles that collapse onto one curve.

Twice the pitch halves the sine of the cone’s half-width, and so does twice the radiator. The constant is 23.6 hertz-metres, measured from every point where $f_0 \cdot a \ge 45$, and none of those points departs from it by more than ten per cent. The seat hardly matters either: at a full critical distance rather than three tenths, the crossings move by a degree or less.

It is no accident that the constant takes this form. The piston’s pattern is a function of $ka \sin\theta$ alone, and $ka$ is proportional to frequency times radius, so any boundary set by the pattern is a boundary in $f \cdot a \cdot \sin\theta$. What makes it a boundary in the fundamental rather than in some particular partial is that the spectrum is a fixed shape in partial numbers. The partials that carry the note’s brightness are always the same few multiples of the fundamental, so fixing where they fall on the pattern fixes $f_0 \cdot a \cdot \sin\theta$. In partial terms, the fourth partial of C3 at the boundary has $ka \sin\theta \approx 1.5$, which is well inside the piston’s main lobe and short of its first null. The width of the cone is therefore set by the main beam, not by the nulls. It is the most robust number in this essay.

Below 45 hertz-metres the law breaks, and it breaks in the direction that means the lid does not beam the note at all. With a 20-centimetre radiator, G2’s release darkens only within three degrees of the axis, and at lower products it does not darken even on the axis. For such a note the radiator is too small to beam anything that matters, so the note behaves nearly as an even source’s does and brightens as it ends everywhere.

To the side, the lid shortens the staccato too #

The earlier essay’s other measure was reach: how far from the source a listener can sit before a damped note’s twenty-decibel fall has taken twice as long as it does at the keyboard. That is where a staccato stops belonging to the player and starts belonging to the hall, which beyond it hears the slower of the two decays. On the axis the lid extended it, a little in the bass and a lot in the treble.

To the side the same trade reverses. C3’s release keeps its articulation to 0.168 of the critical distance on the axis against 0.139 with no lid, but the gain is gone by 47 degrees and at ninety it is 0.130, slightly worse than no lid. C5’s gain on the axis was large, 0.378. It falls to the no-lid value by 33 degrees and to 0.026 at ninety, a fifth of what an even source gives. A listener to the side of a treble passage hears its staccatos blurred by the room almost at the keyboard, because the direct sound that would have carried the articulation is going somewhere else.

For the whole note, the reach is dominated by the fundamental and the partials just above it, the ones carrying most of the power. That is why the reach boundary (47 degrees at C3) sits much wider than the colour boundary (28). The colour is decided by the upper partials, which are beamed narrowly; the reach is decided by the lower ones, which are beamed broadly.

The floor of a hall, seat by seat #

The angle and the distance together decide every seat, and a floor plan puts them together.

The darkening region is the wedge the law predicts, about 28 degrees either side of the axis, running out to the back of the floor. Just outside it lies a thin band of seats that hear a brightening smaller than no lid would give. The rest of the floor, 58 per cent of it, hears the release brighten by more than it would with the lid closed.

The wedge is narrow at C3 and would be narrower for a treble passage: at C5 its half-width is eight degrees, a strip one or two seats wide at the distances drawn. So the finding of the earlier essay, that a raised lid makes a staccato end darker, is true in a region that is a small minority of a hall and shrinks with every octave. Everywhere else the lid pushes the colour of a release the other way from where it pushes it on the axis, and past what the room alone would do.

The nulls and the size of the brightening #

Two things in these figures are the piston’s and not the piano’s, and one of them is large.

The width of the cone is the main beam’s, and it is trustworthy within the model. It sits short of every partial’s first null, as the section on the constant showed, so it depends only on how quickly the main lobe narrows with frequency. Any radiator of roughly that size narrows at roughly that rate, whether it is a piston or not.

The size of the brightening beyond the cone is the nulls’ as well, and a soundboard does not have them. A piston’s pattern has sharp nulls: the eighth partial of C3 has no direct sound at all at 35 degrees, and the fourth, eighth and higher partials of C5 pass through several nulls between five and thirty degrees. That is why C5’s curve on the first figure climbs past a full partial of brightening at 25 degrees and then falls back. A soundboard vibrates in many modes at once, with a lid reflecting part of its sound, and its pattern at any frequency is lobed but not cleanly nulled. The real brightening off the axis should be smoother than drawn and smaller at its peak. It still has to be a brightening, and larger than the no-lid value: the partials beyond the main beam are, on average over the angles outside it, weaker than an even source’s, and that average is what the room’s copy is compared against.

The dial on the last figure lets the release be watched at one seat as the angle changes.

What the model leaves out #

A piano’s radiation is not a piston’s. The piston is the simplest law that points upper partials harder than lower ones; the constant 23.6 hertz-metres belongs to it, and a measured instrument’s pattern would give another number. What should carry over is the form, a boundary in frequency times size times the sine of the angle, because any radiator whose beam narrows with frequency has one.

The lid points up as well as out. A grand’s raised lid reflects sound down and across toward one side of the hall, so its axis is not horizontal and not straight down the hall’s centre line. The angles here are from whatever that axis is. A seat on the axis’s own side of the hall is closer to it than a seat on the other side at the same distance from the stage.

The room is even, and one number. The reverberant field is the same everywhere and the same for every direction of emission. A lid that sends the treble at an absorbent audience and the bass at a reflective ceiling changes what the room returns partial by partial, and that would move the room’s copy, which is the half of every seat’s sound that this essay holds fixed.

The damper is one felt, and brightness is one centroid. Both are the earlier essay’s assumptions. A felt that damps the upper partials faster would push every release darker, shifting the cone’s edge outward a little; and whether a centroid change of a few hundredths of a partial is audible at the end of a staccato is a discrimination question the arithmetic does not answer. The changes outside the cone, tenths of a partial, are the ones most likely to be heard.

Other instruments with narrow beams #

The rule is not special to pianos. The bell decides what gets out of a brass instrument and beams it far more sharply than a soundboard does, so on this arithmetic a trumpet’s staccato darkens inside a very narrow cone straight ahead of the bell and brightens over the whole of the rest of the hall. A woodwind radiates from its holes as well as its bell, and where a woodwind actually sounds from found the radiating aperture growing with the number of open holes, which should widen its cone fingering by fingering. A violin’s body is small against its wavelengths across most of its compass, a filter rather than a beam, and should sit below the 45 hertz-metre line where the law breaks and releases brighten in every direction.

The practice this bears on is the one the earlier essay named: the lid on the long stick for solo playing, on the short stick or closed for accompaniment. The usual reason is balance. This adds a second consequence that depends on where one sits. On this model, a pianist playing with the lid up is heard with darker staccatos in a narrow wedge and brighter ones everywhere else, and lowering the lid, which shrinks the effective radiator, widens the wedge while weakening both effects.

Still open: a lid that is not a piston #

The one number here that stands in for a measurement is the radiator: a flat disc, 35 centimetres in radius, with a textbook pattern. Pianos have been measured — their directivity at each octave band around the instrument with the lid at each of its stops — and published patterns exist in the literature on room acoustics. Replacing the piston’s $2J_1(x)/x$ with such a measured pattern, octave by octave, would turn the constant 23.6 into a piano’s own number. It would also say whether the brightening outside the cone is the smooth rise this essay expects once the nulls are gone. The calculation would be the same; only the source’s pattern would change. Whether the cone keeps its law, a fixed product of pitch, size and the sine of an angle, is the question such a measurement would settle.