Only the player hears a staccato end

A damper cannot reach into the room found that a damper is a loss on the string and not on the hall. Pressed against the wire it makes the string stop feeding the room, and then the room decays at its own rate: in a two-second hall a damper removes 0.90 seconds of audible note, and past four and a half seconds of reverberation it removes nothing. The essay gave its listener the reverberant field and nothing else, and said that was the simplification with most in it.

A listener also receives the direct sound. It arrives first, it carries the string’s own rates, and under a damper it is the one part of the note that really does stop — in the model, its fundamental falls sixty decibels in 0.15 seconds. A damper cannot reach into the room predicted what would follow: that the distance at which the direct sound stops mattering would be uncomfortably short, and that a listener in the fourth row might hear an articulation nobody in the twentieth row hears.

The first half of that prediction is right, and it is shorter than uncomfortable. The second half is wrong, because the fourth row is already too far.

Two copies of one note #

A listener at a distance from a source receives two versions of it. The direct sound arrives along the straight line and its power falls as the square of the distance. The reverberant field has been round the room many times and is nearly the same everywhere. The critical distance is where the two are equal, and how far away the room takes over worked out that it is a few metres in most halls, because a room’s reverberant power is spread over its whole volume and absorbed at a rate set by its reverberation time. At a tenth of the critical distance the direct sound is a hundred times the reverberant power, twenty decibels above it; at the critical distance they are equal; at three times it the reverberant field is nine times the direct.

The flat line in that figure is exactly the reverberant copy in what follows, and nothing in it is changed. What is added is the other copy, which the earlier listener could not hear because they were placed, in effect, infinitely far away. That is why its numbers were about seconds of note removed and these are about milliseconds of release: the room’s copy has a long tail whatever the damper does, and the direct copy has a short one only because of it.

The two copies of a damped note decay differently. The direct sound carries the string’s rates while the key is held and the string’s rates plus the damper’s after it is released, so it falls at once. The reverberant copy, as the room is the slower of the two established and the damper essay confirmed, decays at whichever is slower, the string plus damper or the room — and in a hall the room is slower for every partial. So after the release, a listener’s level is the sum of a steep curve and a shallow one, weighted by where they sit.

The measure of what a listener is given is the time after the release for the whole note, summed over its partials, to fall a stated amount below its level at the release. Twenty decibels is a fall any listener hears as the note having gone; ten decibels is a clear fall but not an end. Both are drawn.

The fall doubles long before the room is equal #

At the source — where a pianist’s ears are, in effect — the note falls twenty decibels 49 milliseconds after the release. That is the damper’s work, the articulation a staccato mark asks for. Far enough away, where the listener hears only the room, the same fall takes 667 milliseconds, the time a two-second room takes to lose twenty decibels by the definition of how long a room rings. The question is how the one becomes the other.

It does not happen at the critical distance. It happens an order of magnitude earlier. At a twentieth of the critical distance the 20 dB fall takes 51 milliseconds; at a tenth, 60; at 0.15, 117; at a fifth, 198; at 0.3, 308. The fall has doubled by 0.14 of the critical distance. By the critical distance itself it takes 568 milliseconds, most of the way to the room’s value.

The reason is arithmetic about the depth of the fall. The damper takes the direct sound down quickly, but the room’s copy is still there at its own level beneath it, and a twenty-decibel fall can only be completed by the direct sound if the room’s copy starts more than twenty decibels down. It does so only while the listener is inside a tenth of the critical distance. Anywhere further, the direct sound drops to the room’s copy before it has fallen twenty decibels, and the rest of the fall happens at the room’s rate. For a ten-decibel fall the same boundary is at a third of the critical distance, and the ten-decibel fall doubles by 0.38.

Drawn as level against time, the mechanism is a bend. At a twentieth of the critical distance the level falls 19.8 decibels in the first fifty milliseconds and 28.7 in the first hundred before the room’s slope takes over. At a fifth of the critical distance, where the direct sound is still 96 per cent of the steady power, it falls 14.4 in fifty milliseconds and then only 2.7 more in the next fifty, because it has met the room’s copy. At the critical distance it falls 4.4 and bends. At three critical distances, where the direct sound is a tenth of the power, the first fifty milliseconds lose 1.9 decibels, and the note simply decays with the hall.

Where the boundary comes from #

The two doubling distances are close to two numbers that can be written down without any decay model at all. A fall of $X$ decibels can be finished by the direct sound alone only if the room’s copy starts at least $X$ decibels below it, and the room’s copy is $20\log_{10}(d/d_c)$ decibels below the direct sound at a distance $d$. So the direct sound can deliver the whole fall only inside $d/d_c = 10^{-X/20}$: 0.10 of the critical distance for a twenty-decibel fall and 0.32 for a ten-decibel one.

The computed doubling distances, 0.14 and 0.38 in a two-second room, sit just outside those boundaries, and the gap between them is set by the room. Just beyond the boundary the direct sound has done most of the fall and the room’s copy has only a few decibels left to take down; how long those few decibels take depends on how slowly the room decays. In a room of 0.8 seconds they are quick, and the release is not doubled until 0.24 critical distances. In a room of eight seconds they are slow, and the release is doubled at 0.08 — slightly inside the boundary, because even before the room’s copy is level with the end of the fall it is strong enough to slow the last decibels.

That gives the result a form that does not depend on the particulars of a piano. An articulation of $X$ decibels belongs to the player within about $10^{-X/20}$ of the critical distance, give or take a factor the reverberation time sets. For any fall a listener would call the end of a note, that is a small fraction of a distance that is itself only a few metres.

What that is in metres #

A critical distance is a property of a room, so the distances have to be read against real rooms to mean anything about seats.

In a concert hall of two seconds and 18,700 cubic metres, the critical distance for an omnidirectional source is 5.51 metres. The 20 dB fall has doubled by 77 centimetres and the 10 dB fall by 2.09 metres. In an opera house of 1.4 seconds and 12,000 cubic metres, 86 centimetres and 2.19 metres. In a large stone church of four seconds and 7,000 cubic metres — a small room with a long tail, so its critical distance is only 2.38 metres — 26 centimetres and 76. In a cathedral, 34 centimetres and a metre. Even in a jazz club, whose 0.8 seconds is short, the 20 dB fall has doubled by 43 centimetres.

The first row of a concert hall is several metres from a piano. A pianist’s ears are about three quarters of a metre from its strings. In every room built for music, the twenty-decibel staccato is heard by the player and by nobody else. The ten-decibel articulation reaches a little further, to a page-turner or a second pianist, and not to an audience.

That inverts the damper essay’s picture of the fourth row and the twentieth. A listener in the fourth row of a concert hall is perhaps eight metres from the stage, about one and a half critical distances, and hears the 20 dB fall take about 614 milliseconds; a listener in the twentieth row, at perhaps twenty-five metres, hears 661. Both hear the hall’s release, and the difference between them is fifty milliseconds out of six hundred. The seats that differ are the player’s and everyone else’s.

A shorter room brings it closer to itself #

The share of the critical distance at which the articulation is lost changes with the reverberation time, and in the direction the arithmetic predicts.

In a room of 0.8 seconds the 20 dB fall doubles by 0.24 critical distances, and in a room of eight seconds by 0.08. A shorter room’s copy decays faster, so the direct sound’s steep drop has to be followed by less of the room’s slope before the fall is complete, and the direct sound can be relatively weaker before the room takes the release over. A longer room works the other way. In metres the two effects partly cancel, because a shorter room of the same volume also has a longer critical distance; what is left is that the jazz club and the concert hall lose a staccato within similar distances of the source, both under a metre.

Which computation produced the numbers #

The note is a C3 of eight partials falling as one over their number, sounded at 80 dB, each partial’s string decay a sixty-decibel time of six seconds divided by its number, held 0.4 seconds and then damped with a sixty-decibel time of 0.15 seconds added to every partial’s rate — the constant added loss a damper changes the clock, not the colour derived. The reverberant field decays, partial by partial, at the slower of the damped string’s rate and the room’s, which is the damper essay’s model; while the key is held it decays at the slower of the string’s rate and the room’s.

The listener’s level is the power sum over partials of the direct copy weighted by $\dfrac{(d_c/d)^2}{1 + (d_c/d)^2}$ and the reverberant copy weighted by $\dfrac{1}{1 + (d_c/d)^2}$, the steady-state ratio at a distance $d$ from the source. The fall time is the time after release for that sum to fall the stated number of decibels below its value at the release, found by bisection. The doubling distance is found by bisection in log distance. Critical distances use $0.057\sqrt{V/T_{60}}$ with a directivity of one, and the room volumes are stated for each named room rather than measured from one.

Where the model stops #

The reverberant field is treated as steady. The direct-to-reverberant ratio used is the one for a source that has been sounding long enough to fill the room. A note held 0.4 seconds in a two-second room has built its reverberant field to about 94 per cent of that, so the room’s copy is slightly weaker than drawn, and the doubling distances are slightly longer than drawn — by a few per cent, not by the factor that would change the conclusion.

A source is not omnidirectional. A piano’s lid throws the direct sound toward the audience and away from the player, which raises its critical distance toward the hall by a factor of about the square root of its directivity. A directivity of four doubles every distance in the figure about rooms, which puts the concert hall’s 20 dB staccato at a metre and a half. That is still inside the stage.

The early reflections are part of the room here. Reflections from the stage floor, the lid and the nearest walls arrive within tens of milliseconds and are heard as part of the direct sound for timing, which is the finding the first eighty milliseconds are a different room is about. They would lengthen the direct sound’s release slightly and move the bend later, and they are not in this model.

The next note is not in it. In a passage the release of one note is overlaid by the attack of the next, and the tail of a staccato quaver at a brisk tempo is masked by what follows within a few hundred milliseconds. What the figures give is the release a listener would hear if nothing followed, which is the last note of a phrase or a note before a rest.

What a fall time cannot say about articulation #

Whether a listener hears articulation as a fall time. A staccato is heard partly through the gap it leaves and partly through the attack of the next note arriving into that gap, and a listener in a hall has learned to hear a short note through a long room. The fall time is the physical fact on which that listening operates, not the percept.

Whether players compensate. A pianist who plays a staccato shorter in a live hall than in a dry studio is adjusting to what they hear, and they hear the direct sound. On this arithmetic the adjustment has almost no effect on what the audience receives, which is the hall’s release in either case — and a staccato is a dynamic mark found that the audible consequence of a shorter note in a hall is mostly a change of loudness.

Whose staccato #

The recording tradition makes the distinction audible in the other direction. A close microphone a few tens of centimetres from the strings hears the player’s articulation, and a hall pair several metres out hears the room’s; a record balances the two, and that balance is a decision about which release the listener is given. On the arithmetic here, a recording made from where the audience sits contains almost none of the articulation the player produced, and a recording that sounds articulated has put a microphone where no member of the audience has ever been.

Still open: the lid, and the directivity that moves the boundary #

Every distance in this essay assumed a source that radiates equally in every direction. A grand piano with its lid open does not: it radiates more toward the audience and less toward the ceiling, and the critical distance along its axis grows with the square root of that directivity. How an instrument points, is already computed band by band, and directivity is strongest in exactly the upper partials that carry the brightness of a note’s attack.

The computation that follows puts a frequency-dependent directivity into the direct share, partial by partial, so that the upper partials’ direct sound reaches further into the hall than the fundamental’s. The prediction is that the articulation of a note’s upper partials survives to two or three times the distance its fundamental’s does, which would make a staccato heard in the hall as a brief brightening followed by a room-length tail — the release of the colour reaching the audience even where the release of the note does not.