A section keeps its pizzicato only inside a cone

A pizzicato violin doubled by a held clarinet owns its note at the strike only by a margin: the composite spectrum is a little nearer the pluck’s than the clarinet’s. A section of violins fills in the note one violin gives away showed that twelve different bodies smooth one violin’s deep notch into a shelf, and in a hall a section of violins loses the notes it saved then took that shelf into a room. It went under as a whole. Over a clarinet the section’s worst note in the corpus region, A♭4 to C♯5, is lost by two metres, and by fifteen metres the whole region has gone.

That essay said what its sources were: omnidirectional, every partial radiating the same in all directions. It said why that might matter, too. The direct-to-reverberant ratio is what costs the section those few decibels, and a violin’s upper partials beam out of its top plate, with all twelve violins in a section facing roughly the same way. So the direct sound reaching the audience should carry more upper partials than the reverberant field does, and this is one thing that averaging over a section cannot remove, since every member shares it. It proposed the calculation and framed the answer as a distance: whether the beam moves the point of loss from two metres to ten, or leaves it where it is.

The answer is neither, because a distance is the wrong shape for it. On the axis the section keeps its worst note past twenty metres. A little way off the axis it keeps it for less than it would have with no beam at all.

Only the direct sound can point #

A beam has a simple effect on the calculation, once it is clear which part of the sound it touches. Each partial reaching a listener has a direct part, which travels straight from the instrument, and a reverberant part, which has been round the room. A source that points does not change how much power it radiates, so the reverberant field is untouched. What changes is the direct part, which is multiplied by the source’s directivity factor and by its pattern at the listener’s angle.

That is exactly the arithmetic how far away the room takes over uses, with one change: the critical distance is now different for every partial. The model for the beam is the one an instrument points already has, a circular piston of radius $a$, with a directivity factor of $1 + (ka)^2/2$ on its axis and a response of $2J_1(x)/x$ at an angle, where $x = ka\sin\theta$. A violin’s top plate is about ten centimetres across its effective radiating region at these frequencies, as a stated figure rather than a measurement, and six and fifteen centimetres are run beside it to see how much depends on that. The clarinet stays omnidirectional in the main figures.

With no radius given the calculation does not take the new branch at all, and it returns the earlier essay’s section exactly at every note and every distance. That is the control.

The fundamental hardly notices; the upper partials get everything. On the axis, the 466-hertz fundamental of B♭4 gains 1.3 decibels, its second partial 3.9, its fourth 8.3 and its eighth 13.9. Those are the partials that decide who owns the note. A pluck’s distinguishing property against a held clarinet lies largely in the upper partials at the strike, and the room’s reverberant field has been covering them. On the axis the beam puts them straight back.

Off the axis, the same figures reverse. At thirty degrees the eighth partial has fallen from a gain of 13.9 decibels to a loss of 6, because the pattern’s first null has swept down the spectrum; at sixty degrees the fourth is down 4. So the direct sound in those seats carries less upper-partial energy than an unpointing source would give them, and the reverberant field, which carries the power that went elsewhere, is the same as ever. A beam is not extra sound. It is the same sound, sent to some seats at the cost of others.

The worst note, against the seat’s angle #

The corpus region’s worst note is what the previous essays used to summarise the margin, and it makes the shape of the answer plain.

Fifteen metres back, the beam helps only inside a cone. With a ten-centimetre radiator the section owns all six notes out to about 20 degrees either side of the axis, at +0.8 there; by 25 degrees it has gone under; and by 30 degrees, at −5.1, it stands below the −3.5 it would have without any beam at all. A larger radiator gives more on the axis and less off it: 4.7 at the centre, and worse than nothing by 25 degrees. A smaller one is a much gentler trade, 0.7 on the axis and worse than nothing only at 50 degrees.

The shape is one that has already appeared in this subject, for a different reason and a different instrument. A lid darkens a staccato only inside a narrowing cone found that a raised piano lid brings a damped note’s release out dark along its axis and does the reverse a little way off, inside a cone whose width follows one product of pitch, size and the sine of the angle. Both results are one pattern, seen from opposite ends. There the beam showed what the note’s end sounds like to a listener; here it decides who owns the note’s beginning. And both make the same point about a beam’s effect on a hall. Beaming does not raise anything on average. It makes the seat’s position matter.

The result’s distance form follows directly.

A section with no beam keeps its worst note for two metres, and on its axis it keeps it for twenty-five. The earlier essay’s own framing asked whether the beam would take two metres to ten, and at the axis it does much more than that, while at 30 degrees it does worse than nothing. There is no single distance to report. Averaging over a section, which was the first thing to check, is indeed no protection: the twelve players’ bodies still differ and smooth the notch, and they all point the same way, so the shelf they make together rides on one beam.

What the beam is worth on the axis #

Set the on-axis margin beside the unpointed one at each distance and a simple regularity shows, one that the earlier figures could not suggest. With a ten-centimetre radiator the worst note reads 6.0 decibels at two metres, 5.5 at five, 4.2 at ten, 2.8 at fifteen, 1.6 at twenty and −0.5 at thirty. Without a beam it reads 0.0, −0.8, −2.3, −3.5, −4.4 and −5.6. The difference is about six decibels at every distance: 6.0, 6.3, 6.5, 6.3, 6.0 and 5.1.

So on the axis the beam acts almost as a fixed allowance. The hall takes what it took before, at the rate it took it, and the beam has simply started the section six decibels further up. Against that allowance the entire loss the earlier essay measured, 3.7 decibels by fifteen metres, is small, which is why the axis seats look so different. Since the dry section’s worst margin was only 0.2, the section on its axis at two metres is better off than the section with no room at all. This is a surprising thing for a hall to do, and it is not the hall that is doing it: the beam needs the reverberant field to be there to matter, since with no reflected sound to compare against, a clarinet’s direct partials and the pluck’s would be scaled only against each other.

It is also the mirror of what a pointing source’s staccato ends dark, not bright found at the far end of a note. There, a beam gives a damped note a release that darkens along the axis, because what it beams is the part the damper stops. Here what it beams is the part that the pluck strikes first. The two together describe a note whose upper partials are unusually present to the seats on the axis at the start and unusually absent at the end, so the listener on axis gets a sharper note in both directions, and the listener off it gets a blunter one.

The notch that began all this has one further turn. A violin’s worst doubling note is wherever its body resonates located the single violin’s weakest note at a corpus resonance, where the fundamental is lifted against everything above it. A beam works the other way on that note, lifting everything above the fundamental and leaving the fundamental alone. That is why the curve in the first figure stays recognisably the same shape on the axis, its lowest point still at B♭4 and B4, and sits above zero all the way across.

A measurement to settle most of this would be cheap, by the standards of these questions. Two microphones in a real hall, one along the section’s axis and one thirty degrees off, at the same distance, recording one pizzicato note over a held clarinet, would give both spectra directly. It needs no listener and no model of perception, and it would say whether the margin’s difference between the two seats is anything like the seven or eight decibels computed here.

How much of a hall that is #

A cone becomes a practical matter once it is laid on a plan. The section does not sit on the hall’s centre line, and no violinist faces the seats squarely. The plan here is a simple one: a hall 24 metres wide, seats from 4 to 36 metres back, the section 5 metres left of centre with its axis turned 15 degrees towards the middle.

With the beam, three seats in ten hear the section own its worst note; without it, none does. In that hall, the seats that matter lie along one band that runs from the section out through the middle of the hall, and most of the audience is outside it. A real violin section’s players are not all oriented alike, its stands and its neighbours interrupt the path, and the first desks sit nearer than the last, so the real band must be blurred. But the result that a section’s upper partials belong to part of the room is the one that survives those details, and an instrument points already said as much of the single instrument.

It also changes the orchestrator’s conclusion the earlier essay drew. That essay reported that a hall takes the same decibels from one violin and from twelve, so a section’s advantage is lost a little way back. That remains true of the section as a sum of omnidirectional sources. With the beam, the statement has a second variable in it: a pizzicato section over a held clarinet in the corpus region is heard as a pluck in a band of seats and as a clarinet with a click in the rest, so the same balance is two different scorings depending on the ticket.

What depends on the piston #

The model. Sixteen partials of each of six sections of twelve, on the earlier essays’ pluck and clarinet, in the same modelled hall. Each pluck partial’s direct power is multiplied by $(1 + (ka)^2/2)$ and by the squared piston response at the seat’s angle, and its reverberant field is what it was. Only the strike is computed, so the margin is what a listener gets in the first moment.

A violin is not a piston. It is the largest simplification here. A real violin’s radiation is lobed and it changes partial by partial; there are measured patterns for it in the room-acoustics literature, and they have no clean nulls like the piston’s. The 30-degree figure in particular is a piston’s null passing through the fifth to tenth partials, and a real instrument’s pattern would soften that into a messy, smaller loss. So the direction of the finding, a rise on axis and a fall off it, is the part that should survive, and the sharp edge at 25 to 30 degrees should not be quoted as a property of violins.

The radius is a stated number. It is doing a good deal: 6, 10 and 15 centimetres move the on-axis margin from 0.7 to 4.7. A pattern measured octave by octave would replace this one number with a table, and the chance is high that any real instrument behaves like a smaller radiator at some frequencies and a larger one at others.

A pointing clarinet changes little. Giving the clarinet a beam of its own, of a radius of three centimetres for its bell and open holes, leaves the section’s on-axis margin at about 2.5 rather than 2.8 at 15 metres and does not move its reach back. The clarinet’s strongest partials are low and beam very little at this size. A radiator as wide as the bell of a brass instrument would be a different case, and it has not been run.

Early reflections are not here. The sound is divided into a direct part and a diffuse reverberant part. Between the two come reflections from nearby walls, a stage floor and a ceiling. In a real hall they can be strong, they can carry upper partials to seats outside the cone, and they sit inside the first fifty milliseconds a listener counts as the note’s beginning. That would soften the band’s edge, perhaps considerably, in a hall designed to do so.

What a margin cannot tell a listener #

The margin compares spectra, not perceptions. It is a distance between a composite spectrum and the two sources, and above-zero is being read as “the pluck owns the note”. Nobody has asked a listener in row M. In particular, a click at the strike can be heard as a separate event while the note is identified as a clarinet, and the margin does not distinguish that from a composite that sounds like one instrument.

Vision is out of the calculation. A listener who sees twelve violinists plucking is unlikely to believe the clarinet has the note, whatever the spectrum says. That is not a quibble, because nearly every audience can see the players, and recordings, on which the whole of this problem is rearranged by microphone placement, are a separate case that the calculation does not touch.

One moment. The strike is the moment the section is strongest. Everything after it hands the note to the clarinet at a rate that one note in the compass loses its pizzicato has already priced, and the beam changes that as well, since the pluck’s upper partials decay fastest and are the ones the beam was helping.

Whose hall, whose section #

The section is a modelled one of modern violins in a Western orchestral seating, plucking in the region of the instrument’s main corpus resonances, in a large modelled shoebox hall. Seating has changed historically, with first and second violins facing each other across the stage in antiphonal layouts: there a second violin section, if it were doubled by the same clarinet, would point its beam across the stage rather than out to the hall, and its share of the seats would be different again. Pizzicato over held winds is a device of the Western orchestral repertoire. None of this extends to a band with amplified strings, where a loudspeaker’s own beam replaces the instrument’s.

Still open: which way the second violins point #

In most modern seatings the first and second violins sit together on the conductor’s left and their f-holes face the audience, but in antiphonal seating the seconds sit on the right, with their instruments turned away from the audience. A section whose top plates point towards the back of the stage is a section with its seats almost all off-axis, and the reflection from a stage wall is then its best route out. The computation is this one with the section’s axis turned, and one image source for the wall behind, and it would say whether antiphonal seconds lose their pizzicato to a doubling clarinet in every seat of the hall, or whether a well-placed wall gives them back a cone of their own.