A wall puts antiphonal seconds back beside the firsts

There are two ways to seat the violins of an orchestra, and the argument between them has been running for a century. For most of the nineteenth century the firsts sat on the conductor’s left and the seconds on the right, facing each other across the stage. In the twentieth, largely for reasons of ensemble and of recording, many orchestras moved the seconds next to the firsts. Advocates of the older plan, often called antiphonal, point out that much of the music was written with it in mind and that it lets a listener hear two lines coming from two places. Everyone concedes one acoustic objection: a violin’s top plate faces its player’s left, so firsts on the left point out towards the audience and seconds on the right point away from it, towards the back of the stage.

A section keeps its pizzicato only inside a cone has made that objection measurable, because it found that a section’s upper partials are beamed and that its pizzicato owes much to them. On the axis, twelve violins plucking over a held clarinet keep their worst note in the corpus region out to twenty-five metres. Thirty degrees off they are worse than if they had not pointed at all. It ended by asking what happens when the whole hall lies behind the axis: whether antiphonal seconds lose their pizzicato to a doubling clarinet in every seat, or whether a well-placed wall behind them gives them back a cone of their own.

Both halves of that turn out to be true, on terms that matter. And the second half comes with a result nobody was looking for, in how many seats that cone adds.

The model is the earlier essay’s with two additions. The plan is the same 24-by-36-metre hall, with seats from four to thirty-six metres back. The firsts sit five metres left of centre, with their axis turned fifteen degrees in. The seconds sit five metres right, in the same posture, and so their axis is fifteen degrees in and backwards, towards the stage wall. The first addition is about the back of the plate. A piston in a baffle says nothing about what lies behind it, so the back is taken as the front’s own pattern mirrored and ten decibels down, a stated figure and not a measured one, and how much depends on it is one of the figures below. The second is a mirror image: a wall behind the section adds one image source, further away and with its axis reflected, so that it points at the audience, its power reduced by the extra distance and by whatever the wall loses. It is added in power since it arrives inside the precedence window, within about forty milliseconds for any wall considered here. A seat is counted if the section’s worst note from A♭4 to C♯5 stays nearer a pluck than a clarinet.

The figure below is that plan.

The dial moves the wall from nothing to eight metres behind the section. With no wall almost no square is shaded, and at two metres there is a large patch in the middle of the hall.

Behind the plate almost nothing is left #

With no wall, the seconds keep their worst note in about one seat in a hundred. The firsts on the other side of the stage keep theirs in three seats in ten, which reproduces the earlier essay’s figure. So the objection that everyone concedes is real and large, a factor of thirty, and on the first half of the question the answer is as near to “every seat” as a computation can reasonably state.

Along a line down the middle of the hall the loss has a size. At fifteen metres the firsts’ worst note stands 2.5 decibels of spectral margin above the clarinet, and the seconds’ stands at −8.8. The difference is not close to the boundary and it does not come and go with the distance.

With the wall in place the line changes in kind. A perfectly reflecting wall two metres behind brings the seconds’ worst note up to within a decibel of the firsts’ from about ten metres back, and in the first few rows the seconds now do better than the firsts. That is not a paradox. The firsts’ axis passes a seat near the middle at a steep angle when the seat is close, and the seconds’ image stands several metres further away, so a close seat lies nearer its axis. Further back, both fall below zero together around twenty-five metres. A wall that loses three decibels and stands four metres off holds the margin near zero to about ten metres and then lets it go.

How hard the wall is matters more than where it stands #

A wall can restore everything, on two conditions. It has to be hard, and it has to be close. With no loss, a wall two metres behind the section gives the seconds 30 per cent of the seats, exactly what the firsts have. Four metres gives 24 and eight gives 8. Because the image’s distance is the seat’s distance plus about twice the wall’s, the falling share is the expected penalty for an image that is further back. The surprise is the other axis. A loss of three decibels at the wall cuts the four-metre share from 24 per cent to 8, and six decibels leaves 1. This is the sense in which the question asked by the earlier essay turned out to be mis-aimed. “A well-placed wall” does not say enough. Placement moves the share gradually, and a reflection loss moves it by a factor of three for an amount that no one would notice as a change in the wall.

The reason for that steepness is that it is a margin being read and not a level. Every seat in the cone holds its worst note by a decibel or two at most, as the margin figure shows. Taking three decibels from the image’s power, in a region where the image is the whole of the upper-partial story, moves a broad population of seats over the line at once. So the shares amount to a statement that the worst note in the corpus region is marginal, wall or no wall, which is what the single violin’s own trouble with that note would lead anyone to expect.

Real walls do lose something. A wooden orchestra shell may reflect the upper partials of a pizzicato almost entirely, while a curtain, a choir stall or an open stage house loses far more, and these figures need a measured number for any particular hall. More seriously, a section’s players have other players behind them. On a real stage, as the arithmetic of a section changing over has reason to notice, the seconds sit in front of violas or winds, and the path to the back wall crosses them twice, so the loss in practice is unlikely to be zero, and the path may not exist at all for the front desks. A stage on which seconds keep their pizzicato by way of the rear wall is one with little behind them. The nineteenth-century platforms on which this seating was developed were often very differently shaped from a modern shoebox hall, and the seating and the room came as a pair. None of that has been computed here.

The number nobody has measured #

The back of the plate is an assumption and everything above stands on it, so it should be swept.

If a violin’s back radiated as strongly as its front, the seconds would have 23 per cent with no wall at all, and the whole objection would shrink to very little. At ten decibels down, the figure used throughout, it is 1 per cent. At fifteen it is none. Measured radiation patterns exist for single violins, and they say that the pattern in the upper partials is complicated and frequency-dependent and not, as is assumed here, a simple mirror. An instrument points collected what can be said about that in general terms. What this figure says is what a measurement of that pattern would decide. Between 6 and 15 decibels lies the difference between an objection that is decisive and one that is merely a preference, and the seating debate has gone on for a century largely without that number being put on the table. With a hard wall behind, the question weakens: 30 per cent at ten decibels and 25 at twenty, since the image carries the load whatever the back does.

The radiator’s size, also a stated figure, does not disturb the ordering in the one way that matters. The seconds with no wall stay at or under 4 per cent at every radius tried, however large a beam is assumed. Larger radiators favour anything with an image on the axis, since the cone is sharper and the axis is what the image offers, and by about 15 centimetres a hard wall four metres behind has given the seconds more than the firsts. A wider cone is more forgiving and a narrower one more demanding, as the earlier essay on the hall would predict, and neither takes the seconds off the floor without a wall.

The cone a wall gives back is one the section already had #

The hope behind antiphonal seating, as far as this measure goes, is that the two sections might between them cover more of the hall. A section on each side, each with its own cone, might reach seats that two sections side by side would both miss. So the seats should be counted by who keeps what.

With a hard wall two metres behind them, the seconds keep what they would have kept sitting beside the firsts. Side by side, with the seconds three metres to the firsts’ right, the shares are 9, 21, 10 and 61 per cent. Antiphonal with that wall, 9, 20, 10 and 61. Those are totals, and the seats themselves are not identical: of the roughly 120 each arrangement serves, about 90, three quarters, are the same seats, and the other thirty have moved a little forward. But the seats kept by at least one section come to 155 of 396 in both cases, to the seat. Neither arrangement reaches further than the other.

This is the essay’s surprising connection, and the geometry behind it is plain once the two patches are drawn. The hope behind two cones is that they would point at different parts of the hall. They do not, because both are turned inward by the same fifteen degrees. A section on the left aimed at the middle and an image on the right aimed at the middle cover the same central patch from opposite sides, and a beam wide enough to serve thirty per cent of a hall cannot help overlapping another one. The seats outside both cones are the same six in ten under every seating that aims at the middle, because what keeps them out is not the position of the section but the distance and angle at which any one beam gives out.

That changes the seating argument in a direction the objection did not anticipate. Seen from a central seat fifteen metres back, the seconds and their image stand within a few degrees of each other, since the image is merely some metres further behind on the same side of the stage. So the energy a wall returns arrives from close to where the section is seated. On a stage with a hard wall behind the seconds, antiphonal seating would cost nothing in seats and keep what it was adopted for, two lines from two places, and the case against it reduces to whether that wall exists in a given hall and whether anything stands in front of it. Where it does not, the cost is thirty to one on these notes. Two ears and one difference was careful to say that its arithmetic is no evidence for or against orchestral seating conventions, and this essay does not change that: the separation itself has not been valued here, only what may have to be paid for it.

It also clarifies what the two sides of the argument are really about. The case for the old seating is one of separation between lines. The objection to it concerns whether one instrument’s worst notes keep their attack against another, in a passage of a particular kind. Those were never measurements of the same thing, so neither side could have answered the other, and on this calculation the answer to the objection lies in the room rather than in the seating plan. A stage is part of an orchestra’s instrument in this respect, as a tube is part of a marimba.

What this plan cannot show #

A pizzicato is the most demanding case. The earlier essays chose it because a held clarinet can take a plucked note’s identity. A bowed note is not at risk in the same way, and most of what second violins do is sustained. The objection this figure measures applies to a passage of pizzicato against sustained winds in the upper-middle register, which is real and is a small part of an orchestra’s output. Nothing here says that a seating is worse in general.

The model has one image and one wall. No side wall, no ceiling and no audience, and a reverberant field that is one number per seat. Early reflections from side walls are what a concert hall is famous for and they would add a further route around the plate, more for seats towards the hall’s edges. The seat shaded in one of these plans and not in the next differs by less than a decibel of margin, and that is well within what a side wall might change. So the shape of the patch should not be taken literally. The shares are the safer reading, and the difference between 1 per cent and 30 is much safer than the difference between 24 and 30.

Twelve players are not one piston. They do not all point the same way or sit the same distance from the wall, and the back desks are closer to a wall than the front. A room’s own contribution to a pizzicato is already in the sum, and the earlier essay gave the reasons that its beam could be blurred; a turned-away section has further ones, since players hold their instruments differently from one another and the back of a violin is shadowed by the player’s body in a way the front is not.

And a seat’s own head has been left out. A listener’s two ears make their own contribution to what is heard, and what is heard in a real seat depends on the head between them and on the partner’s position as well as on a spectrum. The “margin” is a distance in a spectral space between the composite and each part, and that is a stand-in for an identification nobody has tested in a hall.

Which repertoire, and whose hall #

The evidence is the same modelled hall as before, not any actual one. The repertoire in question is the Austro-German and Russian orchestral literature up to about the Second World War for which antiphonal seating is claimed, and the specific texture is a plucked inner line against a sustained wind doubling, the corpus region being where a clarinet and a violin’s plucked pattern come closest. Conductors have made recordings in both seatings for decades, and a careful listening comparison of pizzicato in the two would be easy to arrange and is not reported here.

Still open: the seats the two sections never reach #

The most striking number on the plan is not any of the shares that changed but the one that did not. Six seats in ten are outside both cones whichever way the violins are seated, because the plan’s worst note in the corpus region is lost almost everywhere except along the axis. The obvious repair is to turn the sections. Conductors do sometimes angle the violin desks towards the hall, and a section’s axis aimed at a different part of the hall reaches different seats. The computation is this plan with the two sections’ aims free, searched for the pair that maximises the seats kept by at least one. It would say how much of that sixty per cent is geometry a player could change by turning a few degrees, and how much is simply how far a beam can reach in a hall of this shape.