In a hall a section of violins loses the notes it saved

A pizzicato violin doubled by a held clarinet starts each note as the owner of the sound and hands it to the clarinet as the pluck decays. The margin by which it owns the note at the strike — how much nearer the composite spectrum is to the pluck’s than to the clarinet’s — depends on the note, and for the modelled violin it has one deep notch. A violin’s worst doubling note is wherever its body resonates put the notch at B♭4, where the note’s fundamental lands on the violin’s lower corpus resonance and the pluck’s spectrum stops looking like a violin’s: the margin there is 2.2 decibels below zero, so the pluck never owns B♭4 at all.

A section of violins fills in the note one violin gives away then plucked twelve violins together, each with its corpus resonances a semitone or so from the next one’s, and found the notch smoothed into a shelf: twelve bodies put their resonances at twelve places, and no single note’s fundamental is lifted far above the rest. The section’s worst margin over a clarinet came up to about zero, and over an oboe the section owned every note.

All of that was dry: the margin at the strike, with nothing between the instruments and the ear. The essay named the next thing to compute and did not draw it. One note in the compass loses its pizzicato had found that a hall changes the pluck’s position differently at every pitch. The expectation recorded was that a section, having no deep notch, would give the hall nothing to deepen, and would keep its pizzicato over the corpus region in a hall at about the rate a single violin keeps its ordinary notes.

It does not. In a hall the section’s shelf goes under as a whole.

The margin, in a room #

The pluck, the held note and the measure are the earlier essays’. The violin is a string spectrum through a body of four resonances; a section is twelve such violins, each with its corpus modes moved by its own draw from a spread of one semitone and its air mode by half a semitone, plucked together and summed in power. The pizzicato starts six decibels above the held note and each partial decays at a rate that rises with its number. The composite is owned by whichever player’s spectrum it is nearer, partial by partial, in log-spectral distance; the margin is how much nearer the pluck’s is.

The hall is the one the earlier essay on halls used: a modelled concert hall of 18,700 cubic metres, whose reverberation time is Sabine’s, computed band by band from its plaster, panelling, floor and audience, so that it varies with frequency. A listener a stated distance from the stage receives each player’s direct sound and the room’s reverberant field, in the proportion set by the distance at which the room takes over. The held clarinet’s reverberant field is steady; the pluck’s builds as the pluck decays. At the strike the held note already has its whole reverberant field and the pluck has none, which is why a room counts against the pluck at the moment its margin is measured.

The section’s shelf, dry and in the hall #

Dry, the first figure’s dashed lines tell the earlier essay’s story. The violin’s margin falls from about five decibels in the middle of the octave to −2.2 at B♭4 and is barely above zero at C5. The section’s median follows the violin’s everywhere except across the corpus region, where its notch is filled: 3.7 at A♭4, 2.2 at A4, 0.4 at B♭4, 0.2 at B4, 0.5 at C5 and 2.1 at C♯5. Dry, the section owns every one of the six notes from A♭4 to C♯5; one violin owns four.

Fifteen metres into the hall, both curves have moved down by about four decibels everywhere. The violin’s B♭4 is at −5.9 and the section’s at −3.4. The section still has its shelf where the violin has its notch, and it is still higher by two and a half decibels at B♭4. But it is now a shelf below zero, and it owns none of the six notes. Neither does the violin.

So the section’s advantage over one violin survived the hall in decibels and did not survive it in notes. A shelf a few tenths above zero is exactly the shape a uniform loss of a few decibels takes from above the line to below it all at once. A single violin’s notch is one note lost; a section’s shelf in a hall is six lost together.

What filling the notch costs the note beside it #

The first figure carries a detail that the section’s median hides when it is quoted as one number. At B4, a semitone above the violin’s notch, the section is worse than the single violin: 0.2 decibels dry against the violin’s 1.2, and in the hall −3.5 against −2.6.

That is the price of the smoothing, and it follows from what smoothing is. One violin’s lower corpus resonance lifts B♭4’s fundamental far above the rest of its spectrum and leaves B4’s alone, so B♭4 is very bad and B4 is ordinary. Twelve violins put that resonance at twelve places spread over about four semitones, so each note of the region has a few members whose resonance lifts its fundamental. B♭4 gets fewer than one violin gave it and B4 gets more. The total misfortune is spread rather than reduced, and a note that one violin never troubled becomes one of the section’s weak notes.

Dry, that trade is a good one: no note of the region falls below zero. In a hall, which subtracts its few decibels from every note alike, it stops paying. Both lose the whole band further back, and the section loses B4 nearer the stage than the violin does: at two metres over a clarinet the section’s B4 is at zero and the violin’s still a decibel above it, and at five metres the violin keeps B4 by a fifth of a decibel where the section has lost it by most of one.

The other band the hall takes #

The figure’s left-hand end shows a second band that the hall takes and the dry margins only hinted at. Around C♯4 sits the violin’s air resonance, near 275 hertz, and the same mechanism works there on a smaller scale: dry, one violin has −0.4 at C♯4, and the section, whose air modes are spread by half a semitone, has 1.2. Fifteen metres into the hall, everything from C4 to D4 is below zero for both: −1.9, −4.3 and −1.6 for the violin, −1.9, −2.8 and −0.8 for the section.

So at fifteen metres the second octave of a pizzicato over a held clarinet is owned in two stretches only: E♭4 to G4, by a decibel at most, and D5 upward, where the margin rises steeply and the hall begins to help. The section owns ten of the nineteen notes from C4 to F♯5 and the violin nine. The two resonance regions, the air mode’s and the corpus modes’, are given to the clarinet for every seat past the first few rows, and the section’s twelve bodies make no difference to that.

The hall takes the same from one violin and from twelve #

The reason the shelf cannot hold is visible in what the hall takes from each note. At fifteen metres it takes between 3.5 and 4.2 decibels, varying with pitch — most around E4 and F4, least above E5, where the room decays unevenly across the bands the partials fall in. And it takes the same from one violin as from a section of twelve, to within fifteen hundredths of a decibel at every note.

That is not a coincidence of the draws. What the room does to the margin depends on how much of each player’s energy reaches the listener directly and how much through the reverberant field, which is set by the room, the distance and each partial’s frequency. It does not depend on the shape of the pluck’s spectrum, which is the only thing the section changes. The room’s cost is the room’s, not the body’s, and twelve bodies averaged together do nothing to it.

The earlier essay found that a section is weakest where its members agree — at a resonance they all share. This is the same idea from the outside. A hall is something every member of the section shares exactly, and its effect on the margin passes through the section’s averaging untouched.

How far back the advantage reaches #

If the room takes a fixed amount at each distance, the section’s dry advantage becomes an advantage in distance: it keeps its notes a little further back than one violin does.

Over a clarinet the section’s worst margin in the corpus region is 0.2 decibels dry and falls with distance: 0.0 at two metres, −0.8 at five, −2.3 at ten, −3.5 at fifteen. It loses its first note of the region at about two metres. One violin had already lost B♭4 dry.

Over an oboe the section starts higher, at 1.8, and keeps every note of the region to about six metres: 1.5 at two metres, 0.4 at five, −0.3 at seven. One violin’s worst note over an oboe, −0.3 dry, is lost from the first row.

Two metres is the conductor and the front desks. Six is the first few rows of a large hall. So the section’s smoothing, measured as notes kept, is real for the players and for the listeners nearest them, and gone for most of an audience.

Counting lost notes across the octave from E4 to E5 says the same thing more coarsely. Over a clarinet, one violin never owns two of the thirteen notes dry, five at ten metres, six at fifteen and eleven at twenty. The section loses none dry, none to two metres, two or three from three to seven, and then the violin’s numbers exactly: five at ten metres, six at fifteen, eleven at twenty. Over an oboe one violin loses one note dry and three from five metres; the section loses none to five metres, one at seven, and three at ten.

The section is one violin moved a few metres closer. Past ten metres the twelve bodies and the one have the same count of notes the pizzicato never owns.

Where the hall does the opposite #

The handover times show the other half of what the earlier essay on halls found. Across the corpus region, at fifteen metres, neither the violin nor the section keeps its note for even a millisecond. A little higher, from E♭5, both keep it for 250 milliseconds and more — far longer than dry — because the pluck’s own tail builds in the hall and carries its spectrum after the direct sound has decayed. The section and the violin are indistinguishable there as well: 244 against 250 milliseconds at E♭5, 381 against 386 at F♯5.

So the hall divides the second octave into a band where the pizzicato is the clarinet’s from the start and a band above it where the pluck is helped. Where that boundary falls is a property of the hall, the distance and the two instruments’ spectra. Whether there is one body or twelve does not move it.

What this changes for an orchestrator #

A violin’s worst doubling note and its successor gave an orchestrator two pieces of advice: a single violin has a note at which a held clarinet takes its pizzicato, and a section does not. In a hall the second needs a qualification. A section has no single bad note, but it has a bad region, A♭4 to C♯5 for the violins modelled, and beyond the first rows of an audience that whole region belongs to the clarinet. A doubling’s worst note belongs to one instrument — for a single violin, the violin’s body. For a section in a hall, the worst region belongs to the room.

What does change the answer is the partner. Over an oboe the section keeps the region to six metres, and at fifteen it still owns A♭4 and A4 and loses the four notes above them; over a clarinet it keeps the region to two. A pizzicato doubled by a held oboe is a texture whose plucked part reaches further into a hall than the same pizzicato doubled by a clarinet, which is the dry ordering the earlier essays found, carried into the room.

The model under the margins #

Each violin is the modelled string spectrum through a body with an air resonance near 275 hertz, two corpus resonances near 460 and 540, and a bridge hill, sixteen partials, the pluck six decibels above the held note at the strike. Sections are twelve such violins with corpus modes moved by draws of one semitone and air modes by half a semitone, six sections seeded as in the earlier essay, summed in power; the medians are over the six. The hall is a modelled concert hall of 18,700 cubic metres, its reverberation time computed band by band from its surfaces by Sabine’s formula, with each partial’s direct and reverberant energies combined at the stated distance and an omnidirectional source; the pluck’s reverberant field builds from its own decay. The margin is the composite’s log-spectral distance to the held note less its distance to the pluck, at the strike.

What the margins cannot show #

Directivity. Every player here radiates equally in all directions, so the ratio of direct to reverberant sound depends only on distance. An instrument points, and a section of violins facing the audience sends more of its upper partials forward than the model gives it; a clarinet points differently again. Both would move the distances at which notes are lost, and neither is in these figures.

One hall. The hall is a modelled one, and its reverberation time and the balance of its bands set the few decibels it takes at each note. A drier hall takes less and leaves the section’s shelf above zero further back; a reverberant church takes more.

Whether any listener notices. The margin is a measure of which spectrum the composite is nearer to, at the strike. Whether a listener fifteen metres back hears a pizzicato on B♭4 as a pluck or as a clarinet with a click in front of it is a question about perception that the arithmetic frames and does not answer.

The second violins, deferred #

The earlier essay’s own closing question was about the second violins: two sections plucking an interval apart. The margin used throughout these essays compares spectra partial by partial on one note’s harmonic series, and two sections a third apart do not share one; on that measure a second section would be judged by an artefact of the comparison rather than by what a listener hears. The question needs a measure of resemblance between spectra on different harmonic grids before it can be asked, and it is left open for that reason rather than answered badly.

Still open: a section whose players point #

The model’s sources are omnidirectional, and in a hall the ratio of direct to reverberant sound is what decides the few decibels each note loses. A violin’s upper partials beam forward from the top plate, and a section of twelve violins faces the audience together, so the direct sound reaching the hall carries more upper partials than the reverberant field does. The computation is this one with each partial’s direct energy weighted by a violin’s measured directivity in that band, and it would say whether the violins’ own beaming — which every member of the section shares, and which averaging therefore does nothing to — moves the distance at which the section loses its corpus region from two metres to ten, or leaves it where it is.