A section of violins fills in the note one violin gives away

A violin’s worst doubling note is wherever its own body resonates found the note at which a pizzicato violin, doubled by a held clarinet, oboe or trumpet, comes nearest to never owning the sound. The measure is the margin at the strike: how much closer, in log-spectral distance, the composite of the two instruments is to the pluck’s own spectrum than to the held partner’s. A positive margin is a pluck that starts out owning the note and hands it over as it decays; zero or below is a pluck that never owns it. For the modelled violin every partner chose B♭4, because B♭4’s fundamental lands on the violin’s lower corpus resonance, the mode called B1− near 460 hertz, and a spectrum with its fundamental lifted far above its other partials is the violin spectrum least like a violin’s. Moving that resonance moved the worst note with it, a semitone for a semitone.

Real violins place B1− a semitone or two either side of any modelled value, and that essay drew the orchestrator’s consequence: a section of a dozen players carries its members’ worst notes over about four semitones, from A4 to C♯5, rather than on one. It ended with the question that consequence raises. A section does not play its members one at a time. Plucked together, their twelve spectra sum, and the composite has a body response that is the average of twelve. Does that average have a worst note at all, or a shallow worst region?

It has a worst region, and it is shallow enough to change what the region means. Over a clarinet one violin’s worst note is 2.2 decibels below zero. A section of twelve with typical bodies sits at about zero at its worst: the note one violin gives away, twelve nearly keep.

A section as twelve bodies #

Each member of the section is the modelled violin — the same string spectrum, the same bridge hill, the same air resonance A0 near 275 hertz — with its two corpus modes, B1− and B1+, moved together by an amount drawn from a normal spread of one semitone. That is the spread the earlier essay took as typical of measured violins: a semitone or two either side of the modelled 460 hertz. A second version moves each member’s A0 as well, by its own draw from a spread of half a semitone, since the air resonance depends mostly on the body’s volume and its f-holes and varies less between instruments.

The members pluck together and at one level. Their partials are not phase-locked to one another, so they add as energies: at each partial, the section’s power is the mean of its members’ powers. That composite is then normalised as a single radiator is, so that what differs between a violin and a section is the shape of the spectrum and nothing else. The held partner, the pluck’s decay and the margin are exactly as in the earlier essays, and six sections are drawn for every size so that a result is not one draw’s accident.

Filling in the dip #

The hero shows the worst margin rising as the section grows. One violin drawn from the spread sits between −2.3 and −1.0 decibels — some members, whose corpus modes fall between two notes, have shallower worst notes than the modelled violin. Two players together already help: the median rises to −1.4. By four it is −0.5 to −0.7, and from eight on it levels off. With twelve players whose air modes are varied the median worst margin is 0.0 and the six sections range from −0.5 to +0.5; with their air mode identical, −0.6.

Why the curve levels off from about eight players is a matter of sampling. What smooths the section’s response is that its members’ corpus peaks cover the spread, and a spread of one semitone is covered reasonably well by a handful of draws: with four players the peaks are already at four different notes, with eight they fill the region between A4 and C♯5, and after that a new player mostly lands between two others. Adding players reduces the chance that a section happens to have several bodies bunched together, which is why the range narrows from eight to sixteen more than the median moves. A section of the size an orchestra fields is already large enough; the arithmetic offers no reason to add desks for the sake of the pizzicato, and a reason not to leave a section too small — two or three players sit halfway between one violin and twelve.

Across the compass the mechanism is visible. The single violin’s curve has a sharp notch at B♭4, a semitone wide, where one note’s fundamental sits on one resonance. The sections’ curves lose that notch: twelve bodies put their B1− peaks at twelve places between A4 and C♯5, the summed response there becomes a broad shelf rather than a spike, and no one note’s fundamental is lifted far above the rest. The shelf still makes that stretch the section’s weakest, but its depth is a fraction of the spike’s. Away from it, the three curves agree within a decibel and a half on average.

This is the familiar mathematics of averaging peaks at scattered frequencies, and it answers the earlier essay’s question directly: the composite has a worst region and not a worst note, and the region is shallow. A single violin’s worst note is a note the pluck never owns. A section’s worst region is a stretch where the pluck owns the note only just, or in about half of sections not at all, by a few tenths of a decibel.

Weakest where the members agree #

The second figure hid a surprise that the third draws out. When every member of the section has the same air resonance — the model’s default, since only the corpus modes were varied in the earlier essay — five of six sections have their worst note at C♯4, not in the corpus region at all. C♯4’s fundamental, 277 hertz, lands on A0.

The reason is the averaging again, seen from the other side. Averaging twelve corpus resonances at twelve places flattens them. Averaging twelve air resonances at one place flattens nothing: the section’s A0 is as sharp a peak as any one violin’s. For a single violin A0 is not the worst note, because its corpus peak is deeper; once the section has smoothed the corpus peak away, the one resonance still sharp is the one every member shares. Vary A0 by half a semitone and it smooths too, and the worst note goes back to the corpus region, now shallow.

A section’s worst note is where its members agree, not where any one of them is extreme. For an orchestrator that is a rule about which resonances matter in a section: the ones every instrument has at the same place. For the violins as modelled that is the air mode, and real violins vary it less than they vary their corpus modes, so the air-mode note is the one most likely to survive into a real section. Whether half a semitone of A0 spread is realistic decides whether a real section’s worst doubling note is near C♯4 or in the shallow region around B4. The arithmetic here cannot choose between them; a measurement of A0 across the players of one orchestra could.

Over an oboe, nothing to give away #

The three partners show what the smoothing is worth. Over an oboe a single violin’s worst note is only just below zero, −0.3 decibels, and every section owns every note, its worst between +1.1 and +2.4. The note one violin gives away to the oboe, a section of twelve keeps by a margin a listener has some chance of hearing.

Over a clarinet the sections straddle zero. Over a trumpet nothing changes that matters: a trumpet’s spectrum is so much brighter and fuller than a plucked string’s that the pluck never owns any note in the pair’s shared compass, one violin at −13.8 decibels and a section at about −12. The section is two decibels better and still a long way under.

So the section’s gain depends on the partner, and it is largest exactly where the single violin was borderline. A doubling’s worst note belongs to one instrument found that most pairs’ worst notes are set by the violin; this says that for a section that responsibility is diluted, and what remains is set more by the partner’s overall brightness than by any resonance of the section’s.

The same smoothing, for a note the body will not let start #

The corpus resonance that makes B♭4 a violin’s worst doubling note is the one that makes some notes hard to play on it at all. The note the body will not let start put the body’s admittance into the minimum bow force a string needs to hold a clean note, and found the playable window narrowing fifteenfold at the corpus resonances: the wolf note. A wolf is one instrument’s, at one note, because it lives on that instrument’s B1− and B1+.

A section therefore has twelve wolves at twelve notes, and the same arithmetic that smooths its pizzicato smooths its difficulty. With corpus modes spread by a semitone, the share of players whose resonance lies within half a semitone of any one note is at most 38 per cent — four or five of twelve — and the section’s bowed sound at that note is carried by the seven or eight for whom it is an ordinary note. That was not computed here, and a section’s bowing is a more complicated object than its pizzicato — a wolf is a player’s problem before it is a listener’s — but it is the same fact about spread resonances, and it suggests that the section is the texture in which a violin’s body matters least on both counts.

In a hall #

Everything above is dry: the margin at the strike, with nothing between the instruments and the ear. One note in the compass loses its pizzicato found that a room multiplies the dry answer by a factor that depends on the pitch, by four to nine, and that at the note where the two instruments’ spectra nearly coincide it makes the pluck’s position worse rather than better. For a single violin that note is its worst doubling note, and the room compounds the loss exactly there.

A section offers the room no such note. Where the single violin’s margin is a notch, the section’s is a shelf a few tenths of a decibel either side of zero, and a room multiplying that shelf by its pitch-dependent factor spreads the weakness over the shelf’s four semitones rather than deepening it at one. On the arithmetic of that essay, the section in a hall should keep its pizzicato over the corpus region at about the rate a single violin keeps it at its ordinary notes. The figure that would show it runs the hall’s release over a section’s composite, and it is not drawn here.

How different the bodies have to be #

The last figure turns the spread itself. A section of identical instruments is one violin twelve times over, with its worst note exactly where one violin’s is. A spread of a quarter of a semitone barely helps, since the twelve corpus peaks still overlap into one. At half a semitone the worst margin has come up by a decibel, at one semitone by two, and at two semitones the section owns every note by a decibel.

That makes the variation between instruments, usually thought of as a nuisance for anyone modelling an orchestra, a resource for the orchestra. Twelve different violins give a smoother composite body than twelve copies of the best one would. The body is the filter described one violin’s response as a fixed comb of resonances that stays put while the note moves; a section’s response is the same comb with its teeth blurred, and the blur is proportional to how much the instruments differ. A section is more punctual than a soloist found that dividing a part among a dozen players changes when it is heard; this finds that it also changes which notes it can be heard on.

What the arithmetic rests on #

The violin is the modelled string spectrum through a body of four resonances — A0, B1−, B1+ and the bridge hill — with the corpus pair moved together as the earlier essay’s “both modes” case moved it. Each section member is that violin with its own draws; members sum in power and the composite is normalised to its strongest partial, as every radiator in these essays is. The margin at the strike is the earlier essays’ log-spectral distance from the composite of pluck and held note to each of them, at a pluck six decibels above the held note, and the worst note is the smallest margin within the notes both instruments can play. Six sections of each size and spread are drawn with fixed seeds, and the medians and ranges are over those six.

What the picture cannot show #

Timing. Twelve players do not pluck at the same instant, and a spread of attacks smears the section’s onset as a section is more punctual than a soloist described. The margin is taken at the strike as though every pluck arrived together.

Intonation. Each member plays exactly the written note. A section’s small differences of tuning would add slow beating to the composite, which the margin does not see.

The spread itself. The corpus spread of a semitone and the air-mode spread of half a semitone are the literature’s typical figures, not measurements of one section. The finding that matters most for a real orchestra, whether its worst doubling note is the air mode’s or the corpus region’s, turns on the second.

Directivity. A body’s resonances radiate in different directions, and twelve violins in a section face roughly the same way. What reaches a seat from twelve bodies is not the average of what each radiates on its axis, and the averaging here is the on-axis one.

Whose sections #

The claim concerns a string section’s pizzicato doubled by a sustained wind, the texture the whole doubling sequence has examined: that a section of a dozen violins with bodies as varied as real instruments’ has no note at which the doubling is lost the way a single violin’s is, that its weakest region is shallow and set by the resonances its members share, and that over an oboe it keeps every note. A section has a loudest member found that a section’s balance is dominated by one player; for its spectrum, this finds the opposite, a composite smoother than any member. The measurement that would test it is a recording of a section’s pizzicato over a held clarinet across the violin’s second octave, with each player’s A0 measured beforehand.

Still open: the second violins #

An orchestra has two violin sections, and in much of the repertoire both pluck together, often an octave or a third apart. A section against another section found two choir sections an interval apart beating between partials rather than between fundamentals. The corresponding question here is whether two violin sections, each smoothed by its own spread of bodies, still show a worst note at the interval where one section’s shared air mode lands on the other’s corpus shelf. The computation adds the second section at a stated interval, with its own draws, and asks whether the pair of sections has a worst note that neither section has alone — a note set by where the two sections’ shared resonances meet.