A violin's worst doubling note is wherever its own body resonates

Half the worst notes of a doubling are notes nobody can play read a map of pizzicato doublings inside the compass each pair of instruments can actually play. A plucked note doubled by a held one is heard as the pluck at its attack and as the held note once the pluck has decayed — a doubled pizzicato gives its note away early, and one note in the compass loses its pizzicato in a hall — and the margin at the strike measures how far ahead the pluck starts: how much nearer the combined spectrum sits to the pluck’s than to the held note’s. Most pairs have a worst note, where the margin is smallest, and the worst notes cluster by instrument. Inside the playable compass one cluster survived as advice: a violin plucked over a clarinet, an oboe or a trumpet is worst at B♭4, and a doubling’s worst note belongs to one instrument had already traced that cluster to the violin’s body.

The essay ended on the question that decides what the advice is worth. The cluster sat at the modelled violin’s B1− mode, the lower of the two bending resonances of its corpus, which the model places at 460 hertz. That is a published mean, and real violins place the mode within about a semitone either side. If the worst note tracks the mode, every violin has its own worst doubling note, and a player who knows their instrument could find it. If it tracks something in the partner’s spectrum, it stays at B♭4 wherever the violin resonates.

It tracks the mode. Moved from three semitones below its modelled place to three above, the violin’s worst note over each partner stays within a semitone of the note nearest the mode at 36 of 39 settings, and at every whole-semitone setting all three partners agree on the note that sits on it.

Moving one resonance #

The violin here is the source-filter model the body is the filter built, the same arithmetic a vowel is two resonances applies to a voice: a bowed or plucked string’s spectrum, falling as one over the partial number, passed through a body whose response is a sum of four resonances — the air mode A0 at 275 hertz, the two corpus bending modes B1− at 460 and B1+ at 540, and the broad bridge hill near 2.5 kilohertz. The partners are the same five radiators the margin maps have used: a voice on one vowel, a clarinet whose odd harmonics dominate below its tone-hole cutoff, an oboe, a flue pipe and a trumpet with its bell’s high-pass. Each is drawn inside its stated orchestral compass, and a pair’s worst note is sought in the notes both can play, at least four semitones in from either end so that the end of a compass is not mistaken for a choice.

The only thing that changes is the frequency of B1−, in steps of half a semitone from three down to three up. Its height and sharpness stay as they were, and so does every other resonance. That is deliberately the smallest change the question needs: real violins differ in far more than one mode, but the one mode is what the earlier map identified as the cause, and moving it alone is the cleanest test of that identification.

A semitone for a semitone #

The first figure answers it. At every whole-semitone setting, all three pairs have their worst note on the note nearest the mode: G4 when the mode is three semitones down, then A♭4, A4, B♭4, B4, C5 and C♯5. The one exception at a whole semitone is the oboe at three down, which jumps to C♯5, and the reason for that is the upper corpus mode, taken up below. At the half-semitone settings the worst note falls on one of the two notes either side of the mode.

So the advice the map offers is not about violins in general. It is about a particular violin. A violin whose B1− sits near A4 has its worst pizzicato doubling a semitone lower than one whose B1− sits near B♭4, and a violin at the other end of the usual range would have it a tone higher again. That is the first result in these essays that an individual instrument, rather than a family of instruments, decides — and it is decided by a quantity that a luthier measures routinely, by tapping the body and recording its response.

The dip moves and the rest of the curve stays #

The margin curve across the whole compass shows what the move does and does not do. Against a trumpet the plucked violin’s margin is negative throughout — the trumpet’s strong, bright spectrum owns the composite from the strike on, as the earlier maps found for this pair — and it dips sharply at one note. Moving B1− moves that dip. At two semitones down it is at A♭4 and 12.8 decibels deep; as modelled, at B♭4 and 13.8; at two semitones up, at C5 and 16.1. Everywhere more than a couple of semitones from the dip, the three curves lie nearly on top of one another.

The depth changes a little as well as the place. Moved up, the mode sits closer to B1+, the two resonances begin to reinforce, and the dip deepens. Moved down, it separates from B1+ and the dip is shallower. So a violin’s worst doubling note is not only at a different pitch from another violin’s; it may be worse or less bad. But the change in depth is a few decibels on a margin that is already deeply negative for this pair, and the change of place is a whole note.

Why the worst note sits on the peak #

The body’s response makes the mechanism plain. Each worst note is the note whose fundamental lies on the B1− peak: A♭4 at 415 hertz on a peak at 410, B♭4 at 466 on a peak at 460, C5 at 523 on a peak at 516. A note whose fundamental sits on a strong resonance has its first partial lifted well above the rest, and a spectrum dominated by one partial is the violin spectrum least like a violin’s — the one that most resembles the clarinet’s, the oboe’s or the trumpet’s in the log-spectral distance that measures the margin. The pluck then starts barely ahead of the held note, or behind it.

The ear hears the list, not the shape made the point that a timbre is a list of partial levels rather than a curve, and this is the same point from the other side. The body’s response is a fixed curve, but what a listener receives is the list of values it takes at one note’s partials, and one note in each violin’s compass is the one that puts its fundamental on the curve’s highest point. That note’s list is the odd one out.

Two modes, and a note that lands on one #

Real violins do not move one mode at a time. The two corpus bending modes are properties of the same plates and tend to move together, and the upper one, B1+, is strong too.

Moved together, the two modes produce a cleaner rule than either alone. At every whole-semitone setting the oboe and trumpet pairs are worst on the note under B1−. At every half-semitone setting they jump to the note under B1+, a major second to a minor third higher. The reason is that the two modes are about a semitone and a half apart on the modelled violin, so when B1− falls exactly on a note of the tempered scale, B1+ falls between two, and when B1− falls between two, B1+ falls on one. The worst note is whichever corpus mode lands most squarely on a note. The clarinet pair, whose partner’s odd-harmonic spectrum weights the fundamental most heavily, stays with B1− except at the lowest settings, where it leaves both for C♯4.

That is also the reason for the one exception in the first figure, the oboe jumping to C♯5 with B1− three semitones down: there B1− lay between notes and B1+, left where it was, lay nearly on C♯5.

For a player the rule is still usable, and slightly more interesting. A violin’s worst doubling note is not simply “the B1− note”: it is the note that one of its two corpus modes hits squarely, which on some instruments is the lower and on others the upper. Either way it is a property of that violin, and either way it is found by knowing where the instrument resonates relative to the notes of the scale.

Held, the violin’s body decides nothing #

Turn the pairs round, so that the violin holds and the other instrument decays over it, and the violin’s body stops mattering. A clarinet over a violin has its worst note at F♯6 at every setting of the mode — a note set by the clarinet’s own spectrum, far above anything the corpus modes touch. An oboe or a trumpet over the violin is worst at E4 or F4 across most of the range, and where it wanders, at the extremes, it wanders away from the mode rather than with it.

The asymmetry is the one the earlier map found for every pair: a doubling’s worst note belongs to one instrument, and it is the plucked one’s filter that chooses it. A plucked violin is judged by how unlike itself it sounds at the strike, which is what a body peak on the fundamental does to it. A held violin is only the background, and a background with a slightly different resonance is still a violin.

The same note as the wolf #

The note under a corpus mode has another name among string players, and it is not a flattering one. A cellist’s wolf note is the note whose fundamental lands on the body’s strongest corpus resonance, where, as the other wolf worked out, the string and the body stop being separable and exchange energy, so that the note splits and warbles. On a cello the coupling is strong enough to be heard as a fault. On a violin it is usually weaker, and the same note is heard only as a slightly different colour or a slightly harder note to control.

The doubling result is the same fact from the listener’s side. The note on which the body’s response lifts the fundamental furthest is the note on which the string drives the body hardest, and so both the wolf and the worst doubling note are found at the corpus mode. A violinist who knows where their instrument’s wolf tendency sits — the note that “booms” or catches under the bow — already knows where its worst pizzicato doubling is. Nothing in the doubling model needs the coupling that makes a wolf; it needs only the body’s filter. But the two are one resonance, and they go together from instrument to instrument.

That also says why the result is individual. An instrument is not one timbre: a violin’s radiated spectrum changes at every pitch because its body’s resonances stay where they are while the fundamental moves, and the note at which the change is most extreme is fixed by those resonances, not by the family. Violins differ in their corpus modes by amounts comparable to the shifts drawn here — the semitone or so either way the earlier essay quoted — and so, on this model, differ in their worst doubling note by as much.

How many notes a section holds #

The usual spread of the lower corpus mode, a semitone either side of its mean, maps onto the first figure’s middle three columns: a worst note of A4, B♭4 or B4. The half-semitone settings, where the mode falls between notes, add either neighbour, and when both corpus modes move together the upper mode’s notes join them, up to C♯5. So on the model a section of a dozen violins with typical bodies carries its members’ worst doubling notes over about four adjacent semitones, from A4 to C♯5, rather than all on B♭4.

For an orchestrator the practical reading is the opposite of the one for a single player. A soloist’s worst note is one note, and it can be avoided or exploited. A section’s is a region, and the middle of the violin’s second octave is where a pizzicato doubled by a held wind is least audible as a pluck, whichever players are in the section. Which player on which note found that an orchestrator’s assignment of instruments to notes is not symmetric; this adds that a section’s assignment is smeared by the spread of its instruments’ bodies.

What a player can do with it #

The practical content is small and specific. A violinist who doubles a clarinet, oboe or trumpet in pizzicato has one note in the middle of the compass where the pluck is least audible as a pluck, and on the model it is the note under whichever corpus mode lands squarely on the scale. That is not something the player needs a model to find: tapping the body, or bowing a slow glissando and listening for the note that “booms”, locates the corpus modes on most instruments, and the note that booms is the note to avoid, or to accent, when a pizzicato doubling needs to be heard. On a different violin in the same section the note is different.

The claim is about orchestral pizzicato doublings with a held wind or brass partner in the violin’s middle register, and about the source-filter model’s account of a violin. It is not a claim about any named instrument, whose modes would have to be measured.

Where the model stops #

One mode moved, then two. Real instruments differ in the heights and sharpness of their corpus modes as well as their frequencies, and in the air mode and the bridge hill. Moving the frequencies alone tests the identification of the cause; it does not describe the spread of real violins.

A static spectrum. The margin is measured at the strike, with each partial at its body-weighted level. A plucked note’s partials decay at different rates, and a partial on a body resonance decays faster, because the body radiates it efficiently. That would shorten the time the lifted fundamental dominates, and it could make the worst note’s disadvantage briefer than the strike suggests.

The tempered scale. The worst note is sought on semitones. A player who could play between them would find the true worst pitch on the peak, and the rule about which mode lands squarely is a rule about the tempered notes the orchestra plays, not about the resonances themselves.

What the picture cannot show #

Whether a listener notices. The margin is a distance between spectra at one instant. That a smaller margin makes the pluck harder to hear as a pluck is the premise of every map in this series, and it has not been tested on listeners for these pairs.

Real violins’ modes. The spread of B1− “a semitone or so either way” is the range the earlier essay quoted, and the modelled positions are published means. A survey of real instruments’ corpus modes, joined to this map, would say how many different worst notes a real section holds.

Still open: a section of violins with different bodies #

An orchestral first-violin section is a dozen instruments with a dozen sets of corpus modes, and a pizzicato doubling is played by all of them at once. If each violin’s worst note is its own, a section’s worst note is spread over several notes, and the question becomes whether the section’s composite spectrum has a worst note at all or a shallow worst region. The computation sums a section of violins whose corpus modes are drawn from the measured spread, plucked together over the held partner, and it would say whether a section averages its players’ worst notes away — so that a section has a loudest member, not a colour, even here — or whether the notes where several of its members resonate together stand out.