Acoustics & Music Theory
why it sounds like that
Generator

The same intervals, a clarinet under and over

Each interval scored twice: once with a clarinet on the lower note and a violin above, once the other way round. With one spectrum for both notes these bars would be identical, because the roughness sum is symmetric; with two spectra they are not. The largest disagreement is at the major sixth, where one arrangement is 5.1 times rougher than the other with the same two notes in it.
The same intervals, a clarinet under and over. Each interval scored twice: once with a clarinet on the lower note and a violin above, once the other way round. With one spectrum for both notes these bars would be identical, because the roughness sum is symmetric; with two spectra they are not. The largest disagreement is at the major sixth, where one arrangement is 5.1 times rougher than the other with the same two notes in it.

Drawn above with its standard settings, which is almost never how an essay draws it: an essay states the numbers it is arguing about, so the figure a reader meets there is about that argument rather than about the drawing in general. Every option a placement passes is checked against the ones this function actually reads, because an option it does not read is silently ignored and the figure quietly draws what is above instead.

It makes a noise. 116 of its 116 placements carry sound, built from the same numbers as the drawing, offering these buttons: 12 m back, a hard wall 2 m behind, 2 cents out at the same pitch, 20 m back with no beam, 25 m back on the far side of the hall, 30 m back, the same wall, 45° off the axis, 60° off the axis, 8 m back, a hard wall behind the seconds and 136 more.

Called by 20 essays

the blast radius of changing it

How many intervals a duet is smooth at. Every ordered pairing of 5 radiators, with the number of wells in its dissonance curve over an octave from 262 hertz. Rows are the instrument underneath and columns the one above, so the grid is not symmetric about its diagonal and that asymmetry is the result. The count runs from 1 to 7 across the grid, and a cell and its mirror need not agree: a violin under a clarinet has 1 and a clarinet under a violin has 2. The fifth is a well in every one of the 25 pairings and no other interval is.

Which instrument is underneath

Every roughness curve so far compares two tones of the same timbre, which is a duet nobody plays. Give the two notes different instruments and the sum stops being symmetric: the same written interval, on the same two players, is up to five times rougher depending on which of them takes the lower note. From G3 upward the fifth is a well in all twenty-five pairings and no other interval is; below it, three pairings lose even that, and all three have a clarinet on top.

timbre · Spectrum Six ways to put three players on three notes. The same chord — G3, B♭3, D4 — played by clarinet, oboe, voice in all 6 possible assignments, scored by the roughness each produces. Every bar is the same pitches and the same instruments; only who is on which note changes. The worst is 1.42 times the best, which is a factor a score can control and a chord symbol cannot express at all. Each row is labelled from the bottom note upward.

Which player on which note

An interval's roughness depends on which instrument is underneath, so the pair does not commute. Three players over three notes is the smallest thing that asymmetry has anywhere to go: six assignments, all of them the same chord, and across 450 of them the roughest averages half again the smoothest and reaches six times it. It is orchestration in the only form that can be computed here — not which chord, and not which voicing, but who is on which note.

timbre · Spectrum Every assignment, at equal levels and at its own best balance. The 6 ways of putting 3 players on a chord, each drawn twice: hollow at equal levels, which is what an assignment ranking sees, and filled at the levels that balance the parts and then minimise roughness. Every scoring here is at the same total loudness, 23.3 sones, so two points are comparable. Solving the discrete problem first picks violin · clarinet · oboe; solving both at once picks violin · oboe · clarinet, and the two-stage answer costs 9.0 per cent more roughness. The orderings do not keep their places between the two columns, which is the whole of the argument: a ranking taken at equal levels is not a ranking.

Who plays what and how loud is one question

Two lines of argument, one about spectrum and one about loudness, each stopped at the same wall and each said so. One of them can choose who plays which note and has every player at the same level; the other can choose how loud each part is and has nobody assigned to anything. Put together they are a single problem with two kinds of variable, and solving it in stages picks a different answer from solving it at once — nine per cent rougher, at the same loudness, on an ordinary triad.

form · Orchestration Trumpet at three dynamics, as a spectrum rather than a level. The radiated partials of a trumpet at 45, 70, 95 decibels, each normalised to its own strongest partial so that only the SHAPE is compared. A linear source would give three identical pictures. This one does not: the spectral centroid moves from partial 2.19 to 6.41, a factor of 2.92, because the excitation is nonlinear and blowing harder steepens the pressure front rather than scaling it. The tilt used is 3 decibels per octave of partial number per ten decibels of level, referred to 70 dB — a stipulated, ordinal number, not a measurement of any instrument.

A dynamic mark changes what a note is

Every spectrum until now is a shape with a level in front of it, so that playing ten decibels louder raises every partial by ten. That is true of exactly one instrument in an orchestra. Everybody else steepens their own spectrum as they lean on it, and a trumpet's centre of gravity moves from the second partial to the sixth across a dynamic range while an organ flue pipe's does not move at all.

timbre · Orchestration Intonation is a unison problem and nothing else. The roughness between two instruments on one note, against how far apart they are in cents, drawn for a unison and for the intervals beside it. A perfect unison is 0.0007 — the partials coincide and there is nothing to beat. Five cents apart it is 0.0465, 65 times as rough, and ten cents apart it is rougher than a major third played exactly. The mechanism is that partial n of a note mistuned by c cents is mistuned by c cents as well, which is n times as many hertz — so the top of the spectrum enters the critical band long before the fundamental does. The other curves are flat, because a third's roughness is set by which partials nearly coincide and a few cents does not change which.

Two players on one note

Six essays have put one instrument on each note of a chord, and the commonest thing an orchestrator actually does is put two on the same note. Two independent sources add in power, so the composite is neither of them — except that it nearly always is one of them, because the level at which ownership changes hands is rarely at zero. And a unison ten cents out is rougher than a major third dead in tune.

timbre · Spectrum Who owns clarinet, oboe, voice at every balance. The composite of three players at 392 hertz belongs to whichever of them it is nearest in log-spectral distance, and here that is drawn over the whole plane of balances a conductor could set — the second and third players from 24 decibels below the first to 24 above. voice owns 79 per cent of the square. The three regions meet where all three distances are equal, which is the only balance at which the composite belongs to nobody: it is at -0.3 and -19.1 decibels, inside the square and therefore a balance an ensemble could actually be asked for. A trio has a colour of its own at one point, not over a region.

A section has a loudest member, not a colour

Two players on one note have a balance at which the composite belongs to neither, and that is what blending means. Three should have three such balances and no reason for them to agree — a trio with a rock-paper-scissors ownership would have no strongest member at all. Twenty trios, sixty pairwise comparisons, and not one disagreement: the possibility is real, arbitrary spectra do it once in twenty, and instruments never do.

timbre · Spectrum Which pairs blend is a question about the note. The level at which a doubled pair's composite changes owner, drawn for all 15 pairs of 6 radiators over 2.6 octaves from 131 to 784 hertz. A pair blends when that level is inside the shaded band, which is the twenty-four decibels either way two players can manage; a curve outside it, or absent, is a pair one instrument owns at every balance. 6 of 15 pairs blend at the bottom of the range and 12 at the top. Every filter in this collection is fixed in frequency and the fundamental is not, so a radiator's shape is a function of pitch and so is everything computed from two of them — the blend ranking at the bottom and at the top disagree on 70 of 105 comparisons, which is more than half, so the order has turned over rather than merely shuffled.

The blend table has a row for every note

Eight earlier essays sound their instruments at one note, and one of them says why that cannot be innocent: every filter here is fixed in frequency and the fundamental is not. Swept over four octaves, the number of pairs that blend doubles from six to twelve, the ranking turns over rather than shuffles — seventy of a hundred and five comparisons swap — and a clarinet with an oboe goes from the best pair in the collection to the eleventh.

timbre · Spectrum Four players on three notes, every arrangement. The 36 ways of putting 4 players on a 3-note chord so that every note is covered, ranked by roughness, all at one total loudness of 26.9 sones. Each row is shaded by which note carries the pair. The best is flue | clarinet+violin | oboe and the worst is clarinet | oboe+flue | violin, a factor of 2.21. Every earlier essay puts exactly one player on each note, which is a permutation; a doubling makes the arrangement a surjection instead, and the doubled note sounds neither of its two players but the composite they make. Which note gets the pair explains 7 per cent of the spread here and which players sit on the lowest note explains 89: the fourth player is a much smaller decision than the three that were already there.

The fourth player is a spectrum, not a decision

Six earlier essays put exactly one instrument on each note, which makes an arrangement a permutation — and the commonest operation in orchestration is a doubling, which does not. Four players on three notes give thirty-six arrangements instead of six, and the extra choice turns out to be the smallest thing on the page: which note carries the pair explains three per cent of the spread and which players sit on the bass explains eighty-nine. A doubled note can be priced as one player, and which one is not the one a spectral account would have named.

instruments · Orchestration One doubling, held down a phrase. Where a single held arrangement of 4 players on 3 notes stands among the 36 at each chord of a 5-chord passage, best at the top, with what each chord would rather have named along the bottom. The held answer is flue pipe · trumpet · clarinet+violin, and it is the chord's own first choice at 4 of 5 of them. Holding it costs 16.2 per cent of the passage's roughness against re-scoring every chord — which is 2.4 per cent of the range the choice actually spans, since the arrangements at one chord differ by a factor of 7.8 on average. The cost is not spread over the passage: 1 chord carries nearly all of it.

An orchestrator doubles a line, not a chord

Three earlier essays made the objective a functional over a passage and a later one went back to holding one chord still. Put the doubling back into time and the retreat turns out to have been cheap: one arrangement held down a five-chord phrase is that phrase's own best answer at four of its five chords and costs 2.4 per cent of the range the choice spans — while the forward mask named earlier as the third temporal constant reaches for twenty milliseconds rather than two hundred, and cannot change the answer at any pace at all.

instruments · Orchestration The attack is the balance dial, turned by the clock. The level of a violin against a clarinet on one note at 392 hertz, moment by moment through the attack, with both players starting together. Two envelopes rising at different rates are a balance, so this axis is the same dial a conductor turns — and its whole travel is 6.02 decibels, which is twenty times the log of the ratio of the two attack times, 45 against 90 milliseconds, and nothing else. The pair does not begin as one player alone: both envelopes leave zero at the same slope ratio, so the dial starts at a finite offset rather than at silence. The dashed line is the balance at which the composite changes owner, -3.48 decibels — inside the travel, so the note belongs to a clarinet for its first 29 milliseconds and to a violin for the rest of its life.

The blend arrives before the note does

Nine essays on spectrum draw a steady state, and the strongest cue that two instruments are two instruments is that they do not start together. Two envelopes rising at different rates turn out to be a balance — the same dial an earlier essay swept — so the attack is that dial moved by the clock, and its whole travel is fixed at twenty times the log of the two attack times. It is six decibels for a clarinet with a violin against a crossing twelve to twenty-two decibels out, so one pair in ten changes hands during its own attack, and which one depends on a convention rather than on the instruments.

timbre · Spectrum A doubled pizzicato gives its note away while it is still the louder. The power of a violin plucked, against a flue pipe holding the same note at 392 hertz, through the first 600 milliseconds of the pluck, with the pluck starting 12 decibels up and its fundamental decaying over 1 second. With each partial losing level in proportion to its number, the composite stops resembling the pluck at 70 ms, when the pluck is still 5.2 decibels the louder. With every partial fading together it would keep the note until 543 ms. The dashed line is the balance at which the steady-state doubling changes owner, minus 20.6 decibels: the release crosses the owner long before its balance gets there, because what hands the note over is the pluck's upper partials going, not its level.

A doubled pizzicato gives its note away early

The attack turns the balance between two players on one note by a few decibels and stops. A pluck does not stop — every partial of it decays, so a pizzicato doubled by a held instrument walks the balance for the whole note, and the expectation was a handover as slow as the decay. It is fast. A one-second pizzicato over a flute loses its note in 70 milliseconds, while it is still five decibels the louder, because what hands the note over is its upper partials going first. A uniform fade would have kept it eight times as long.

timbre · Spectrum How long a doubled pizzicato keeps its note, seat by seat, in two rooms. How long a doubled violin pizzicato on 392 hertz keeps its note against the metres from the players, the pluck starting 12 dB up and decaying over 1 s with a loss exponent of 1. a concert hall, a flue pipe: 1 → 86 ms, 1.5 → 123 ms, 2 → 226 ms, 3 → 359 ms, 5 → 445 ms, 7 → 481 ms, 10 → 506 ms, 15 → 522 ms, 20 → 528 ms, 30 → 532 ms; a concert hall, an oboe: 1 → 52 ms, 1.5 → 55 ms, 2 → 59 ms, 3 → 77 ms, 5 → 149 ms, 7 → 195 ms, 10 → 224 ms, 15 → 242 ms, 20 → 248 ms, 30 → 254 ms; a concert hall, a clarinet: 1 → 44 ms, 1.5 → 45 ms, 2 → 45 ms, 3 → 47 ms, 5 → 53 ms, 7 → 65 ms, 10 → 86 ms, 15 → 105 ms, 20 → 112 ms, 30 → 118 ms; a large stone church, a flue pipe: 1 → 440 ms, 1.5 → 578 ms, 2 → 651 ms, 3 → 728 ms, 5 → 784 ms, 7 → 803 ms, 10 → 814 ms, 15 → 820 ms, 20 → 822 ms, 30 → 824 ms; a large stone church, an oboe: 1 → 56 ms, 1.5 → 65 ms, 2 → 89 ms, 3 → 207 ms, 5 → 281 ms, 7 → 303 ms, 10 → 315 ms, 15 → 322 ms, 20 → 325 ms, 30 → 326 ms; a large stone church, a clarinet: 1 → 44 ms, 1.5 → 43 ms, 2 → 43 ms, 3 → 44 ms, 5 → 53 ms, 7 → 67 ms, 10 → 80 ms, 15 → 89 ms, 20 → 92 ms, 30 → 94 ms. The mid-band critical distance is 5.3 m in a concert hall and 2.3 m in a large stone church. In none of the 60 cases does the note return to the pluck once it has left.

A room keeps a pizzicato from giving its note away

Doubled by a flute, a one-second pizzicato loses its note in 70 milliseconds dry, because its upper partials go first. The question left open was whether a room, whose reverberation keeps those partials alive, gives the note back afterwards. It does not give it back. It stops the note going: ten metres into a concert hall the pluck keeps it for 506 milliseconds, in a stone church for 814, and the room's own uneven decay takes back between a quarter and two fifths of that. In a room the loss law that decided everything dry matters a tenth as much, because the room's decay has become the clock.

timbre · Spectrum A room pulls the compass apart rather than evening it out. How long a pizzicato entering 6 decibels above a held note keeps the composite spectrum, at eight pitches across two and a half octaves, heard 15 metres from the stage. no room: 0.07, 0.05, 0.06, 0.06, 0.02, 0.05, 0.05, 0.04 seconds; a concert hall: 0.61, 0.27, 0.50, 0.38, 0.01, 0.25, 0.27, 0.19 seconds; a large stone church: 1.03, 0.39, 0.79, 0.56, never, 0.33, 0.35, 0.23 seconds. Dry the figures barely move — a spread of 3.0 across the whole compass — because a room is the thing that varies with frequency and there is none. In a hall the spread is 44. The room does not scale the dry answer by a constant: it multiplies it by between four and nine times depending on the note, and at C6 it makes the pluck's position worse rather than better, because the two instruments' spectra nearly coincide there and the pluck starts only 4.0 decibels ahead instead of twelve.

One note in the compass loses its pizzicato

Dry, how long a pluck keeps the composite spectrum barely depends on which note it plays: three-hundredths of a second at the worst pitch and seven at the best, a spread of three. In a concert hall the same eight notes spread by a factor of forty-three, and in a stone church one of them never gets the note at all. The room does not scale the dry answer by a constant — it multiplies it by between four and nine times depending on the pitch, and at the one note where the two instruments' spectra nearly coincide it makes the pluck's position worse instead of better.

timbre · Spectrum Every pair has a worst note, and for most of them it is at one end of the compass. The margin at the strike — how much nearer the composite spectrum sits to the decaying instrument's own spectrum than to the held instrument's, in decibels of log-spectral distance — for every ordered pair of six radiators at every semitone from C3 to C7, with the decaying instrument entering 6 dB above. violin over voice: worst −34.9 dB at E♭3, best 14.9 dB at C♯6, behind at 35 of 49 pitches; violin over clarinet: worst −2.2 dB at B♭4, best 19.6 dB at F♯6, behind at 3 of 49 pitches; violin over oboe: worst −0.3 dB at B♭4, best 11.4 dB at E3, behind at 1 of 49 pitches; violin over flue pipe: worst 2.7 dB at C7, best 16.9 dB at E3; violin over trumpet: worst −13.8 dB at B♭4, best −3.4 dB at D3, behind at 49 of 49 pitches; voice over violin: worst −9.2 dB at C♯6, best 41.8 dB at E3, behind at 8 of 49 pitches; voice over clarinet: worst 1.4 dB at C♯6, best 43.1 dB at E3; voice over oboe: worst −8.2 dB at C♯6, best 44.5 dB at E3, behind at 6 of 49 pitches; voice over flue pipe: worst −14.7 dB at C7, best 47.5 dB at E♭3, behind at 8 of 49 pitches; voice over trumpet: worst −24.2 dB at C7, best 39.5 dB at E3, behind at 20 of 49 pitches; clarinet over violin: worst −14.0 dB at F♯6, best 7.0 dB at B♭4, behind at 26 of 49 pitches; clarinet over voice: worst −36.6 dB at E3, best 3.5 dB at C♯6, behind at 46 of 49 pitches; clarinet over oboe: worst −14.2 dB at F♯6, best 5.5 dB at C3, behind at 27 of 49 pitches; clarinet over flue pipe: worst −14.5 dB at G6, best 13.3 dB at C3, behind at 18 of 49 pitches; clarinet over trumpet: worst −25.5 dB at F♯6, best −6.5 dB at C3, behind at 49 of 49 pitches; oboe over violin: worst −7.5 dB at E3, best 5.8 dB at B♭4, behind at 43 of 49 pitches; oboe over voice: worst −37.7 dB at E3, best 13.8 dB at C♯6, behind at 37 of 49 pitches; oboe over clarinet: worst −0.8 dB at C3, best 19.2 dB at F♯6, behind at 10 of 49 pitches; oboe over flue pipe: worst 2.4 dB at C7, best 14.3 dB at C3; oboe over trumpet: worst −14.2 dB at F4, best −12.8 dB at F♯6, behind at 49 of 49 pitches; flue pipe over violin: worst −12.7 dB at D3, best 3.0 dB at C7, behind at 43 of 49 pitches; flue pipe over voice: worst −40.4 dB at E♭3, best 19.4 dB at C7, behind at 34 of 49 pitches; flue pipe over clarinet: worst −7.7 dB at C3, best 18.9 dB at G6, behind at 24 of 49 pitches; flue pipe over oboe: worst −10.4 dB at C3, best 2.1 dB at C7, behind at 45 of 49 pitches; flue pipe over trumpet: worst −21.0 dB at C4, best −16.3 dB at C7, behind at 49 of 49 pitches; trumpet over violin: worst 8.7 dB at C3, best 17.3 dB at B♭4; trumpet over voice: worst −29.3 dB at E3, best 29.9 dB at C7, behind at 26 of 49 pitches; trumpet over clarinet: worst 10.7 dB at C3, best 30.6 dB at F♯6; trumpet over oboe: worst 16.9 dB at D5, best 18.8 dB at C7; trumpet over flue pipe: worst 21.2 dB at C7, best 24.7 dB at B♭3. A pale cell is a pitch at which the two spectra nearly coincide and the room has almost nothing to multiply; an unfilled cell is one at which the decaying instrument never leads at all.

A doubling's worst note belongs to one instrument, not to the pair

The essay before this one found one pitch at which a violin and a flue pipe nearly share a spectrum and asked which pitch that is for every pair. Run over six radiators, thirty ordered pairs and forty-nine semitones, the answer is that seventeen pairs choose a worst note and thirteen simply run out of compass — and that the seventeen do not choose thirty different notes. They cluster in fours and threes, and each cluster is one instrument's own filter: the violin's worst note is B flat 4, its strongest body resonance; the clarinet's is F sharp 6, its cutoff; the voice's is C sharp 6, where its second partial meets its second formant. A doubling fails at a note that belongs to one of its players.

timbre · Spectrum Inside the compasses the instruments have, 8 of the map's seventeen worst notes cannot be played by both. The margin at the strike for every ordered pair of six radiators from C3 upward, with every note outside the pair's shared compass greyed: the violin from G3 to E7, a soprano from C4 to C6, a B♭ clarinet sounding D3 to B♭6, the oboe B♭3 to A6, a B♭ trumpet sounding E3 to B♭5, a flue pipe throughout. violin over voice: shared C4 to C6, worst inside it −30.6 dB at C4; violin over clarinet: shared G3 to B♭6, worst inside it −2.2 dB at B♭4; violin over oboe: shared B♭3 to A6, worst inside it −0.3 dB at B♭4; violin over flue pipe: shared G3 to C7, worst inside it 2.7 dB at C7; violin over trumpet: shared G3 to B♭5, worst inside it −13.8 dB at B♭4; voice over violin: shared C4 to C6, worst inside it −5.9 dB at C6; the full map's worst, C♯6, is outside; voice over clarinet: shared C4 to C6, worst inside it 6.9 dB at F♯5; the full map's worst, C♯6, is outside; voice over oboe: shared C4 to C6, worst inside it −3.4 dB at C6; the full map's worst, C♯6, is outside; voice over flue pipe: shared C4 to C6, worst inside it −2.5 dB at C6; voice over trumpet: shared C4 to B♭5, worst inside it −12.6 dB at F♯5; clarinet over violin: shared G3 to B♭6, worst inside it −14.0 dB at F♯6; clarinet over voice: shared C4 to C6, worst inside it −31.3 dB at C4; the full map's worst, E3, is outside; clarinet over oboe: shared B♭3 to A6, worst inside it −14.2 dB at F♯6; clarinet over flue pipe: shared D3 to B♭6, worst inside it −14.5 dB at G6; clarinet over trumpet: shared E3 to B♭5, worst inside it −22.7 dB at A♭5; the full map's worst, F♯6, is outside; oboe over violin: shared B♭3 to A6, worst inside it −6.0 dB at E4; the full map's worst, E3, is outside; oboe over voice: shared C4 to C6, worst inside it −33.0 dB at C4; the full map's worst, E3, is outside; oboe over clarinet: shared B♭3 to A6, worst inside it 0.2 dB at B♭3; oboe over flue pipe: shared B♭3 to A6, worst inside it 4.2 dB at A6; oboe over trumpet: shared B♭3 to B♭5, worst inside it −14.2 dB at F4; flue pipe over violin: shared G3 to C7, worst inside it −11.3 dB at G3; flue pipe over voice: shared C4 to C6, worst inside it −34.8 dB at C4; flue pipe over clarinet: shared D3 to B♭6, worst inside it −7.7 dB at D3; flue pipe over oboe: shared B♭3 to A6, worst inside it −10.4 dB at B♭3; flue pipe over trumpet: shared E3 to B♭5, worst inside it −21.0 dB at C4; trumpet over violin: shared G3 to B♭5, worst inside it 9.1 dB at F4; trumpet over voice: shared C4 to B♭5, worst inside it −22.4 dB at C4; the full map's worst, E3, is outside; trumpet over clarinet: shared E3 to B♭5, worst inside it 11.0 dB at E3; trumpet over oboe: shared B♭3 to B♭5, worst inside it 16.9 dB at D5; trumpet over flue pipe: shared E3 to B♭5, worst inside it 22.7 dB at B♭5. Of the seventeen worst notes the full map chose, 8 lie outside a compass and 9 inside.

Half the worst notes of a doubling are notes nobody can play

A map of where two instruments' spectra come closest found seventeen pairs with a worst note of their own, gathered into four clusters, each belonging to one instrument's filter. It drew every instrument over the same five octaves. Read inside the compass each pair can actually play, eight of the seventeen worst notes are out of reach — the voice's C♯6 lies a semitone above a soprano's top C, and its E3 below her lowest note — and two pairs that seemed only to run downhill turn out to have a worst note after all. Of the four clusters, only the violin's B♭4 is advice every pair in it can use.

timbre · Spectrum A violin's worst doubling note follows its own body resonance. A violin plucked over a clarinet, an oboe and a trumpet holding the same note: the note within the pair's shared compass at which the pluck's margin at the strike is smallest, against how far the violin's B1− corpus mode is moved from its modelled 460 Hz, in semitones. The diagonal is the note nearest the mode itself. -3: mode near G4, worst notes clarinet G4, oboe C♯5, brass G4; -2: mode near A♭4, worst notes clarinet A♭4, oboe A♭4, brass A♭4; -1: mode near A4, worst notes clarinet A4, oboe A4, brass A4; 0: mode near B♭4, worst notes clarinet B♭4, oboe B♭4, brass B♭4; +1: mode near B4, worst notes clarinet B4, oboe B4, brass B4; +2: mode near C5, worst notes clarinet C5, oboe C5, brass C5; +3: mode near C♯5, worst notes clarinet C♯5, oboe C♯5, brass C♯5. As modelled, all three pairs' worst note is B♭4. Moved, it moves with the mode, a semitone for a semitone, within a semitone at 36 of 39 settings.

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

A pizzicato violin doubled by a held clarinet, oboe or trumpet has one note in the shared compass at which its pluck comes nearest to never owning the note, and for the modelled violin all three partners choose B♭4. That is where the violin's lower corpus mode sits, and real violins place that mode a semitone or two either way. Move it and the worst note moves with it, a semitone for a semitone, at 36 of 39 settings — so every violin has its own worst doubling note, and it can be found by knowing where the instrument resonates. Held rather than plucked, the violin's body decides nothing.

timbre · Spectrum A section of twelve violins nearly fills in the note one violin never owns. The margin at the strike at the worst note in the shared compass for a section of violins plucked together over a held clarinet, against the number of players, each player's two corpus modes moved by its own draw from a spread of one semitone; the line is the median of six sections and the bar the range. The modelled violin alone: −2.2 dB at B♭4. air mode shared: 1 −1.7 dB (-2.1 to -1.0), 2 −1.4 dB (-2.0 to -0.6), 4 −0.7 dB (-1.1 to -0.1), 8 −0.5 dB (-0.8 to -0.2), 12 −0.6 dB (-0.8 to -0.4), 16 −0.6 dB (-0.7 to -0.3). air mode varied too: 1 −1.7 dB (-2.3 to -1.0), 2 −1.4 dB (-2.0 to -0.3), 4 −0.5 dB (-1.0 to -0.1), 8 −0.1 dB (-0.8 to 0.3), 12 0.0 dB (-0.5 to 0.5), 16 0.2 dB (-0.6 to 0.5).

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

A pizzicato violin doubled by a held clarinet has one note — the note whose fundamental sits on its own lower corpus resonance — at which the pluck never owns the sound, and real violins put that resonance a semitone or two either way. A section plucks with a dozen different bodies at once. Summed, their resonances smear into a broad shelf, and the section's worst note over the clarinet rises from 2.2 decibels below zero to about zero: the note one violin gives away, twelve nearly keep. Over an oboe the section owns every note. And if every member shares the same air resonance, that shared resonance becomes the section's worst note instead — a section is weakest where its instruments agree, not where any one of them is extreme.

timbre · Spectrum In a hall a section's shallow shelf goes under all at once. The margin at the strike for a pizzicato over a held clarinet, note by note from C4 to F♯5, dry (thin) and 15 metres into a hall (thick): the modelled violin, and the median of six sections of twelve with corpus modes spread by a semitone and air modes by half. C4 2.1/2.1 dry, -1.9/-1.9 in the hall; C♯4 -0.4/1.2 dry, -4.3/-2.8 in the hall; D4 2.6/3.3 dry, -1.6/-0.8 in the hall; E♭4 4.2/4.5 dry, 0.0/0.3 in the hall; E4 4.9/5.2 dry, 0.7/0.9 in the hall; F4 5.1/5.3 dry, 1.0/1.1 in the hall; F♯4 5.0/5.1 dry, 0.9/1.0 in the hall; G4 4.4/4.6 dry, 0.4/0.6 in the hall; A♭4 3.3/3.7 dry, -0.7/-0.3 in the hall; A4 0.5/2.2 dry, -3.3/-1.7 in the hall; B♭4 -2.2/0.4 dry, -5.9/-3.4 in the hall; B4 1.2/0.2 dry, -2.6/-3.5 in the hall; C5 -0.1/0.5 dry, -3.9/-3.3 in the hall; C♯5 0.3/2.1 dry, -3.5/-1.7 in the hall; D5 5.0/5.0 dry, 1.1/1.2 in the hall; E♭5 11.1/11.0 dry, 7.3/7.2 in the hall; E5 12.2/12.1 dry, 8.5/8.4 in the hall; F5 12.7/12.6 dry, 9.1/9.0 in the hall; F♯5 13.0/12.9 dry, 9.5/9.4 in the hall. Dry, the section fills the violin's notch at B♭4 into a shelf a few tenths above zero. In the hall the room lowers every note by the same few decibels for both, and the shelf goes under with the notch: the section owns 0 of the six notes from A♭4 to C♯5, the violin 0.

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

Dry, twelve violins with different bodies fill in the note one violin's pizzicato gives away to a held clarinet, leaving a shallow shelf a few tenths of a decibel above zero. In a hall that shelf goes under all at once. The room takes the same few decibels from each note whether it comes from one violin or twelve, so a section's smoothing survives only to the front of the hall — about two metres over a clarinet, six over an oboe — and from ten metres back a section loses exactly as many notes as one violin does.

timbre · Spectrum A section whose upper partials beam keeps its pizzicato deep into a hall on its axis, and loses it faster than an unpointing one a little way off. The margin at the strike for a pizzicato section of twelve over a held clarinet, 15 m into a hall, from C4 to F♯5, with each partial's direct sound scaled as from a piston of 10 cm radius. Above zero the pluck owns the note. no beam: A♭4 to C♯5 -0.3, -1.7, -3.4, -3.5, -3.3, -1.7; on the axis: A♭4 to C♯5 5.2, 4.1, 2.8, 3.0, 3.5, 5.3; 15° off: A♭4 to C♯5 4.4, 3.2, 1.8, 1.8, 2.2, 3.9; 30° off: A♭4 to C♯5 1.9, 0.3, -1.6, -2.2, -5.1, -2.1.

A section keeps its pizzicato only inside a cone

In a hall a section's pizzicato loses its corpus notes to a held clarinet within two metres, on the assumption that a violin radiates equally in every direction. It does not, and twelve players sharing a beam cannot average it away. Directly on the players' axis, the same section keeps every note for twenty-five metres. Thirty degrees off, with the beam's upper partials thrown elsewhere, it is worse than if it had never pointed. A question about how far back a pizzicato carries turns out to be a question about which way the listener's seat faces.

timbre · Spectrum Antiphonal second violins own their worst pizzicato note almost nowhere, unless a hard wall stands close behind them. A plan of a 24 by 36 m hall with the second violins at the right of the stage, plates turned away from the audience, and a perfectly reflecting wall 2 m behind them. A seat is shaded if the section's worst note from A♭4 to C♯5 stays a pizzicato rather than a clarinet with a click. The seconds keep 30 per cent of seats, the firsts on the other side 30.

A wall puts antiphonal seconds back beside the firsts

Second violins seated opposite the firsts have their plates turned away from the audience, and in a hall a section's pizzicato depends on its beam. With nothing behind them they keep their worst note in about one seat in a hundred. A perfectly hard wall two metres behind gives them back a cone as large as the firsts' own, but a wall that loses three decibels takes most of it away again. And the seats that cone serves turn out to be, within a seat or two, the seats the same section would serve sitting beside the firsts.

timbre · Spectrum

All figures · What can be heard