Acoustics & Music Theory
why it sounds like that
Timbre and acoustics

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.
15 min read 6 figures The instrument is not the modelThe ear is not a microphoneSomebody measured it

Assumes: A doubling's worst note belongs to one instrument, not to the pair · One note in the compass loses its pizzicato

A doubling’s worst note belongs to one instrument built a table the essay before it had asked for. For thirty ordered pairs of six radiators — a violin, a voice, a clarinet, an oboe, a flue pipe and a trumpet — at every semitone from C3 to C7, it computed the margin at the strike: how much nearer the two instruments’ combined spectrum sits to the decaying instrument’s own spectrum than to the held one’s, at the instant of the attack. Where the margin is small, a room has little to multiply and the decaying instrument loses its note almost at once. Seventeen pairs had a worst note inside the map rather than at an end of it, and the seventeen notes gathered into four clusters: B♭4 for the violin, where its body resonates; F♯6 for the clarinet, at its cutoff; C♯6 for the voice, where its second partial meets its second formant; and E3 for anything held against a voice.

That essay drew every instrument over the same five octaves and named the simplification: an oboe does not play C7 and a trumpet does not play C3. It claimed that the clusters all fell in registers their instruments do have, and asked for the join — each radiator given its own compass, and the table read again. The claim was wrong for three of the four clusters. Read inside the compass each pair can actually play, eight of the seventeen worst notes are notes the two instruments cannot both sound.

The compass each instrument has

The same essay also said the ranges were computed elsewhere in these essays rather than looked up. They are not: nothing in the collection derives an oboe’s lowest note or a trumpet’s highest from a bore. So they are stated here, as the ordinary orchestral compasses at sounding pitch — the violin from G3 to E7, a soprano from C4 to C6, a B♭ clarinet sounding D3 to B♭6, the oboe from B♭3 to A6, a B♭ trumpet sounding E3 to B♭5, and a flue pipe wherever its builder cut it. The voice is a soprano because the soprano is the voice that doubles violins and clarinets in the treble; that choice matters, and a later figure shows how much.

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.
Fig. 1 The margin at the strike for every ordered pair from C3 to C7, with every note outside the pair’s shared compass greyed. Rings are the full map’s worst notes; dots are the worst notes inside the compass. Eight of the seventeen worst notes the full map chose are outside a compass both instruments have: the voice’s three C♯6s, the four E3s under a voice, and the clarinet’s F♯6 over a trumpet.

The grey is the part of the full map that no ensemble can reach. For a pair including the voice it is most of the map: a soprano’s two octaves, C4 to C6, share at most twenty-five of the forty-nine semitones. For a pair including the trumpet it is the top octave and a half. Only the flue pipe, with the whole map, never greys anything on its own account.

Eight worst notes out of reach

Of the seventeen worst notes, the eight outside a shared compass are not a random eight. They are three whole clusters and part of a fourth.

Of the map's four clusters of worst notes, only the violin's lies inside every compass that meets it. The compass of each of the six radiators, as stated, on the pitch axis of the margin map, with the four notes at which three or more ordered pairs had their worst margin: B♭4, the worst note of violin over clarinet, violin over oboe, violin over trumpet, playable by both instruments in 3 of 3; C♯6, the worst note of voice over violin, voice over clarinet, voice over oboe, playable by both instruments in 0 of 3; F♯6, the worst note of clarinet over violin, clarinet over oboe, clarinet over trumpet, playable by both instruments in 2 of 3; E3, the worst note of clarinet over voice, oboe over violin, oboe over voice, trumpet over voice, playable by both instruments in 0 of 4. The compasses: violin G3 to E7; voice a soprano, C4 to C6; clarinet a B♭ clarinet, sounding D3 to B♭6; oboe B♭3 to A6; flue pipe whatever the builder cut; trumpet a B♭ trumpet, sounding E3 to B♭5.
Fig. 2 Each radiator’s compass on the pitch axis of the map, with the four notes at which three or more pairs had their worst margin. B♭4, the violin’s, is playable by both instruments in all three of its pairs. F♯6, the clarinet’s, in two of three. C♯6, the voice’s, in none of three. E3, the held voice’s, in none of four.

The violin’s cluster survives whole. B♭4 is inside the violin’s compass and inside the clarinet’s, the oboe’s and the trumpet’s, so all three pairs whose worst note it is can play it, and it stands as what the earlier essay took all four clusters to be: a note an orchestrator doubling the violin should know about.

The voice’s cluster does not survive at all. C♯6 is the worst note of a voice decaying over a violin, a clarinet and an oboe, and it lies one semitone above a soprano’s written top. The E3 cluster, where anything is worst over a held voice, lies a sixth below her lowest note. The clarinet’s F♯6 survives with the violin and the oboe and is lost with the trumpet, whose top is B♭5.

So the earlier essay’s summary sentence — four clusters, each one instrument’s filter — is still true of the model and is advice about only one of them. The mechanism it described is unchanged: the voice’s second formant meets its second partial where it meets it, and the violin’s body resonates where it does. What the compasses change is whether those meetings happen on notes anyone can sing or play together.

Where each worst note falls now

Reading the worst note inside the compass rather than across the map changes more than the lost eight.

Read inside the compass, 11 pairs choose a worst note, and two of them are pairs that chose none before. For every ordered pair, the pair's shared compass as a bar, the full map's worst note as a ring and the worst note inside the compass as a dot. flue pipe over clarinet: D3 to B♭6, worst D3 at an end; trumpet over clarinet: E3 to B♭5, worst E3 at an end; flue pipe over violin: G3 to C7, worst G3 at an end; oboe over clarinet: B♭3 to A6, worst B♭3 at an end; flue pipe over oboe: B♭3 to A6, worst B♭3 at an end; violin over voice: C4 to C6, worst C4 at an end; clarinet over voice: C4 to C6, worst C4 at an end, the full map's E3 unplayable; oboe over voice: C4 to C6, worst C4 at an end, the full map's E3 unplayable; flue pipe over voice: C4 to C6, worst C4 at an end; flue pipe over trumpet: E3 to B♭5, worst C4 inside; trumpet over voice: C4 to B♭5, worst C4 at an end, the full map's E3 unplayable; oboe over violin: B♭3 to A6, worst E4 inside, the full map's E3 unplayable; oboe over trumpet: B♭3 to B♭5, worst F4 inside; trumpet over violin: G3 to B♭5, worst F4 inside, where the full map ran out of compass; violin over clarinet: G3 to B♭6, worst B♭4 inside; violin over oboe: B♭3 to A6, worst B♭4 inside; violin over trumpet: G3 to B♭5, worst B♭4 inside; trumpet over oboe: B♭3 to B♭5, worst D5 inside; voice over clarinet: C4 to C6, worst F♯5 inside, the full map's C♯6 unplayable; voice over trumpet: C4 to B♭5, worst F♯5 inside, where the full map ran out of compass; clarinet over trumpet: E3 to B♭5, worst A♭5 at an end, the full map's F♯6 unplayable; trumpet over flue pipe: E3 to B♭5, worst B♭5 at an end; voice over violin: C4 to C6, worst C6 at an end, the full map's C♯6 unplayable; voice over oboe: C4 to C6, worst C6 at an end, the full map's C♯6 unplayable; voice over flue pipe: C4 to C6, worst C6 at an end; clarinet over violin: G3 to B♭6, worst F♯6 inside; clarinet over oboe: B♭3 to A6, worst F♯6 at an end; clarinet over flue pipe: D3 to B♭6, worst G6 at an end; oboe over flue pipe: B♭3 to A6, worst A6 at an end; violin over flue pipe: G3 to C7, worst C7 at an end.
Fig. 3 For every ordered pair, its shared compass as a bar, the full map’s worst note as a ring and the worst note inside the compass as a dot. Eleven pairs now have a worst note inside their compass rather than at an end of it; two of them — the voice over the trumpet and the trumpet over the violin — had none on the full map.

Eleven of the thirty pairs now have a worst note that the two spectra choose — at least four semitones inside the range they share rather than at one end of it. Seven of them keep the worst note the full map gave them: the violin over the clarinet, the oboe and the trumpet, the clarinet over the violin, the oboe over the trumpet, the flue pipe over the trumpet, and the trumpet over the oboe. Two have lost their full-map worst note and found another inside the compass: the oboe over the violin, whose E3 is below the oboe’s compass and whose margin is least inside it at E4, an octave up; and the voice over the clarinet, whose C♯6 is beyond the soprano and which is worst inside her range at F♯5. The last two are pairs that had no chosen worst note on the full map at all.

The other nineteen pairs have their worst note at an end of the shared compass — which in most cases is simply the lowest note the higher instrument has. Six of the eight lost worst notes do not reappear anywhere inside the compass. The voice over the violin, for instance, is worst at C6, the top of the compass, which is the nearest the soprano can get to the C♯6 the spectra wanted.

Two pairs that turn after all

The earlier essay counted thirteen pairs whose margin simply ran one way across the whole map and was worst where the compass ended. The earlier essay predicted that some of them were monotone only because the map was drawn wider than either instrument could reach — that a margin running downhill from C3 to C7 might well turn inside the range the pair shares. Two of them do.

Two pairs that only ran downhill across the full map turn inside the compass they share. The margin at the strike against pitch for voice over trumpet and trumpet over violin, with each pair's shared compass shaded. voice over trumpet: across the full map its lowest margin is at C7, the end; inside C4 to B♭5 it is lowest at F♯5, −12.6 dB; trumpet over violin: across the full map its lowest margin is at C3, the end; inside G3 to B♭5 it is lowest at F4, 9.1 dB.
Fig. 4 The margin at the strike against pitch for the voice over the trumpet and the trumpet over the violin, with each pair’s shared compass shaded. Across the full map the first is lowest at C7 and the second at C3, both ends. Inside C4 to B♭5 the voice over the trumpet is lowest at F♯5, −12.6 dB; inside G3 to B♭5 the trumpet over the violin is lowest at F4, 9.1 dB.

The trumpet over the violin has a margin that is lowest at C3 on the full map, where neither instrument plays, and rises almost steadily to C7. Inside G3 to B♭5 it dips to a minimum at F4 before rising again. The dip was always there; on the full map it was shallower than the margin’s value at C3, so it did not register as the worst note. The voice over the trumpet is the mirror case: its full-map worst note is C7, far above both, and inside C4 to B♭5 it has a genuine minimum at F♯5, a few semitones below where the voice’s cluster would have been.

So of the thirteen monotone pairs, two turn inside their compass and eleven do not. The earlier essay’s conjecture holds in part: a margin that looks monotone across five octaves can hide a local worst note in the two octaves a pair actually shares.

Why the violin’s is the one that survives

The pattern has a reason, and it is the same mechanism the earlier essay found, read against the compass instead of against the map. Three of the four clusters sit where a note’s fundamental lands on a filter frequency. The violin’s worst note, B♭4, is 466 hertz, and the body is the filter put its strongest plate mode there. The clarinet’s, F♯6, is 1,480 hertz, and its cutoff — the cutoff that is a list explains why a clarinet has one — is 1,500. The voice’s, C♯6, is 1,109 hertz against a second formant of 1,090, which a vowel is two resonances places there for “hod”.

A filter’s worst note is therefore the note whose fundamental sits on the filter, and whether an instrument can play that note depends on where its filter is relative to its range. The violin’s body resonance is low in the violin’s compass: 466 hertz is an octave and a minor third above the open G, and two and a half octaves below the top. So its worst note is a middle note, inside every partner’s range too. The clarinet’s cutoff and the voice’s second formant are high: they are what shapes the upper partials of the instrument’s middle notes, and a note whose fundamental reaches them is at the top of the compass or beyond. An instrument’s worst doubling note is playable when its filter sits low in its own range, and a filter that shapes the top of the spectrum makes a worst note at the top of the compass or past it. The trumpet is the check on the rule from the other side. Its filter is a high pass — the bell decides what gets out, radiating what lies above about 1,150 hertz and reflecting what lies below — so there is no note whose fundamental lands on a peak, and the trumpet has no cluster of its own anywhere on the map.

The fourth cluster, E3, is a different case and fails for a different reason. It is the held voice’s, and it arises where the voice’s fourth and fifth partials straddle its first formant — a low note for a voice, and below a soprano’s range by a sixth. Changing the singer does not bring it back. A tenor has E3 comfortably, but of the four instruments whose margin fails there over a voice, the violin’s lowest note is G3 and the oboe’s B♭3, so neither can double a tenor on it at all; the trumpet’s written bottom sounds E3 exactly; and the clarinet has it only a whole tone above its lowest note, where every instrument’s tone is at its least representative. The E3 cluster is a place where the held voice’s spectrum is unusual, and it is also a place where almost nothing that could double the voice exists. It is a finding about the model’s coverage rather than about any doubling a composer could write.

The violin has one bad note and it is the same note against every partner. The margin at the strike for a violin decaying over each of five held instruments, at every semitone from C3 upward. over voice: worst −34.9 dB at E♭3, best 14.9 dB at C♯6; over clarinet: worst −2.2 dB at B♭4, best 19.6 dB at F♯6; over oboe: worst −0.3 dB at B♭4, best 11.4 dB at E3; over flue pipe: worst 2.7 dB at C7, best 16.9 dB at E3; over trumpet: worst −13.8 dB at B♭4, best −3.4 dB at D3. 3 of the five put the worst note at B♭4 exactly, and the rest have no interior minimum at all — their margins fall or rise across the whole compass and are worst at an end of it.
Fig. 5 The violin’s margin at the strike against each partner across the map. Over the clarinet, the oboe and the trumpet the violin has its worst margin at B♭4, 466 hertz, where its strongest body resonance puts a peak in its spectrum; over the voice and the flue pipe the margin runs one way. B♭4 is more than an octave from every edge of every compass it shares.

Drawn for the violin alone, the survivor is plain: three partners, one dip, at the one note the violin’s own body picks, and that note is in the middle of every compass that meets it. It is the only cluster on the map with that property, and so the only one that is advice rather than a curiosity.

A semitone above the soprano

The one cluster that is lost for a reason worth a figure of its own is the voice’s, because it is lost by one semitone and that semitone depends on which voice.

The voice's worst note, C♯6, is a semitone above a soprano's top CThe margin at the strike against pitch for a voice decaying over a violin, a clarinet and an oboe, with the voice's compass shaded from C4 to C6. All three pairs have their worst margin at C♯6; a voice whose top is C6 does not reach it. Over violin: worst within the compass −5.9 dB at C6; over clarinet: worst within the compass 6.9 dB at F♯5; over oboe: worst within the compass −3.4 dB at C6.C♯6voice over violinvoice over clarinetvoice over oboeC3C4C5C6C7-10010203040pitchmargin at the strike, dB
Fig. 6 The margin at the strike against pitch for a voice decaying over a violin, a clarinet and an oboe, with the voice’s compass shaded from C4 to a top the dial sets. All three pairs have their worst margin at C♯6. A soprano whose top is C6 stops a semitone short of it; an alto whose top is F5 stops eight semitones short; a coloratura reaching F6 has it.

The voice here is modelled on one vowel, “hod”, whose second formant sits at 1,090 hertz, and C♯6 is the note whose fundamental, at 1,109 hertz, lands almost exactly on it. For a soprano whose top is C6, that is one semitone out of reach. For an alto, whose top is nearer F5, it is eight. For a coloratura soprano who sings to F6 it is inside the compass, and for her the voice’s worst doubling note is a note she has. The dial moves the top of the voice’s compass and nothing else, and shows the cluster appearing only once the top passes it.

That makes the voice’s cluster a statement about the model’s vowel rather than about the instrument. A different vowel puts the second formant elsewhere, and on the earlier essay’s arithmetic that moves the cluster with it. A singer who changes vowel changes where her worst doubling note is, and for most sopranos on most vowels it will fall inside her range somewhere; for this vowel it falls just outside.

The table, and what it is joined to

The margin at the strike is the earlier essay’s: each radiator’s partials at each pitch, sixteen of them, the decaying instrument entering six decibels above the held one, and the two log-spectral distances compared. Nothing about it is changed here. The compasses are joined to it by greying every note that either instrument of a pair cannot play, and a worst note counts as chosen by the spectra if it lies at least four semitones inside the shared compass — the earlier essay’s own rule, applied to the pair’s range instead of to the map’s. Two worst notes are in one cluster if they are the same semitone; the four clusters are the pitches at which three or more pairs have their full-map worst note.

The compasses are stated, not computed, and they are the one free parameter the join adds. A narrower violin or a lower trumpet would grey a little more or less and move nothing that the figures above depend on, since the clusters that are lost are lost by at least a semitone and the one that survives is at least an octave from any edge of the compasses it sits in.

What the compasses do not settle

The voice is one singer on one vowel. The cluster the compasses remove is tied to the vowel’s second formant, and a real soprano sings many vowels, modifying them toward the top of her range precisely to keep her partials and formants in useful places. A soprano on a vowel with a higher second formant might have the voice’s worst note inside her compass.

Ranges have soft ends. A compass is where an instrument is written for, not a wall. A professional violinist plays above E7 and a trumpeter above B♭5; the lost clusters are at, or just past, those soft ends, which is exactly where they would be heard least often.

Transposing instruments are drawn at sounding pitch. The clarinet and trumpet compasses are the sounding ranges of B♭ instruments. An A clarinet or a C trumpet shifts them a semitone or two and could bring F♯6 within the trumpet’s reach, though not comfortably.

The flue pipe is given everything. An organ’s flue ranks run through several octaves and are cut to the builder’s plan, so the flue pipe’s rows are not greyed at all; any real rank would grey some of them.

Which doublings, and whose manuals

The pairs are the doublings of the orchestral and choral repertoire, and the margin is the quantity a doubled pizzicato gives its note away early found a room multiplies, so a small margin is a doubling in which the decaying instrument is lost in the hall — an instrument points is the reason the hall hears each partner differently in the first place. The doublings are a voice doubled by an instrument, two woodwinds, a string and a wind, a trumpet reinforcing a line. The earlier essay predicted that orchestration manuals should warn about the violin’s B♭4, the clarinet’s F♯6 and the voice’s C♯6. The compasses sharpen the prediction. A manual has reason to warn about the violin’s B♭4, which every pair that meets it can play; about the clarinet’s F♯6 only with the violin and the oboe; and about the voice’s C♯6 hardly at all, since the voice that has the note is rare and the note is at the top of her compass. That is the one piece of the prediction a reader could now check in a library, and it is narrower than the one it replaces.

Still open: whether the violin’s note is the body’s or the doubling’s

The cluster that survives is the violin’s, and the violin’s body response put it at the B1− plate mode, near B♭4. That resonance is a property of one modelled violin; real violins place it within about a semitone either side. If the cluster tracks the body mode, then every violin has its own worst doubling note and a player could find it; if it tracks something about the doubling partner’s spectrum, it is fixed wherever the violin’s body puts its resonance.

The computation is the margin map with the violin’s body mode moved by a semitone or two either way, joined to the compasses as here. It would say whether the one piece of advice the map still offers is advice about violins in general or about a particular violin — and if it moves with the body, it would be the first result in these essays that an individual instrument, not a family, decides.

Part 15 of 20

One essay in the series on spectrum. The essays either side of this one:

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 Part 14 — 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. 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 Part 16 — 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.

Where this goes next

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. 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. Where each frequency goes, from a source 18 cm across. Polar response of a circular radiator of radius 9 cm at 200 Hz (ka = 0.3), 800 Hz (ka = 1.3), 2000 Hz (ka = 3.3), 5000 Hz (ka = 8.2). Zero degrees is straight ahead. The low frequency is a circle — it goes everywhere — and the high one is a narrow lobe with nulls either side of it, so a listener off to the side hears the same note with its top missing. An instrument points A source radiates evenly while it is small compared with the wavelength and beams once it is not, and the crossover is one number. So the same instrument is omnidirectional in its bottom octave and a searchlight in its top one — which means its spectrum depends on where the listener is standing, and a microphone position is a choice about what the instrument sounds like.

The objects named here

The third way in, after the field and the series: the things themselves, and every essay that touches each one.

BlendDoublingFormantOrchestrationRegisterSpectrum