Acoustics & Music Theory
why it sounds like that
Concept

Doubling — where it appears

Which member of a chord appears twice when there are more parts than notes. The choice is the oldest question in part-writing and it is a ranking rather than a magnitude, which is why it survives a change of dynamic unaltered.

Named by 15 essays across 5 fields — each of them below, with the objects they name alongside it.

Which note a chord would rather have twice. Every complete four-part voicing of each chord inside the SATB ranges, grouped by which member sounds twice and scored for roughness — 480 voicings for a triad. The order for a major triad is root < fifth < third, which is the rule every part-writing treatise states. For a minor triad it is fifth < root < third, which is not. The numbers printed under each bar are the mean error, in cents, with which the four sounding notes fit a single harmonic series, and that measure separates the three far more sharply than roughness does.

The note that sounds twice

A triad has three notes and a four-part texture has four voices, so one note is doubled — and the voicing model used here leaves the choice free because the rules have an opinion about it. Asked properly, the arithmetic agrees with the treatises for the first time in nine essays: root, then fifth, then third. For a minor triad it does not agree, and for a symmetric chord it correctly has nothing to say.

intervals · The triad Roughness and loudness do not rise together. One four-note chord, played at levels from 35 to 95 decibels, with both quantities drawn as multiples of what they are at the quietest. Roughness is quadratic in pressure, so 60 decibels multiply it by 1.0e+6. Loudness is compressive — about ten phons to a doubling of sones — so the same range multiplies it by 96. The gap between the two lines is the quantity: roughness per sone rises by a factor of 1.0e+4 between a pianissimo and a fortissimo of the same chord.

The ranking survives the dynamic and the chord does not

Roughness is quadratic in pressure and loudness is compressive, so sixty decibels multiply a chord's roughness by a million and its loudness by ninety-six. Roughness per sone therefore rises ten thousandfold between a pianissimo and a fortissimo of the same four notes — and yet the ranking of which doubling is smoothest, over four hundred and eighty voicings, does not move by a single place.

harmony · 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 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 Scored as one arrival, a pair enters as a loud part carrying one that cannot be heard. Six parts entering, leaving and re-entering a five-chord passage over four sounding players, in the walk through all 64 sets of sounding parts that makes the least audible entrance as audible as possible, with no memory of the chord before, at least 2 of the six sounding at the last chord. I, open: clarinet on G4 enters at -10.9 dB, violin on G5 enters at 19.6 dB; vi, close below: oboe on E4 enters at -1.5 dB, voice on C5 enters at 14.5 dB, and clarinet and violin leave; IV, close above: nobody enters, and oboe and voice leave; V, bracketing: brass on E3 enters at -0.0 dB, flue pipe on C6 enters at 25.0 dB; I, hollow: nobody enters. Scored with the parts entering on a chord taken together, at most 2 at a time, as one arrival against what was already sounding: the weakest arrival is 10.93 dB, and the most buried single entrant, measured as itself, is -10.94 dB.

A pair entering as one hides the part that cannot be heard

Scored as two entrances, a pair of parts coming in together is charged for masking itself; scored as one arrival — a new colour against what was already sounding — it is not, and the best schedules change completely. Allowed two at a time, the weakest arrival rises from 0.6 dB above the mask to 10.9, and every pair the search chooses is a loud part carrying a quiet one: a violin 19.6 dB clear beside a clarinet 10.9 under. Allowed any number, the six parts enter at once. The staggered build-up of every earlier schedule exists only because each part was meant to be heard as itself.

form · Orchestration Every decibel taken off the passenger comes off the arrival. The best walk with parts entering together scored as one arrival, at most two at a time, against a floor under each member: no entrant, heard as itself, more than the floor below its mask. Solid lines are the weakest arrival, dashed the most buried member, for two parts required at the end and for all six. 2 at the end: floor 0: arrival 0.59, most buried 0.59; floor 1: arrival 0.59, most buried -0.86; floor 2: arrival 0.59, most buried -1.57; floor 3: arrival 0.59, most buried -2.87; floor 4: arrival 0.59, most buried -3.47; floor 5: arrival 0.59, most buried -3.47; floor 6: arrival 4.50, most buried -5.42; floor 8: arrival 4.50, most buried -6.16; floor 10: arrival 10.93, most buried -9.90; floor 12: arrival 10.93, most buried -10.94; every part heard as itself, the earlier walks' score, 0.59. 6 at the end: floor 0: no walk; floor 1: no walk; floor 2: no walk; floor 3: arrival -2.65, most buried -2.65; floor 4: arrival 0.50, most buried -3.27; floor 5: arrival 0.59, most buried -4.24; floor 6: arrival 4.50, most buried -5.42; floor 8: arrival 4.50, most buried -6.13; floor 10: arrival 4.50, most buried -6.13; floor 12: arrival 4.50, most buried -6.13; every part heard as itself, the earlier walks' score, -2.65.

A floor under each member is paid for in steps

Score parts entering together as one arrival and the best pairs are a loud part carrying one that cannot be heard. Put a floor under each member — no part more than so many decibels under its own mask — and the prediction was a schedule between the two, neither staggered nor buried. There is none. The arrival's margin climbs in three steps as the floor is loosened, each step bought with a passenger buried about as deep as the step is high, and below a floor of five decibels nothing beats bringing the parts in one at a time, because the clarinet can clear its mask only alone.

form · Orchestration 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 How far under its mask a passenger is, and how much it still changes. Every pairing of two parts entering together on any chord of the passage in which the quieter part — the passenger — sits under its mask on average: 71 of them. Across, the passenger's mean margin, the measure every earlier walk used; up, how far its partials together change the arrival above the threshold of detection. Every one of the 71 is above zero: the lowest is 0.11 dB. 35 of the 41 buried more than 3 dB by the mean still change the arrival by more than 3 dB. The few near the zero line are the clarinet on the bracketing dominant, where every one of its partials is just under.

A passenger under its mask is still heard in the arrival

The best way to bring parts in together was a loud part carrying a quiet one, and the quiet one was scored as buried — ten decibels under its mask, on average over its partials. An average is the wrong question. A masked threshold is the level at which a sound becomes a detectable change, and nearly every buried passenger has partials well clear of it. Scored as a change to the arrival rather than as a line of its own, every one of seventy-one passengers is heard, and the carriers' walk passes almost untouched.

form · Orchestration 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 Two parts entering together mask each other unequally: the carrier loses a partial or two, the passenger up to eleven. Every pairing of two parts entering together on a chord of the passage in which the quieter part sits under its mask on average, 71 of them, sorted. Up: how many of the carrier's partials that are clear of the mask with the carrier alone over the chord are pushed under it by the passenger. Down: how many of the passenger's are pushed under by the carrier. In all, carriers lose 19 partials and passengers 135. 54 carriers lose none, 15 lose one and 2 lose two; passengers lose 0 in 25, 1 in 25, 2 in 6, and three or more in 15, the most being 11.

A carrier gives up room, not partials

A passenger masks its carrier as the carrier masks it, so the arrival might be a new spectrum belonging to neither part. Across the seventy-one pairings of the passage it almost never is. Carriers lose nineteen partials in all to their passengers and passengers lose a hundred and thirty-five to their carriers; a voice beside a violin takes thirteen decibels of the violin's room and not one of its partials. The nine pairings that leave neither part a majority are mostly on one chord, and there the chord had already done it.

form · Orchestration A third part takes from the carrier what it would have taken alone; the room the passenger spent adds almost nothing. Every pairing of carrier and buried passenger, with each other part of the roster entering beside them as a third: 284 arrivals. For each third part, the carrier's partials — clear of the mask over the chord alone — that are under it with all three sounding, split into those the passenger alone already took, those the third part would take on its own, and those lost only to the two together. clarinet: 46 arrivals, 19 partials (14 to the passenger, 3 to the third alone, 2 only together), the third alone costing 1.5 dB of room; oboe: 50 arrivals, 20 partials (9 to the passenger, 9 to the third alone, 2 only together), the third alone costing 1.3 dB of room; brass: 46 arrivals, 23 partials (8 to the passenger, 14 to the third alone, 1 only together), the third alone costing 1.0 dB of room; voice: 47 arrivals, 40 partials (14 to the passenger, 20 to the third alone, 6 only together), the third alone costing 7.3 dB of room; violin: 47 arrivals, 127 partials (17 to the passenger, 108 to the third alone, 2 only together), the third alone costing 8.8 dB of room; flue: 48 arrivals, 209 partials (14 to the passenger, 187 to the third alone, 8 only together), the third alone costing 14.1 dB of room. In all 438 partials: 76 to the passengers, 341 to the thirds on their own, 21 only to the combination.

A third part takes what it would have taken alone

A passenger costs its carrier room rather than partials, and the obvious worry is that the room compounds: that a third part, entering beside a carrier already ten decibels nearer its limit, finds it fragile. Across 284 three-part arrivals it does not. Of the 438 carrier partials lost with three parts sounding, 21 are lost only to the combination; the rest are taken by one of the two parts alone. Masking adds in power, so a threshold the passenger has raised rises less when a third part adds to it. What decides the loss is which part enters third, and two parts — the flue and the violin — take five sixths of it.

form · Orchestration

Named alongside it

The objects these essays reach for when they reach for this one.

OrchestrationMaskingRegisterSpectrumTimbreTextureResonanceRoughnessSource-filterDecayEnvelopeOptimisation

All concepts