Orchestration — the series · Acoustics & Music Theory

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.

part 1 · form

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.

part 2 · harmony

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.

part 3 · timbre

A written dynamic is an instruction to the listener's impression. Every earlier scoring holds one chord still. A passage is a succession, and the running impression of loudness carries a chord into the one after it, so what a marking asks for and what playing the marking produces are different things. Here is a five-chord passage with a written shape. Playing each chord at its own written loudness gives the running impression 2.4, 3.0, 4.2, 5.6, 4.0 sones against the 2.4, 3.0, 4.2, 5.6, 2.0 that were asked for — right until the last chord, where it misses by 2.0. Solving for levels that make the impression arrive at the marking does not fix it: the last chord's target is I, two parts, and it is unreachable — the correction runs to silence and the impression still sits 1.1 sones above. A subito piano after a full chord is not a level a player can produce. It is a rate of change, and the smoother's two-second release is what refuses it.

A subito piano is a rate, not a level

All three earlier essays score one chord held still. An orchestration is a succession, and the running impression carries a chord into the one after it — so a written dynamic is an instruction to the listener's impression rather than to the instantaneous sound, and there are markings that cannot be produced at all. The correction runs to silence and the impression still sits above the target.

part 4 · form

One contrast survives every tempo anybody plays and the other does not. How much of each quantity's contrast between chords a listener still has at the end of each chord, against how long a chord lasts. The roughness curve is flat at one down to 45 milliseconds a chord and then falls off a cliff, because its window is 37 milliseconds and a boxcar either fits inside a chord or does not. The loudness curve is already losing at a second a chord and keeps 83 per cent at the slowest pace here, 39 at the fastest. Nothing in music is faster than the roughness window and a great deal of music is faster than the loudness one, so a passage delivers its dissonance and averages its dynamics.

The dissonance arrives and the dynamic does not

A scoring decides two things at once and both of them have to be integrated by a listener before they exist. The loudness smoother's release is two seconds and the roughness window is thirty-seven milliseconds, and that ratio of fifty decides which of the two survives at the pace music is actually played. Nothing anybody performs is fast enough to blur a dissonance, and a great deal of it is fast enough to average a dynamic.

part 5 · form

What the chord before takes out of the chord after. Five chords at 1.2 seconds each, with the roughness each one has on its own — its simultaneous masking and the threshold of hearing already applied — and the roughness it actually has once the chord in front of it has raised the threshold. Four of the five are untouched. The fifth, i, two parts, follows the only step in this passage that falls more than fifteen decibels, and it arrives into a hole: it is entirely below threshold for its first 13 milliseconds and takes 240 to get all of itself back. Masking can only remove partials, so it can only lower a roughness — and the chords it can reach are the ones a written dynamic has just made quiet, which are already the smooth ones. Across the passage the dissonance contrast goes from 3021 to 3113: the mask widens it by 3.0 per cent rather than eating it.

A soft chord has to fade in

Forward masking sits between the two integration times already in play — two hundred milliseconds against a thirty-seven millisecond roughness window and a two-second loudness release — and it was owed as the term that might eat the dissonance contrast. It does not. It widens it, by three per cent at a chorale's pace and fifty-nine at four chords a second, because it can only ever remove partials and it can only reach the chord a dynamic has already made quiet. What it does instead is stranger: one chord in the passage is entirely inaudible for its first twelve milliseconds and takes a quarter of a second to arrive whole.

part 6 · form

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.

part 7 · instruments

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.

part 8 · instruments

An entrance is a change of colour

Eight essays on orchestration move the assignment and hold the ensemble still, and a score does the opposite: it brings players in and takes them out. Loudness is a sum over parts and roughness is a sum over pairs, so the player who joins adds one term to the first and one to the second for everybody already there. What the entrance is worth in phons falls by a factor of forty-eight across the range an ensemble spans and crosses the difference limen at five players; what it is worth in roughness rises by twelve, and by a further factor of ten for every ten decibels the passage is played at.

part 9 · form

The release is on the wrong side

Whether the loudness model's two-second release makes an entrance inaudible has the answer no, for a reason the question did not anticipate. The smoother is asymmetric — ninety-nine milliseconds going up and two seconds coming down — so a rise is tracked twenty times faster than a fall, and an entrance is received promptly by every one of a listener's three readings. The colour of it arrives first, at twenty-five milliseconds against ninety, and the reading that moves with the ensemble is the one nobody would have picked.

part 10 · form

A part that leaves is not a part that arrives

The same player, the same note, the same level, and the only difference is which way round it happens. A listener's loudness reading takes 1.43 seconds to receive half of a departure and 90 milliseconds to receive half of an arrival — a factor of sixteen with nothing asymmetric in the sound at all, since both readings integrate the same two states in the same order. The colour reading receives the two identically, because a window has no direction, so a departure is a change whose grain arrives at once and whose level takes most of two seconds.

part 11 · form

The chord that has room for an entrance

Three essays have made the ensemble something a score can change and none of them has asked when. The ensemble already sounding puts a masked threshold over whatever register an entering part takes, and that threshold is set by the voicing rather than by the dynamic — so the five chords of one passage differ by 8.7 decibels in how much of an entering oboe survives them, and the quietest chord of the five is the worst place in the passage to bring somebody in. Swept over the entrant's own pitch, the choice of moment is worth as much as the choice of register.

part 12 · form

Room is used up by whoever enters first

The chord with the most room for a part entering alone is a fact about that chord. It stops being a fact the moment two parts want it, because each part that comes in raises the mask over everybody after it. Given six parts waiting to enter a five-chord passage, choosing each part's moment the way one part's moment is chosen puts three of them into the same chord and lands in the bottom fifth of all 15,625 schedules. Placing them one at a time does no better. The schedule under which the least audible entrance is heard best is unique, and it brings the low and middle parts in while the texture is thin and holds the three highest back for the last three chords — because a high part keeps its room over a full texture and a middle part does not.

part 13 · form

An exit is worth nothing until the tutti is given up

Six parts entering a five-chord passage have a best schedule when each enters once and stays, and letting parts leave and come back was supposed to improve it. Searched over every set of sounding parts at every chord, it improves it by exactly nothing, with or without the chord before still masking — as long as all six must be playing at the end. Let one part be missing from the final chord and the weakest entrance gains 1.4 decibels; let two be missing and it gains 1.9, by a relay in which the parts with least room come in, are heard for one chord, and give way.

part 14 · form