Every assignment, at equal levels and at its own best balance
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. 145 of its 145 placements carry sound, built from the same numbers as the drawing, offering these buttons: 0.15 s a chord: 100% of the roughness contrast kept, 76% of the loudness, 2.4 s a chord: 100% of the roughness contrast kept, 83% of the loudness, 45 dB — the shape, not the level, 5 parts, then the flue pipe on C5 entering: its loudness halfway there in 90 ms, 70 dB — the shape, not the level, 95 dB — the shape, not the level, A cadence with one doubling held: 16.2% rougher than re-scoring every chord, A rising sequence with one doubling held: 0.0% rougher than re-scoring every chord and 113 more.
Called by 23 essays
the blast radius of changing it
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.
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.
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.
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.
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.
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.
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.
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.
A general pause is spent by the note after it
Whether a composer should write the pause before the diminuendo or after it looks like a question about how big the ensemble is. It is not. Forty decibels of ensemble are worth one decibel of silence, and the order is worth seven — because a running impression rises twenty times faster than it falls, so half a general pause is spent by ninety-seven milliseconds of sound.
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.
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.
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.
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.
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.
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.
A part that must stay changes over in shifts
The best way to keep every entrance audible, when the passage need not end with everybody in, was a relay: a part enters, is heard for one chord, and gives way. Make every part stay at least two chords once it is in and the relay does not collapse into overlaps, as predicted. It changes over in shifts: two parts enter together, one chord passes in which nobody enters, and both leave on the chord their successors arrive. Half of what exits were worth survives a two-chord stay and nearly two fifths a three-chord one — and with the chord before still masking, the stay costs less, not more, because it was the one-chord handover that memory had already made leaky.
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.
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.
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.
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.
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.
A quieter third part gives back only as deep as it took
A high part entering third costs its carrier partials, and the masking rule suggests that marking it down should return them all at once. It does not. Each partial comes back when the third part has been lowered by about as far as it had pushed that partial under, so the return is as gradual as the depths are spread. The violin's are shallow: twelve decibels down it gives back nine-tenths and is still heard in two arrivals of three. The flue's run to thirty decibels, and there is no level at which it is both heard and harmless.
An octave down a third part is heard and harmless, with nothing to spare
A flue pipe cannot be marked down, but it can be moved, and the expected answer was upward: an octave higher clears the carrier's main region and spreads over partials that are already faint. The count says the opposite. Up an octave a third part takes half as many partials again. Down an octave it takes a quarter of what it took, gives a voice back its whole upper spectrum, and is still heard in nearly every arrival. No level reaches that. The price is that the part has spent its spare level, and three decibels soft takes it out of a third of them.