A floor under each member is paid for in steps

A pair entering as one hides the part that cannot be heard changed how a pair of parts entering on the same chord is scored. Every earlier walk had heard each entering part as itself, masked by everything else sounding including its partner, and so charged a pair for masking itself. Scoring the pair once, as a single arrival against what was already sounding, made pairs look ten decibels better. It also changed what a good pair is. Every best pair was a loud part carrying a buried one: a violin clearing its mask by 19.6 decibels with a clarinet 10.9 under its own.

That essay’s closing section treated the two scores as bracketing the answer. One charges a pair for masking itself and produces staggered entrances; the other charges nothing and produces carriers and passengers. An orchestrator who wants every part heard, and is willing to bring parts in together, wants something between them: the pair scored as one arrival, but with a floor under each member, so that no passenger may sit more than a stated number of decibels under its mask. The prediction was that a tight floor would bring back the staggered schedule, a loose floor would bring back the carriers, and somewhere between would be a schedule that neither staggers every part nor buries any — doubling as a way of making an entrance audible rather than of hiding one.

There is no such schedule in this passage. Swept from nothing to twelve decibels, the floor moves the best walk in three steps. Each step up in the arrival’s margin is bought with a passenger buried roughly as deep as the step is high. Below a floor of about five decibels nothing does better than bringing the parts in one at a time, and the reason is one part: the clarinet.

The walk and the floor #

The passage and parts are the earlier essays’. Four players sound five chords — an open tonic, a close submediant below, a close subdominant above, a bracketing dominant and a hollow tonic — and six further parts (flue pipe, brass, clarinet, oboe, voice and violin, each on one note) may enter, leave and re-enter. A walk is a sequence of the sets of parts sounding at each chord. Every part must enter at least once, a stated number must be sounding at the end, at most two may enter on one chord, and the walk is scored by its weakest entrance. The best walk is found exactly, by dynamic programming over the sixty-four possible sets.

An entrance is scored as one arrival. Its margin is the mean, over every partial of every part entering, of how far that partial clears the threshold that the four players and the parts already sounding put over it; the entering parts do not mask each other. The new constraint is the floor. Each entering part is also scored as itself, masked by everything else sounding including its partner, exactly as the earlier walks scored every entrance, and an arrival is allowed only if no member’s own margin is below minus the floor. A floor of zero requires every entering part to clear its own mask. With no floor, the walk is the earlier essay’s.

Three steps and a flat #

With two parts required at the end, the arrival’s margin does not rise smoothly as the floor is loosened. It sits at 0.59 decibels for every floor up to 5.4. Then it jumps to 4.50, then to 7.05 at a floor of about 8.6, and to 10.93 at about 10.9, where it reaches the earlier essay’s no-floor value and stops.

Each step is bought with a passenger. The step to 4.50 comes when the floor admits a clarinet 5.42 decibels under its mask. The step to 7.05 comes when it admits one 8.63 under, and the step to 10.93 when it admits one 10.94 under. Every decibel the arrival gains above the one-at-a-time schedule is a decibel, or near enough, that some passenger is allowed to lose. No setting of the floor produces a pair in which both members are comfortably above their masks and the arrival is also strong, which is what the prediction’s middle schedule would have been.

The walk at a floor of six shows the shape. Three chords, three pairs. On the open tonic, the oboe at 3.5 decibels over its mask carries the clarinet at 5.4 under. On the submediant, the violin at 7.6 over carries the voice at 3.3 under. On the subdominant, the flue pipe at 18.9 over carries the brass at 3.1 under. The pairs are still carriers and passengers, only milder. The floor has limited how far under a passenger can be; it has not made anyone into a partner.

Why nothing happens below five decibels #

The flat stretch, from no floor at all up to 5.4 decibels, is where the prediction’s middle schedule should have been, and the arrival there is exactly the one-at-a-time walk’s: 0.59 decibels. The walk inside that stretch still uses pairs, but its weakest entrance is always the same one, the clarinet entering alone on the open tonic.

The clarinet is the part the whole walk turns on. Entering on the open tonic by itself, it clears its mask by 0.59 decibels, and that is its best anywhere. With its least costly partner, the brass, it is 1.4 under. On the submediant it is 3.7 under alone, and on the three later chords, where the four players’ voicing puts their partials across its register, it is ten to twelve under whatever it does. So the clarinet has exactly one audible entrance in the passage, and it is a single entrance on the first chord.

That fixes the flat. Any walk whose weakest arrival is to beat 0.59 must bring the clarinet in some other way, and every other way puts it at least 1.4 decibels under its own mask — in practice the walk that gains most puts it 5.4 under, with the oboe. Until the floor admits that, the best the walk can do is let the clarinet in alone and make every other arrival stronger than 0.59, which it easily can. The prediction imagined a floor as a gentle pressure on every pair. In this passage it is a door for one part: the floor decides when the clarinet may become a passenger, and nothing else about the walk changes until it does.

The walk inside the flat #

The walk the flat stretch produces is worth drawing, because it shows what the floor does to everything except the clarinet.

At a floor of three the clarinet enters alone on the open tonic, clear of its mask by 0.59 decibels, which is the walk’s weakest arrival. On the submediant the violin enters with the voice, 8.0 decibels over its mask against the voice’s 2.9 under, an arrival of 19.0. On the subdominant the flue pipe enters alone at 19.3. On the dominant the brass and the oboe enter together as the flue leaves, 2.6 and 2.7 under their masks, an arrival of 0.9.

That last pair is the nearest thing in any walk here to the prediction’s middle schedule: two parts entering together, neither buried by more than three decibels, making an arrival a little stronger than either would be alone. It exists, and it does not raise the walk’s score, because the walk is only as good as its weakest arrival and the clarinet’s is weaker. The prediction’s schedule is available on some chords of the passage and never on all of them at once. On a passage without a part like the clarinet, the pair on the dominant would be typical rather than exceptional.

The staggered schedule was already here #

The earlier essay’s other bracket was the staggered schedule, and the floor recovers it only at the bottom. A part that must stay changes over in shifts found that a walk made to keep each part two chords becomes a change of shift, pairs in and pairs out, spread in register so that neither member masks the other. At a floor of zero, where every entering part must clear its own mask, the walk here is exactly the one-at-a-time walk of the earlier essays with one pair at the end, weakest entrance 0.59. The staggered schedule is what a floor of zero forces, and the floor has to be loosened past five decibels before anything else is better.

So of the two brackets the earlier essay drew, the floor moves between them in the way a switch does, not the way a dial does. Tight, it gives the staggered walk. Loose, it gives the carriers. The interval between is not a third kind of schedule but a region in which the second kind is not yet allowed. An exit is worth nothing until the tutti is given up found the same shape for exits: a constraint that does nothing until it suddenly does everything. That is what an objective set by its single worst member tends to produce. It moves only when the worst member changes, and in this passage there are only a handful of members that could be worst.

A floor tight enough admits no walk at all #

Requiring more parts at the end makes the floor bite differently.

With all six parts required at the last chord, the walk must build to a tutti, and a tutti cannot be built with every entering part near its own mask. At floors of two decibels or less there is no walk at all: some part, entering somewhere, must be more than two decibels under. At a floor of three the only walk is the earlier essays’ one-at-a-time build-up, whose weakest entrance is 2.65 under. At four it is 0.50 over, and from six upwards it is 4.50 however loose the floor, because with a tutti at the end the carriers cannot all leave to make room.

The intermediate cases fill in the grid. With four parts required at the end, a floor of nothing admits no walk and a floor of one admits one at 0.30. With five, floors of one or less admit none. So the more of the ensemble must be playing at the end, the looser the floor has to be before any schedule exists, and the tightest the floor can be is set by how buried the last parts to enter are bound to be.

Carriers are not a scoring artefact #

The earlier essay could have been read as showing that the one-arrival score was too generous, since it produced passengers because it did not care about them. The floor rules that reading out. The walk cares about passengers here, exactly as much as the floor says, and it still produces carriers. The carriers were never there because the score ignored the passengers. They are there because in this passage a strong arrival requires a buried member: the arrival’s margin is a mean over both parts’ partials, and the partials that raise it belong to a part that is well clear of the mask, while the partner that makes it a pair has to be somewhere, and the only places left are under a mask.

An orchestrator doubles a line, not a chord found that doubling pays when the second instrument supports the first rather than adding its own voice. The steps here are that finding stated as an exchange rate. In this passage a doubling makes an entrance about as much more audible as the doubled part is made less audible, until the doubled part is fully buried.

The chord that has room, and the part that does not #

The clarinet’s single audible entrance is on the open tonic, and it is worth saying why that chord and no other. The chord that has room for an entrance asked which chord of a passage leaves room for a given part, and the answer depends on where the players already sounding put their partials. For the clarinet’s G4, the open tonic is the only chord that leaves enough: its masking threshold there is lower than on any other chord by four decibels or more. On the three close chords that follow, the players’ partials lie across the clarinet’s register and it is ten or more decibels under whatever it does.

Room is used up by whoever enters first then found that a gap can be spent only once, and the floor makes that literal. On the open tonic, a pair spends the gap on two parts; alone, the clarinet has it to itself. Every walk in the flat stretch gives the gap to the clarinet alone, because it is the only place in the passage where the clarinet can be heard at all, and the other five parts can be heard somewhere else. The floor does not create that constraint. It exposes it, by forbidding the one move — burying the clarinet under a partner — that the unfloored walk used to get round it.

That points to the practical reading. When an orchestrator wants a weak part heard at its entrance, the arithmetic here says to find the chord whose voicing leaves its register open and bring it in there alone. Pairing it with a strong part helps only if the pairing is meant to hide it.

What the arithmetic assumes #

One passage and one roster. Every number is for the five chords and six parts of the earlier essays, with the clarinet on G4. The flat stretch belongs to the clarinet, and a roster without such a part — one whose every member has an audible entrance with a partner — would have no flat and might have the middle schedule the prediction imagined.

A mean over partials. An arrival’s margin averages every partial of its members. That average is what lets a loud carrier lift a buried passenger’s score, and a score that took the worse member, or weighted the passenger’s partials more, would be a different arrival rule with fewer carriers.

Masking without time. As in the earlier walks the parts start together and hold, and the masking is simultaneous. A part entering is not a change of level found an entrance heard by its own onset, and an onset asynchrony of a few tens of milliseconds between carrier and passenger would give the passenger a moment on its own that the arithmetic here does not credit.

Whose doublings #

The practice at issue is the orchestral habit of bringing a weak instrument in with a strong one: the clarinet with the violins, the horns doubled by the bassoons, the voice supported by the strings at an entry. It is usually justified by blend or by security. The fourth player is a spectrum found that what an orchestrator adds is partials, not a player. On the arithmetic here, it is a trade, and in this passage the trade is fixed: the weak part can be made part of a strong arrival or it can be heard, and the floor sets the exchange rate between the two without ever removing it.

Still open: whether a passenger two decibels under its mask is heard in the arrival #

Every floor above treats a passenger’s margin as a threshold: over it the part is heard, under it the part is not. But a part 1.4 decibels under a simultaneous mask is not silent. It changes the timbre of the arrival it is in, and an entrance is a change of colour was about exactly that kind of hearing. The next computation asks not whether each member is heard as itself but whether the arrival with the passenger is distinguishable from the arrival without it — a spectral difference against a discrimination threshold rather than a masking margin. If a clarinet five decibels under its mask still audibly colours the oboe it enters with, then the carriers above are doing what orchestrators say doublings do, and the floor was measuring the wrong kind of audibility.