An exit is worth nothing until the tutti is given up

Room is used up by whoever enters first searched every way of bringing six waiting parts into a five-chord passage — a brass part on E3, an oboe on E4, a clarinet on G4, a voice on C5, a violin on G5 and a flue pipe on C6, over four players already sounding — and found the schedule under which the least audible entrance is heard best. It brings the brass and the clarinet in on the first chord, the oboe on the second, and the voice, the violin and the flue pipe one a chord after that; the oboe’s entrance is the weakest, 2.65 decibels under the mask the others put over it.

That search had two restrictions it named itself. A part entered once and stayed, so nobody ever left. And every entrance was scored against what was sounding at that chord alone, with no memory of the chord before, although a sound hides what came before it for a fifth of a second after it stops. The essay observed that removing the first restriction costs nothing computationally — six parts can be sounding in sixty-four combinations, and any schedule of entrances and exits is a walk through those combinations chord by chord — and asked whether, with exits and a memory both in, the best way to make room for a part would be to take somebody else out.

The walk has now been run, and the answer comes in two halves that point in opposite directions.

The walk, and what it scores #

A state is the set of the six parts sounding at one chord. A walk is a state for each of the five chords, and every change from one chord to the next is some parts entering and some leaving. Each entrance — a part absent at one chord and present at the next — is scored exactly as the earlier essay scored one: the mean, over the entering part’s partials, of how far each sits above the threshold that everything else sounding puts over it. The objective is the least of those margins over the whole walk, so a walk is only as good as its worst-heard entrance.

Two constraints make it a question about orchestration rather than about a parade of soloists. Every part must enter at least once, so no part can be silenced to spare the others. And a stated number of parts must be sounding at the last chord; six reproduces the earlier essay’s texture built up to a full tutti, and fewer lets the passage end thinner.

The search is exact. The best least-margin of any walk ending in a given state, having brought in a given set of parts so far, depends only on that state and that set, so it is carried forward chord by chord over sixty-four times sixty-four combinations. The entrance scores are computed once for every pair of consecutive states, and the whole space — which counted naively is over a thousand million walks — takes a few hundredths of a second.

The memory is one number. An entrance is masked by what sounds with it and also by what sounded at the chord before, reduced by a stated number of decibels: at infinity the chord before is forgotten, which is the earlier essay’s model and reproduces its schedule and its 2.65 exactly; at zero the previous chord masks at full strength.

With the tutti required, an exit buys nothing at all #

When all six parts must be sounding at the end, allowing exits changes nothing.

Not a small improvement, and not a different schedule of the same quality: the best walk with exits allowed is the same six entrances on the same six chords, with no part leaving. That holds at every strength of memory tried. With the chord before forgotten, the weakest entrance is 2.65 decibels under; with it masking twenty decibels down, 2.67; twelve down, 2.82; eight down, 3.06; four down, 3.60; and at full strength, 4.65. The schedule does not move at any of them.

The reason is a counting argument and it is worth having because it generalises. An exit is only worth making if the part that leaves makes room for somebody’s entrance. But a part that leaves must come back, since it has to be playing at the end, and it comes back into the passage later — and later, in a texture that is being built up to a full tutti, means into a fuller texture than the one it left. Every exit creates an extra entrance, and in a build-up the extra entrance is into a fuller texture than the part’s first one. Under an objective that is only as good as the worst entrance, a trade of one good entrance for an extra worse one never pays.

The memory makes each entrance harder without changing the order in which parts are hardest to hear. That is why every margin falls with it and no decision does: it lowers the oboe’s entrance by two decibels and the flue pipe’s by 1.2, and the oboe is still the weakest at every strength.

Each part let off the final chord is worth something #

Relax the ending and exits start doing work immediately.

With no memory, requiring five of the six at the end raises the weakest entrance from 2.65 decibels under the mask to 1.27 under. Requiring four raises it to 0.72 under; three, to 0.26 above; two, to 0.59 above, and one is no better than two. Every part released from the final chord is worth something, and the first is worth the most.

The constraint does not say which part is let off, and the walk’s choice is informative. Asked to end with five, it releases the oboe — the part whose entrance was the weakest in the build-up. The clarinet now enters alone on the open tonic, 0.6 decibels above the mask; the oboe enters on the submediant with only the clarinet from the roster sounding, 0.6 under; and on the subdominant the brass and the voice enter while the oboe leaves. The weakest entrance becomes the brass’s, 1.27 under. So the first part given up is not the quietest or the lowest but the one the build-up could least afford to keep adding to.

The dashed line is the other half of the argument. If nobody may leave, a thinner ending is worth nothing: the schedule must still bring every part in, and a part that is in stays, so the final chord is full whatever the constraint says. Exits and a thin ending are each worthless alone and worth more than three decibels together — the constraint on the ending is what an exit is paid out of.

The relay #

The walks that use exits all have one shape, and it is visible in the figure at the top of the essay.

With four parts required at the end, the clarinet enters alone on the open tonic, 0.6 decibels above the mask. On the next chord the oboe enters, 0.3 above, and the clarinet leaves. On the third the brass and the voice enter and the oboe leaves. The violin and the flue pipe follow, and the passage ends with brass, voice, violin and flue pipe sounding. The two parts that leave are the clarinet and the oboe, each sounding for exactly one chord: it enters, is heard, and gives way.

With only two required, the relay runs the length of the passage. Each chord has one new part and loses the one before it, until the oboe and the flue pipe finish together. The violin, entering on the submediant with nobody else from the roster sounding, is 23.9 decibels above the mask — there is nothing in that chord’s register to hide it — and the weakest entrance anywhere is the brass’s, 0.59 above.

Two things about the relay are consistent with the earlier essay and one is new. The parts released first from the ending are the clarinet and the oboe, which that essay found have room over a thin texture and lose it quickly as the texture fills; the parts kept to the end are the ones that keep their room over a full one. That is the same register fact the chord that has room for an entrance measured one part at a time, read from the other side. What is new is the handover: a leaving part leaves on the chord its successor enters, never a chord before. In this model a part that left a chord early would have sounded for no chords at all, and every part has to be heard entering once. So “take somebody out before the next part comes in” is not a strategy the walk has available unless the leaving part has more than one chord to sound.

What the chord before costs a relay #

A relay is exactly the texture the memory should hurt, because a part that leaves on the chord its successor enters was sounding one chord earlier and so is still in the forward mask.

It does. The build-up to a full tutti loses 2.00 decibels from its weakest entrance as the memory goes from absent to full strength. The best walk with four required at the end loses 2.73, from 0.72 under the mask to 3.45 under, and its advantage over the schedule in which nobody leaves shrinks from 1.92 decibels to 1.20. At every one of the six strengths of memory the best walk still hands parts over on the chord of the entrance, so the memory does not change the relay’s shape; it makes each handover leakier, because the part that has just given way is still in the listener’s ear for the first moments of its successor.

That answers the question the earlier essay left, and the answer is narrower than the question. The best way to make room for a part is sometimes to take somebody else out — but only when the passage is not obliged to end with everybody in, and the somebody is taken out on the chord of the entrance, not before it. A memory of the chord before makes that trade worth about a third less, and never makes it worth nothing.

The arithmetic #

The passage, the four players under it and the six waiting parts are the earlier essay’s, each part a radiator from the collection’s measured set at seventy decibels, fourteen partials a part renormalised to unit power and tested against the threshold of hearing. A partial’s threshold is the power sum of the spreading functions of every component sounding with it — the same ones by which one sound hides another — plus, when the memory is on, the spreading functions of every component of the chord before, including the four base players, reduced by the stated decibels; the result is floored at the threshold of hearing. An entrance’s margin is the mean of its partials’ levels over that threshold.

The walk is found by dynamic programming over the chord, the set sounding and the set that has entered at least once, maximising the least entrance margin so far, and the ending is the best state with every part entered and at least the stated number sounding. A walk with nobody leaving is the same programme restricted to states that only grow, which at six required at the end reproduces the earlier essay’s exhaustive search of 15,625 schedules to the hundredth of a decibel.

Where the walk stops being a score #

The memory is a flat reduction. Forward masking decays over about two hundred milliseconds, so how much of the chord before is still in the ear depends on how long the chords are and where in the new chord the entrance is heard. A single number for the whole previous chord treats every entrance as heard at the onset; the release is on the wrong side found a listener takes about a tenth of a second to receive one, by which time most of the forward mask has gone. The sweep across strengths is there because the right value is a fact about tempo.

The running loudness is not in it. The earlier essay named two memories and only one is here. A part that leaves is not a part that arrives found that the running loudness receives an arrival in a tenth of a second and a departure over more than a second, which would make a relay sound thicker than its score, and it would not change which entrance is weakest unless the loudness enters the objective as well.

A part that sounds for one chord is a gesture, not a line. The relay’s clarinet and oboe each play one held note for one chord, which is a colour, a fill, an interjection — real devices, but not what a composer bringing in a woodwind section means by an entrance. A constraint on how long a part must sound once it enters would turn the relay back into something like the build-up.

And the objective sees only entrances. It is one term of the larger decision who plays what and how loud is one question set out, solved alone. A walk is scored on the moments parts come in, not on whether the texture between those moments is balanced, harmonically complete or playable, and not on whether any part can be heard once it has been in for a chord.

What a walk through states cannot show #

Whether a listener tracks entrances at all. The objective assumes the thing an orchestrator is protecting is each part’s arrival being heard. In much orchestral writing an entrance is meant to be felt as a change of colour rather than heard as a new line, which is what an entrance is a change of colour is about, and for those entrances a negative margin is the intention.

Whether a thinner ending is the price or the point. The arithmetic says every part released from the final chord buys the weakest entrance room. Whether a composer would rather hear every entrance clearly or end with the full ensemble is a decision about what the passage is for, and neither choice is a better score.

Whose relays #

Handing a line or a colour from one instrument to the next on the chord where the next comes in is a standard orchestral technique — the melody passed through the woodwinds, a sustained note taken over by a different colour, a Klangfarbenmelodie made of single notes in different instruments. The arithmetic says this is the best way to keep every arrival audible when the ensemble does not have to be complete at the end, and that it is worth nothing when it does. That is consistent with where such relays appear, in textures that thin and change rather than in the build-up to a tutti, but nothing here has counted scores, and the consistency is a property of this passage, this roster and this model.

Still open: a part that must stay once it is in #

The relay found here depends on parts that sound for a single chord and leave. The natural constraint that removes that — a part, once in, must stay for at least two chords, or for a stated fraction of the passage — is one more number in the state, since the walk must remember how long each part has been sounding. With six parts and a two-chord minimum that is still small enough to search exactly.

The prediction the figures above make is specific. A minimum stay should destroy the one-chord relay and force the leaving parts to overlap their successors for a chord, which puts each successor’s entrance under both the leaver’s simultaneous mask and the fuller texture — so the gain an exit buys should shrink sharply, perhaps to nothing, at a two-chord minimum. If it does not, there is a strategy the walk has not yet shown; if it does, then exits in a texture of sustained lines are worth nothing whether the ending is full or not, and the relay is a device only for textures made of short gestures.