A part that must stay changes over in shifts

An exit is worth nothing until the tutti is given up searched every way six parts can enter, leave and come back over a five-chord passage, scoring each walk by its least audible entrance. With all six required at the last chord, letting parts leave bought nothing. With fewer required, every part released from the ending bought room, and the best walks all had one shape: a relay, in which a part enters, is heard for one chord, and leaves on the chord its successor arrives. With only two required at the end, every entrance of the relay was above its mask, the weakest by 0.59 decibels.

The essay’s own list of limits named what was wrong with that as orchestration. A part that sounds for one chord is a gesture — a fill, an interjection — not a line. Its closing question added the constraint that turns gestures back into lines: a part, once in, must stay for at least two chords. And it predicted the result. The leaving parts would have to overlap their successors, each entrance would come in under both the leaver and a fuller texture, and the gain exits had bought would shrink sharply, perhaps to nothing.

The gain shrinks, and by about half. The overlaps never happen.

The walk with a memory of its own stints #

The search is the earlier essay’s with one more thing remembered. A state is the set of parts sounding at a chord and the set that have entered at least once; to that it adds, for each chord of the minimum stay, the set of parts that entered that recently and so may not yet leave. A part in that set can stay or be joined by others, but it cannot drop out. A stint that runs to the last chord is allowed to be shorter, since the passage ends under it.

Every other part of the scoring is unchanged, and it is the scoring room is used up by whoever enters first built for parts that entered once and stayed, with the thresholds of one sound hides another underneath it. An entrance is a part absent at one chord and present at the next, and its margin is the mean, over its partials, of how far each sits above the threshold that everything sounding with it — the four players of the passage and whichever of the six are in — puts over it. The walk is only as good as its worst entrance. Every part must enter at least once, and a stated number must be sounding at the end.

With a minimum stay of one the search is the earlier essay’s exactly, and the figures check that it reproduces its walks. Only a few thousand states are reachable at each chord, far fewer than a table over every combination would hold, so the search stays exact.

The relay becomes a shift change #

With two parts required at the end and a two-chord stay, the best walk is the hero figure’s, and its shape is not the one predicted.

The clarinet enters alone on the open tonic, 0.6 decibels above the mask, as in the free relay. On the submediant nobody enters — the clarinet is serving its second chord, and nobody else is brought in to share it. On the subdominant the brass and the voice enter together and the clarinet leaves. The violin enters on the dominant, over the brass and the voice. On the last chord the oboe and the flue pipe enter together and the brass and the voice leave together.

So the walk does not keep the relay and force it to overlap. It keeps the relay’s handover — a part leaves on the very chord its successor enters, never earlier, never later — and halves its rate. Parts come and go in pairs, and one chord of five is spent with no entrance at all, so that the part already in has served its minimum by the time anyone replaces it.

What the minimum stay removes is the relay’s pace, not its handover. A chord that nobody enters is a chord in which nobody’s entrance can be masked, and pairing two entrances puts each under the other; those are the two prices the walk pays instead of overlapping. The pair on the subdominant is the brass and the voice, at opposite ends of the register — the brass’s E3 is two octaves under the voice’s C5, so neither is much in the other’s way. The pair on the last chord is the oboe and the flue pipe, again an E4 against a C6. The walk does not pair parts at random. It pairs the ones that cost each other least, which is the register fact the chord that has room for an entrance measured one part at a time.

What the empty chord gives up can be read off the two walks. In the free relay the submediant was where the violin entered, with nothing from the roster in its register, 23.9 decibels above its mask — the easiest entrance in the whole passage. The shift change leaves that chord empty and brings the violin in on the dominant instead, over the brass and the voice, where it is 7.2 decibels above. The walk is willing to spend its easiest entrance, because the violin was never going to be the weakest one; what it cannot spend is room for the oboe and the brass, whose entrances decide the score. So the minimum stay costs the passage its most comfortable arrival long before it costs the least comfortable one, and only the second is on the scoreboard.

The weakest entrance of the shift change is the oboe’s, entering on the last chord beside the flue pipe with the violin still sounding: 1.07 decibels under. In the free relay the oboe also came last, with the flue pipe, and was 1.43 above; the difference is the violin, which the shift change keeps sounding into the final chord because it entered only on the dominant and has not served its two chords. That is 1.66 decibels worse than the free relay and 1.58 better than a schedule in which nobody leaves at all.

What each minimum costs, at each ending #

The same search at every ending and every minimum puts a number on the constraint.

With all six required at the end, nothing changes at any minimum: the build-up has no exits to constrain. With five required, a two-chord stay costs almost nothing — 1.40 decibels under the mask against 1.27 free — because the free walk released only the oboe, and a single early exit is easy to delay. With four required, the two-chord stay drops to 1.40 under against 0.72 free; with three and two, to 1.07 under against 0.26 and 0.59 above.

The three-chord stay is worse again, but not much, and it has a floor: from four required at the end down to two it sits at 1.40 under the mask. At a three-chord minimum in a five-chord passage a part that enters on the first chord can leave on the fourth at the earliest, so there is room for only one change of shift, and releasing more parts from the ending cannot buy a second.

As a share of what exits were worth — the free walk’s gain over nobody leaving — a two-chord stay keeps 49 per cent with two parts required at the end, and a three-chord stay 38. The prediction was that the gain would shrink sharply, perhaps to nothing. It shrinks, by about half, and the reason it does not vanish is the shift change: a strategy the free walk never needed and so never showed.

Why the minimum stay is cheaper with a memory #

The earlier essay added a memory: the chord before an entrance still masks it, at some number of decibels below full strength, because the ear keeps a sound for a moment after it has stopped. The relay was the texture that memory hurt most, since a part that leaves on the chord its successor enters is still in the ear for the first moments of that entrance.

The earlier essay measured how much the memory cost the relay against the build-up, and found the relay the more exposed of the two: 2.73 decibels lost from its weakest entrance across the range of memories, against 2.00 for the build-up. The handover was the reason. A part that leaves is not a part that arrives had shown that the ear does not treat a departure as the mirror of an arrival, and here the asymmetry is in the masking: the part that has just left is still masking, the part that has just arrived is not yet established.

With that memory at eight decibels down, the free relay loses more than the shift change does. Its weakest entrance falls by 1.27 decibels, to 0.68 under the mask; the two-chord stay’s falls by only 0.33, to 1.40 under. As a share of what exits are worth, the two-chord stay now keeps 70 per cent, and the three-chord stay 53.

The reason is the chord nobody enters. In the free relay every chord after the first has a handover, and every handover puts the leaving part in the memory of the entering one. The shift change has fewer handovers — it changes over twice where the relay changes over four times — and it spends one chord with no entrance, whose memory masks nothing. A constraint that slows the relay down removes exactly the handovers the memory was charging for. The more the ear holds over from the chord before, the less a minimum stay costs, down to a point: at four decibels down the shift change is within a third of a decibel of the relay, and only at full strength, where every walk is badly masked, does the gap open again to seven tenths.

A three-chord stay, and the pair it cannot separate #

A longer minimum forces the pairing further forward.

With four parts required at the end and a three-chord stay, the walk has one change of shift, and it puts both of its early entrances on the very first chord. The brass and the clarinet enter together on the open tonic, each 1.40 decibels under the mask: a pair whose registers, E3 and G4, are close enough to cost each other, and whose shift has to start on the first chord to be over by the fourth. They leave together on the dominant as the oboe and the violin arrive.

That pair is the price of the three-chord stay made visible. The two lowest parts could each have entered alone over the open tonic, and in the free walk the clarinet did, 0.6 decibels above its mask. Made to stay three chords, they have to come in early and together or not have left by the time the passage ends, and the walk judges that a shared, masked entrance is the least bad of the options.

The arithmetic #

The passage, the four players under it and the six parts are the earlier essays’: five chords voiced for flue pipe, brass, clarinet and violin, and six entrants — a clarinet on G4, brass on E3, a voice on C5, a violin on G5, an oboe on E4 and a flue pipe on C6 — each a radiator from the collection’s measured set at seventy decibels, fourteen partials renormalised to unit power and tested against the threshold of hearing. A partial’s threshold is the power sum of the spreading functions of everything sounding with it, and with a memory the spreading functions of the chord before reduced by the stated decibels. The walk is found by dynamic programming over the chord, the sounding set, the entered set and the sets of parts still inside their minimum, maximising the least entrance margin; reachable states are kept in a map.

Where the constraint stops being orchestration #

A stay counted in chords is a stay of no fixed length. Two chords of a slow passage are a long line; two chords of a fast one are a gesture again. The constraint is the right shape and its unit is the passage’s, which in a phrase is a number of seconds would be the first thing to question.

The chord nobody enters is invisible to the objective. The walk counts only entrances, so a chord with no entrance costs it nothing. To a listener it is a chord in which the texture does not change, which may be exactly what the passage needs or exactly what it does not; the objective has no way to prefer one.

The memory is a flat reduction. How much of the chord before is still in the ear depends on how long a chord lasts and on when in the new chord an entrance is heard; the release is on the wrong side found a listener takes about a tenth of a second to receive an arrival, by which time much of the forward mask has decayed. The finding that a memory makes the minimum stay cheaper holds across the strengths swept, but the strength that applies to a given passage is a fact about its tempo.

Entrances are one term of a larger decision. Who plays what and how loud is one question set out the whole of it — balance, register, doubling, dynamics — and the walk optimises only whether arrivals are heard. A shift change that keeps every arrival audible may leave the texture between arrivals thin or unbalanced, and nothing here would notice.

Pairs are judged only by their entrances. Two parts entering together are scored as two entrances, each under the other. Whether a listener hears them as two arrivals or as one new colour is the question an entrance is a change of colour asked, and for a pair at opposite ends of the register, like the brass and the voice, the answer is probably two.

Whose shifts #

Changing an ensemble’s colour by relieving one group with another is the commonest large-scale gesture of orchestration: the strings handing a passage to the woodwinds, a horn pair replaced by a trumpet pair, antiphonal choirs. What the arithmetic adds is that, once each group must play for more than a moment, changing over in pairs on the chord of arrival is the best way to keep every arrival audible, and that the pairs it forms are the ones spread furthest in register. That is consistent with how instruments are grouped in orchestral practice — horns and bassoons, flutes above clarinets — but the consistency is a property of this passage and this roster, and nothing here has counted scores.

Still open: a pair that enters as one #

The shift change depends on two parts entering on the same chord and being scored as two entrances, each masked by the other. If a pair entering together is heard as one arrival — a new colour rather than two new lines — then its entrance should be scored once, by how well the pair as a whole stands above what was there before, and the mutual masking inside it should not count against it.

The computation is the walk with a second kind of entrance: a set of parts entering together, scored as one object whose margin is the combined level of its partials over the threshold everything else puts over them. The prediction is that pairs would become more attractive than single entrances — the walk would change over in larger groups and earlier — and that the weakest entrance would rise, perhaps back above the mask. If it does, then the shift change is not a compromise forced by the minimum stay but the better way to orchestrate an arrival, and the relay of single entrances was only ever the best of the gestures.