Room is used up by whoever enters first

The chord that has room for an entrance asked where in a passage one part should come in, and answered with a margin: how far the entering part’s partials sit above the threshold the ensemble already sounding puts over them. Over five chords voiced by the same four players, an oboe entering on E4 has 9.0 decibels of room at the open tonic chord and 0.3 at the close-position submediant, and the moment is worth as much as the register.

That essay scored an entrance it was handed. Its last section named what nothing had yet done, which is choose. A score brings in not one part but several, and the objective stops being separable as soon as it does: a part that has entered raises the mask for every part that enters after it. The question it left was whether an exhaustive answer — every possible schedule scored and the best one kept — looks like anything a score does.

Six parts and one passage #

The passage is the one the entrance essay used: five chords — an open tonic, a submediant in close position low, a subdominant close above, a dominant bracketing the middle and a hollow tonic — each voiced for a flue pipe, a brass part, a clarinet and a violin. Waiting to enter are six more parts, each on one held note and all on tones of the tonic chord so that harmony is not a term: 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.

A schedule gives each waiting part the chord at which it enters, and a part that has entered keeps sounding to the end. That is five choices for each of six parts, 15,625 schedules. Each entrance is scored exactly as the single entrance was — the mean margin of the part’s partials over the threshold set by everything sounding at that chord — except that everything sounding now includes every part that entered at that chord or earlier. Two numbers summarise a schedule: the sum of its six margins, and the least of them.

Each part alone, and each part after the others #

The first thing the search needs is how much room each waiting part has, and it turns out to be two different tables depending on whether anybody else has come in.

Alone, the room follows the register trend the entrance essay found, which is the critical band’s doing: a low chord stops being a chord because a band is many semitones wide at the bottom of the range and under three at the top, so energy low in a texture masks a great deal and energy above it very little. The flue pipe on C6 has between 5.3 and 35.8 decibels of margin, the violin on G5 up to 23.9, the voice on C5 up to 16.2; the clarinet on G4 and the brass on E3 have almost none anywhere. And alone, three of the six parts have their most room at the same chord — the close submediant, whose four notes sit low and leave the upper register clear.

With the other five already in, the table changes shape rather than merely shifting down. The flue pipe keeps between 4.0 and 10.4 decibels at every chord, because it is above almost everything the others put into the spectrum. The violin falls to between −5.8 and −7.9. The voice and the clarinet fall to between −15.8 and −18.4: a middle-register part entering after five others is buried by them at every chord, whatever the voicing underneath. The room a chord offers is a property of what is sounding, and what is sounding is the schedule.

Choosing each moment alone puts three parts in one chord #

The obvious schedule is the one the single-entrance essay implies: give each part the chord that has most room for it.

That schedule puts the voice, the violin and the flue pipe all at the close submediant, the oboe and the clarinet at the open tonic, and the brass at the subdominant. It is the schedule nobody would write, and it is what choosing each entrance independently produces. Three parts arriving together at the chord that was roomiest for each of them alone mask one another. The voice, which had 16.2 decibels of margin there by itself, enters 15.6 decibels under the mask; the violin, which had 23.9, enters at −6.4.

Its six margins sum to −19.2 decibels, and among all 15,625 schedules that ranks 12,267th — in the bottom fifth, and below the median schedule chosen at random. Its least audible entrance, −15.6, ranks 3,845th. Choosing each moment by the rule that is right for one part is, for six, worse than not choosing.

Placing them one at a time does not rescue it #

The next obvious method is to place the parts in turn, each at the chord best for it given the ones already placed. That respects the interaction the independent choice ignored, at least in one direction, and it still fails.

Placed from the lowest part up, the six margins sum to −30.3 and the schedule ranks 14,065th, worse than choosing independently. Placed from the highest part down, they sum to −9.5 and rank 7,320th, close to the median. The same greedy rule gives schedules nearly seven thousand places apart depending on which part it starts with. A part placed early takes the room without knowing who will arrive into it, and a part placed late finds the room already gone. Placed from the lowest, the voice takes the close submediant, where it has most room; the violin and the flue pipe, placed after it, take the chord before, so they are already sounding when the voice arrives, and it enters 15.6 decibels under the mask.

The best summed margin and the best least entrance are different schedules #

Searched exhaustively, the schedule with the largest summed margin reaches 21.8 decibels. It gets there by giving three parts a great deal of room — the voice 12.3 decibels at the close submediant, the flue pipe 9.7 at the dominant, the violin 6.8 at the subdominant — and giving up on two. The brass enters 3.2 decibels under the mask and the oboe, left to the last chord, 4.4 under it. By its least audible entrance that schedule ranks only thirty-ninth.

The other summary asks that every entrance be heard, and it is harsher: the median schedule’s least audible entrance is 16.8 decibels under the mask, because in most schedules some middle-register part arrives into a crowd. The best is 2.6 decibels under it, and only one schedule reaches it. Its summed margin is 7.6, which ranks 577th on that measure. A score that wants three conspicuous entrances and a score that wants six heard ones are asking for different schedules, and nothing in the mask decides between them; it is a decision about what the passage is for.

The schedule that hears every entrance holds the high parts back #

The schedule at the head of this essay is that unique best. The brass on E3 and the clarinet on G4 enter together on the first chord, each 1.4 decibels under the mask; the oboe on E4 enters on the second, 2.6 under it, which is the least audible entrance and the best any schedule can make its least audible entrance. Then one part a chord, rising: the voice at the subdominant with 2.1 decibels of room, the violin at the dominant with 6.9, and the flue pipe at the hollow tonic with 4.0.

The three highest parts enter at the last three chords, in order of register, and the three lower parts all enter in the first two. The strict order by register fails only by a swap — the clarinet on G4 comes in before the oboe on E4 — and the reason for the shape is the table of room. The high parts keep room over a full texture and the middle parts do not, so the schedule that looks after the weakest entrance brings the vulnerable parts in while few others are sounding and saves the parts that can still be heard for when the texture is thickest. It is the same fact that makes a piccolo audible over a tutti and a second viola not, applied to an order rather than to one entrance.

The schedule also staggers. Among the best one per cent of schedules by their least audible entrance, 138 of 156 use four or five different entry chords; among all schedules, 61 per cent do. Bringing parts in together is costly for exactly the reason the independent choice was: each is part of the other’s mask.

A rule of thumb is not the schedule #

It would be tidy if the best schedule were a rule — one part a chord, in order of register — and it is not. That rule, brass on the first chord and flue pipe on the last, with the violin and flue pipe sharing the fifth, has a least audible entrance 13.1 decibels under the mask and ranks 1,900th. The same rule run from the top down ranks 14,026th, and bringing all six in on the first chord ranks 5,271st.

The rule fails on one part, and the failure is the room table again. The clarinet on G4 has room at exactly one chord of the five, the open tonic, and even there only 0.6 decibels of it by itself; at the subdominant it has −10.6 with nobody else in. A rule that gives it the third chord because it is the third lowest sends it into one of its worst places, and it enters 13.1 decibels under the mask. The best schedule brings the clarinet in on the first chord beside the brass, which is out of register order and is the only chord where it had any room to lose.

So the shape of the best schedule is two things at once: an order, which the register decides because the high parts keep their room as the texture fills, and a set of particular chords, which each part’s own room profile decides because a voicing that is open for one register is closed for another. A rule of thumb captures the order and misses the chords, and on this passage the chords are worth more than ten decibels to the weakest entrance.

The search was never the hard part #

The entrance essay described the search space as the obstacle, and counted it as two to the thirtieth — each of six parts on or off at each of five chords. Exhaustive search over the narrower space here, where a part enters once and stays, is 15,625 schedules and takes a few hundredths of a second once each part’s contribution to every other part’s threshold has been computed, because masking intensities add.

The larger space is not harder, and the reason is worth stating because it is what the next question has to respect. An entrance’s margin depends only on which parts are sounding at the chord where it happens. So a schedule of any complexity — parts entering, leaving and returning — is scored by walking through the passage one chord at a time and knowing only who was sounding at the chord before and who is sounding now. Six parts can be sounding in sixty-four combinations, so the whole two-to-the-thirtieth space is a walk through sixty-four states over five chords, and the best walk is found by carrying the best score for each state forward: sixty-four states times sixty-four possible predecessors at each of five chords is about twenty thousand comparisons, against a thousand million schedules. The search is hard only for an objective that remembers further back than one chord, and the two quantities this subject has found that do — the running loudness with its two-second release and the forward mask, which an orchestrator doubles a line, not a chord found reaches about twenty milliseconds on a dense texture — are exactly the ones left out here.

The arithmetic #

Every part is a radiator from the collection’s measured set at seventy decibels, with its partials renormalised to unit power and tested against the threshold of hearing, fourteen partials a part. A partial’s threshold is the power sum of the spreading functions, the same ones by which one sound hides another, of every component sounding with it — the base chord’s and every entered part’s — floored at the threshold of hearing, and an entrance’s margin is the mean over the entrant’s partials of its level above that threshold. The base chord and each other part contribute to each partial’s threshold once, so every schedule is scored by adding precomputed contributions; the summed margin and the least margin are taken over the six entrances, whichever chords they fall on.

What a schedule of held notes cannot show #

The parts hold one note each. A part enters on a tone of the tonic and keeps it through a dominant and a subdominant, which no part in a score would do. The note fixes the entrant’s register, which is what the mask reads; a real line would move through registers with different room, and harmony, which is not in the objective at all, would forbid most of these held notes on most of these chords.

Nobody leaves. A score takes parts out as well as bringing them in, and a part leaving gives room back to everything that stays — a part that leaves is not a part that arrives measured how differently a listener receives the two. The walk through sixty-four states described above is the computation that includes exits, and it has not been run.

The mask is simultaneous. An entrance is masked less at its own onset than once established, because the previous chord’s forward mask is weaker than the current chord’s simultaneous one, and a listener receives an entrance within a tenth of a second. None of that timing is in these margins.

A mean margin is not an instrument heard. A clarinet’s weak even partials drag its mean down while its strong odd ones carry it, so its negative margins say less than the flue pipe’s positive ones. The ranking of chords survives a change of summary, which is what the single-entrance essay checked; whether the ranking of schedules does has not been checked.

And no score is consulted. The optimum is compared with nothing a composer wrote. That the schedule looking after the weakest entrance holds the high parts back is a property of this passage, this roster and this model, and whether scores tend to do the same is a count over scores rather than a search.

Whose entrances #

The practice in view is orchestral and chamber writing from the late eighteenth century on, in which the order parts come in is composed — a texture built up for a climax, a fugal exposition, a tutti reached by addition. An entrance is a change of colour found that each such entrance is a colour event before it is a loudness one, and who plays what and how loud is one question found that solving parts of a scoring one at a time chooses differently from solving it whole. The search here is the same finding one level up. Each entrance chosen well is not a schedule chosen well, and the difference is not small: the schedule assembled from six good single decisions is worse than most schedules picked blind.

Still open: a schedule with exits and a memory #

The walk through the ensemble’s sixty-four states is exact for any objective that looks back one chord, and it makes the full space of entrances and exits searchable in the time the narrow space took. What it cannot do unchanged is the objective this subject has spent five essays building: the running loudness, whose release lasts two seconds and so remembers several chords, and the forward mask, which carries a chord’s partials into the next. What the extended search would settle is whether the best schedule with those terms still holds the high parts back and staggers the rest — or whether, once a departure’s slow fade and an arrival’s quick one are both in the objective, the best way to make room for a part is to take someone else out a chord before it comes in.