Form and structure

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.

14 min read 6 figures The design was forced The ear is not a microphone What the listener supplies

Assumes: A part that must stay changes over in shifts · An entrance is a change of colour

A part that must stay changes over in shifts made every part stay at least two chords once it entered, and the best schedule for keeping every entrance audible stopped being a relay of single parts and became a change of shift: two parts entering together, a chord with no entrance, two leaving together as their successors arrived. The pairs it formed were spread far apart in register, so that each cost the other little.

It scored each member of a pair as a separate entrance, masked by everything sounding with it — including its partner. Its closing section questioned that. If two parts entering on the same chord are heard as one arrival, a new colour rather than two new lines, then the pair should be scored once, by how well it stands above what was there before it, and the masking inside it should not count. The prediction was that pairs would become more attractive and the weakest entrance would rise.

Both happen, by a mechanism the prediction did not have in mind. Scored as one arrival, the best pair is not two parts that leave each other room. It is a part that can be heard and a part that cannot.

Fig. 1 The best walk when parts entering on the same chord are scored as one arrival, at most two at a time, with two parts required at the end. The clarinet and the violin enter together on the open tonic; the oboe and the voice on the submediant as those two leave; the brass and the flue pipe on the dominant. Each arrival clears the mask by at least 10.9 dB. Measured as itself, the clarinet is 10.9 dB under it.

Two ways of scoring the same entrance

The walk is the earlier essays’ exactly: a state is the set of the six parts sounding at a chord and the set that has entered at least once; every part must enter; a stated number must be sounding at the end; the walk is only as good as its worst score. What changes is the score.

Each heard as itself is the rule every earlier walk used. Every entering part is scored by the mean, over its partials, of how far each clears the threshold that everything else sounding puts over it — the four players of the passage, every part already in, and every other part entering on the same chord. A pair pays for masking itself.

As one arrival takes the parts entering on a chord together. The score is the mean over all their partials of how far each clears the threshold that the passage’s players and the parts already sounding put over it; the partners in the arrival do not mask each other. That is the reading an entrance is a change of colour gave an entrance that is meant to be heard as a change in the texture rather than as a new line, extended to a group. A further limit sets how many parts may enter on one chord.

With one part at a time the two scores are the same thing, and with six parts, five chords and at most one entrance a chord no walk can bring every part in, so the comparison starts at two.

The best pair is a carrier and a passenger

With two parts required at the end and at most two entering a chord, scored each as itself, the best walk is the earlier essay’s free relay, and its weakest entrance is the clarinet’s on the first chord, 0.59 decibels above the mask. Scored as one arrival, the same limit produces the hero figure’s walk, and its weakest arrival clears the mask by 10.93 decibels — ten more than any schedule of single entrances has managed on this passage.

The walk gets there by pairing. On the open tonic the clarinet enters with the violin: the violin, a fifth above the clarinet and with nothing from the roster in its register, clears its mask by 19.6 decibels; the clarinet, measured as itself, is 10.9 under. On the submediant the oboe enters with the voice, the voice 14.5 clear and the oboe 1.5 under. On the dominant the brass enters with the flue pipe, the flue pipe 25.0 clear and the brass exactly at its threshold. Every arrival is a loud part and a quiet one, and in two of the three the quiet one is buried.

That is what “one arrival” means once it is taken literally. The mean over a pair’s partials is dominated by whichever partner stands clear, so a pair’s score is its carrier’s, and the search, looking for the best score, pairs every part it cannot make audible with one it can. Scored as a colour, an entrance does not need its weakest member to be heard at all.

The more that enter together, the deeper the passenger

The limit on how many parts may enter together decides how far the pattern goes.

The more parts may enter together as one arrival, the better the arrival and the deeper the part it hides. With 2 of six parts required at the last chord, the best walk scored two ways against the most parts that may enter on one chord: every entrant heard as itself, and the entrants on a chord taken together as one arrival against what was already sounding; and, for the second, how far the most buried entrant sits under the mask measured as itself. 2 at a time: each heard 0.59 dB, as a group 10.93, most buried member -10.94; 3 at a time: each heard 0.59 dB, as a group 13.44, most buried member -14.74; 4 at a time: each heard 0.59 dB, as a group 13.44, most buried member -14.74; 6 at a time: each heard 0.59 dB, as a group 13.59, most buried member -16.37. — Fig. 2 With two parts required at the end: the best walk’s weakest arrival scored as a group, its weakest entrance scored each as itself, and the most buried member of any group arrival measured as itself, against the most parts that may enter on one chord. Each heard: 0.59 dB at every limit. As a group: 10.9 at two, 13.4 at three and four, 13.6 at six. The most buried member: 10.9 dB under at two, 14.7 under at three and four, 16.4 under at six.

Scored each as itself, the best walk’s weakest entrance is 0.59 decibels above the mask whatever the limit, because the relay of single entrances is already its best walk. Scored as groups, the weakest arrival rises with every part allowed in together — 10.9 decibels at two, 13.4 at three, 13.6 at six — and the most buried member sinks with it: 10.9 under at two, 14.7 at three, 16.4 under at six.

The two curves are one mechanism read from its two ends. A larger group has more room to carry a buried member, because the group’s score is a mean and the loud members raise it; and a larger group gives the search more freedom to put the part it can do least for into the arrival where it will be noticed least. The group score improves exactly as fast as the group finds places to hide its weakest part.

With the tutti required, one chord

With all six parts required at the end the effect is starker, because nothing can leave.

Fig. 3 The best walk with all six parts required at the end, scored as arrivals of at most two. The oboe carries the clarinet on the open tonic; the voice carries the brass on the submediant; the flue pipe carries the violin on the dominant. Every arrival clears its mask by at least 4.5 dB; the violin, measured as itself, is 6.1 under, and the clarinet 5.4.

The build-up to a full tutti, scored each as itself, is the schedule room is used up by whoever enters first found: brass and clarinet on the first chord, then the oboe, the voice, the violin and the flue pipe one after another, the oboe’s entrance 2.65 decibels under. Scored as arrivals of at most two, the build-up becomes three pairs — oboe with clarinet, voice with brass, flue pipe with violin — each a loud part and a quieter one, and the weakest arrival clears its mask by 4.5.

The more parts may enter together as one arrival, the better the arrival and the deeper the part it hides. With 6 of six parts required at the last chord, the best walk scored two ways against the most parts that may enter on one chord: every entrant heard as itself, and the entrants on a chord taken together as one arrival against what was already sounding; and, for the second, how far the most buried entrant sits under the mask measured as itself. 2 at a time: each heard -2.65 dB, as a group 4.50, most buried member -6.13; 3 at a time: each heard -2.65 dB, as a group 10.47, most buried member -16.46; 4 at a time: each heard -2.65 dB, as a group 10.83, most buried member -16.46; 6 at a time: each heard -2.65 dB, as a group 11.77, most buried member -16.90. — Fig. 4 The same comparison with all six parts required at the end. Each heard as itself, the build-up’s weakest entrance is 2.65 dB under the mask at every limit. As groups, the weakest arrival is 4.5 dB clear at two, 10.5 at three, 11.8 at six; the most buried member is 6.1 under at two, 16.5 at three, 16.9 at six.

Lift the limit and the build-up disappears. At three at a time the walk brings the parts in as two trios; at six it brings all six in on one chord, an arrival 11.8 decibels clear with its most buried member 16.9 decibels under its mask. Every staggered schedule these essays have found — the build-up, the relay, the shift change — exists because each part was scored as itself. Score arrivals as colours and there is no reason to stagger anything: the texture that changes most at once is the texture whose change is heard most.

The schedules the single-entrance score produced

It is worth putting the earlier walks beside these, because the difference between them is the whole of what the choice of score does.

Fig. 5 The earlier essay’s shift change, scored each as itself: two parts required at the end, each staying at least two chords. Pairs enter and leave together here too — brass and voice, oboe and flue pipe — but they are chosen far apart in register so that each costs the other little, and the weakest of them, the oboe, is 1.07 dB under its mask.

The shift change also enters parts in pairs, and at first sight it looks like the group walk. It is its opposite. Its pairs were chosen so that each member would be as audible as possible: the brass on E3 with the voice on C5, the oboe on E4 with the flue pipe on C6, two octaves apart, each partner leaving the other’s register clear. The group walk’s pairs are chosen so that the pair’s mean is as high as possible, which puts a loud part beside a quiet one and lets the quiet one sink. Same entrance count, same passage, same masking functions, and the reason for pairing reversed by one line of the scoring.

Fig. 6 The build-up to a full tutti scored each as itself, from the earlier essays: brass and clarinet on the first chord, then the oboe, the voice, the violin and the flue pipe, one a chord. The weakest entrance, the oboe’s, is 2.65 dB under the mask. Scored as arrivals, the same six parts come in as three carried pairs, or all at once.

The build-up shows the same reversal at full scale. Scored each as itself it spaces six entrances across five chords and puts the brass and clarinet together on the first, the one chord where both have room. Scored as arrivals it collapses to three pairs, or one tutti, because nothing in the group score rewards spacing: the build-up was a device for hearing each part begin, and a score that does not ask for that does not produce it.

What the arithmetic says about covering

That result has a familiar name in orchestration. An entrance that would not be heard alone is often covered — doubled at the unison or octave by an instrument that will be, or brought in under the attack of a louder part — so that the listener hears the moment of arrival and the weaker instrument slips in under it. The walks here are covering by construction, and the figures put numbers on what it costs: the covered part can be ten or more decibels under its mask and still be part of an arrival heard ten decibels clear.

The size of the numbers matters here. A part ten decibels under its mask is not faintly heard; on the masking model every earlier essay has used, it contributes nothing a listener could pick out, and its partials are there only as energy the louder part’s partials already cover. One sound hides another set out the spreading functions that make that so, and at the margins in these figures they are not in doubt. A covered clarinet at −10.9 decibels is not a soft clarinet. It is a violin with a little more energy in its lower partials.

What the arithmetic cannot say is whether that is what an orchestrator wants. If the clarinet’s entrance is meant to add a colour to the violin’s, it has succeeded, whatever its own margin; if it is meant to begin a line that the listener follows afterwards, it has failed, because the listener never heard it begin. Who plays what and how loud is one question set out the whole decision; the two scores here are two answers to one part of it, about what an entrance is for, and they lead to schedules that have nothing in common.

The earlier essays can be reread in that light. Each of them assumed every part’s entrance was a line beginning, and each found a schedule that spaced entrances out so each could be heard. That assumption was not neutral. It was the whole reason the schedules looked like orchestration.

Carriers and passengers by register

The pairs the search forms are not random, and they say something about the roster. The carriers are the violin on G5, the voice on C5 and the flue pipe on C6 — the three highest parts, with little from the passage’s players above them. The passengers are the clarinet on G4, the oboe on E4 and the brass on E3 — the middle and low parts, sitting among the players’ partials.

That is the register fact the chord that has room for an entrance found one part at a time, turned into a pairing rule. A part in the crowded middle of the texture is hard to make audible by any schedule; a group score lets the search stop trying and attach it to a part in the clear register above. Whether the pairing is a good orchestration depends on whether the middle part matters as itself — a new line in the tenor, which it would bury, or a thickening of the upper part, which is what doubling is for.

Whose doublings

Doubling a weak instrument with a strong one is among the oldest devices of orchestration: a flute line doubled an octave below by a clarinet, a horn entrance reinforced by a bassoon, a solo line that begins on a unison with the violins and then emerges from them. Orchestration manuals present it in two ways that correspond exactly to the two scores. As reinforcement, the weaker instrument is there to change the colour of the stronger one, and it succeeds whether or not it is separately audible; as support, the stronger instrument is there to carry the weaker one’s entrance, and the weaker one is supposed to be heard afterwards.

The arithmetic separates those cleanly. Reinforcement is what the group score optimises, and the best reinforcements it finds put a middle-register part under a high one. Support needs something the group score does not measure — that the weaker part becomes audible once the entrance is over — and no walk here, under either score, has a term for what happens after the chord of arrival.

How the scores are computed

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 at seventy decibels with fourteen partials, each partial’s threshold the power sum of the spreading functions of the components masking it, floored at the threshold of hearing. For a group arrival the masking components are the passage’s players and the parts sounding at the chord that did not just enter; the score is the mean of all the arriving partials’ margins over those thresholds. The walk is the exact dynamic programme over chords, sounding sets and entered sets, maximising the least score, with transitions bringing in more than the stated number of parts excluded.

Where the two scores stop being enough

A mean is a choice. The group score averages over partials, and an average is exactly the statistic a loud member can carry. A group score that took its weakest member, or required every member to clear its mask by some amount, would forbid passengers and would lie somewhere between the two scores here; which is right is the question of what the arrival is for, not an arithmetic one.

Doubling is not only masking. A doubled part at the unison fuses with its double — the collection’s essays on what makes two partials one note are about exactly that fusion — and a fused pair is heard as one timbre whose colour both members change. The group score treats the arrival as a set of partials above a threshold, not as a new timbre, and so says nothing about whether the passenger changes the colour it rides in.

Leaving is scored the same way under both rules. An exit costs nothing in either score, and a part that leaves is not a part that arrives found the ear treating a departure very differently from an arrival. A group walk that brings in a carried pair and later lets the carrier leave alone would uncover its passenger, and neither score here would notice the moment the passenger is suddenly the only one of the two sounding.

One passage, one roster. The pairings are properties of these six parts over these five chords. The rule — carriers from the clear register, passengers from the crowded one — is general; the particular pairs are not.

Still open: a group score that asks each member to be heard

The two scores bracket the answer. One charges a pair for masking itself and produces staggered entrances; the other charges it for nothing and produces carriers and passengers. An orchestrator wanting every part heard and willing to bring parts in together wants something between them: the pair scored as one arrival, but with a floor under each member — no passenger allowed below, say, three decibels under its mask.

The computation is the same walk with the group score and a per-member floor, swept. The prediction is that a floor near the single-entrance walk’s own weakest margin recovers the shift change of the earlier essay, and a floor well below it recovers the carriers — and that somewhere between is a schedule that neither staggers every part nor buries any, which would be the first result in these essays that describes doubling as a way of making an entrance audible rather than a way of hiding one.