A carrier gives up room, not partials

A passenger under its mask is still heard in the arrival settled one side of a question the orchestration essays had circled for a while. Two parts entering together on a chord had been scored by the mean, over each part’s partials, of how far each clears the threshold everything else sounding puts over it, and by that measure the best schedules were full of buried parts: a clarinet 10.9 decibels under its mask beside a violin, an oboe under its mask beside the voice. Read partial by partial, every one of the seventy-one passengers under its mask on average changed its arrival detectably, almost always through two or three partials standing clear. The carriers were not hiding their passengers. They were turning them into colour.

It ended by turning the question round. Masking is mutual. A clarinet entering beside a violin raises the threshold over the violin’s partials too, and some of them may go under. The arrival would then be not “the violin, coloured” but a new spectrum in which some of the carrier’s partials have gone and some of the passenger’s have arrived — and if neither part kept most of its partials in the clear, a listener might hear a third sound rather than either instrument. The calculation is the same one run the other way, over the same seventy-one pairings.

It comes out lopsided, in a way that says something about what makes a carrier.

What each part takes from the other #

For each pairing, each part is read twice: once over the chord’s four players alone, and once with the other part entering beside it. A partial clear of the mask in the first reading and under it in the second is a partial the other part took.

The two halves of the figure are not mirror images. Carriers lose nineteen partials in all, across seventy-one pairings; passengers lose a hundred and thirty-five. Fifty-four carriers lose none, fifteen lose one and two lose two. Passengers lose none in twenty-five pairings, one in twenty-five, two in six — and three or more in fifteen, up to eleven: the voice beside the flue pipe on the open tonic keeps thirteen of its fourteen audible partials clear over the chord alone and two beside the flue.

So the first answer is that the reverse effect exists and is small. A passenger leaves its carrier’s partials almost exactly where they were. The arrival is the carrier with the passenger’s surviving partials added, which is what the earlier essay assumed without checking, and which turns out to be right in all but a handful of pairings.

Room, not partials #

The more interesting finding is in how the carrier loses so little, because it is not that the passenger leaves it alone.

The voice beside the violin on the open tonic is the pairing where the passenger presses hardest. Over the chord alone the violin’s partials clear their thresholds by between 5.8 and 32.4 decibels, a mean of 21.7. With the voice beside it, its third partial falls from 29.1 to 12.9, its sixth from 27.8 to 9.5, its seventh from 32.4 to 12.6. Its mean falls to 9.1: the voice has taken thirteen decibels of the violin’s room. It has taken none of the violin’s partials. The lowest, the fundamental, was 5.8 decibels clear with or without the voice, because the voice’s partials do not reach down to it; every other partial is still between 4.7 and 12.9 decibels clear.

That is the distinction the whole question turns on. A partial’s margin over its mask is room it has to spare. A carrier is, almost by definition, a part with a lot of room — it was chosen as carrier because its mean margin was the higher of the two — and a passenger entering beside it spends that room without exhausting it. The voice, meanwhile, keeps only four of its fourteen partials clear beside the violin, against thirteen over the chord alone. Masking between the two is symmetric in mechanism and opposite in effect, because one of them had room to give and the other did not.

Why a carrier keeps what it has #

The lopsidedness is not a coincidence of this passage, and it is worth saying why before going further. A pairing’s carrier is defined as the part with the higher mean margin, and the walks that produced these pairings — a pair entering as one hides the part that cannot be heard and a floor under each member is paid for in steps — found their best arrivals by bringing a loud part in beside a quiet one. So the carriers are, by selection, the parts with margin to spare and the passengers the parts without it. Masking takes decibels of margin, and a partial is lost only when the decibels taken exceed the decibels it had. The same mutual masking therefore costs the carrier room and the passenger partials, for the same reason that a tax of ten per cent leaves a large income comfortable and pushes a small one under a line.

There is a second, quieter reason in the spectra themselves. The passenger’s partials that matter most to its carrier are the ones that fall close to the carrier’s own, and a passenger’s strongest partials are usually its lowest. A clarinet’s are its first, third and fifth, the closed tube’s odd series that a tube that skips every other partial describes; they land between a violin’s partials more often than on them, and so raise the threshold over the violin’s partials by a few decibels rather than by tens. The voice is the exception precisely because its energy does not fall off with partial number in that way: its formants put strong partials across the region where a carrier’s upper partials live.

So the earlier picture of an entrance — an entrance is a change of colour, not a new line — gains a qualification rather than a correction. The colour is added to a carrier that stays itself, but it is added at a price the carrier pays in room, and how large that price is depends on which instrument the passenger is.

Which passengers cost room #

Not every passenger presses on its carrier as the voice does.

The brass, clarinet and oboe as passengers cost their carriers a decibel or a decibel and a half on average, and never more than five and a half. The voice costs 8.2 on average and up to 20.1; the violin, in the seven pairings where it is the passenger, 8.9 and up to 21.5. Neither takes many partials: the voice averages 0.42 of a partial taken, the violin 0.29.

The difference is in the spectra the earlier essays used. The reeds and the brass entering at their pitches put most of their energy into a few partials in their own register, and those overlap only part of a carrier’s spectrum. The voice and the violin have strong partials spread across the whole region where the carriers’ upper partials sit — the voice through its formants, the violin through its body resonances — so they raise the threshold over nearly all of a carrier’s partials at once. The fourth player is a spectrum found that what an entering part adds is partials rather than a player; what it takes, the figure says, is room in proportion to how widely those partials spread.

A part that costs its carrier thirteen decibels of room is changing the carrier, even though it takes nothing. The violin beside the voice is still the violin, with every partial audible, but each of them now stands ten decibels nearer the point at which it would be lost. The next part to enter, or a slight drop in the violin’s dynamic, finds a carrier with much less to spare. That is a cost the earlier walks did not count, and it is the one that might matter in a texture of three or four parts.

Where neither part is in charge #

The question the earlier essay posed was whether any pairing leaves neither part with the majority of its partials in the clear.

Nine do. Sixty of the seventy-one carriers keep a majority of their partials clear — the cloud at the top of the figure — and the passengers, as the earlier essay found, mostly keep between a fifth and a half. The nine pairings in the shaded corner are the brass with the oboe on the close subdominant, the oboe with the clarinet on the bracketing dominant, and seven pairings on the hollow tonic: the brass with the clarinet and with the voice, the oboe with the brass and with the violin, the voice with the clarinet, and the violin with the clarinet and with the voice.

The colours make the pattern plain. Seven of the nine third-sound pairings are on one chord, and the reason is in the second reading of each part, not the first.

On the hollow tonic the violin, which clears its mask by twenty decibels on the open tonic, has only three of its eight partials clear over the four players alone: its second by 16.4 decibels, its third and sixth by under one. The others are already between 1.7 and 5.4 under. The clarinet entering beside it takes none of them and costs the violin three tenths of a decibel of mean margin; the clarinet itself keeps three of eight clear, exactly what it had alone. The arrival is a third sound, and the passenger did not make it one. The chord did. Of the seven hollow-tonic pairings in the corner, every carrier had at most four partials clear before its passenger arrived — the voice three of fourteen — and in six of them the passenger took one partial or none.

The two pairings off that chord are the ones where a passenger really does tip its carrier over. The brass on the close subdominant has four of eight partials clear alone and three beside the oboe, and the oboe keeps one of seven; the oboe on the bracketing dominant has four of seven alone and three beside the clarinet, whose partials on that chord are, as the earlier essay found, all just under. Both carriers are parts with almost no room to begin with — means near zero over the chord alone — which is the same lesson from the other side. A carrier that is barely a carrier can be pushed into a draw.

What the arrival is made of #

The share of each part’s own partials clear is one way to ask whose sound the arrival is. The other is to ask whose partials make up what stands clear of the mask in the arrival as a whole.

In the median arrival two thirds of the clear partials are the carrier’s. Twenty-two arrivals are three quarters carrier or more; eight are split evenly; two are mostly passenger. The arrival is the carrier’s sound in the sense that matters for identity — most of what reaches the ear above the mask comes from it — with a third of its clear partials belonging to the passenger, which is a substantial colouring rather than a trace.

Those two numbers put the earlier essay’s conclusion on a firmer footing. It found that a passenger is heard as colour rather than as a line. This one finds that the colour is laid on a carrier that keeps its partials, keeps most of the clear spectrum, and gives up only room. The doubling the treatises describe — a clarinet brightening the violins, a bassoon darkening the horns — is a description of exactly this arrangement: the stronger instrument’s identity intact and the weaker one’s partials added in the gaps. An orchestrator doubles a line, not a chord found doubling paying when the second instrument supports the first; the arithmetic here says what “supports” amounts to in the spectrum.

The surprising part #

The expectation the question was written with was that mutual masking would sometimes merge the two parts, and that the merging would depend on the pair. It depends on the chord. Change the pairing and the carrier keeps its partials almost always; change the chord and a whole group of pairings falls into the third-sound corner together, because the chord’s own players have already masked most of what both parts would bring.

That makes the hollow tonic the place in the passage where orchestration cannot do what it does elsewhere. The chord that has room for an entrance asked which chords leave a given part room and found the answer varied by part; this is the same question asked of a pair, with a sharper answer. On the hollow tonic no pairing produces “the violin, coloured” or “the brass, coloured”, because there is no violin or brass there to colour — only three or four partials of each, clear between the chord’s own. Whatever enters on that chord is heard as part of a blend the chord already owns.

What the model assumes #

The masking model of the earlier walks. Each partial’s threshold is the sum of the masking every other sounding partial puts over it, from the spreading function every orchestration essay has used, with the threshold of hearing as a floor. The parts enter together at the nominal level and hold; nothing here depends on onsets, which would only make each part easier to hear, as a part entering is not a change of level found.

Clear means above its masked threshold. A partial with a positive margin is counted as present in the arrival, one with a negative margin as absent. That is a hard line drawn across what is really a gradual change, and a partial half a decibel clear is counted the same as one thirty clear. The room figure exists to show the difference the count hides.

A majority of partials as the line between “coloured” and “a third sound”. The question was posed that way and answered that way. Identity is not a vote of partials. The fundamental and the lowest few partials carry pitch and much of an instrument’s character, and an instrument that kept its first three partials and lost five upper ones might still be recognised; one that kept five upper partials and lost its fundamental might not. In the third-sound pairings on the hollow tonic, the violin keeps its second, third and sixth partials and loses its fundamental, which is the second case.

What the count cannot settle #

Whether the room a voice takes is heard. A violin whose partials have all moved ten decibels nearer their thresholds sounds different from one that has not, in the sense that some of its fine detail is now less salient. Whether a listener hears that as the violin changing, or as the voice arriving, is the attribution question the arithmetic cannot reach.

What happens with three parts. The room a passenger takes is the cost that would compound in a fuller texture: the third part to enter finds a carrier already ten decibels nearer losing its partials. The walks have scored arrivals of at most two parts together, and nothing here says where in a growing texture the carriers’ room runs out.

Whether the hollow tonic is heard as one object. Seven pairings land in the third-sound corner on that chord because the chord masks both parts heavily. Whether a listener then hears a blend, or hears the chord and two faint instruments, is a question about grouping that the masking count can raise and not answer.

Still open: the room a third part finds #

The carriers here give up room rather than partials, and the room is what a further part would spend. The walks have only ever let two parts enter together. The computation is this one extended to a third entrant on each chord, beside the best carrier and passenger already sounding: how many of the carrier’s partials survive the third part when the second has already taken ten decibels of its room, and whether there is a number of parts — three, or four — at which the carriers that survived every pairing start losing partials outright, which would say how many doublings a line can carry before it stops being the line.