A third part takes what it would have taken alone

A carrier gives up room, not partials settled what a passenger costs the part it enters beside. Across the seventy-one pairings of the orchestration passage in which one part sits under its mask on average, the carriers lost nineteen partials in all and the passengers a hundred and thirty-five. The carriers were not untouched, though. A voice entering beside a violin on the open tonic took none of the violin’s partials and thirteen decibels of their margin: every partial still clear of its masked threshold, each now ten decibels nearer the line.

That essay ended on the cost it had not counted. Room is what a further part would spend. A carrier whose partials all stand ten decibels nearer their thresholds looks fragile, and the natural expectation is that a third part entering beside it would push those partials under where it could not have pushed a carrier with its room intact — that the passenger’s spending and the third part’s would compound, and that somewhere around three or four parts a line would stop being the line.

The calculation is the same one with a third entrant. Each of the seventy-one pairings is read again with each of the four other parts of the roster entering beside the carrier and passenger, which makes 284 three-part arrivals, and for every carrier partial clear of the mask over the chord alone it records three things: whether the passenger alone pushes it under, whether the third part alone does, and whether the two together do.

The expectation turns out to be wrong in an instructive way. The room a passenger spends does not make a carrier fragile. It makes it, if anything, harder for the next part to hurt.

What the third part takes, and from whom #

With three parts sounding, the carriers lose 438 partials across the 284 arrivals. Seventy-six of them the passengers had already taken, in the pairings the earlier essay scored. Of the rest, 341 are partials the third part takes on its own — it would push them under if it entered beside the carrier with no passenger at all. Twenty-one are lost only to the combination: partials that survive the passenger alone and survive the third alone and go under when both are sounding. That is the compounding the question was about, and it is under five per cent of the loss.

The bars are sorted by the part entering third, and they divide sharply. The clarinet, the oboe and the brass as third parts cost the carriers nineteen, twenty and twenty-three partials across all their arrivals, most of them partials the passenger had already taken. The voice costs forty. The violin costs 127 and the flue 209. The flue and the violin, entering third, account for 336 of the 438 partials — and almost all of their share is partials they take alone.

So the first answer is that the number of parts is the wrong variable. What a third part costs the carrier is very nearly what that part would cost on its own, and what a part costs on its own depends overwhelmingly on which part it is.

Two maskers are not twice the masking #

The reason the room does not compound is in how masking adds, and it shows plainly in one arrival.

This is the pairing the earlier essay chose as the passenger that presses hardest: the voice beside the violin on the open tonic. Over the chord alone the violin’s eight partials clear their thresholds by 5.8 to 32.4 decibels. Beside the voice they clear by 4.7 to 12.9, every one still audible, the mean down by 12.6 decibels. Beside the flue alone — no voice — the picture is different in kind: the violin’s second and third partials keep most of their room, and its fourth to eighth go under, the eighth by more than twenty decibels. The flue costs the violin 26.1 decibels of mean margin and five partials.

Beside both, the violin loses the same five partials, and each of its eight sits within a decibel of the lower of its two single margins. The fourth partial is 4.7 clear beside the voice and 1.7 under beside the flue; beside both it is 2.5 under. The sixth is 9.5 clear beside the voice and 15.0 under beside the flue; beside both, 15.0 under. The two parts together cost the violin 27.0 decibels of room, not the 38.6 their separate costs would add to.

The arithmetic behind that is the standard rule for combining maskers, which every orchestration essay here has used: the intensities each masker puts over a partial add, and the threshold is that sum expressed in decibels. Where one masker puts thirty decibels more intensity over a partial than the other, the weaker contributes a thousandth of the total and moves the threshold by a few hundredths of a decibel. Where two maskers put equal intensity over it, the threshold rises by three decibels above either. Two maskers together cost a partial at most three decibels more than the stronger of them alone, and usually much less. One sound hides another set the rule out for a pair of tones; here it decides how a texture grows.

Over all 284 arrivals the pattern holds as a rule rather than an instance. In 252 of them, what the two parts cost together is within a decibel of what the larger costs alone. The points that matter are the few far to the right, where both parts cost the carrier real room: in the fourteen arrivals where the smaller of the two would cost five decibels or more on its own, it adds a fifth of its own cost on top of the larger. A cost that simply added would sit on the diagonal. Every arrival sits under it, most of them on the floor.

Room spent is not room lost twice #

The earlier essay’s picture was of room as a reserve, spent by each part that enters, so that a carrier with less room left would be pushed under sooner. The rule above says the opposite about the mechanism. Room is not drawn down by a fixed amount for each part; each part raises the threshold to what the strongest masker in the texture sets, and a part weaker than the strongest adds almost nothing.

Read partial by partial, the effect is graded. Where the passenger had taken under a decibel from a carrier partial, the third part takes all of what it would have taken alone. Where the passenger had taken between one and three decibels, the third part takes 82 per cent of its own cost; between three and six, 78; between six and ten, 51; and where the passenger had already taken more than ten decibels, the third part takes 39 per cent of what it would cost on its own. The more of a partial’s room has been spent, the less there is for the next part to spend, because the threshold over that partial is already set by something the next part does not outweigh.

That is why the compounding the lead expected is so rare. A partial lost only to the combination needs two maskers of nearly equal strength over it, each leaving it a little clear on its own and the pair pushing it three decibels further. In the passage that coincidence happens twenty-one times in 284 arrivals, and it is spread across every kind of part and every chord.

So the earlier essay’s tax analogy needs turning round. A tax of ten per cent leaves a small income nearer the line, and a second tax takes ten per cent of what is left. Masking is more like a set of competing claims on the same partial, of which only the largest is paid in full.

Which carriers are fragile #

If the parts entering decide the loss, the carriers they enter beside decide how much there was to lose.

The voice as carrier has most to lose and loses most. It has 10.7 partials clear over the chord alone, keeps every one beside its passenger, and is down to 6.6 when a third part enters — four partials taken, nearly all by the flue and the violin. The violin as carrier goes from 6.7 to 5.6. The flue as carrier hardly moves: 5.5 alone, 5.3 beside its passenger, 5.1 with a third. The brass and oboe start with four partials clear and end with two and a half.

The voice is fragile for the reason the earlier essay found it costly as a passenger: its partials are strong across the whole region from a few hundred hertz to several kilohertz, where its formants sit. That puts many of them clear of a chord’s own masking, and it puts many of them where the flue’s and the violin’s spectra are strong. The flue is robust because it is the highest part in the roster and the purest: its few partials sit above most of what the others play, in a region masking reaches mostly from below. The fourth player is a spectrum argued that what an entering part adds is partials rather than a player; what a carrier loses is likewise decided by where its partials are, relative to the strongest of what enters.

Why the flue and the violin take so much #

The flue pipe here sounds C6, the violin G5 — the two highest entrants in the roster — and the arrivals in which they take most are carriers whose upper partials lie at and above those pitches. The example above shows the shape: the flue left the violin’s second and third partials nearly alone and took everything from the fourth up, the partials at and above the flue’s own fundamental near a kilohertz. Masking spreads upward from a masker far more than downward, so a strong high part covers the region above it with a heavy threshold and leaves the region below largely free.

That reading is consistent with every arrival examined but has not been decomposed partial by partial across all 284, and it is offered as the likeliest one. What the arithmetic shows directly is the ranking: the two parts that cost a carrier most room entering alone — the flue at 14.1 decibels on average, the violin at 8.8 — are the two that cost it most partials, and the voice at 7.3 decibels follows them. The clarinet, oboe and brass entering third cost a decibel or a decibel and a half, and they take a partial only where the passenger had already left one within a decibel or two of the line.

An orchestrator doubles a line, not a chord found doubling paying when a second instrument supports the first; an entrance is a change of colour found that an entering part is heard as colour more often than as a line. Both are about the second part. A third part’s effect on the carrier is mostly about its register and its spectrum’s reach upward, and in this passage that means a high flute stop or a high violin line is the thing that strips a carrier, not the number of instruments already doubling it.

Is there a number of parts at which a line stops being the line? #

The lead asked for one. The arithmetic says there is not, and shows why the question looked as if it had an answer.

Averaged over every choice of parts, carriers do erode as the texture grows. With two parts entering, 84.5 per cent of carriers keep a majority of their audible partials clear; with three, 58.5; four, 45.8; five, 37.7; all six, 32.4. The mean number of partials lost goes from 0.27 to 1.54, 2.62, 3.45 and 4.00. Read that way, there is a gentle decline with no knee in it, and no number at which the line goes.

Restrict the added parts to the ones that are not the flue or the violin and the decline nearly vanishes. With three parts, 76.2 per cent of carriers keep a majority and they lose 0.54 partials on average; with four, 82.4 per cent and 0.70. Those are close to the pairs’ own figures. The averaged curve falls because, the more parts are chosen from a roster of six, the more likely the flue or the violin is among them — and beyond four parts nearly every choice includes one. The count of parts is a proxy for whether the high, strong ones have entered.

That answers the lead’s question in a form an orchestrator can use. A line carried by one instrument and doubled by two or three others keeps its partials if none of the doublers is a strong part whose spectrum covers the carrier’s upper range, and loses them to the first such part that enters, whatever the count. The chord that has room for an entrance asked which chords leave a part room; the companion question for a growing texture is which parts take it.

How the texture behaves as a whole #

Put together, the figures describe a texture that grows in an unexpected way. Each part added raises every other part’s thresholds to whatever the strongest masker over each partial dictates, and once that strongest masker is present, further parts weaker than it add almost nothing. A texture of a violin, a voice, a flue and two reeds masks the violin almost exactly as the flue alone does, because the flue is the strongest masker over most of the violin’s spectrum and the others’ contributions are swamped.

That has a consequence for how orchestrators speak about thick textures. A texture is often described as getting muddier with every part added, and the arithmetic here supports that only in the sense that more parts raise the probability of including a strong masker. The number itself does little. A pair entering as one hides the part that cannot be heard and a floor under each member is paid for in steps built their walks two parts at a time; the three-part reading says that those walks lose little by stopping at two, so long as the parts they leave out are not the strong high ones.

It also explains a regularity the earlier essays reported without accounting for it: that the passengers’ losses were concentrated in a few pairings while the carriers’ were almost nil. A carrier is the stronger of a pair, so over most of its spectrum it is its own strongest neighbour, and a passenger weaker than it adds little to the thresholds over its partials. The carriers’ immunity and the third parts’ small joint cost are one fact seen twice.

The masking model, restated #

Thresholds from summed intensities. Each partial’s masked threshold is the sum, in power, of the masking every other sounding partial puts over it, from the spreading function every orchestration essay here has used, floored at the threshold of hearing. That rule is what makes two maskers worth at most three decibels more than the stronger, and the whole result rests on it. It is the standard assumption for simultaneous maskers of different frequencies; some measurements find maskers combining more than additively in power, which would make joint losses more common than the twenty-one found here.

Clear means above threshold. A partial with a positive margin is counted as present, as in the earlier essays. The room figures report the margins themselves so that the hard line can be read past.

One passage, one roster. Five chords and six parts entering at fixed pitches, all at the same nominal level. A roster with a different high part — a piccolo, a trumpet in its upper register — would put the strong masker somewhere else, and the ranking of third parts would change with it.

What the masking count cannot say #

Whether a listener tracks the line. Partials clear of the mask are audible as components; whether a listener hears the carrier as one line through a thick texture is a question about grouping that the count can inform and not settle. A part entering is not a change of level is a reminder that onsets do much of that work and are not modelled here.

Dynamics. Every part enters at the same nominal level. An orchestrator who marks the flue down would shrink its upward reach, and the arithmetic says that one change would remove most of the loss in this passage, but no figure here varies it.

Still open: a texture with one strong part kept quiet #

The ledger names the culprits: two parts whose spectra cover the carriers’ upper partials. The obvious remedy is to play them more softly, and the masking rule says the gain should be large and sudden — each decibel taken from the strongest masker is a decibel returned to every partial it dominates, until a second masker takes over as the strongest. The computation is this one with the third part’s level swept below the others’, and it would say how far a high flute or violin line has to be marked down before a carrier keeps its partials, and whether that level is one an orchestra would actually write.