A passenger under its mask is still heard in the arrival

Three essays on this passage have been about one trade. A pair entering as one hides the part that cannot be heard scored two parts coming in on the same chord as one arrival, and found that the best pairs were always a loud part carrying a quiet one: a violin clearing its mask by 19.6 decibels beside a clarinet 10.9 under its own. A floor under each member is paid for in steps then forbade any member to be more than so many decibels under, and found the arrival paying for every decibel of floor with a decibel of margin, in three steps, until nothing beat bringing the parts in one at a time.

Both essays measured a passenger the same way: the mean, over its partials, of how far each clears the threshold everything else sounding puts over it. On that measure a clarinet at −10.9 is buried, and the carriers’ walk was a walk of hidden parts.

The last of those essays ended by doubting the measure. A part two or five decibels under its mask is not silent; it changes the timbre of the arrival it is in. The question it left was whether the arrival with the passenger is distinguishable from the arrival without it — and if so, whether the floor had been measuring the wrong kind of audibility.

It had.

What a threshold is a threshold of #

A masked threshold is not the level below which a component stops existing. It is the level at which a listener can just tell that the component has been added: the change it makes to the sound is detected about as often as not. So a partial five decibels over its masked threshold is a clearly audible change to the sound, and a partial five decibels under is not — on its own.

That last qualification is the whole of the matter. A passenger is not one partial but eight or fourteen, and an average over them treats a partial thirty decibels under the same way it treats one ten decibels over, as if the two cancelled. They do not. The partial thirty under contributes nothing; the partial ten over is heard. An average of the two is not the right number to ask for.

The clarinet above is the case every earlier walk hinged on. Its mean margin beside the violin on the open tonic is −10.9 decibels, the most buried member of the best walk. Its individual partials run from +12.0 — the fifth, at 1,960 hertz — to −32.5, the sixth. The first, third and fifth are clear of the mask by 8.8, 7.7 and 12.0 decibels; the seventh is 2.0 under; the second, fourth, sixth and eighth are 22 to 32 under.

Look at why those four are so far under, and the mean stops looking like a measurement of anything. The clarinet’s even partials are weak to begin with — that is the tube that skips every other partial, a closed cylinder’s own acoustics — and in this spectrum they sit at 41, 38, 27 and 22 decibels against 69, 63 and 55 for the first, third and fifth. They are under the mask chiefly because there is almost nothing of them. Averaging them in charges the clarinet for partials it barely makes, and it is those that pull the mean ten decibels under.

A listener who hears the violin arrive with the clarinet beside it therefore hears the clarinet’s three strongest partials added to the arrival, each well above the level at which a change is noticed. Three partials — including the fundamental — are not enough to make the clarinet a line in its own right against a violin nearly twenty decibels clear, but they are more than enough to make the arrival a different sound from the violin alone. That is what an orchestrator means by a doubling colouring a line.

Adding up chances to notice #

To put a number on “a different sound”, each partial can be treated as an independent chance to notice the change. An energy detector’s sensitivity grows with the power of the thing to be detected, so a partial at its masked threshold gives a sensitivity of one, a partial three decibels over gives two, and one three decibels under gives a half. Independent chances combine as the root of the sum of their squares. The result, expressed in decibels, is zero when the whole passenger is just detectable and positive when it is more than that.

For the clarinet with the violin it is +12.7 decibels. Nearly all of that comes from the fifth partial alone: combining the others adds seven tenths of a decibel. That is typical. Across every pairing computed here, the median difference between the combined figure and the single clearest partial is a third of a decibel. The combination matters only when no partial is clear, which is one case, and it comes up below.

Every pairing in the passage #

Take every pair of the six parts entering together on every one of the five chords, with nothing else of the roster sounding, and call the one with the lower mean margin the passenger. In 71 of those pairings the passenger is under its mask on average; in 41 it is more than three decibels under — buried, on the old measure.

All 71 change the arrival detectably. The smallest change is 0.11 decibels over detection, and 35 of the 41 buried passengers change it by more than three. The passengers with the most negative means — a voice sixteen decibels under beside the flue pipe on the subdominant, a clarinet sixteen under beside the violin on the dominant — sit at +1.8 and +0.4. The rest are spread between five and twenty.

The figure also shows why the mean was misleading. Along the bottom axis the passengers range over sixteen decibels; up the side they range over twenty, and the two are barely related. A passenger’s mean is set by its worst partials, the ones sitting under the carrier’s and the players’ strongest; its audibility is set by its best, the ones that fall between them.

Two or three partials, not the whole part #

The typical buried passenger has two or three partials clear of the mask: 47 of the 71 have two or three, 14 have four, one has five. Only 9 have fewer than two, and only 5 have none.

Two or three partials in the clear is a specific kind of audibility. It is enough to change the arrival’s spectrum — its brightness, its body, the weight of one region against another — and it is usually not enough for the ear to follow the passenger as a line of its own, since the partials that survive are scattered and rarely the lowest ones that carry a note’s pitch most strongly. On the clarinet with the violin, the fundamental survives, but it survives alone among the lower partials, beside a violin whose own fundamental is the loudest thing in the arrival.

So the earlier measure and this one are both right about something. The mean margin was a fair proxy for whether a part is heard as itself: a passenger ten decibels under on average has lost most of what would let a listener pick it out and follow it. It was not a proxy for whether a part is heard at all. The carriers were never hiding their passengers. They were turning them from lines into colour.

The one place a passenger is nearly lost #

The five passengers with no partial in the clear are all the same part on the same chord: the clarinet on the bracketing dominant, with each of the other five as its carrier. That chord’s voicing — G2, D4, G4, B4 — puts one of the players on the clarinet’s own G4 and the others’ partials across its register; the chord that has room for an entrance is about exactly this, which chord of a passage leaves a given part room, and the earlier walks found the clarinet more than ten decibels under on the chords after the open tonic whatever it did.

There, the clarinet’s clearest partial is 0.8 decibels under its threshold with most carriers and 1.4 under with the oboe. Every other partial is further under. Combined, its eight partials come to between +0.11 and +0.40 decibels — just detectable, by the barest margin, on the strength of eight near-misses adding up.

This is the only place in the passage where adding the partials up is what decides whether the passenger is there. It is also the only place where the earlier essays’ word “buried” is close to right. A clarinet entering on that chord with anything else is at the threshold of being a change nobody notices, which is the practical reason an orchestrator would not bother. Everywhere else in the passage a passenger colours its arrival by several decibels at least.

What each kind of floor costs #

The earlier essay’s floor asked every member of an arrival to be within so many decibels of clearing its mask on average. Asking instead that every member change its arrival by so many decibels over detection gives a very different price.

On the mean margin, as the earlier essay found, the weakest arrival falls from 10.93 decibels with no floor to 7.05, then 4.50, then 0.59 — the one-at-a-time schedule — as the floor tightens from twelve decibels under the mask to five.

On colour, requiring every member to be at least just detectable costs nothing: the carriers’ walk already meets it, with its clarinet at +12.7, oboe at +3.7 and brass at +10.6. Requiring three decibels over detection costs nothing either. At four, the oboe’s +3.7 is too little and the weakest arrival drops to 10.15, where it stays until the floor reaches ten. At ten it is 8.97, at twelve 1.18, and past about thirteen decibels over detection there is no walk at all, because some part has no pairing on any chord in which it changes the sound that much.

The two floors do not merely charge different prices; they disagree about which passenger is the problem. The first pairing the colour floor removes is the oboe entering with the voice on the submediant — a passenger only 1.5 decibels under its mask on average, the least buried of the walk’s three, but one whose partials all sit close to their thresholds so that none stands far clear. The clarinet, 10.9 under on average and the whole reason the mean floor stepped, survives every colour floor up to twelve decibels, because one of its partials is twelve decibels clear. Ranked by mean, the clarinet is the most buried passenger; ranked by how much it changes the sound, the oboe is.

The walk at a colour floor of ten is still a walk of carriers and passengers. The clarinet still enters beside the violin at a mean margin of −10.9; the brass beside the flue pipe; the oboe beside the violin, which has left and come back. Every one of them is heard in its arrival by at least ten decibels, and the weakest arrival is 8.97. Asking that every passenger be heard in its arrival costs the carriers almost nothing; asking that it be heard as itself costs everything they were for.

The connection the earlier essays were circling #

An entrance is a change of colour argued, before any of the walks were computed, that what a listener hears when a part enters over a texture is often not a new line but a change in the texture’s timbre. The walks then scored entrances as lines — by the mean margin — and found their best schedules full of parts that could not be heard. This essay closes the loop: scored as changes of colour, the same parts are heard, and the schedules were colouring the texture exactly as that essay said entrances do.

The same shape turned up in an orchestrator doubles a line, not a chord, which found doubling paying when the second instrument supports the first rather than adding its own voice. A supporting voice is a passenger in the sense here: heard in the colour of the line, not as a line. And the fourth player is a spectrum found that what an orchestrator adds with a part is partials rather than a player. This essay is that finding measured: what a passenger adds is two or three partials, and two or three partials clear of the mask are a change anyone can hear.

What the model assumes #

Partials are independent chances to notice. The rule that combines them — each partial’s power over its threshold as a sensitivity, combined as the root of the sum of squares — treats every partial as a separate look at the sound. Partials of one instrument are not independent to a listener; they are grouped, and a listener may be better at noticing a coherent set than the rule credits, or worse at noticing a change in one partial of a sound they have grouped with the carrier. The rule matters only on the dominant, since everywhere else the clearest partial alone decides.

The masked threshold is the collection’s spreading function. Each partial’s threshold is the sum of the masking every sounding partial puts over it, from the spreading function every orchestration essay has used, together with the threshold of hearing. A different spreading function would move every number; it would not easily make a partial twelve decibels clear into one that is buried.

Everything is simultaneous and steady. As in every earlier walk, the parts start together and hold. A passenger starting a few tens of milliseconds after its carrier would be heard more readily still, because its own onset marks it; one whose attack is smeared into the carrier’s would be heard less.

And detectable is not the same as noticed. Every number here is about whether a change could be detected by a listener comparing the arrival with and without the passenger. In a performance there is no comparison. What makes a passenger worth writing is not that a trained listener could tell it was there, but that it changes the sound everybody hears — and whether a change of a few decibels over detection does that is a question this arithmetic stops short of.

Whose doublings #

The practice at issue is the orchestral habit of bringing a weak instrument in beside a strong one: a clarinet with the violins, a bassoon with the horns, a flute an octave over the oboe. Treatises tend to describe such pairings by their effect on colour — a brighter, darker, rounder line — rather than by whether the weaker instrument is heard. This essay supports that description with numbers, in this passage and roster: the weaker instrument is almost always audible as a change to the stronger one’s sound and rarely audible as a line in its own right.

Still open: whether the carrier is changed as much as the arrival #

Everything here asks what the passenger adds. The reverse question is just as computable. A passenger masks its carrier too, and a violin beside a clarinet has some of its own partials pushed under by the clarinet’s. The arrival is then not the carrier plus a colour but a new spectrum in which some of the carrier’s partials have gone and some of the passenger’s have arrived, and whether a listener would describe the result as “the violin, coloured” or as “a third sound” depends on how much of each survives. The calculation is this essay’s, run the other way round over the same seventy-one pairings: for each carrier, how many of its partials the passenger takes below their threshold, and whether any pairing in the passage leaves neither part with the majority of its partials in the clear.