An octave down a third part is heard and harmless, with nothing to spare

Some parts cannot be told to play more quietly. A quieter third part gives back only as deep as it took found how much a high entrance has to be marked down before it stops stripping the carrier beside it of its partials, and it ended with an awkward result. The violin can do it: twelve decibels down, it gives back nine tenths and is still heard. The high flue pipe cannot. It needs thirty decibels to be harmless, and it is heard in under half its arrivals by twenty-four, and on a pipe organ the level of a stop is settled by its voicing and is not something a player can ask for at all.

What an organist has, and what an orchestrator has too, is a different octave. A line given to the eight-foot stop can go to the four or the sixteen; a flute’s phrase can be written where it lies, or below or above. The obvious reasoning, with which that essay closed, ran as follows. An octave higher, the third part clears the carrier’s main region and spreads over partials that are already faint. An octave lower, it comes closer to the carrier’s strong lower partials. On that reasoning, up is the natural way out, and it agrees with long habit: a part that needs to be clear of the texture is taken up above it.

The count says the opposite, and by a wide margin. The reason is the most basic asymmetry in hearing, which every essay on masking has quoted and which the reasoning above overlooks at the moment it matters.

The method is the same arithmetic as the essays before it. The roster is six parts, a brass line, oboe, clarinet, voice, violin and flue, on a five-chord passage. In 284 arrivals, a carrier and its buried passenger enter with a third part beside them, each at a nominal 70 decibels per fundamental with a spectrum of fourteen partials. The count is the number of carrier partials that stood clear of their masked threshold beside the passenger and go under when the third part enters. The only change is that the third part now sounds some whole number of octaves from where it stood. The figure below carries that move on its dial, from two octaves down to two up, with the part’s old cost in outline behind it.

At the third part’s own register the total is 362 partials taken, the same 362 as before. Moved up an octave it is 539. Moved down an octave it is 95. The flue goes from 195 to 50, and the violin from 110 to 3.

Going up costs more and going down costs less #

The second figure is the same count against the shift, and it shows that this is not an accident of one direction.

The total rises at every step, and the two steps either side of the third part’s own register are not equal: the 267 partials saved by going down an octave are more than the 177 lost by going up. Reading one part at a time adds something. The flue, which is already at the top of everybody’s register, goes from 195 to 176 to 147 as it rises further. Its cost is being taken away by a different effect: a part high enough leaves carrier partials below itself and is soon above the top of those it can reach. The oboe shows the whole shape. It stands at E4, at the same height as most carriers’ strong partials, and it takes 11 where it stands. At E5 it takes 94 and at E6 it takes 164, which is more than the violin ever takes.

So the principle at work is not “move away”. It would be natural to suppose that any move away from the carrier reduces its effect, and in Hertz up and down are both away. It is the opposite of that in practice, because masking does not treat up and down alike. A sound covers partials above itself far more than partials below it. A sound’s skirt runs upward at a shallow slope and downward at a steep one, which is how a bass hides its tenor, and this has been in the sum since the masking pattern was first drawn. What decides the cost of a third part is therefore not whether it is near the carrier’s strong partials. Those can defend themselves. The cost is decided by how much of the third part’s strength lies just below the carrier’s weak ones.

Where the voice’s partials come back from #

The clearest view of that is a single arrival read partial by partial.

With the flue on C6, the voice’s first seven partials are in the state they were in: some just clear, some a few decibels under. From the eighth partial up, every one of them is between 22 and 32 decibels under. Those are the partials above 4 kilohertz, and they are weak, because a voice’s spectrum falls towards the top. With the flue on C5 instead, those same seven partials are all clear, by 3 to 18 decibels, and the first seven hardly move at all. The flue at C6 has its fundamental and its first few strong partials sitting at one, two and three kilohertz, directly under the voice’s weak top. Move it down an octave and the same strength moves to half a kilohertz, one and one and a half. Its energy is then a long way below the voice’s top in the only unit that matters here, the Bark scale, and at the slope of an upper skirt a few Bark is tens of decibels.

The reasoning that opened this essay was half right. An octave higher does spread over partials that are already faint. That is exactly the problem. Faint partials are the ones with no defence: they have nothing above them for a masked threshold to compete with, and they are what a carrier’s count is mostly made of. Their contribution to the level of the note is negligible, and their contribution to the question “is this partial above its threshold” is one each, the same as any other.

There is a caveat in that, and it should be stated. The count weighs every partial equally. Whether a voice’s thirteenth partial is worth as much as its second depends on what the listener is doing with them; an arrival’s character may be in its first few. Measured by level, a flue moved down an octave might well be worse for the voice’s lower partials, and the figure above does show it taking a few decibels from the first. So the claim here is about the count of partials clear of the mask, which has been the measure across this whole line of argument, and it is not a claim that a low third part is better for the carrier by any measure. It may be worth noticing that an arrival’s brightness in particular lives in exactly the partials this figure is restoring.

No level does what the octave does #

The result can be put against the alternative directly, because the two moves both take partials back and both may silence the part.

One octave does for the flue what about twenty-two decibels of marking down would do, without any of the cost. On the level path a flue reaches 56 partials at 21 decibels and is heard in only 79 per cent of arrivals. By 24 it is at 33 and 48. The octave puts it at 50 partials, heard in all of them. For the violin the difference is greater still. Marked down fifteen decibels, the violin reaches three partials, and is by then heard in 40 per cent of its arrivals. An octave down at full level it also takes three, and is heard in 98. The earlier essay asked whether there is a level at which the flue is both heard and harmless, and showed there is none. The answer to the question it left is that there is an octave. And it is the one that reasoning from the carrier’s strong partials had set aside.

The passenger does not pay for any of this.

The buried passenger, which is still heard in an arrival despite being under its mask on average, might have been expected to suffer. Its position is low in the texture, and a third part an octave lower might land on it. It does not move: 95.4 per cent of arrivals with the third part an octave down against 95.1 with it in place. A third part that takes nothing from the carrier is not taking it from the passenger instead, and that is exactly what makes the octave a real gain rather than a transfer. Nothing here counts what the moved part does to the chord already sounding, since only the entering parts are being scored, and a part moved down into the chord’s own register is a change to the voicing that the entrance essays did not have to consider. That is left open below.

The octave has spent the part’s spare level #

So far the octave has looked free, and it is not. The third part is still heard because at full level it just clears the mask of the chord it has moved into. What it cannot then do is lose any level.

A flue an octave lower drops out of a third of its arrivals at just three decibels softer. Where it stood it could lose fifteen decibels without leaving any. The violin shows the same thing: it could lose six in its own register, and an octave down three decibels takes it out of nearly half. That is the price of the octave, and it is a real and quite specific one. The harmony of register and strength that makes the moved part harmless is the same one that makes it barely audible. A high part stood clear of the texture by a large margin, which it was using to take partials from others. Low down, the texture covers it nearly to the threshold, so it has no margin left and it can no longer take anything from anybody.

That gives the essay’s surprising connection, and it lands on something an organist knows from the opposite side. A part that can be marked down has a margin, and a part that has been moved has spent it. The two levers are not two ways of doing one job. Marking down keeps a part’s register and trades its presence gradually. Moving it down trades all its presence at once for all of its cost. A flue pipe has no dynamic, so it is exactly the part for which the octave is the right lever: its level will not wobble at the wrong moment. A violinist, who has a bow and an intention and whose level moves through every phrase, is at risk from the octave in a way a pipe is not. So the part that the earlier essay found could be managed by level does not need the octave, and the part that needed it can bear it. It would be hard to design that coincidence on purpose.

There is also a plain consequence for scoring, though no orchestration manual has been asked about it here. The usual reasoning about doubling and spacing warns against a low part and encourages a part placed well above a voice to give it room. Measured as partials taken, the second is the more expensive of the two: an entrance above a voice takes its top and an entrance beside or just below it does not. Both claims can be true together. The warning is about level and the balance of a chord’s weight, which this count does not measure, and who plays what and how loud is one question is the reminder that deciding the one without the other has already failed once.

The move also says something about a thing organists do every day, which is to add an octave and not to replace one. A four-foot stop drawn beside an eight-foot one is a doubling, and the commonest operation in orchestration is priced in those terms. A doubling an octave above a line adds its upper reach to the line’s without taking the line away, and what this count says of a part moved upward it says of that addition too: its strong partials now lie beneath everyone else’s weak ones. The custom of building a registration upwards is a custom of brightening the whole sound, and on this measure the brightness is partly paid for by the other parts’ tops. That was fine for the reason a third part takes what it would have taken alone records: the losses do not compound, so a registration is not ruined by its fifth stop. Whether a listener at a real organ can follow an inner solo line more easily against a sixteen-foot accompaniment than against a four-foot one is the obvious test, and nothing here has made it.

What the count leaves out #

The compass. Every part is moved by whole octaves whether or not it could go there. A flue from C6 to C4 is fine, and a violin to G3 is a low open string. But the brass going down two octaves to E1, and the clarinet to G2, are off any plausible instrument or into a poor register, and a real part an octave down does not keep the same spectrum, as a violin’s low string is not its high string and a flue pipe’s spectrum changes with its scale. Each part’s spectrum here is one curve moved bodily, so the two-octave readings are a limit and not a suggestion. The one-octave ones hold up best for the flue, which is the part the question was about, because a stop an octave down is an honest pipe with its own voicing.

Consonance. Every move here is by octaves, which keeps the part’s pitch class, and so its partials still land near those of the chord in the same pattern. A part moved by a sixth would be a new line and would have new coincidences. Those are not read here.

Time. All of this is the moment of arrival. A part that leaves is not a part that arrives, and room is used up by whoever enters first, and neither is folded into a calculation in which three parts arrive at one moment and hold. In a real score a third part is often already sounding, and the count against an established part is another question.

And a count is not a percept. A partial crossing its masked threshold is a statement of detection in a stationary tone. Whether a voice with its top back is a voice that sounds better, or even different, in a passage is a listening question nobody has answered with these stimuli. The buttons under the figures are those stimuli, and they are the honest end of this argument.

Whose scoring it is #

The roster is a mixed ensemble of six, and the passage is five chords of a tonal harmony in four parts at a fixed dynamic. It is a stand-in for the entrance of a solo line over an accompaniment, as in a good deal of nineteenth-century orchestral and organ writing. It is not a claim about what orchestrators do. Whether they prefer to bring a third line in below a voice rather than above it would have to be counted in scores, and that corpus has not been built here.

Still open: what the moved part does to the chord #

A third part an octave down has moved from above the chord to inside it, and on this count the only thing it has changed is the fate of the carrier. It has also become one more thing in the chord’s own register. If it now stands where a chord tone already stands, the two reinforce each other and the part is hardly a part. If it stands between two chord tones, it may be covering one of them. The computation is this one with the base chord’s own voices scored as carriers, against the moved part, which would say how much of the 267 partials saved at the entrance has been taken from the accompaniment instead. It would also say whether there are chords, such as the open tonic with its wide space in the middle, where the octave is really free, and chords, such as the close voicings, where the saving is a loan from the parts nobody was counting.