An entrance is a change of colour

Every figure this anchor has drawn puts the same players on every chord. The chord moves, the assignment moves, the level moves — and the list of who is playing is fixed at the top of the figure and never touched again. Eight rungs have optimised inside that constraint, and the last of them ended by naming it: the commonest thing a score does with a player is not to give them a different note but to bring them in, and to take them out again.

An entrance is a different kind of object from an assignment, and the difference is arithmetic rather than aesthetic. This anchor’s objective is a pair — a loudness that has to arrive at a written mark and a roughness that is then minimised — and the two are sums over different things. Loudness is a sum over parts. Roughness is a sum over pairs of parts. So a player joining an ensemble of n adds one term to the first quantity and n terms to the second.

That is the whole rung, and the sizes are not close. What the entrance contributes to the loudness falls from 15.5 phons when it doubles a soloist to 0.32 when it joins ten — a factor of forty-eight. What it contributes to the roughness rises by a factor of 12.3 over exactly the same range.

The player who is not a permutation #

It is worth being precise about what the ladder above has been holding still, because the constraint is invisible in every drawing that obeys it.

Thirty-six arrangements, and every one of them has all four players in it. That is what makes the problem an assignment: the search is over which position each player takes, and a candidate in which somebody is silent is not in the set at all. The seventh rung says so explicitly — a note left empty is a different chord and not a different orchestration — and it is the right rule for the question that rung asks.

It is also the reason the ensemble has never been a lever. A permutation cannot represent a rest, and neither can a surjection. The moment a player is allowed not to play, the object stops being a map from players to notes and becomes a schedule, and the objective stops being a number and becomes a number per chord with a boundary between them.

One term against n terms #

The arithmetic underneath is elementary and it is worth writing out, because everything below is a consequence of it and nothing below is a fit.

A scoring’s loudness is computed by putting every audible component of every part onto one excitation pattern along the Bark axis and integrating it. Adding a part adds its components to that pattern. Two things then work against the addition: the pattern is compressed, so a component landing where there is already energy raises the integral by much less than it would in silence, and the ensemble’s own components put a masked threshold over the entrant’s, so some of them do not enter the integral at all.

A scoring’s roughness is computed by taking every pair of components belonging to different parts, and summing a kernel that depends on how far apart in a critical band they are and on the product of their two pressures. Adding a part adds every pair between its components and everybody else’s. There are n such new parts to pair with, and nothing about them is compressed, because roughness is not a perceptual integral — it is a sum with no saturation in it anywhere.

So the two quantities do not merely differ in size. They differ in what they are functions of: the entrance’s loudness contribution is a function of what is near it in the spectrum, and its roughness contribution is a function of how many others there are for it to be near.

The consequence is visible in the census the hero draws, and the numbers are worth reading in order. A second player is worth 15.5 phons and almost no roughness at all, because there is only one other part for it to be rough against. The fifth is worth 0.90 phons and eleven times the roughness the second was. The tenth is worth 0.32 phons and twelve times.

The point where a part stops being heard as an arrival #

The loudness limen is about one phon, and a phon is a decibel at these levels — so an entrance worth less than a phon is not a loudness event at all. It is an addition a listener cannot detect as an addition, however well they can hear the sound it is part of.

That crossing falls, on this texture, at five players already sounding. A sixth player joining a quintet contributes 0.90 phons, and the seventh contributes 0.70.

The loudness ladder measured the same quantity in a completely different way and arrived at the same place.

Its texture is a different one, its parts are all the same string timbre rather than five different radiators, and its level is eight decibels lower. Its median answer is 0.89 phons against the 0.90 this census reads at the same ensemble size. The agreement is looser than the third digit makes it look — two models this unlike each other agreeing to a hundredth of a phon is a coincidence rather than a confirmation — but the order of magnitude is not a coincidence, and the order of magnitude is the whole claim. A part joining a quintet is worth about nine tenths of a phon, and the limen is one.

What that says about a score is uncomfortable and specific. A composer who adds a woodwind to a five-part texture to make the passage grow has not made it grow. They have changed its colour, by an amount the next section prices, and the growth they wrote has to come from somewhere else — from the players already there playing louder, or from the entrant taking a register nobody is in.

Where the two models disagree about the sign #

There is a defect in the machinery here that has to be stated before any of the above is trusted, because it is large and it points the wrong way.

The grouping model reports that the entrance made the ensemble quieter, at eight of the nine sizes drawn, by as much as 6.7 phons. Nothing in the ear does that. It is a property of a greedy grouping rule: components are gathered into bands, a band’s loudness is not the sum of its members’, and a new component sitting between two bands can merge them, at which point the total loses everything the two bands were counted separately for.

The loudness ladder found the same thing when it wrote its own entry census, and recorded it in the same words: the sum is not monotone in the number of parts, the excitation model cannot fall, and the disagreement belongs to the grouping rule rather than to hearing. Its census falls at twenty-four of the sixty-one pitches it tries; this one falls at eight of ten sizes.

So the headline above is the excitation reading, and this figure is here rather than in a footnote because a reader has no other way to know that the choice was made. Both are in the collection, both are used elsewhere, and on this particular question they do not merely differ in magnitude — they differ about whether the effect exists.

Ten decibels multiply one objective by ten and the other by nothing #

The second rung of this anchor established the shape of the level dependence, and it turns out to decide the character of an entrance.

Roughness is a product of two pressures. Raise every part by ten decibels and every pressure in every pair has gone up by the same factor, so the sum goes up by a factor of ten exactly. Loudness in sones is compressive — a doubling of loudness costs about ten decibels, not three — so the same ten decibels multiply it by two.

Applied to an entrance, that produces a result the arithmetic makes inevitable and nobody would guess from the practice.

Across the fifty decibels drawn, what the entrance adds to the roughness rises by a factor of a hundred thousand and what it adds to the loudness falls, from 1.25 phons to 0.73. Both directions matter. The fall is small and is real: a louder ensemble puts a higher masked threshold over the entrant, so fewer of its components reach the excitation pattern at all.

The crossing point barely moves — six players at the two quietest dynamics and five at the other four — which is the honest reading of a curve that is flat. What changes with the dynamic is not whether an entrance is a loudness event but how much colour arrives with it, and that changes by five orders of magnitude.

So the rule a score could take from this has a dynamic in it. An entrance into a quiet texture is nearly a pure addition of loudness and an entrance into a loud one is nearly a pure addition of roughness, and the two are the same operation written on the same page.

Which is not the lever the ladder has been pulling #

It would be easy to read the above as saying that an entrance is a bigger decision than an assignment. It is not, and the seventh rung’s own decomposition says why.

An assignment moves the roughness by a factor of about eight between its best and worst candidates, and eighty-five per cent of that range is decided by one thing — which instrument is on the bass, because roughness lives in the low notes of a chord wherever the low note happens to be. An entrance moves it by rather less than that at any single chord.

The two are different levers on the same quantity and neither dominates. What separates them is that the assignment’s lever is available at every chord and costs nothing, while the entrance’s lever is available once and is heard as an event. A score that wants a colour change without an event has the assignment; a score that wants the event has the entrance, and gets the colour whether it wanted it or not.

The entrance that lands on a note somebody already has #

There are two kinds of entrance and the arithmetic separates them cleanly, which is worth setting out because a score does not distinguish them typographically at all.

An entrance onto a new note is what the census above computes: a set of components arriving where the ensemble had none of its own, making n new pairs with everybody. An entrance onto a note already sounding is a doubling, and the seventh rung priced that — with the ladder’s own convention that the pair splits the note’s power between them, so that nothing is won by being louder.

The convention is right for an assignment and wrong for an entrance. When four players are arranged over three notes, splitting is what keeps thirty-six candidates comparable; when a fifth player joins four who were already there, nobody redistributes anything. The horns do not play quieter because the trumpets have arrived.

Run without the split, a doubling entrance and a new-note entrance are worth very different things in the two objectives, and in opposite directions. The doubling adds three decibels to one note and creates pairs only with the other parts, since two components at the same frequency are not rough — roughness needs a difference, and a coincident pair has none. So a doubling entrance is more of a loudness event and less of a colour one than a new-note entrance of the same player at the same level.

That is the sharpest statement this rung can make about a decision an orchestrator actually faces. Bringing a player in on a line that is already being played buys loudness; bringing them in on a line nobody has buys colour. Both are the same mark on the page.

What a part is here, and what an orchestra has instead #

The model of a part throughout this anchor is one player, at one level, with one radiator’s measured spectrum, sounding a single steady note. An orchestral entrance is not that in three ways that all push the same direction.

A section is several players, and a section is more punctual than a soloist precisely because their small asynchronies average; they are also mistuned against each other by a few cents, which turns every coincident pair into a slow beat rather than a null. The entrance also has an attack, and the parts already sounding do not — a listener meets the entrant’s transient against a steady background, which is the one condition under which an onset is maximally informative. And an orchestral entrance is very often doubled at the octave, which is neither of the two cases the section above separates.

None of these is a correction to the two directions. Each of them changes how much colour arrives with an entrance and none of them changes the fact that the loudness contribution divides by the number of parts while the roughness contribution multiplies by it. But every number quoted here is a number for one player on one note, and a score does not often write one.

What the pictures cannot show #

The census fixes the entering player, the entering note and the entering level, and sweeps only the size of what is already there. That is deliberate — a texture that grew by adding the entrant higher each time would be sweeping register and ensemble size together, and the next rung but one shows that register is worth twenty-six decibels of masked margin on its own — but it means every number here is this oboe on this E, and the shape of the two curves rather than their heights is what carries.

Nothing here has a listener in it. The one-phon limen is a published quantity and the crossing is computed against it; there is no published limen for roughness, so this rung says where the loudness event stops and says nothing whatever about where the colour event begins. The claim is that one quantity falls and the other rises, and that is a claim about arithmetic. Whether a listener notices the rise is a listening experiment this collection cannot run.

And the roughness sum has no masking in it. Every pair of components between two different parts contributes, including pairs in which one member is under the threshold the rest of the ensemble sets — which the masking anchor found is most of the components in a dense texture. Putting the mask into the roughness sum would lower every number in the right-hand curve and would lower the later ones most, because that is where the masking is heaviest. It would not reverse the direction, because the pair count still rises, but it would flatten it, and by how much is not known here.

Whose music, and where the practice already knows this #

The scoring practice this prices is European orchestral writing from about 1800, where an entrance is a device with a name — a horn that enters under a held string chord, a trombone section that arrives for a single phrase — and where the treatises discuss entries almost entirely in terms of when and almost not at all in terms of how much louder.

That emphasis now has an arithmetic behind it. In a texture of five or more, an entrance is not audible as an increase; it is audible as a change in the sound’s grain, and the treatises’ silence about loudness is not an omission. Berlioz’s advice about the trombone’s entrance is about the moment and the register, and both of those are exactly the variables that survive when the loudness variable has gone under the limen.

The place the practice and the arithmetic disagree is the tutti. A score that adds three players to a twelve-part chord to mark a structural arrival is spending three parts on an event that this census says is worth under a fifth of a phon in loudness and a great deal in roughness — which is a description of what a nineteenth-century tutti actually sounds like, and is not usually how it is described.

The clock this has not touched #

Everything above is a comparison of two states: the ensemble before the entrance and the ensemble after it. A listener does not receive a state. They receive a change, over some interval, and this anchor already carries two integration windows of very different lengths — the roughness window at a twentieth of a second and the long-term loudness smoother whose release is two seconds and which the fourth rung is entirely about.

The eighth rung’s closing question was whether that two-second release makes an entrance on a loud chord audible at all. The next rung answers it, and the answer is no in a way the question did not anticipate: the release is on the wrong side of the smoother. Its attack is 99 milliseconds and its release is two seconds, so a rise is tracked twenty times faster than a fall, and what a two-second release hides is not an entrance but a departure. Three of a listener’s readings receive the same entrance at three different times, and the earliest of them is the one this rung has just shown carries all of the change.