The chord that has room for an entrance

Three rungs have now made the ensemble a variable. An entrance changes both objectives in opposite directions; a listener receives it in a tenth of a second and a departure over two; and the colour component carries the moment in both directions.

Each of them holds one thing fixed that a score does not: where in the passage the entrance happens. In every figure so far the ensemble is sounding, a player joins, and which chord it joins over has not been a variable at all.

It is the variable an orchestrator actually chooses, many times a page, and it is not free. The parts already sounding put a masked threshold over whatever register the entrant takes, and that threshold is a fact about the voicing — which notes are where — rather than about the dynamic.

The best chord gives the entrant 9.0 decibels of margin over the mask and the worst gives it 0.3, and the difference between them is nothing but the spacing.

The quietest chord is the worst place to come in #

The chord that leaves least room is the quietest of the five — 21.4 sones against 30.8 for the loudest — and that inverts the rule anybody would state from the practice.

The reason is that a mask is not a level, it is a level at a frequency. The quiet chord in this passage is the close-position one, whose four notes sit within a seventeenth and whose upper partials therefore pile up in the same critical bands the entrant’s fundamental is in. The loud chords are spread, so their energy is distributed across the Bark axis and the entrant’s own region is comparatively empty.

The masking anchor’s sixth rung established the same fact about a chord masking itself, in the register direction: a critical band is thirty-six semitones wide at the bottom of the keyboard and under three at the top, so the same close triad is a different object depending on where it is put. This is the same statement asked about a stranger arriving rather than about the chord’s own members.

Room for an entrance is spectral, not dynamic, and the two are decorrelated enough on an ordinary passage that a score choosing the quiet chord is choosing badly about half the time.

A hole in the notes is not a hole in the spectrum #

The passage contains one chord written with a deliberate gap in the middle — a bass at C2 and G2 and a pair of high notes at G5 and C6, with nothing between them — and if room were a matter of where the notes are, that chord would be the best moment in the passage by a long way.

It is not. It comes second, at 3.0 decibels against the best chord’s 9.0, and the reason is visible in the same computation. The gap is empty of notes and it is full of partials: the fourth, fifth and sixth partials of a C2 flue pipe land at E4, G4 and B♭4, and the entrant’s fundamental is at E4. A masking threshold does not know that those components belong to a note two octaves below; it knows only that there is energy in that band.

This is the defect a score cannot see, because a score draws notes and the ear hears partials. The stave is not a ruler makes the general version of the point about pitch and this is its version about occupancy: an orchestral texture has no holes in the register where its bass has partials, which on a four-part chord with a real bass is most of the middle.

What the register is worth, and what the moment is worth #

The choice of moment can be set beside the choice of register, which is the other lever a score has over the same quantity, and the two turn out to be the same size.

The trend belongs to the ear. The mean margin rises from −2.6 decibels at G3 to +23.8 at G6, because a masking threshold falls away above the components that set it and an orchestral ensemble’s energy is mostly below. An entrance high above the texture has more room than one inside it, at every chord, without exception — which is why a piccolo entering over a tutti is audible and a second viola is not.

The distance between the curves belongs to the score. It reaches 29.2 decibels at C6, which is larger than the whole twenty-six-decibel trend. At one pitch, in one passage, the difference between the best chord to enter on and the worst is bigger than the difference between entering in the bass and entering three octaves up.

That is the result this rung exists for. Choosing the moment is worth as much as choosing the register, and it is the choice a score makes without appearing to make one — because a register is written into the part and a moment is written into the rests before it, which nobody reads as a decision about audibility.

Where the curves are ragged, and why that is not noise #

The curves in that figure are not smooth, and the roughness in them is the mechanism rather than a numerical artefact.

A masking threshold at a given frequency is set by the nearest strong component and falls away on both sides of it — steeply below, gradually above. The components of four instruments are not evenly spaced: they are five interleaved harmonic series with different amplitude envelopes. So the threshold along the Bark axis is a run of overlapping skirts with gaps between them, and moving the entrant by a semitone can move it from the top of one skirt into a gap.

Read as advice, that says a semitone is sometimes worth several decibels of audibility, and it says which semitone: the one that lands between two partials of the loudest low part rather than on one. Nothing in the practice is stated that way, and something adjacent to it is — the advice against doubling an entering line at the unison with something already sounding, which is the extreme case of landing on a partial rather than between two.

The honest limit is that the raggedness depends on the timbres. Five radiators with the measured spectra this collection carries produce this particular comb; a section of eight violins slightly mistuned against each other produces a smeared one, and every gap in it is shallower.

One passage is an anecdote #

Five hand-voiced chords are five chosen examples, so the same computation is run on a second passage in the minor, voiced by different principles.

The spread is smaller here, 2.7 decibels, which is what a passage of more evenly voiced chords should produce and is itself a useful reading: the moment is worth a great deal in a passage that changes its spacing and very little in one that does not.

Both of the findings survive. The quietest chord of the five is not the best moment; and the chord with the largest written gap — a bass at A1 and A2 and a pair at A4 and A5, with three octaves clear in the middle — comes last of the five, at 0.32 decibels, with two of the entrant’s seven partials audible against five for the best. Written on a page it is the most open chord in the passage. Measured on the Bark axis it is the most crowded, because two low notes with strong partials fill exactly the region the gap looks like.

That figure is the mechanism seen directly, and it is drawn by the masking anchor’s machinery rather than this one’s, on the same voicing. What an entering part meets is that curve, and where it can be heard is the places the curve dips — which are between components rather than above the notes.

What the entrant’s own spectrum does to the number #

The margin is a mean over the entrant’s partials, and swapping the entrant shows how much that summary hides.

A clarinet entering on the same E at the same level has a mean margin that is negative at every chord in the passage, from −1.0 at the best to −10.4 at the worst, against the oboe’s +9.0 and +0.3.

That is not a statement that a clarinet cannot be heard entering. It is a statement about the mean. A clarinet’s even partials sit some twenty-five decibels below its odd ones, so half of its components are weak by construction and are under any threshold at all; averaging over them drags the mean down while saying nothing about the strong odd partials, which are what a listener hears the instrument by.

What is preserved is the ranking. The best chord and the worst chord are the same chords for the clarinet as for the oboe, because the ranking is a property of the ensemble’s threshold and not of the entrant. The moment is chosen by the ensemble and the margin is set by the entrant, and those are two separable decisions, which is the most useful thing this section produces.

What the ensemble does to itself first #

There is a control this rung needs and it is on the anchor already.

The ensemble’s mask over the entrant is a special case of the mask the ensemble puts over everything, itself included — and that self-mask is severe. On the largest texture drawn, only 23 of 80 components belonging to the parts that stay are above the threshold the rest of them set.

Which bounds how much this rung can matter as a texture grows. In a quartet the entrant’s margin is decided by four other players and the choice of chord moves it by a lot; in a full orchestral tutti it is decided by twenty, the threshold is high everywhere below the top octave, and the choice of chord moves it less because there is no chord in which the middle register is empty.

So the rule this rung produces is strongest exactly where a score has the most freedom to use it, which is chamber and small-ensemble writing, and weakest in the tutti, where the only lever left is the register one — and the register lever is worth twenty-six decibels, which is why a tutti’s entrances are piccolos and trumpets and not violas.

The rest before the entrance is doing work #

There is one more thing a score controls that this computation makes legible, and it is the thing a score writes instead of a dynamic.

A part that has been silent for a bar enters against the ensemble’s simultaneous mask, which is what everything above computes. A part that has been silent for a beat enters against that plus whatever forward masking is left from its own previous note — and against the ensemble’s forward mask too, which the eighth rung found reaches about twenty milliseconds on a dense texture rather than the two hundred it is quoted at.

Twenty milliseconds is short enough that the length of the rest before an entrance is almost irrelevant to its audibility. A quaver’s rest and a four-bar rest deliver the entrant into the same threshold. That is a null result and it is worth having, because the opposite is a plausible piece of folk advice — that a part which has been out for longer arrives more strongly — and the arithmetic says the mechanism it would need does not reach.

What the rest does do is something the mask cannot see. A listener who has not heard an instrument for four bars has no current auditory object for it, and the entrance therefore forms a new stream rather than continuing one; a part re-entering after a beat is heard as the same line resuming. The auditory scene anchor is where that belongs and it is not a masking question at all. So the length of the rest decides whether the entrance is a new voice or a returning one, and the mask decides how much of it arrives — two different questions with two different answers, and a score writes one symbol for both.

What the pictures cannot show #

The masked threshold here is the same simultaneous-masking model the collection uses everywhere: a spreading function per component, powers summed, thresholded against the quiet. It has no temporal component in it at all, so it says what happens once the entrant is established and says nothing about the first tens of milliseconds — which is precisely the interval the tenth rung showed carries the change. An entrance is masked less at its own onset than afterwards, because forward masking from the preceding chord is weaker than simultaneous masking by the current one, and none of that is computed here.

The margin reported is a mean over the entrant’s partials, which is a summary of a distribution and not a quantity a listener has. Two chords with the same mean can differ in whether it is the fundamental that survives or the upper partials, and those sound entirely different — the first is the instrument, the second is a whistle. The figure carries the audible count beside the mean for that reason and neither number decides it.

Every chord in the passage is voiced by hand to make the point, so the 8.7-decibel spread is a spread over five chosen chords rather than a statistic about music. What is not hand-chosen is the ranking’s mechanism, which is the same computation at every chord and would rank any other five the same way.

And the entrant is one player. A section entering is several slightly mistuned copies, which smears its own partials across a few cents and changes nothing about the threshold it meets, but does change how much of it is a single component that can be masked as one thing.

Whose music, and what four rungs have added up to #

The practice is orchestral and chamber scoring from about 1780, where entries are placed rather than merely allowed to happen, and where the treatises’ advice about entries is about register and moment and almost never about level. That emphasis now has three separate arithmetics behind it: the level component is under the limen in any texture of five or more, the moment is worth up to twenty-nine decibels of margin, and the register is worth twenty-six.

Four rungs have made the ensemble a variable this anchor can move, and the summary is short. An entrance is a colour event, not a loudness one. It is received in a tenth of a second while a departure takes two. The moment it happens is worth as much as the register it happens in, and the quietest chord is not the best moment.

Where this ladder goes next #

Twelve rungs. Who plays what and how loud is one question; the ranking survives the dynamic and the chord does not; a dynamic mark changes what a note is; a subito piano is a rate rather than a level; the other quantity succession has arrives whole; the mask arrives with it and pushes the same way; more players than notes; the doubling put back into time; and then four rungs in which the ensemble itself became something a score could move.

What the anchor owes now is the search. Every figure on all twelve rungs evaluates a scoring that is handed to it. The assignment rungs enumerate — thirty-six candidates, scored exhaustively — and that was affordable because an arrangement is a permutation of a small set. An entrance schedule is not: a passage of five chords and six available players has two to the thirty candidates, and the objective is no longer separable across chords because the running impression carries one chord into the next and the mask carries a voicing into the entrance over it. The machinery for scoring one schedule is now complete and nothing has ever been asked to choose one. What would come out is whether the exhaustive answer resembles the practice — whether an optimiser given a passage and a roster brings players in where a composer does — and it needs arithmetic and a passage and no listener, which is the third time this anchor has been able to say that and the first time the search space has been the hard part rather than the model.