One note in the compass loses its pizzicato

The essay that put a pluck and a held note into a hall put a pizzicato and a held note into a hall and swept three dials: where the listener sits, how fast the pluck’s upper partials leave, and how long the room rings. It found that a room roughly quintuples how long the composite spectrum stays the pluck’s — the dry version of which the dry measurement of the same handover had already measured at a few hundredths of a second, that the seat matters for the first few metres and then stops, and that the loss law matters about a tenth as much as the room does.

The dial it did not turn is the note. Every figure on that essay is one pitch, and pitch is the variable a room does the most to — because a room’s own decay time falls with frequency, because the two instruments’ spectra overlap differently at every fundamental, and because those two things interact rather than adding.

Dry, the note hardly matters #

With no room in the arithmetic the eight figures run from 0.023 seconds at C5 to 0.068 at G3 — a spread of three across the whole compass, and no particular shape to it. That flatness is not an accident of the instruments chosen. It is what the dry model can produce: the only frequency-dependent quantity in it is each instrument’s own spectrum, and two spectra compared at eight fundamentals give eight numbers scattered inside a factor of three.

A player would not hear that as a property of the instrument. Twenty-three milliseconds against sixty-eight is the difference between two attacks, and the whole range sits inside the time it takes to say the word.

In a hall the same eight notes spread by forty #

Put the same eight notes into a concert hall and the figures become 0.607, 0.263, 0.496, 0.376, 0.014, 0.254, 0.272 and 0.189 seconds. The spread is forty-three, and it is no longer inside anybody’s attack. Six-tenths of a second is a held sound; fourteen milliseconds is nothing at all.

The obvious reading is that the room multiplies everything by about five and the spread comes along for the ride. It does not. The multipliers, note by note, are 8.9, 5.7, 7.6, 6.5, 0.6, 5.2, 5.4 and 4.5 — so the room is not applying a gain, it is applying a different gain to every note, and at one of them the gain is less than one.

The note where it reverses #

C5 is the note the room makes worse, and the reason is visible before the room enters at all.

At the instant of the pluck, the composite spectrum sits 3.59 decibels from the pluck’s own and 16.03 from the held note’s at G3 — a margin of 12.4 decibels. At C5 the two figures are 8.49 and 10.07, a margin of 1.6. The two instruments’ spectra nearly coincide at that fundamental, so the composite is nearly equidistant from both and the pluck starts with almost no lead at all.

The room does not cause that; it converts it. A small margin and a large one are both margins in a dry room, where nothing has time to happen: the pluck loses the note in a few tens of milliseconds either way. In a live room there is time for a small margin to be spent, and spending it is what the extra tenths of a second are for.

In a stone church the same note goes further and the pluck never owns the composite spectrum at all — the only cell in everything computed here where that happens. It is not a dramatic failure and nobody would describe it as the pizzicato vanishing; what it means is that a listener fifteen metres back has no moment at which the sound is nearer a plucked string’s spectrum than a held instrument’s.

Two halves of one account, measuring two different things #

The flatness of the dry line is worth more than a sentence, because it says something about the eight essays below this one that none of them could say about itself.

The earlier essays here are dry. They ask which of two instruments a listener attributes a composite spectrum to, and they answer it by comparing spectra — which instrument is underneath, which player on which note, a blend table with a row for every note. Every quantity in them is a property of the two instruments.

The essay that put the pair into a hall, and this one, are not. Once a room is in the arithmetic the dominant term is the room’s, and the two instruments enter mainly through how big a margin they start with — a single number at the strike, before any time has passed. At G3 that margin is 12.4 decibels and at C5 it is 1.6, and the room then multiplies whatever is there by five or so and lets time spend it.

So the dry essays measure a pair of instruments and the room essays measure a room, with the pair entering as one initial condition. That is a cleaner division than it looked from inside either half, and it explains why the dry line is flat: a quantity that varies by a factor of three across the compass is being asked about at a time resolution where nothing has happened yet.

It also says where the useful work is. A blend table computed dry is a table about two instruments and is the right object for a scoring decision made at a desk. The same table computed in a hall is mostly a table about the hall, and its instrument content is one number per cell — which is a much smaller thing than it appears, and much easier to compute.

The room’s tilt costs the pluck, and costs it everywhere #

There are two things a room does and they are worth separating, because only one of them is about frequency.

A room adds level, by the ratio of reverberant to direct energy at a seat, and that ratio is roughly the same for every partial. And a room decays unevenly: its decay time falls with frequency, from about three seconds at 125 hertz to a second and a half at four kilohertz in the hall drawn here.

Giving every band the room’s mid-band decay time raises the handover everywhere: 0.72 seconds instead of 0.61 at G3, 0.51 instead of 0.38 at G4, 0.39 instead of 0.28 at G5. A room’s tilt costs a pizzicato between fifteen and twenty-five per cent of its ownership, and it costs it for the reason that keeps recurring in this collection: the bands that carry a pluck’s identity are its upper ones, and the upper bands are the ones a room absorbs fastest.

So the room helps the pluck and takes back a fifth of the help with the same mechanism that supplied it. That is the same double action a struck note’s colour drain runs into from the other direction, and the two essays were computed on the same afternoon from the same band tables.

What a seat does to a pitch #

The seat and the pitch are not independent, and the grid is the way to see it.

The dry panel’s columns are identical by construction and the figure leans on it rather than remarking on it, because it is what makes the other two panels readable. In a room the ratio of reverberant to direct energy is the square of the distance over the critical distance, so a listener further back is given proportionally more of the room’s copy of both instruments — and since the room’s copy is worth more to the pluck, a further seat is a longer ownership.

That growth is not unbounded. Past the critical distance, almost everything a listener receives is already reverberant and a further seat converts nothing. Where the room takes over computed that radius, and it turns out to be where this effect stops growing — which is a pleasant cross-check, since the two quantities were derived for different purposes.

That growth interacts with the entry level the same way the blend arrives before the note does found an attack interacting with a balance. A quiet pizzicato at the back of a very reverberant room is the one corner where the pluck never gets the note, and it is the same failure as C5’s arriving by a different route: a margin too small to survive a room that gives it time to be spent.

What it means for a passage #

The practical shape of this is narrow and it is worth stating narrowly rather than broadly.

A pizzicato doubling that carries in one register does not carry in another, and the difference is far larger in a hall than in a rehearsal room. On the numbers here a pizzicato doubling at G3 delivers its colour for six-tenths of a second at a seat fifteen metres back and the same doubling at C5 delivers it for fourteen milliseconds. Those are not adjacent quantities and the orchestrator choosing between them has no way to hear the difference in a dry room, because dry they are twenty-three and sixty-eight milliseconds.

So a scoring decision that sounds identical in a studio can be a factor of forty apart in the hall it is written for, and the direction of the error is that the studio flatters the treble.

That is a claim about a model rather than about a rehearsal, and the one reason to take it seriously is that the mechanism is not subtle. A pluck is upper partials that leave quickly; a room holds them; a room holds the treble less well than the bass; and how much there is to hold depends on how far the pluck’s spectrum sits from its partner’s at that particular note.

The margin is the quantity to carry #

If the room multiplies an initial margin, then the margin is what an orchestrator is actually choosing when they choose a register, and it is the cheapest quantity on this page to compute: one log-spectral distance between two static spectra at one fundamental, with no room, no time and no seat in it.

Across the eight pitches drawn, the margins are 12.4 decibels at G3, 12.1 at E4, 8.7 at G5 and 1.6 at C5 — and the handovers in a hall follow them in the same order, which is the claim reduced to a single sentence: the room’s contribution is nearly the same shape at every note and the pair’s contribution is not, so the pair decides the ordering and the room decides the scale.

That is testable inside this collection and against something outside it. Inside, the prediction is that the handover in any room is a monotone function of the dry margin, at fixed seat and fixed entry level — so a single scatter of one against the other, over every pair and every semitone, either lies on a curve or does not. Outside, it says which doublings should be unreliable in a hall, and orchestration manuals have opinions about that which were arrived at by ear.

The margin also explains the entry level, which essay twelve swept and treated as a separate dial. Entering a pizzicato six decibels above the held note rather than level with it is a way of buying margin directly, and it buys it in the same units: a louder entry moves the composite nearer the pluck’s spectrum at the strike and gives the room more to multiply. A player’s dynamic and the pair’s register are the same variable seen twice, which is why the grid above is nearly symmetric between its rows and its columns.

Which computation produced the numbers #

The pluck is this collection’s plucked-string model at the stated pitch with sixteen partials, partial $n$ decaying at a rate rising as $n$ to the stated exponent. The held note is a steady spectrum a stated number of decibels below the pluck’s attack, and it has been sounding long enough that its reverberant field is in equilibrium.

The room’s reverberant response is computed band by band from Sabine’s equation on a stated surface list, with the critical distance per band from it, so that the treble is both shorter-lived and more quickly reverberant than the bass. A source decaying at rate $g$ in a room decaying at rate $k$ delivers a reverberant power proportional to the difference of the two exponentials, normalised so that a steady source gives the textbook ratio $(d/d_c)^2$ at a seat $d$ metres away. The held note being steady, it arrives with its direct power times $1 + (d/d_c)^2$; the pluck’s tail starts at zero.

Ownership is decided as the dry essays decide it: the composite spectrum is compared with each instrument’s own by log-spectral distance over the partials, and the note belongs to whichever it is nearer. The handover is the last moment it is the pluck’s.

The flat comparison is the same arithmetic with every band given the room’s mid-band decay time. It is not a room anybody has built; it is the room’s reverberation without the room’s frequency response, and it exists to separate two effects that arrive together.

Where the model stops #

The margin at the strike is an artefact of two particular instruments. Which pitch reverses depends entirely on where a violin’s and a flue pipe’s spectra happen to cross, and a different pair crosses somewhere else. What is general is that some pitch does, because two spectra compared across a compass will be nearest somewhere — and that the room amplifies whatever is smallest.

The diffuse field has no early reflections in it. Sabine’s model gives a field that surrounds a listener evenly, and the first eighty milliseconds of a real room are a handful of discrete arrivals from particular surfaces. That window is where the handover happens at most pitches, so the statistical model is being used at the place it is weakest, and the first eighty milliseconds are a different room is the essay that says how different.

And the two instruments are at the same point. A pluck at the back of the string section and a held note at the front excite the room differently and reach a seat with different direct-to-reverberant ratios. Every number here assumes one source position for both.

What the picture cannot show #

It cannot show a listener with two ears. Two sources at different places on a stage are far easier to tell apart than a single-channel sum of them, and everything above understates what a listener can do by exactly that amount.

Nor a vibrato. A held note that fluctuates and a pluck that does not are two streams before any spectral comparison runs, and a sustained instrument in a hall almost always fluctuates.

It cannot show the passage. A pizzicato in a real score is one of a string of them, and the second arrives while the first is still in the room. Whether a run of plucks accumulates a reverberant field that helps the later ones is a question this model could answer and this essay does not ask — and it is the same accumulation two players on one note runs into when the two are not quite together.

Still open: which pairs of instruments have a note like C5 #

The finding above has a general half and a particular half, and only the particular half has been computed. That some pitch has a small margin follows from two spectra being compared across a compass. Which pitch, for which pair, and how small, is a table the essays here do not have.

It is a cheap table to build: the margin at the strike is a single log-spectral distance between two static spectra at a fundamental, with no room and no time in it, and the radiator models for five instruments are already here. Running every pair at every semitone across the orchestral compass would produce a map of where each pairing’s spectra come nearest — and the claim this essay makes is that those are the notes at which a hall will do the most damage to a doubling.

What would make that worth having is that it is checkable against something outside this collection. Orchestration manuals name registers where particular doublings are said not to work, usually without a reason. If the pitches where two spectra nearly coincide line up with the registers the manuals warn about, the map explains a piece of received practice. If they do not, the margin is not what anybody was hearing.