A doubling's worst note belongs to one instrument, not to the pair

One note in the compass loses its pizzicato found that a hall multiplies a single number, and that the number is fixed before any time has passed: how much nearer the composite spectrum of two instruments sits to the decaying one’s own spectrum than to the held one’s, at the instant of the attack. Where that margin is large the room has a great deal to multiply and the pluck keeps its note for six-tenths of a second; where it is small the room has almost nothing, and at one pitch — the one at which a violin’s and a flue pipe’s spectra nearly coincide — the pluck kept its note for fourteen milliseconds.

It closed by pointing out that the whole table was cheap and that nobody had built it:

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.

Six radiators make thirty ordered pairs, and four octaves make forty-nine semitones. Fourteen hundred and seventy margins take sixteen milliseconds to compute, and they are not thirty unrelated curves.

Two kinds of worst note #

The first thing the map separates is two things a single example could not distinguish.

A pair has a worst note in the interesting sense when its margin falls and then rises again — when there is a pitch at which the two spectra come nearest and are further apart on either side. Seventeen of the thirty pairs are like that. The other thirteen have margins that are monotone in pitch: the trumpet over a flue pipe leads by more the higher it goes, the voice over a violin by less, and each is worst at whichever end the compass happens to stop at.

That distinction matters for the practical claim the earlier essay made, because the two shapes give an orchestrator different advice. A pair with an interior minimum has a register to avoid and registers on both sides of it that are safe. A pair with a monotone margin has a direction: it gets better upward, or worse, and there is nothing to step around.

It also matters for the blend table, which is the dry object these essays built before any room entered it: a table with a row for every note is the right shape for a pair with an interior minimum and an over-elaborate one for a pair whose margin only slopes.

It also cautions against reading that essay’s C5 as typical. Violin over flue pipe, as it happens, is one of the thirteen — its margin falls slowly across the whole compass and never turns. The eight pitches that essay sampled found the smallest of the eight and called it the reversal note, which it was; the map says it is not a minimum in the compass, and that the violin’s genuine interior minimum is against other partners and at a different pitch.

The seventeen do not choose seventeen notes #

The second thing the map shows is the one worth the essay.

Written out, the seventeen interior worst notes are E3, E3, E3, E3, C4, F4, B♭4, B♭4, B♭4, D5, C♯6, C♯6, C♯6, F♯6, F♯6, F♯6 and G6. Seventeen pairs, seven distinct pitches, and four clusters of three or four.

And the clusters do not group by pair. They group by instrument:

• B♭4 is where the violin is worst — over a clarinet, over an oboe and over a trumpet alike.
• C♯6 is where the voice is worst — over a violin, over a clarinet and over an oboe.
• F♯6 is where the clarinet is worst — over a violin, over an oboe and over a trumpet, with G6 for the flue pipe a semitone away.
• E3 is where anything is worst over a voice — a clarinet, an oboe and a trumpet all fail there, and that is the one cluster organised by the held instrument rather than the decaying one.

So the question “at which note does this doubling fail” mostly has an answer that does not mention the doubling. It is a property of one of the two players, and the other one is along for the ride.

Each cluster is one instrument’s own filter #

The reason is not a statistical regularity. Each of the four pitches is a number already in the model, put there for reasons that had nothing to do with blend.

The violin’s B♭4 is 466 hertz and its body model’s B1− plate resonance is at 460 — twenty-two cents apart. A fundamental sitting on that resonance is lifted far more than any of its own partials, so at that one pitch a violin’s normalised spectrum is dominated by its first partial, which is what a held wind instrument’s spectrum looks like. The composite is then nearly equidistant from both, which is what a small margin is.

The clarinet’s F♯6 is 1480 hertz and its radiator’s cutoff is 1500. Above the cutoff every partial is attenuated as the square of frequency, so a clarinet playing at its own cutoff has a spectrum that is almost entirely its fundamental — for exactly the same reason, arrived at from the opposite direction. Its worst doubling note is the frequency at which its tube stops radiating.

The voice’s C♯6 is 1109 hertz and the second formant of its vowel is at 1090 — twenty-nine cents apart. A sung fundamental sitting on a formant is the loudest a sung fundamental gets, and the partials above it at 2218, 3327 and 4436 hertz are past every formant the vowel has and fall away. So a voice at C♯6 is, again, a spectrum that is almost entirely its first partial.

Three instruments, three quite different mechanisms — a wooden plate, a hole in a tube, a resonance in a throat — and one consequence. A spectrum is most fundamental-dominated at the pitch where its own filter peaks, and a fundamental-dominated spectrum is the one hardest to tell from anything else, because it is what every instrument looks like when only its first partial is left.

And E3, the cluster that belongs to the held voice, is the same mechanism run backwards. At 165 hertz the voice’s fourth and fifth partials, at 659 and 824, straddle the first formant at 730, so a held voice down there has a spectrum with a strong bump in the middle rather than at the bottom — which is as far from fundamental-dominated as this set of instruments gets, and correspondingly hard to hide behind. The margin at E3 is negative for every partner, which means the decaying instrument does not lead at all: the composite is nearer the voice’s spectrum from the first instant. Which instrument is underneath is the essay that set up this comparison, and this is its sharpest case.

Four clusters, four numbers that were already in the instruments’ definitions, three of them agreeing to within a third of a semitone. Nothing in the margin computation knows about body modes, cutoffs or formants; it compares two lists of partial amplitudes and takes a difference of two distances.

That is worth one more sentence about what kind of result it is. The four numbers were put into these models for four separate arguments — the violin’s plate modes for what a body does to a string, the clarinet’s cutoff for the tube it belongs to, the trumpet’s for what a bell lets out — and none of them was chosen with blend in mind. A measurement that lands on four of them without being told they exist is either a real consequence or a circularity, and it is not a circularity, because the margin is computed from the amplitudes after every filter has been applied and has no access to the filters themselves.

Which pairs are safe and which are hopeless #

Depth is a separate question from position, and the map answers it too.

Seven of the thirty lead at every pitch in the compass, and four of the seven are the trumpet over something. Four never lead at any pitch, and all four are something over the trumpet — which is the same fact twice: a trumpet’s spectrum in this model is a sawtooth with everything below 1150 hertz reflected back down the bore, so it has more energy high up than any of the other five and a composite containing it resembles it.

The exception is instructive. A voice does lead a trumpet, by as much as thirty-nine decibels at the bottom of the compass, and it is the only one of the five that does. A formant filter is the only spectrum in the set with peaks of its own rather than a slope, and a peaked spectrum is distinguishable from a bright one in a way a smooth bright spectrum is not.

The nineteen in between are the interesting ones for a score, because they are the doublings that work in one register and not in another — and the map says which, by how much, and where the boundary is.

The test the earlier essay named #

One note in the compass loses its pizzicato made a prediction about what this table would be good for and stated the test in a sentence:

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.

It lies on a curve, loosely. Among the two hundred and fourteen points whose decaying instrument leads at the strike, the margin correlates with the logarithm of the handover at 0.72. Among the hundred and seventy-six that do not lead, the handover is zero to three decimal places in every case: a pair that starts behind never gets in front, whatever the room does for it.

Loosely is the operative word and it is the part that essay could not have known. A correlation of 0.72 on a log scale is a prediction good to within about a factor of two, which is enough to rank two doublings and not enough to price one. The residue is the room’s own frequency tilt — a room does not decay evenly, and its treble goes first — bearing differently on different spectra. It was measured at fifteen to twenty-five per cent for one pair; across thirty pairs it varies by more than that, because how much of a spectrum sits in the bands a room drops is exactly what distinguishes one radiator from another.

There is a second residue and it is the seat. How far away the room takes over computed the distance at which the reverberant field overtakes the direct sound, and the scatter figure is drawn at one distance; a pair whose energy is high is reverberant at a nearer seat than one whose energy is low, so the same ten metres is a different fraction of the critical distance for each row of the map.

So the margin is a first-order instrument. It says which doublings have a register problem and roughly where, and it does not say how bad the problem is in a particular hall. That is a fair description of what an orchestration manual claims, which is the comparison that essay wanted to make.

What an orchestrator would take from it #

The map is a table of thirty rows and its practical content is three lines.

A doubling that has to carry a melodic line should avoid the decaying instrument’s own peak. For a pizzicato violin that is B♭4 and the semitone either side of it; for a clarinet dropping out under a held note it is F♯6; for a voice it is C♯6. Those are narrow bands — the margins recover within two or three semitones in every case — so the instruction is to move the line rather than to change the scoring.

A doubling whose margin only slopes has no bad note and a bad half. Thirteen of the thirty are like that, and for them the question is which end of the compass the passage sits in. The trumpet over a flue pipe improves upward by three and a half decibels across four octaves, which is a small enough slope to ignore; the voice over a flue pipe falls by thirty-four across the same span, which is not.

And four doublings do not work at any pitch. A violin, a clarinet, an oboe or a flue pipe decaying over a held trumpet is behind from the first instant at every one of the forty-nine pitches, so nothing about register or dynamic rescues it. That is a stronger statement than the map’s other rows make and it is the one a reader should be most suspicious of, because it rests entirely on the trumpet’s radiator being a high-pass with a sharp corner — which the bell decides what gets out says is a model output rather than a measurement.

Which computation produced the numbers #

Each instrument is the radiator model these essays use: a source spectrum through a filter — a violin’s body as four resonances, a clarinet’s and an oboe’s tube as a low-pass at 1500 and 3000 hertz, a trumpet’s bell as a high-pass at 1150, a voice as three formants, a flue pipe as its source alone. At a stated fundamental each gives sixteen partial amplitudes.

The decaying instrument’s spectrum is raised six decibels and added in power to the held one’s, partial by partial. The margin is the log-spectral distance from that composite to the held instrument’s spectrum, less the distance to the decaying one’s, both computed over the sixteen partials with each vector normalised — so the margin is in decibels and a positive one means the composite is nearer the decaying instrument.

The hall figures use the same room model as the two essays before them: Sabine’s decay band by band, the critical distance per band, and the handover as the last moment the composite is nearer the decaying instrument than the held one.

Where the model stops #

Every instrument is modelled at every pitch. An oboe does not play C7 and a trumpet does not play C3, and the map draws both. The clusters all fall in registers their instruments do have, which is the reason to report them, but the rows around them include cells nobody could play.

A filter is not an instrument. A real violin’s body response has dozens of modes rather than four, and a real clarinet’s cutoff is a soft knee across an octave rather than a corner. What the map gets right is the shape of the claim — a spectrum is most fundamental-dominated where its filter peaks — and the pitches it names are as precise as the filters are.

The margin is computed at one entry level. Six decibels above is the earlier essay’s setting, and the map at a different one is a different map, because the composite moves toward whichever instrument is louder. What does not change is where the interior minima are, since they are set by the shape of the two spectra rather than by the balance.

Nothing here is a doubling in unison with vibrato, an attack, or two positions on a stage, all of which the essays before this one record as things the model leaves out.

What a table cannot say about a score #

Whether an orchestrator hears the clusters. The prediction is that the violin’s B♭4 doublings, the clarinet’s F♯6 doublings and the voice’s C♯6 doublings should be the ones manuals warn about. Checking that is a reading exercise in a library rather than a computation, and it is the one piece of evidence that would move this from a model to an explanation.

Whether the margin survives a real attack. Every figure here is the composite at the instant the decaying note starts, and a real attack takes tens of milliseconds during which the partials arrive in their own order — which the blend arrives before the note does priced for a pair of onsets and did not price for a whole spectrum.

Whether a listener uses spectral distance at all. It is the measure these essays have used since which instrument is underneath, it is a reasonable proxy for what a listener attributes a sound to, and nothing in any of these essays tests it against a listener.

Still open: the compass each instrument actually has #

The map is drawn over one compass for six instruments and three of them do not have it. An oboe’s lowest note is around B♭3 and a trumpet’s around F♯3; a flue pipe’s range is whatever the builder cut; a violin’s top is bounded by the player rather than the instrument.

Giving each radiator its own range is a small change to the same table and it would sharpen two things. It would say which of the thirteen monotone pairs are monotone only because the compass was drawn wider than either instrument can reach — a margin that runs downhill from C3 to C7 may well turn inside the range they share. And it would say whether the four clusters fall inside their instruments’ ranges or at their edges, which decides whether they are advice an orchestrator can use or curiosities about a model.

The ranges are computed elsewhere in these essays already, rather than looked up, and the join has never been made.