The blend table has a row for every note

There is a way of finding what a ladder has not asked, and it is to look for the parameter every one of its figures sets to the same value. On this one it is not hidden. Eight rungs have put instruments together on a note, and with two exceptions every figure among them sounds that note at G above middle C — one pitch, in the middle of the treble staff, chosen once and never revisited.

That would be a detail if a radiator’s spectrum were a property of the radiator. An instrument is not one timbre is the rung that says it is not: every filter in this collection is fixed in frequency — a violin’s body modes, a woodwind’s radiation cutoff, a bell’s high pass, three formants on a vowel — and the fundamental is not, so moving the note sweeps the partials through the filter rather than moving the filter with them.

Six pairs of fifteen at the bottom of the range and twelve of fifteen at the top is the shortest statement of the result, and it is worth saying immediately what it is not. It is not that pairs blend better up high in some loose sense. It is that the specific quantity two players on one note offered as a definition — is there a balance a section could actually be asked for at which the composite belongs to neither player — has a different answer for half the pairs depending on which note they are on.

And the count is not an artefact of the twenty-four decibels the definition allows. Widened to thirty it is unchanged, six and twelve; narrowed to eighteen it falls to four and eleven, which is nearly a factor of three rather than a factor of two. The doubling survives every range that leaves the definition meaning anything.

The mechanism, which is one rung old and was never applied here #

A violin’s lowest body resonance sits near 280 hertz and does not move when the player stops a higher note. So the first partial of an open G string lands squarely inside it and the first partial of the same instrument two octaves up lands nowhere near it, and the two spectra — same instrument, same bow, same everything — have different shapes.

That is the fourth rung’s finding, and every rung after it has quietly assumed it away by working at one note.

The instruments are not equally exposed to it. An organ flue pipe has no filter at all in this model — its partial list is the same shape wherever it is played — and a clarinet’s cutoff at 1,500 hertz is high enough that at a low note the cutoff is above the sixteenth partial and does nothing. Those two barely move. The voice and the trumpet move enormously, for opposite reasons: one has resonances the fundamental climbs past, and one has a cutoff the partials climb over. Everything between is a mixture, and a pair is the product of its two members’ exposure, which is why the fifteen curves on the drawing above have so little in common with each other.

What that does to the table of pairs #

The seventh rung ranked every pair of the ladder’s six radiators by how much of the composite belongs to neither of them, and named the winner: a clarinet with an oboe, at 0.575, crossing at −3.4 decibels. That is the pair the orchestration manuals put at the top too, which was the rung’s one external check.

Asked two octaves lower the same table has the same winner, more emphatically: clarinet and oboe at 0.528, and now crossing at +8.2. But the bottom of it has been rebuilt. Nine of the fifteen pairs have no crossing anywhere in the range down there, against seven at the working pitch — and the three that lose it are all pairs containing the voice.

And at the top of the range it is a different table #

An octave and a fifth above the working pitch it is not the same ranking with the numbers moved. It is very nearly the ranking turned over.

The clarinet and oboe — first of fifteen at the bottom, first at the working pitch — are eleventh, at 0.096. The oboe and voice — last of fifteen at the bottom, at 0.038 — are second, at 0.484. Counting every pair of pairs whose order swaps between the ends of the range gives seventy of a hundred and five, and the number to compare that against is fifty-two and a half, which is what two unrelated rankings would give. More than half of the comparisons swapping means the table has not been shuffled; it has been inverted.

The voice is most of that movement and the reason is the plainest thing on the page. A vowel is three fixed resonances, and the lowest of them on this vowel sits at 730 hertz. Below that the fundamental is under the first formant and every partial above it is being shaped, so the voice covers the whole spectrum and owns any composite it is in. Above it the fundamental has climbed past the first formant, the partial list collapses to a handful, and the voice becomes the thinnest spectrum in the collection rather than the fullest.

That is the mechanism a vowel is two resonances built and the sound a listener knows best measured, arriving in a table about orchestration by a route nobody laid.

The order changes and does not become a cycle #

A section has a loudest member found that ownership across the six radiators is a total order — no three of them go round in a circle — and that the possibility of a circle is real rather than excluded, since arbitrary spectra produce one about once in twenty triples. It found that at one note.

Asked at nine pitches over four octaves, the order changes seven times and is a total order at every one of them: no cycle among any of the one hundred and eighty triples.

So the eighth rung’s result is robust and its content is not. There is always a strongest member; which instrument it is depends on the note. A trumpet is fifth from the top two octaves below middle C and first two octaves above it, which is the same statement as the bore ladder’s about what a bell lets out: below the cutoff the bell reflects and above it the bell radiates, so a trumpet’s partial list at a low note is short and at a high note is long.

That is worth separating carefully. The structural claim — ownership is transitive, so a section has a loudest member — held at every pitch tried and is the kind of claim a family of spectra makes rather than a particular six. The specific claim — the voice owns everything — was a claim about G above middle C, presented as a claim about the voice.

It is a useful distinction to have because the two claims fail in different ways. A structural claim of that shape survives a change of instruments: hand the same arithmetic six spectra from another family of rolloffs with resonances on them and it will still almost never produce a cycle. A specific claim survives nothing — not a change of vowel, not a change of note, not a change of dynamic, since a dynamic mark changes what a note is puts a spectral tilt under every one of these lists that this page holds still. What the sweep does is sort the eight rungs’ findings into the two piles, and most of them go into the second.

The other quantity that was measured at one note #

The seventh rung’s other finding was about intonation, and it is the one this ladder has been quoting since: two players on one note, ten cents apart, are rougher than the same two playing a major third exactly. It listed the register sweep among the things it could not show, and predicted the sweep would be large, because a critical band is a fixed width in hertz and a cent is a different number of hertz at every pitch.

It is larger than that.

Nine cents at the pitch the rung quoted. Two cents two octaves above it. Thirty-nine cents a fifth below middle C. And below that there is no crossing anywhere inside sixty cents: two players on a low C can be a third of a semitone apart and still be smoother than the same two playing a major third in tune.

A factor of twenty, and the direction is the one that matters to a player. Everything an orchestra says about unison intonation is said about the register where the sweep is steepest.

Why the low end inverts, which is not the same mechanism #

The high end is the rung’s own explanation running as predicted: partial n of a note mistuned by c cents is mistuned by c cents as well, which is n times as many hertz, so the top of the spectrum enters the critical band long before the fundamental does — and the higher the pitch, the fewer cents it takes.

The low end is something else, and it is visible in the partial lists rather than in the mistuning.

At a low fundamental the partials of a perfectly tuned unison are already close enough to each other, in hertz, to be inside one critical band. The roughness of a unison at the bottom of the range is 0.138 with nothing wrong with it at all, against 0.00004 four octaves up — a factor of three thousand, produced by nothing but the width of the band.

So down there a unison is not a quiet baseline that a mistuning disturbs. It is already rough, on its own account, and five cents changes it by nothing measurable — the ratio between the perfect unison and the five-cent one is 1 at the bottom of the range, 65 at the working pitch and 1,551 at the top. Intonation as a roughness problem does not exist in the bass and dominates in the treble, and both halves of that come from one fixed number of hertz.

Which is the same shape as a third is rougher in the bass, pointed the other way: a fixed band makes wide intervals rough down low and makes near-unisons smooth down low, because the two are on opposite sides of the same curve.

There is a practical reading and it is the sort of thing a section leader would recognise. The instruction play in tune with each other costs almost nothing to obey on a low unison and is nearly unobeyable on a high one, and the exchange rate is a factor of twenty over the range of an orchestra. A pair of piccolos two cents apart is at the boundary this figure draws; a pair of double basses a sixth of a semitone apart is comfortably inside it. Rehearsal time is not distributed that way — it is distributed by how exposed the line is — and the arithmetic says it could be.

There is one thing the drawing does not settle, and it is a real one. A crossing computed against a major third is a crossing against one reference; asked against a fifth or an octave the curve moves, because which instrument is underneath showed the reference interval’s own roughness is not the same for every pairing. The shape of the sweep does not depend on the choice and the numbers on the axis do.

Which computation produced the numbers #

Every figure is the seventh and eighth rungs’ own arithmetic with the fundamental as a variable and nothing else changed.

The radiators return sixteen partial amplitudes at a stated pitch, normalised to the strongest. The composite of two is the root of the sum of the squares, partial by partial, with the second scaled by the balance. The log-spectral distance is the root mean square difference in decibels between two spectra each normalised to its own total power. The crossing is the balance at which the two distances are equal, found by sweeping from −24 to +24 decibels in steps of two and interpolating.

The ownership order at a pitch is read off the fifteen pair results at equal level; the cycle count enumerates every triple; and the two are checked against each other, since a tournament is a total order exactly when it has no cycle and exactly when its win counts are all different.

The intonation curves are the roughness between two radiators’ full partial lists through the Plomp–Levelt curve scaled to the critical bandwidth, with the second note displaced by a stated number of cents, and the crossing is the first mistuning at which the unison exceeds the major third played exactly.

Where the model stops #

Nine pitches is not a function. The sweep is a set of points four octaves apart at the ends and a fifth apart in the middle, and the movements are large enough that something between two of them could be missed. Nothing here claims the curves are monotone.

The radiators do not change how they are played. A real instrument’s spectrum at the bottom of its compass differs from the top by more than the filter accounts for, because the player is doing something different — a dynamic mark changes what a note is prices one part of that and the rest is not modelled. The sweep here is the filter alone.

And some of these pitches are outside some of these compasses. An oboe does not play two octaves below middle C and an organ flue pipe of a given rank does not play four octaves. The figures ask every radiator at every pitch because the question is about the filter, and the answer at a pitch an instrument cannot reach is arithmetic rather than orchestration.

What the picture cannot show #

It cannot show two notes. Everything here is a unison, and the interval sweep is the fifth rung’s. A pair playing a third has two fundamentals in two places on the filter, which is a two-dimensional version of this page and is not drawn.

Nor can it show a section. The trio table is the rung below’s and was computed at one pitch, so it inherits the whole of this problem and none of the correction.

It cannot show what a player does about the low end. If a unison in the bass is forgiving to a third of a semitone, players in the bass should be measurably less accurate than players in the treble, and whether they are is a question about recordings that this collection has no corpus for.

And it cannot show loudness. Every spectrum here is normalised to its own strongest partial, so a pitch at which an instrument is simply quieter looks the same as one where it is not — which is the one thing an orchestrator would have asked about first.

Whose instruments, and when #

The radiators are generic, so nothing here measures an orchestra.

The observation that fits a practice is about where the doublings are. Nineteenth-century orchestration doubles most heavily in the middle of the range, and the two things this page says about the extremes are that at the bottom almost nothing blends and at the top almost everything does — which would make the middle the region where a choice between pairs exists at all. That is consistent with the practice and it is not evidence for it, since the middle of the range is also where the instruments are loudest, easiest and most often written.

Where this ladder goes next #

Nine rungs. A spectrum is a list; a mouth filters it; a body filters it; a fixed filter under a moving note makes the list a function of pitch; two lists do not commute; three lists over three notes have six arrangements; two lists on one note make a composite one of them owns; three lists make one it owns more emphatically; and the whole of that, asked at every note rather than one, is a different table at each end of the range.

What is owed after this is the attack. Every figure on all nine rungs is a steady state — a partial list, a composite, a balance held still — and the single strongest cue that two instruments are two instruments is that they do not start together. The onset ladder already has the numbers: it puts the difference between a doubled flute and a doubled violin at tens of milliseconds and it knows that a low note cannot start on time. What it has never been asked is what the composite does during that time, which is a spectrum sweeping from one player’s shape to the pair’s over some tens of milliseconds, at every note. This ladder has the composite and that one has the clock, and neither has looked at the other. It needs no listener and no corpus — only the two pieces of machinery in the same figure.