Which instrument is underneath

The previous rung of this ladder ended by naming the thing it had made both possible and necessary: a dissonance curve for two different instruments. Every roughness figure on this site — and, as far as the literature goes, very nearly every roughness figure anywhere — compares two tones of the same timbre. That is a duet nobody plays.

The obstacle was one line of code. The function that computes a dissonance curve takes a spectrum and uses it for both notes; the general case takes two, and the sum runs over every pair of partials exactly as before. What comes out is not a generalisation. It is a different object, because the sum with one spectrum is symmetric under swapping the notes and the sum with two is not.

What each of them is putting into the air #

The rung rests on five spectra, and not one of them is a new table. Each is a source this collection already carries multiplied by a filter it already carries, because inventing a partial list to make an argument come out is exactly the failure this site is arranged against.

The crucial property is not the shape of any one row. It is that the filters do not move with the note. A body resonance sits at 460 hertz whatever is being played, exactly as a vowel’s first formant sits at 730 hertz whatever the pitch. So the filter weights the fourth partial of one note and the first partial of a note two octaves up, and the two notes of an interval are therefore filtered differently even when the same instrument plays both.

The symmetry that was never noticed because it was never tested #

Sensory dissonance between two tones is a double sum: every partial of the first against every partial of the second, each pair contributing an amount that depends only on the two frequencies. Swap the two tones and the sum is unchanged, because addition commutes.

That is true when both tones have the same spectrum. When they do not, the swap is not a swap of the summation order — it is a swap of which spectrum is at which frequency, and the answer changes.

Where a clarinet’s missing partials come from #

The clarinet is the sharpest case in the set and it is worth saying why its spectrum is what it is, because the reason is not a fact about clarinets but a fact about a stopped tube.

A cylinder closed at one end supports only the odd modes: the even ones would need a pressure antinode where the reed forces one and a node where the open end forces one, and no wave can do both. So the second, fourth and sixth partials are not weak, they are structurally absent, and the site’s own partial list has them at four hundredths.

The other half of the clarinet’s row is the top end, and that is the tone-hole lattice. Above a computable cutoff the wave runs straight past the open holes and leaves from the bell, so the radiated spectrum rolls off above that frequency whatever the fingering. Both features are computed elsewhere in this collection for reasons that had nothing to do with roughness, and both turn out to decide which intervals a clarinet is smooth at.

The same is true in the other direction for the oboe. A strong reed driving a conical bore produces the complete series with slowly falling amplitudes, which is why the oboe’s row is the densest in the figure and why the pairings involving it have the most wells. An instrument with more partials has more chances to coincide, and more chances to be rough.

There is a third radiator in the set whose row is decided somewhere else again, and it is the one whose filter moves. A singer choosing a vowel is choosing a pair of formant frequencies, and unlike a violin’s body those frequencies are under continuous control — so the voice is the one instrument here whose radiated spectrum at a fixed pitch is a decision rather than a property. That has a consequence for every count in this essay: the voice’s row is not one row but a family, and the pairings it appears in are scored against whichever vowel the figure’s filter was built from. A duet result involving the voice is therefore a result about a vowel, and a singer who changes vowel on a held note changes which intervals are smooth underneath the note being held.

The same caution applies more weakly to every wind instrument, since a player’s embouchure and breath pressure move the upper partials by several decibels without moving anything else. What none of them can move is the positions of the filter’s peaks, which is the quantity all of these results turn on. A performer controls how much of the spectrum is radiated and not where the radiation is possible, and it is the second that makes a duet curve register-dependent.

How large the asymmetry is #

Scoring intervals both ways round puts a number on it.

Five to one, at the major sixth, with the same two notes and the same two players. The mechanism is visible in the partial rows above. A clarinet’s second partial is almost absent, so a clarinet on the lower note contributes nothing at the octave above itself — which is precisely where the violin’s fundamental sits when the interval is a sixth or a seventh. Put the violin underneath and its second partial, which is strong, lands in the critical band around the clarinet’s fundamental and beats with it.

The pattern is not confined to that pair, and one interval stands out for a different reason.

An octave doubling is not one sound. It is rougher with the spectrally rich instrument underneath than with the spectrally rich instrument on top, by a factor of three to four across the pairs tested. That is an orchestration rule with an arithmetic behind it, and it is the reverse of the shape most doubling advice takes.

The fifth, and only the fifth #

Across all twenty-five ordered pairings the well counts run from one to seven. Something has to be said about which intervals appear and which do not, and the answer is unusually clean.

At middle C the fifth is a well in every one of the twenty-five. No other interval is — the major sixth, which is next, manages seventeen.

That is worth putting beside three results this collection already has. Three is the largest agreeable number scored every chord of every size for roughness and found the fifth in the answer at every size. A spectrum chooses its own scale found that the minima of a dissonance curve move when the spectrum does, which is the argument that a gamelan’s scale follows its bars. The previous rung of this ladder found that the count of minima changes with register on one instrument — three at the bottom of a violin’s range, two in the middle, four at the top. This one adds the third axis, the pairing, and in all three the fifth is the invariant.

The reason is arithmetic rather than aesthetic. A fifth puts the upper tone’s fundamental at 1.5 times the lower one’s, so the upper tone’s second partial coincides with the lower tone’s third — and the second and third partials are the two that survive nearly every filter, because a filter has to be extraordinarily selective to remove both. Every other interval’s coincidences involve a partial high enough for some filter in the set to have taken it out.

And the invariance is a fact about this register #

That explanation makes a prediction the claim above does not survive, and finding out costs one line: run the census at other roots.

The fifth is a well in all twenty-five pairings from G3 upward and in twenty-two of them below it, so the invariant this section names holds over most of the range and is not a property of the pairing axis on its own. It is one more thing that is true at middle C.

And the three pairings that lose it are the three the explanation predicts. At C3 they are clarinet under clarinet, oboe under clarinet and flue under clarinet; at E3, clarinet, flue and voice under clarinet. Every one of them has a clarinet on the upper note, and the fifth’s coincidence is the upper note’s second partial against the lower note’s third. The clarinet is the one radiator in the set whose second partial is not merely filtered but structurally absent, at four hundredths of the fundamental — so the argument that the second and third partials survive nearly every filter is exactly right, and the exception it allows for is the one instrument here whose second partial was never there to be filtered.

Why the exception appears only low down is the register mechanism again. Above G3 the lower note’s own third partial is strong enough to make a well by itself against whatever little the clarinet has at its second; at C3 the lower note’s partials sit under the fixed filters differently and the coincidence has nothing left to be a coincidence with.

So the honest statement is narrower and better founded than the flat one. The fifth is the only interval that is a well in every pairing at any register tested, it fails in three of twenty-five in the bottom fifth of the range, and the three failures are all the same instrument in the same position for a reason the model gives in advance.

It is also a function of register #

Because the filters are fixed and the notes are not, a duet curve is a different curve at every pitch — which means the asymmetry is not a constant of a pair of instruments either.

The count for this one pairing goes five, one, two as the lower note rises from C3 to C4 to C5. Whether a duet has any smooth intervals at all beyond the fifth is a fact about where it is playing, and a scoring decision made at one register does not transfer to another.

That is a strong claim and it should be read with the model’s limits in mind, which the last section does. What survives the limits is the direction: a fixed filter and a moving note cannot produce a register-independent curve, whatever the filter is.

Which computation produced the numbers #

The roughness of a pair of partials is Plomp and Levelt’s, in the form this site has used from the beginning: for two partials at frequencies a and b with amplitudes weighting the pair, the contribution falls off with the separation scaled by a critical bandwidth at the lower frequency. Nothing in that has changed.

What has changed is the two lists it runs over. The lower note’s partials are computed once, at its own frequency, through its own filter. The upper note’s partials are recomputed at every step of the sweep, because a fixed filter applied to a moving note gives a different spectrum at every point — that is the previous rung’s whole finding, and leaving it out would have produced a smooth curve that was wrong everywhere except at one pitch.

The well-finding is a windowed local-minimum search: a sample counts as a well if it is no higher than any of the twelve samples on either side of it, which at 360 steps to the octave is a window of forty cents. That is a weaker test than a prominence rule and it is what the census figure runs, so the counts here are counts of dips of any depth. It matters most where it is least visible — a pairing whose curve is nearly flat contributes wells that a prominence rule would refuse — and it is the reason the well counts should be read against each other rather than as a number of consonances.

The amplitudes are normalised so that each radiator’s strongest partial is 1. That choice matters and is not neutral: roughness is quadratic in amplitude, so an unnormalised comparison would report which instrument had been set louder. Normalising per instrument answers the question “which pairing is rougher at equal prominence”, which is the orchestration question. It does not answer “which pairing is rougher as actually balanced”, which needs a balance, and a balance is a decision rather than a measurement.

Whose music, and when #

The claim that doubling at the octave sounds better one way round than the other is a claim about a repertoire, and it is a claim European orchestration made long before anybody computed anything. Rimsky-Korsakov’s Principles of Orchestration prescribes which instrument takes the upper note of a doubling and gives the reason as blend; Berlioz gives lists of good and bad combinations with no reason at all. The arithmetic here does not confirm those lists — several of them turn on carrying power and attack, which nothing above models — but it supplies a mechanism that has to be present in whatever the right explanation is.

The one place the result is testable against practice without any interpretation is the octave doubling, and there the tradition and the arithmetic agree: the standard advice is to put the richer, reedier instrument above and the simpler one below, and the computation says that arrangement is three to four times smoother.

Where it says nothing at all is any repertoire in which two instruments of the same family play together, which is most chamber music. Two violins are a symmetric duet again, and every figure in this rung collapses back to the one this site already had.

What the picture cannot show #

The body model has four resonances and a real violin has hundreds. Between the four, the modelled response falls to a floor, which is why the violin’s partials above the third are so weak at middle C in the partial figure — weaker than a real instrument’s. Every violin result above therefore overstates how much the body removes, and the direction of the error is known even though its size is not.

The filters are static and instruments are not. A player’s dynamic changes the source spectrum, and on several of these instruments it changes it more than the filter does. Nothing here varies level at all.

Roughness is not consonance. Consonance is half learned, and the two models this site carries disagree about the other half, and roughness is the one that knows nothing about harmony, expectation or context. A pairing with fewer wells is not a worse duet; it is a duet whose smoothness is less determined by the interval.

The register sweep is over roots and not over intervals. Every census in this essay sweeps the upper note across one octave above a fixed lower note, so a “pairing at C3” means a lower note at C3 and an upper note anywhere from C3 to C4. A duet whose two players are three octaves apart — a piccolo over a bassoon, which is an ordinary orchestral texture — is outside every figure here, and the fixed filters guarantee it is a different curve again rather than an extrapolation of these.

And the well count is a count of a model’s features, not of anything heard. Whether a listener can tell that a clarinet-under-violin sixth is smoother than a violin-under-clarinet sixth is a listening question, and the prediction — a factor of five in a quantity whose relation to judgement is itself a model — is the kind that has been wrong before.

Where this ladder goes next #

Five rungs. The first said a spectrum is a list of frequencies and strengths, with the relative timing of the partials discarded. The second and third found the filter — a mouth, then a body — and the fourth found that a fixed filter under a moving note makes the list a function of pitch. This one puts two such lists together and finds that the pair does not commute.

The rung after it is the one the asymmetry makes available and nothing here has taken. If an interval’s roughness depends on which instrument is underneath, then a chord of three different instruments has an assignment problem: six ways to distribute three players over three notes, with a roughness for each, and the same machinery that scored the seven intervals above can score all six. That is the orchestration question in its smallest complete form — not which chord, and not which voicing, but which player on which note — and it is a computation this collection can now run.

The rung after that is the one the register result points at from the other side: an assignment that is smooth at one pitch need not be smooth at another, so the answer is a function of where the chord sits as well as of who is in it. That is a surface rather than a ranking, and drawing a surface honestly is a harder problem than computing one.