An instrument is not one timbre

This site has a table of timbres. A string is 1, ½, ⅓, ¼ and so on; a clarinet has its even partials suppressed; a bell has its own measured set. Every roughness computation, every dissonance curve and every chord comparison in the collection uses one of those lists, and uses the same list whatever note is being played.

The ladder’s own third rung is the reason that cannot be right. The body is the filter: a violin string radiates almost nothing on its own, and what reaches a room is the string’s sawtooth multiplied by the body’s response — a comb of measured resonances that stays put while the note moves.

If the comb stays put and the partials move, the pattern that comes out is different at every pitch. This rung is about how different.

Five notes, five spectra #

Take the same source spectrum, put it through the same body, and change only the fundamental.

Two things in that table are worth separating.

The variation is large. A centroid of 1.02 is a note that is almost entirely fundamental; 1.64 is a note with real energy in its second and third partials. Those are different timbres in the ordinary sense of the word, on one instrument, from one bow stroke, three or four notes apart.

And it is not monotonic. The centroid does not fall smoothly as the fundamental rises: it dips at 440 and comes back up at 660 and 988. That is the comb showing through — at some fundamentals a strong resonance lands on the second partial and at others it lands between two, and the pattern of hits and misses is not ordered by pitch at all.

What it does to the dissonance curve #

The consequence lands on a result this collection is fond of. A spectrum chooses its own scale: compute roughness between two tones of a given timbre across an octave and the minima fall where the scale’s intervals are.

That computation is done with a fixed partial list. Doing it with the radiated one makes it a function of the pitch it is done at.

If the body can take seven minima down to two, it can also give some of them back, and where it does is decided by which partials its resonances happen to catch. The bottom of the instrument’s range is where that is easiest to see, because there the body’s strongest resonances sit on the second and third partials rather than above them.

So the scale a spectrum chooses is a scale per note. That is not a refutation of the earlier rung — the fifth is present in every version, which is the finding that mattered — but it is a sharp limit on how far the argument can be pushed. A claim of the form this instrument’s spectrum implies this scale has a hidden argument in it, and the argument is which note the instrument was playing.

Reading the four curves together gives the honest summary. The fifth at 702 cents is a minimum in every one of them, including the fixed-timbre reference; the fourth at 498 is in three of four; and everything else comes and goes.

The count of minima is the quantity to watch rather than their positions, and it moves the way the partial count does. Seven minima with the raw sawtooth, two at A4 where the body has stripped the upper partials hardest, three at the bottom and four at the top: the curve has as many dips as the spectrum has strong partials to make coincidences with, because every minimum is a coincidence between one partial of each note. So the essay’s four pictures are four spectra of different lengths, and the scale each one implies is short or long accordingly. So the part of the spectrum-chooses-a-scale result that survives the variation is the part that was least surprising, and the part that made the result exciting — the thirds and the sixths falling out of a spectrum — is exactly the part that is a function of which note is being played.

What this does to every roughness number here #

There is a general correction implied and it is worth stating rather than leaving as a mood.

Every roughness figure in this collection is computed at one root, usually middle C or the C below it, with a fixed timbre. Two of the three ingredients are named in the caption — the timbre and the interval — and the third, the pitch it is evaluated at, is named as an argument nobody reads as load-bearing. It is load-bearing whenever the instrument has a body.

The correction does not run one way. At some pitches the body strips the partials and roughness falls; at others it boosts the second and third and roughness rises. What it removes is the idea that a chord has a roughness on an instrument. It has one per pitch — and how large that is against the spread between chords is a comparison worth making rather than asserting.

Scoring a major triad at sixteen roots across a violin’s range with the radiated spectrum at each, against seven chord types scored at one root:

The first three factorise exactly: 5.8 × 2.7 = 15.7, so the compass spread is the register effect this collection already had multiplied by a body effect that is new here. And the two answers to the original question are different:

The total spread across the compass is larger than the spread between chords — sixteen against nine — which is a stronger statement than “comparable” and is mostly not this rung’s doing. The body’s own share is 2.7, which is smaller than the spread between chords and smaller than the register effect it multiplies. So the correction is real, it is about a third of the size of choosing a different chord, and it sits underneath a larger effect that was already known.

There is a second thing in that column worth naming, because it runs against the essay’s own emphasis. The body’s contribution falls almost monotonically with pitch — 0.90 of the fixed-timbre value at G3, 0.50 in the middle, 0.33 at B5 — so what it is mostly doing to a summed roughness is rolling the partials off, not combing them. The non-monotonicity that makes the centroid interesting largely washes out of a sum over pairs, because a resonance that boosts one partial of one note is as likely to boost a partial that reduces roughness as one that raises it.

So the comb matters most where the essay first found it, in the centroid and in the dissonance curve’s minima, and least in a total. A quantity that adds over many partials is insensitive to which particular ones the body caught; a quantity that reports where the curve dips is not.

The same effect is a vowel #

There is a version of all this that this site has already drawn, and drawing them side by side is the point.

The site has already drawn that arrangement, in a vowel is two resonances: two fixed formants, and the partials of the voice sliding underneath them as the singer changes pitch. Put a violin body where the mouth is and the diagram is unchanged in every particular. A soprano taking her fundamental from 523 to 1046 hertz drags her whole ladder up through two stationary peaks, and nobody thinks this means a sung “ee” has one spectrum; everybody knows it has one filter, and that the spectrum is what the filter and the pitch produce together. The violin’s case differs only in that its filter was never given a name, so there is no word to hang the variation on.

The source–filter idea is completely standard for the voice and is not applied to instruments with the same firmness, and the reason is probably that a voice changes its filter deliberately and an instrument does not. But a fixed filter under a moving source produces just as much variation as a moving filter under a fixed one, and the variation here is the fixed-filter kind.

The one asymmetry between the two cases is worth stating, because it is why the voice’s version is uncontroversial and the violin’s is not. A singer changing vowel moves the filter by hundreds of hertz and holds the pitch; a violinist changing note moves the pitch by a few per cent and holds the filter. So the voice’s variation arrives labelled — a different vowel is a different word — and the violin’s arrives as a property of the note, which is exactly the thing a listener attributes to the instrument rather than to the pitch.

Which suggests the right way to say what an instrument’s timbre is. Not a spectrum. A filter, plus a rule for the source. The spectrum is what those two produce at a particular pitch, and a table of spectra is a table of instants.

The violin’s three entries in that same table — string, clarinet, bell — are of two different kinds, and only the last is measured. string is 1/n and clarinet is odd harmonics: both are source models, partial lists for a vibrating thing with no body applied to them at all. Everything above says the violin’s entry should be of the bell’s kind and cannot be, because unlike a bell a violin does not have one spectrum to measure.

The number the ladder should have been carrying #

If a timbre is a filter plus a source rule, then the quantity worth tabulating for an instrument is not a partial list but the variation — how much the radiated spectrum moves across the compass.

The centroid range does that in one number. For this violin body it is 1.02 to 1.64, a spread of 0.62 partials, against a source that would give 1.77 at every pitch with no body at all. So the body both darkens the instrument on average and makes the darkening pitch-dependent, and the second effect is the one no table can hold.

A useful way to read that: the spread across the compass is comparable to the difference between two of the site’s stock timbres. A string at 1.77 and a clarinet at its own value are treated as different instruments; a violin at 1.02 and the same violin at 1.64 are treated as one. That is the size of what the fixed-list convention discards.

Which computation produced the numbers #

The source is the site’s own string timbre, 1/n, unchanged. The filter is bodyGain over VIOLIN_BODY, which is the measured resonance table this ladder’s third rung is built on and which is the same function that figure calls.

The radiated amplitude of partial n at fundamental f₀ is therefore (1/n)·bodyGain(n·f₀), and every number here is that expression evaluated. Nothing is fitted and no new model is introduced: this rung’s whole content is applying one existing function inside another.

The centroid is the power-weighted mean partial number over the first twenty-four partials.

The dissonance curves are the site’s own roughness model, run over the radiated amplitudes rather than the source’s. The minima are found by scanning the curve at two-cent resolution and taking local minima, so a shallow minimum near a steep one may be counted or not depending on the resolution — the count is therefore softer than the positions.

What a player does about it, which is most of technique #

If an instrument’s timbre varies this much from note to note, somebody has to even it out, and the somebody is the player.

The compensations are all familiar and none of them is usually described this way. Vibrato sweeps the fundamental by a few tens of cents several times a second, which drags every partial back and forth across the comb and averages the hits and misses — so a vibrato note has a smoother spectral envelope than any of the steady notes inside it. Bow position and pressure change the source spectrum, and a player who finds a note dull has the option of feeding the body more upper partials. And which string a note is taken on is a choice between two different sources under the same filter, which is why the same written note sounds different on the D string and the A.

Vibrato is the one worth drawing out, because the mechanism is not the one usually given. A singer’s or a violinist’s vibrato sweeps the fundamental by a few tens of cents several times a second — the site drew that trace for the voice, a periodic excursion about a mean the ear hears as a single pitch. Put that excursion under a fixed comb of body resonances and it is also a periodic sweep of every partial across every resonance: the fourth partial of a note whose fundamental wobbles by 30 cents wobbles by 30 cents too, in absolute terms four times as far, and at the top of the ladder the excursion is wide enough to cross a resonance and come back twice a period. So the radiated spectrum of a vibrato note is an average over the comb rather than a sample of it, and the averaging is broader the higher the partial — which is exactly the correction the fixed-comb problem calls for, applied more strongly where the comb is finest. That gives a mechanical reason for something usually explained as expression, and it predicts that vibrato should do least for an instrument whose resonances are widely spaced and most for one whose are dense, which is the difference between a violin’s low register and its high one.

That reframes a good deal of string technique as filter management. It is not the only thing a player is doing, and it is a use of the model rather than a result of it, but it is the kind of consequence a fixed-partial-list account cannot even state.

One more thing about the numbers deserves a line, because it is the sort of detail a fixed table cannot represent at all. The centroid at 440 hertz — 1.02 — means that note is very nearly a pure tone as far as its power distribution goes. An A4 on this body radiates almost nothing above its fundamental. That is not a defect and it is not a general fact about violins; it is where that body’s resonances happen not to be, and a different instrument would put the hole somewhere else.

Which is the most concrete form of the argument. Two violins with the same strings, the same bowing and the same note produce measurably different spectra, and the difference is not in anything either player is doing. It is in where two combs sit relative to one ladder.

Whose instrument, and when #

The body table is one measured violin, and violins differ from each other more than most instruments do — that is what a violin is famous for. A different body gives a different comb and therefore a different pattern of variation across the compass, so the specific centroids here belong to that instrument.

What generalises is the arrangement rather than the numbers. Any instrument whose radiator is fixed and whose source pitch moves is in this situation: the violin family, the guitar, the piano’s soundboard, a marimba’s resonators, and every wind instrument whose bell is a fixed filter on a moving harmonic series. The exceptions are the instruments where the resonator moves with the pitch, and there are few — an organ, where each pipe is its own resonator, is the clearest.

That exception is worth noticing because it predicts something. An organ should have a more consistent timbre across its compass than a violin, since each note is its own instrument and no partial has to slide under anything. Whether it does is a listening question this site cannot settle, but the mechanism says it should.

What the picture cannot show #

Only the body is varied. A real violin’s source also changes with pitch: bow force and speed relative to the string’s impedance change what corner the bow makes, so the sawtooth is not the same sawtooth at G3 and E5. Schelleng’s diagram is where that lives and none of it is here.

The body response is a magnitude. Phase is discarded, which is defensible for a roughness calculation — the ear hears the list, not the shape — and not defensible for anything about transients.

Radiation direction is ignored. A body’s resonances radiate in different directions, so the spectrum in front of a violin differs from the spectrum beside it, and an instrument points. The centroids here are for an unspecified listener.

And the roughness model is the same one throughout, with the same limits it always has: a critical-band width, pairwise partials, and no account of what the ear does with a spectrum that has been shaped by a resonance rather than by a source.

Where it lands on the collection’s other arguments #

Three results elsewhere on this site have this as an unstated premise, and they are affected unequally.

A third is rougher in the bass survives and gains a second mechanism. That rung explains register-dependent roughness by the width of a critical band, which is a fact about the listener. The body adds a fact about the instrument pointing the same way at the bottom of the range, where the resonances catch the low partials and there are more of them to be rough with. Two mechanisms, one conclusion, arrived at independently.

A chord is a register is unaffected in its ranking and moves in its numbers. Its comparisons are between voicings at one pitch level, so the filter is nearly common to all of them and divides out; the absolute roughness values move and the ordering does not.

And roughness cannot choose a scale is strengthened. That rung already found the spectrum-to-scale argument weaker than it is usually presented. This adds a reason it had not used: the spectrum the argument needs is not a property of an instrument but of an instrument at a pitch, so a tradition’s scale would have to be derived from an average over the notes of that scale, which is circular.

Where this ladder goes next #

Three rungs of this ladder have said what a spectrum is, that a vowel is a filter and that a body is one too. This one puts the third against the site’s own habit of treating an instrument as a partial list, and finds that the list is a snapshot — with enough variation across a compass to change how many minima a dissonance curve has.

The rung after it is the one this makes both possible and necessary: a dissonance curve for two different instruments. Every roughness figure here compares two tones of the same timbre, which is a duet nobody plays. Two instruments with different fixed filters produce partials that are attenuated differently at the same frequency, so their coincidences are weighted unequally — and whether an interval is smooth might depend on who is playing which note.