The body is the filter

A violin string, stretched on a rigid frame with no body under it, is nearly inaudible. It is thin, it displaces almost no air, and what it produces would not carry across a room. Everything a listener hears has been through the box.

That is a source-filter arrangement, and the site has met it before.

The same argument, on a different object #

A vowel is two resonances. The vocal folds produce a buzz with a full set of harmonics; the vocal tract’s resonances — the formants — shape which of them come out; and because the formants are a property of the shape of the mouth rather than of the note, the same vowel is recognisable across a wide range of pitches.

A violin is the identical arithmetic with the vocal folds replaced by a bowed string and the mouth replaced by a box. The string’s sawtooth is the source, the body’s resonances are the filter, and because the body does not change when the note does, the instrument keeps a recognisable character across its whole range.

This is the general reason an instrument has a voice rather than a tone. It is not specific to violins or to voices; a woodwind’s tone-hole lattice does the same job by a different mechanism, fixing the character of everything above its cutoff regardless of the fingering.

What is actually resonating #

Four features do most of the work on a violin, and each has a name and a measured frequency.

A0, near 275 Hz, is the air resonance — the enclosed air in the box breathing in and out through the f-holes. It is a Helmholtz resonator, the same object as a bottle blown across the top, and it is why an f-hole is a hole rather than decoration. It sits near the bottom of the instrument’s range and it is what gives the open G and the low register their power.

B1− near 460 Hz and B1+ near 540 Hz are corpus bending modes — the whole box flexing, top and back moving in opposite directions. They are the strongest resonances the instrument has and they are what a maker is adjusting when graduating the plates.

The bridge hill, a broad rise around 2.5 kHz, is not a body mode at all. It is the bridge’s own resonance: the bridge is a small piece of maple with legs, and it has a rocking mode of its own that boosts a band of frequencies on its way from string to top. That band is where a violin’s brilliance lives, and it is why a bridge is cut and thinned as carefully as it is.

The comparison with a vowel, made exactly #

The two cases are close enough that the differences are worth listing, because they are what distinguishes an instrument from a voice.

A vowel’s filter is variable and a violin’s is not. A singer changes formants continuously — that is what articulating words is — while a violinist’s body is a fixed object. So a voice can sing several vowels on one note and a violin has one voice.

A vowel’s formants are broad and a violin’s resonances are narrow. A vocal tract is lossy, its resonances have low Q, and a formant is a hundred hertz wide or more. A violin’s A0 and B1 modes have Qs of around twenty, so they are tens of hertz wide, which is why individual notes fall on and off them and why a singer’s vowel does not vary from note to note in the same way.

And a singer can move the source. A trained singer adjusts the fundamental to put a partial on a formant — the whole technique of a soprano’s high register — while a violinist’s fundamental is fixed by the fingerboard position and by the tuning.

The consequence for how a violin plays #

Because the resonances are fixed and the notes are not, some notes fall on resonances and some fall between them. That is not a defect; it is a property of the arrangement.

A note whose fundamental lands on A0 or B1 is louder and easier to produce than its neighbours. One that lands between them is weaker. Players know these as strong and weak notes and adjust for them constantly, and the pattern is different on every instrument because every body’s resonances are in slightly different places.

This is also why a violin is loudest in a band rather than at a note, and why the instrument’s projection is a matter of where its resonances sit relative to the frequency region a hall and an audience are most sensitive to.

Why two identical violins are not identical #

The frequencies above are means. An individual instrument’s resonances depend on the arch of its plates, the graduation of their thickness, the wood’s stiffness and density, the setup of the bass bar and sound post, and the humidity in the room.

That is the whole content of the claim that instruments differ. Not a mystery, not an unmeasurable quality: a comb of resonances with measurable positions, widths and heights, differing from instrument to instrument by amounts that are large compared with the differences a listener can hear.

The measurement is routine and the prediction from it is not. Makers and researchers have measured thousands of instruments; what nobody can do reliably is say from a measured response whether an instrument is a good one. The response is what a violin sounds like; what makes a response desirable is a separate question that has resisted a great deal of effort — which is the honest state of the subject and is worth saying plainly rather than gesturing at.

The source is not quite independent of the filter #

The source-filter model assumes the two are separable: the string does what it does, the body colours it, and the string is unaffected. That is a very good approximation and it is not exact.

The string’s terminations are the bridge and the nut, and the bridge sits on a body that moves. Where the body is stiff, the bridge is nearly rigid and the string’s termination is nearly fixed. Where the body has a strong resonance, the bridge moves — and a string whose end is moving is not the same oscillator as one whose end is fixed.

Almost everywhere this is a small effect. At one place it is not, and that place is where the string’s own mode frequency coincides with a strong body resonance. There the two stop being separable altogether and the note breaks up, which is a wolf note and is the next essay.

What a guitar and a piano do instead #

The same architecture, with different pieces, is worth a paragraph each because the differences are informative.

A guitar has a large flat top with its own modes and an air resonance near 100 Hz through the sound hole, which is much lower relative to its range than a violin’s A0. Its plates are flat rather than arched and it is driven at a bridge glued to the top rather than pressed against it, so the coupling is stronger and the resonances are broader.

A piano has a soundboard — an enormous plate with a huge number of closely spaced modes, driven at many points by many bridges. So many modes, so densely spaced, that the response is nearly smooth rather than a comb, which is one reason a piano’s timbre changes less across its range than a violin’s does and why a piano has no wolf notes.

The direction of the trend is worth naming: the more modes a body has, the less of a voice it imposes, and the more the note’s own spectrum comes through. A violin is at the strongly voiced end, a piano at the weakly voiced end, and the family of instruments in between.

Why the filter does not disturb consonance #

There is a question this arrangement raises that is worth answering, because the answer is not obvious and it matters for the rest of the site.

Roughness between two notes is computed from their partials, and the body changes the amplitude of every partial. Does putting two violins through two different bodies change which intervals are consonant?

Barely, and the reason is instructive. A filter changes partials’ amplitudes and not their frequencies. Roughness depends on frequency differences between pairs of partials — whether they fall inside a critical band — and on their amplitudes only as a weighting. So the positions of the wells in a roughness curve are untouched by any filter, and only their depths move.

So the wells in a roughness curve are fixed by where the partials of a harmonic spectrum fall, and every one of them sits at a simple ratio because that is where partials coincide. A body multiplies those partials’ amplitudes by a curve; it cannot move a partial, so it cannot move a well. What it can do is make a well shallower, by attenuating the two partials whose coincidence produced it — and a shallower well is a weaker preference, not a different one. Which is why the whole tuning half of this collection can proceed without knowing what instrument is playing: the arithmetic of consonance is a fact about frequencies, and an instrument’s body is a fact about amplitudes.

That is a satisfying separation of concerns, and it is worth running rather than resting on, because “barely” is a word and the curve can be drawn both ways.

Computed at the open G with sixteen partials, once bare and once through the body, the shared wells are in identical positions to a hundredth of a semitone — 4.98, 7.02 and 8.13, the fourth, the fifth and the minor sixth. The argument holds exactly where it applies: a filter multiplies amplitudes, a well is a coincidence of frequencies, and a coincidence cannot be moved by a multiplication.

Two things the argument does not cover turn up in the same computation.

The filter destroys wells. The bare curve has eleven local minima across the octave and the filtered one has four. The shallow features between the named ratios — at 1.38, 3.87, 5.83, 9.69 semitones — are produced by coincidences among high partials, and the body attenuates those partials enough that the coincidences stop registering. One minimum appears at 2.87 that has no counterpart in the bare curve at all.

And the depths do not move together. Normalised to each curve’s own maximum, the octave’s well deepens by a factor of six through the body and the fourth’s by 1.2. That is a reweighting rather than a scaling, and it has a consequence the separation-of-concerns statement does not allow for:

The two thirds change places. On a bare sawtooth the major third is the smoother of the two; through this body the minor third is, by three per cent. Three per cent is small and the swap is real, and it is exactly the thing a filter was supposed not to be able to do.

So the separation is between positions and depths, and only the first half is exact. The instrument cannot decide which ratios are the smooth ones and it can decide how smooth they are relative to each other, which is why a string quartet and a choir agree about the fifth and can disagree about whether a major or a minor third is the easier interval to tune. And an inharmonic instrument disagrees about the positions too, because there the frequencies themselves have moved.

What the picture cannot show #

The response is drawn as a smooth curve and is not one. A real measured violin response is dense and jagged above about 1 kHz — hundreds of overlapping modes, not four resonances — and what the figure draws is the envelope of it. The four named features are the ones that are robust across instruments; everything between them differs from violin to violin.

Radiation is not the same as response. A body’s mechanical response says how much the top moves; how much of that reaches a listener depends on how well each mode radiates and in which direction, and the two are not the same function. A mode that moves a lot of wood in a pattern that cancels itself radiates little.

The reweighting result is one body and one spectrum. The two thirds change places by three per cent through this collection’s violin response, and three per cent is inside what a different measured instrument, a different partial count, or a different roughness model would move. What is not inside that margin is the mechanism — the filter attenuates high partials, high partials are what make the shallow wells, so the shallow wells go and the deep ones are reweighted — and the mechanism says a swap of two nearly equal wells is possible on any instrument rather than that it happens on every one.

The numbers are means over instruments. They are measurements of full-size violins in the literature, and any individual instrument’s are different by tens of hertz.

And no player is in the figure. A violinist’s left hand, chin and shoulder are in contact with the instrument and damp it; a violin measured on a stand and the same violin under a chin have measurably different responses.

What a maker is actually adjusting #

It is worth connecting the measured resonances to the operations a maker performs, because the connection is direct and is often described as though it were not.

Plate graduation — thinning the top and back to a thickness map — sets the plates’ stiffness and therefore the frequencies of the corpus modes. A maker tapping a free plate and listening is measuring a mode frequency by ear, and the target frequencies for free plates before assembly are a standard part of the craft.

The bass bar is a strip glued under the top on the bass side. It stiffens the top asymmetrically, which splits and shifts the corpus modes, and its dimensions are one of the parameters most often adjusted when an instrument is reworked.

The sound post is a dowel wedged between top and back under the treble foot of the bridge. Moving it by a fraction of a millimetre measurably changes the response, which is why it is the first thing a luthier adjusts and why players describe its position in terms that sound superstitious. It is not superstition; it is a stiff coupling between the two plates whose position changes which modes are driven.

The f-holes set A0 — their area and the enclosed volume are the two terms in a Helmholtz resonator’s frequency — and their length also relieves the top’s stiffness along the middle, which affects the corpus modes as well. One feature doing two jobs, which is why their shape has been so stable for four hundred years.

Every one of those is an adjustment to a curve that can be measured before and after. What cannot be done reliably is the inverse problem: deciding which curve to aim at.

Whose instruments, and when #

The measurement tradition is long — Felix Savart in the 1830s, Hermann Backhaus in the 1930s, Carleen Hutchins and the Catgut Acoustical Society from the 1960s, and a large modern literature centred on Stockholm and on the Violin Acoustics group. The modal names used here (A0, B1−, B1+) are the current convention and are not universal in older papers, which is a real hazard when reading across the literature.

The bridge hill was identified as a distinct phenomenon in the 1960s and 70s and its mechanism was argued about for decades. The claim here — that it is largely a bridge resonance rather than a body one — is the current consensus and was not always.

The claim that measured response does not predict quality is a claim about the state of the subject as it stands, and it is worth restating that it is a claim about prediction: the correlation between measured features and judged quality is real, weak, and not strong enough to select instruments by.

Where this goes #

Immediately next is the exception the source-filter model has: a wolf note, where the string and the body stop being separable and begin exchanging energy, which is a coupled oscillator rather than a filter and which shares a word with something in the tuning field that has nothing to do with it.

Beyond that, the filter’s output still has to get to a listener, and it does not go in all directions equally. An instrument’s directivity changes with frequency, so the spectrum this essay computes is the one available at a point rather than the one radiated — and a recording is a choice of seat.