One cut cannot place two partials

A bar free at both ends has partials at 1, 2.756, 5.404 and 8.933, which support no fundamental and make an octave and a tritone with the note the bar is loudest at. A maker who wants that bar to carry a melody has one variable left, and it is not the material and not the length. It is the shape of the cross-section along the bar.

Turn a marimba key over and there is an arch cut into the underside, deepest at the middle and running out to nothing perhaps two-fifths of the way toward each end. It is the only tuning operation the instrument has, and it is asked to put two partials where a listener wants them.

What removing material does #

The mechanism is a competition between two quantities, and it is decidable without solving anything.

A mode’s frequency is set by the ratio of the stiffness it feels to the mass it has to move — Rayleigh’s quotient, which is the integral of bending stiffness times curvature squared, over the integral of mass times displacement squared. Thin the bar at a station and both integrals lose something. Bending stiffness goes as the cube of the thickness and mass goes as its first power, because stiffness depends on the second moment of the cross-section and mass on its area. So a thickness reduction of a few per cent removes three times as much stiffness as mass, in proportion.

Which of the two matters at a given station depends on what the mode is doing there. Where a mode bends hard, the stiffness term dominates and thinning lowers the frequency. Where a mode merely translates without bending — near a free end, where the bending moment must vanish — the mass term dominates and thinning raises it.

That last observation is the whole lever, and it is exact rather than approximate. The deepest point of the arch is the one station on the bar where the fundamental bends hardest and the second partial does not bend at all. A cut deepest at the middle therefore takes stiffness from the fundamental and almost none from the mode above it, so the fundamental falls and the second partial nearly stays. Since only ratios are audible, a fundamental that falls further than the partial above it is the same thing as a partial that has risen — and the arch is not a mysterious craft gesture but the one cut that exploits a symmetry the bar was born with.

It also explains why an arch has to be centred. A cut of the same volume placed a quarter of the way along would sit where the second mode bends hard and the fundamental bends less, and would move the ratio the wrong way. The position is not a matter of appearance or of leaving the ends thick enough to survive; it is the only place the operation works.

One knob, and it moves both partials #

The arch has parameters — how deep it goes, how far along the bar it reaches, how flat its floor is — but for tuning it behaves as one. Deepen it and every partial above the fundamental rises together; there is no setting at which the second partial moves and the third stays.

Here is where the two instruments part company, and it is worth stating the numbers before the interpretation.

A xylophone bar is tuned so that its second partial is a twelfth above the fundamental — a ratio of 3 — and its third partial two octaves and a fifth above that, a ratio of 6. Cut the arch until the first target is met and the second is met too. At the arch length used in the figure the third partial lands at 5.9998, which is a tenth of a cent from 6. Swept over arch lengths from half the bar to the whole of it, it runs from 5.856 to 6.040 — from 42 cents flat to 11 cents sharp, and the whole span is smaller than a semitone.

A marimba bar is tuned so that its second partial is two octaves above the fundamental — a ratio of 4 — and its third is usually quoted at 10. Cut until the first target is met and the third partial is at 9.029. Swept over the same arch lengths it runs from 8.283 to 9.176, which is between 326 and 149 cents under 10. No single arch closes that gap.

The trade the numbers force #

Two targets and one degree of freedom is a solvable problem only when the targets happen to lie on the curve the knob traces. The xylophone’s do. The marimba’s do not, so a maker with one arch has to choose which one to hit.

Given that choice, a maker takes the second partial, and the reason is about hearing rather than about arithmetic. The second partial is the loud one — it is the next mode up and carries far more energy than the third — and a partial 110 cents from a whole ratio produces beating against the fundamental’s own second harmonic that is heard as a note out of tune rather than as a colour. A third partial 177 cents flat is nearly three octaves above the fundamental, where the ear’s resolution is poor and no harmonic of the fundamental is available to beat against it.

So the observed practice — second partial exact, third partial approximate and variable between makers — is what a one-knob problem with two targets predicts. The variability is the diagnostic: a quantity a maker controls exactly does not vary, and a quantity that arrives wherever it arrives does.

There is a second reason to take the second partial, and it belongs to the field this essay sits in rather than to the ear. Tuning a bar is not a calculation performed once; it is a bar held up, struck, filed, struck again, and the operation converges because each stroke of the file moves the pitch a little in a known direction. A maker can hear the second partial and cannot easily hear the third. The second partial of a marimba bar at middle C sits at just over a kilohertz, in the region where pitch discrimination is at its best; the third sits above two kilohertz and is buried under whatever the strike put there. A target nobody can hear is not a target a file can converge on, whatever the arithmetic says about its reachability.

That is worth separating from the arithmetic because the two conclusions are independent and they agree. The geometry says one arch cannot place both. The listening says that even if it could, only one of them could be verified.

The result the model will not give #

There is an honest discrepancy here and this essay does not intend to talk it away.

Published accounts of marimba tuning say 1 : 4 : 10, and this computation says 1 : 4 : 9.03. The gap is 177 cents, which is far too large to be rounding. Three explanations are available and they are not equally likely.

The first is that the arch is not the family drawn here. A single smooth arch has effectively two parameters and traces a one-dimensional path; a profile with a second, separately shaped relief has more freedom and can reach places the path does not. Searching over a freer family of symmetric profiles — one that only ever removes material, described by thicknesses at six stations along the bar — finds shapes that hold the second partial at 4 while pushing the third past 15. The shapes it finds are not arches: they are a plateau at the centre with deep notches on either side of it, at about a tenth of the length in from the middle. So 1 : 4 : 10 is reachable and it is not reachable with an arch. If real marimba bars hit 10, their undersides have a feature the word “arch” does not describe.

The second is that the published figure is looser than it reads. The third partial’s target is quoted variously as 9.2 and as 10 in different accounts, and there is no reason to expect makers to agree about a partial they cannot place exactly.

The third is that the model is wrong in the direction that would make things worse rather than better. The Euler–Bernoulli beam ignores shear and rotary inertia, and both of those lower the upper modes of a real bar relative to this calculation. Correcting for them moves the third partial further from 10, not nearer.

What would settle it is a measured profile. A tracing of the underside of one instrument’s bar, fed through the same computation, would say at once whether the shape is an arch and whether that shape gives 9 or 10. That is a measurement this collection does not have, and it is named as a debt rather than guessed at.

Nine is a number with a shape #

One thing is worth noticing about where the computation lands, because it may not be an accident. The model puts the marimba’s third partial at 9.029, which is six cents from 9 — and 1, 4 and 9 are the square numbers.

They are also, exactly, the partials of a bar pinned at both ends. A pinned bar’s wavenumbers are exact multiples of $\pi/L$, so its frequencies are $1 : 4 : 9 : 16$ with nothing rounded, and the first rung of this ladder drew it as a curiosity nobody builds. The arch turns out to be a way of approaching it: cutting away the middle of a free bar makes the centre floppier and the ends stiffer, which is a step in the direction of a bar that is held rather than free.

The correspondence stops at the third partial — the computed fourth is 14.97 where a pinned bar’s would be 16 — so this is a resemblance rather than a limit being approached. But it is a reason to take 9 seriously as a place a bar naturally goes, and the next rung produces an entirely independent argument for the same conclusion.

Worth noticing too is what the arch does not do to the pattern. Deepening it does not move the partials toward the whole numbers in general; it moves them along a particular one-dimensional path, and the whole numbers it happens to pass are the ones on that path. Nothing about a bar wants its partials to be small integers, and the two instruments that exist are the two places where a cut a maker can execute crosses a ratio a listener can use. There is no third instrument here waiting to be found by cutting harder — past a ratio of 4 the second partial climbs into a region where it is neither a recognisable interval nor safely out of the way, and the fundamental has fallen so far that the bar has almost no stiffness left to be loud with.

What the cut costs #

Tuning is not the only thing the arch does, and the second effect is larger than the first.

The arch is a size device as much as a tuning device. A marimba’s lowest bars would be half again as long without it, and a four-octave instrument would not fit in a hall’s pit or a player’s reach. The xylophone’s shallower cut buys correspondingly less, which is one reason a xylophone’s range starts higher.

This is the same kind of bargain a wound bass string strikes and the same kind a brass bell’s flare strikes: a shape chosen for one reason turns out to decide the instrument’s dimensions, and the maker cannot buy the two separately. What is unusual about the bar is the size of the second effect. A wound string’s winding changes its pitch by a tone or two per layer; an arch changes its bar’s pitch by fourteen semitones. On a marimba the tuning operation is also the largest single determinant of how big the instrument is.

The two effects cannot be separated, because they are the same cut. A maker who wanted the tuning without the pitch drop would have to make the blank shorter to compensate — which is exactly what happens, and it means the arch depth and the bar length are chosen together rather than in sequence. That coupling is why a set of marimba bars is not a set of scaled copies: the deep bass bars need the deepest arches, which drop their fundamentals furthest, which is partly why they are so much wider than the treble bars in relation to their length.

What the arch is actually for #

The three buttons in that figure make a point the ratio list does not. The xylophone’s twelfth is added to the sound: 3 is a consonant ratio, it is the second partial of the harmonic series a fundamental would have, and putting it there gives the bar something that sounds like a note with a bright upper partial on it.

The marimba’s two octaves is not added, it is removed. A partial at exactly 4 is the fundamental two octaves up, which is the least conspicuous place a partial can be — it reinforces the pitch and contributes nothing a listener can name. The marimba’s arch does not put a partial somewhere useful; it takes one out of the way.

That reading resolves what would otherwise be a puzzle about the depth of the cut. Removing 58 per cent of a bar’s thickness is a violent operation that costs an octave and a fifth of pitch and a great deal of the bar’s stiffness and loudness. Nobody would do it to gain a two-octave partial. It is worth doing to lose an octave-and-a-tritone one, which is the interval roughness is computed from at its worst against a fundamental.

The reading also predicts something about which instruments bother. A tradition that wants its struck metal to shimmer has no reason to spend an octave and a fifth of pitch removing a partial, and Javanese and Balinese practice does not: a saron bar is a plain bar over a trough, its second partial is where the equation put it, and the tuning of the whole set is fitted to those spectra rather than to a scale imported from elsewhere. Whether that fit is a cause or a coincidence is not settled here, but the design decision is unambiguous: the arch is what a maker cuts when a bar has to agree with a violin, and not when it has to agree with the gong beside it.

The next rung tests that reading, because if the second partial is being removed rather than placed, the rest of the instrument should be indifferent to it or hostile to it. It turns out to be hostile.

Where the account stops #

The thickness law is stated, not traced. Every profile here is a raised cosine of a chosen depth and reach, and the whole finding about the marimba’s third partial is a statement about what that family can reach. A different family reaches different places, which is exactly the escape route the discrepancy section above takes seriously.

The bar is treated as a beam and not as a plate. A marimba bar is perhaps six times as long as it is wide, which is beam-like enough for the first three modes but not for everything: real bars also have torsional modes and transverse bending modes across their width, both of which are audible in the attack and neither of which is in this calculation. They are not affected by the arch in the way the longitudinal modes are, which is one reason a struck bar’s first fifty milliseconds sound different from its steady ring.

And nothing here has anything to say about loudness. Removing 58 per cent of a bar’s thickness at its centre takes out a great deal of its stiffness, and a floppier bar radiates less. That cost is real, it is paid on every marimba bar, and the instrument’s answer to it is the tube — which is the subject of the next rung and turns out to have an opinion about the tuning as well.

Where this goes #

An arch is one cut with one degree of freedom, aimed at two targets, and it hits both on a xylophone and one on a marimba. The choice a marimba maker makes — second partial exact, third wherever it falls — is what the arithmetic says the choice has to be.

The bar is now tuned and it is still nearly inaudible, because a bar is a poor radiator: it is a small, stiff object moving very little air. Every tuned percussion instrument therefore hangs a tube under each bar. The next rung asks what that tube does to the partials the arch has just placed, and the answer is decided by something neither the arch nor the maker chose: whether the ratio the partial was tuned to is odd or even. That is the argument the tube shuts on the partial the arch placed is built on.