A bar's partials are the odd numbers, squared

Every object this field has taken apart so far has been one-dimensional or nearly so. A string’s modes are sines with a whole number of half-waves along it; an open pipe’s are the same; a stopped pipe’s are the odd members of that set. A circular membrane broke the pattern by being two-dimensional, and its Bessel-zero ratios of 1, 1.593, 2.136 support no fundamental at all.

A struck bar breaks it a second time, and for a reason that has nothing to do with dimension. A marimba key is as one-dimensional as a string: a long thin thing, fixed at neither end, doing all its interesting work along a single axis. Its partials are 1, 2.756, 5.404, 8.933 — further from whole numbers than a drum’s — and the cause is in the equation rather than in the geometry.

Second order and fourth order #

A string resists being displaced because it is under tension: pull a point of it sideways and the tension in the two halves pulls back. The restoring force depends on how sharply the string is bent, which is the second derivative of its shape, and the wave equation that results is second order in space.

A bar under no tension at all still resists being displaced, because bending it stretches the material on the outside of the curve and compresses it on the inside. That restoring force depends on how sharply the curvature varies, which is the fourth derivative. The equation is

$$\frac{\partial^2 y}{\partial t^2} = -\frac{E\kappa^2}{\rho}\,\frac{\partial^4 y}{\partial x^4}$$

where $E$ is the material’s stiffness, $\rho$ its density and $\kappa$ the radius of gyration of the cross-section — for a rectangular bar, its thickness over the square root of twelve.

Feed a travelling wave into a second-order equation and frequency comes out proportional to wavenumber. Feed one into a fourth-order equation and frequency comes out proportional to wavenumber squared. That single line is the whole of what follows, and it is worth stating in the form that makes its consequence obvious: a bar’s frequencies are the squares of a ladder that is itself evenly spaced.

Squaring an evenly spaced ladder produces a ladder whose gaps grow. The differences between consecutive squares are the odd numbers, so nothing about the result can be a set of multiples of anything. There is no fundamental for which 1, 2.756, 5.404 are even approximately consecutive multiples, and there is no fundamental for which any three of them are.

The point generalises past bars. Whole-number partials are not what a vibrating object normally does; they are what a second-order equation does. Strings and air columns are the exceptions in this collection precisely because they are governed by second-order equations, and a piano string’s stiffness is a small fourth-order term leaking into a second-order object and pulling its partials off the whole numbers by a few cents. A marimba key is what that term looks like when there is nothing else present.

The equation the ends impose, and the one root that misses #

Knowing that frequency goes as the square of wavenumber does not yet give the ratios. The wavenumbers themselves come from the boundary condition, and for a bar free at both ends the condition is that the bending moment and the shear force both vanish at each end — four conditions, which is what a fourth-order equation needs. Combined, they reduce to one transcendental equation:

$$\cos(\beta L)\cosh(\beta L) = 1$$

The roots converge on $(2n+1)\pi/2$ so quickly that everything after the first is that value to five decimal places. The first is not: it is 4.73004 where $3\pi/2$ is 4.71239, a difference of 0.37 per cent.

That difference is the only irregularity in the whole set, and it survives into the frequencies doubled, because frequency goes as the square of wavenumber. Take the ratios as the odd squares over nine — 1, $25/9$, $49/9$, $81/9$, which is 1, 2.778, 5.444, 9 — and every one of them is 12.9 cents sharp of the true value, and the same 12.9 cents each time. The error is a constant, because it lives entirely in the reference the ratios are taken against.

So there are two ways to state a bar’s spectrum and they differ by one interval smaller than most listeners can hear. From the second partial upward, a free bar’s partials are exactly the odd numbers squared. Only the fundamental declines to join in.

What those numbers are as intervals #

A ratio list is not yet an argument about sound. Converted into the units the rest of this collection argues in, the plain bar’s partials are 1755, 2921 and 3791 cents above its fundamental.

The two buttons under that figure are the argument in the form it is easiest to check, and it is worth pressing them before reading on. Both play the same fundamental with the same envelope and the same falling amplitudes; only the ratios differ. One is a note that could be sung back and named. The other is an event with a register — it can be called high or low, and a set of them can be ordered — and no pitch a listener could match. The distinction is the same one the categorical essay draws from the other direction: a pitch falls into a category and can be an interval, and a brightness is a continuum with no categories on it.

The second partial is the one that decides how a plain bar sounds, and where it falls is worth dwelling on. An octave and a tritone is the least helpful interval available. It is not close enough to a twelfth or to two octaves to be heard as reinforcing the fundamental, and it is not far enough away to be heard as a separate event. What it does instead is sit in the same region of the spectrum as the fundamental’s own upper harmonics would, without coinciding with any of them, which is the exact recipe for roughness — pairs of partials falling close together without landing on each other.

This is why a glockenspiel plate struck on its own has a bright clang rather than a pitch, and why a set of them, played together, produces something that does not behave like harmony. The dissonance curve computed from a bar’s own partials has its minima at 6.94, 8.70 and 11.66 semitones — not one of them at an interval a keyboard has — because the curve is built from a partial list, and this is the partial list.

No fundamental fits, and it is the worst case in the collection #

The claim that a set of partials “supports no fundamental” can be tested rather than asserted, and this collection has the machinery to test it. The procedure is the one the missing-fundamental ladder uses: search every candidate fundamental that any partial could be a small multiple of, and count how many of the partials land within one per cent of a whole multiple of it. That is the arithmetic behind a pitch inferred from a pattern, applied to a partial list rather than to a laboratory tone.

Run over five partials each, the answers separate the objects sharply. An ideal string fuses five out of five on a fundamental of one, which is the premise. A piano string with its stiffness fuses five out of five on 1.005, which is why a stretched octave is the tuner’s answer rather than a failure of fusion. A church bell fuses four of five on a fundamental half its prime — that is the strike note, and it is what six centuries of shaving metal off castings bought. A kettledrum fuses three of four, which is what the enclosed air bought.

A free bar fuses two of five, and the fundamental that does it is 4.466 times the bar’s own lowest mode. The two partials it accounts for are the fourth and the fifth; the lowest three, including the loudest thing the bar produces, do not belong to it at all. An untuned membrane does better — three of five — which is a result worth stating plainly, because the drum essay already put an untuned membrane at the far end of an axis running from a string through a bell. A bar is further along it than the drum is.

The difference between the last two is the one this ladder is about. A membrane’s ratios are what a disc gives, and the timpani’s kettle shows they can be moved by loading the object. A bar’s ratios are what a fourth-order equation gives, and the only free parameter left in the equation is the shape of the cross-section along the bar — which is why a maker’s response is a cut rather than a cavity, a patch or an added mass.

The mode shapes behind the numbers #

Each root carries a mode shape, and the shapes matter as much as the frequencies once the object stops being an abstraction and becomes a key that has to be held and hit.

Two features carry into everything that follows. The first is that the nodes are irrational and have to be computed rather than constructed. The fundamental’s node at 0.2242 of the length is where the holes are drilled in every marimba key ever made, and it is a number nobody could have found by dividing a length into equal parts.

The second is a symmetry. A bar free at both ends is symmetric about its middle, so its modes are alternately symmetric and antisymmetric about that point. The odd-numbered modes have an antinode at the centre; the even-numbered modes have a node there, exactly, and by symmetry rather than by arithmetic. The consequence for how a bar is played is large and is taken up at the top of this ladder.

Nothing on a bar travels at the same speed as anything else #

A second-order equation has one more property that is easy to take for granted: its wave speed does not depend on frequency. Every partial of a string travels at the same speed, which is why a plucked corner is still recognisably a corner after a round trip, and why the excitation ladder had to work quite hard to find any dispersion on a piano string at all.

On a bar the speed is $\omega/k$, and since $\omega$ goes as $k^2$ the speed goes as $k$ — which is to say, as the square root of the frequency.

A factor of 3.65 across five partials is not a correction, it is a different regime. The practical consequence is that the modal description of a bar is the only useful one. On a string it is often easier to think in travelling waves — a corner going round and coming back — and the excitation ladder’s whole account of a piano hammer is built that way. On a bar there is no corner to follow, because whatever shape the mallet leaves behind has smeared into something unrecognisable within a fraction of a period. What a strike does to a bar is set the amplitude of each mode once, and after that the modes go their own ways.

This also removes a term the piano ladder spent two rungs on. The corner that does not come back a corner is about a disturbance returning to a hammer still in contact with the string, degraded by the phase error the partials have accumulated. A bar has enormous dispersion and no returning corner worth speaking of, so the term does not shrink — it stops applying.

What the ends decide #

One last comparison makes the point that none of this is about wood, or metal, or being a bar. It is about where the ends are.

The clamped case explains why a tuning fork is a usable reference and a marimba key is not. Its second partial is at 6.267, which is nearly two and a half octaves up and falls away quickly; what is left is very nearly a pure tone, and that is what a fork is for. It is the same material and the same equation with one end held.

The pinned case is the one to remember, because it is exact. A bar simply supported at both ends has wavenumbers at exact multiples of $\pi/L$, so its partials are exactly $1 : 4 : 9 : 16$. That is not a harmonic series and it does not have a fundamental in the ordinary sense — but it is a set of small whole numbers, which is a different kind of thing from 2.756, and the ear’s machinery for finding a fundamental has more to work with. Nobody builds an instrument out of pinned bars, because pinning a bar at two points means damping it at two points and a supported bar rings for no time at all.

What is worth carrying forward is that $1 : 4 : 9$ exists, is reachable by a fourth-order equation, and is a long way from where a free bar starts.

It is also worth saying plainly that a plain bar is not a defective object waiting to be repaired. A glockenspiel is a set of plain steel bars, a celesta is a set of plain steel bars struck by a keyboard action, and a Javanese saron is a set of bronze bars over a trough. All three are played for their own sake, all three are perfectly usable in ensembles, and none of them is undercut in the way the next rung describes. What they give up is the ability to carry a line that another instrument has to agree with in tune — which is why they double melodies rather than state them, and why a gamelan’s tuning is fitted to its own instruments instead of to a standard anything else could join.

The bright, ringing, slightly unplaceable quality those instruments have is the 2.756 in the drawing. It is a sound, and the only reason to change it is that a maker wants a bar that can play a bass line.

Where the account stops #

The model is Euler and Bernoulli’s and it ignores two things. A real bar has shear deformation and rotary inertia, both of which matter more the thicker the bar is relative to a wavelength, and both of which lower the upper partials. On a bar whose length is twenty times its thickness the third partial is perhaps one or two per cent below what is drawn here; on a stubbier bar the departure is larger. Every ratio in this essay is therefore a ceiling rather than a prediction.

The material has been assumed uniform and isotropic, which rosewood is not. Wood’s stiffness along the grain is ten or more times its stiffness across, and a bar cut from a plank has whatever variation the plank had. That is a source of scatter between bars rather than a systematic shift.

And the bar has been assumed to be a rectangular prism of constant section, which no instrument uses. That assumption is the one the next rung removes, and removing it is the whole craft of making a tuned percussion instrument.

Nothing here says anything about how loud each partial is. A ratio list says where the partials are and not what a strike puts into them, and the two are independent: a mode can be exactly where the arithmetic says and carry no energy at all because the mallet landed on its node. Every figure in this essay draws the modes a bar has, and the modes a bar uses on any given stroke are a smaller set decided by where and with what it was hit. That is a separate computation with its own ladder behind it — the piano’s excitation-point account is nine rungs deep — and it is taken up at the top of this one.

And decay has been ignored entirely. The partials of a real bar do not last equally long: internal friction in wood rises roughly in proportion to frequency, so the upper partials are gone while the fundamental is still ringing, and a spectrum measured a second after the strike is a different list from a spectrum measured at the strike. That matters more for a bar than for most objects here, because a bar’s partials are so widely spaced that their decay times differ by factors rather than by percentages.

Where this goes #

A plain bar rings and does not have a note, which is a description of a glockenspiel plate and not of a marimba key. What a maker does about it is cut an arch into the underside, and the arch is a single deformation asked to move two partials at once onto whole-number ratios.

The next rung computes what one arch can and cannot do. The answer turns out to divide the two instruments that use it: for a xylophone’s targets the second partial and the third arrive together, and for a marimba’s they do not. What that costs, and what a maker has to do about it, is the argument an arch cannot place two partials at once is built on.