Instruments and their design

The tube shuts on the partial the arch placed

A stopped tube tuned to a bar's fundamental resonates at every odd multiple of it and presents a rigid lid — an infinite input impedance — at every even one. A xylophone's arch puts its second partial on 3, which is a resonance, and the tube passes it within five decibels. A marimba's puts it on 4, which is an antiresonance, and the tube takes it thirty-eight decibels down. Same tube, opposite answers, and the difference is parity.

17 min read 7 figures The design was forced The instrument is not the model

Assumes: One cut cannot place two partials

A tuned bar is a quiet object, and the reason is worth a sentence because it decides everything that follows.

Its fundamental mode has the two ends going one way and the middle going the other — three lobes of alternating sign, which is what the shapes drawn at the foot of this ladder look like. A source of that shape is a quadrupole, and a quadrupole radiates very badly when it is small compared with a wavelength. A middle-C marimba bar is 400 millimetres long and the wavelength at 262 hertz is 1.3 metres, so most of the air the bar pushes at one lobe simply moves across to the next lobe and back, rather than leaving. Struck alone on a bench, a rosewood key is barely audible across a hall — and the arch has made it worse, since removing more than half the thickness at the middle leaves a floppier bar with less force behind it.

Every tuned percussion instrument therefore hangs a tube under each bar. The tube is closed at the bottom and open at the top under the bar, and its length is set so that its first resonance is the bar’s fundamental. Its function is described in every account of the instrument as amplification, and that description is right as far as it goes.

What no account states is what it does to the partials the arch was cut to place, and the answer turns out to be decided by a property nobody chose.

The tube fixes the radiation problem by changing the shape of the source. Its mouth is a single small opening breathing in and out — a monopole, which radiates far better at low frequency than a quadrupole of the same size — and the bar’s job is reduced to driving it. That is a different architecture from a violin body, which takes what a string produces and filters it, and different again from a timpani’s kettle, which changes the head’s own modes. A marimba’s tube sits downstream of the bar and does not move the bar’s modes appreciably; what it changes is which of them get out.

What a stopped tube tuned to the fundamental does to a marimba bar's partials. Power radiated from the mouth of a 311 millimetre stopped tube of 56 millimetre bore, tuned so that its first resonance is the bar's fundamental at 262 hertz, plotted against multiples of that fundamental and referred to what it does at the fundamental itself. Its quality factor there is 80. A stopped tube resonates at the odd multiples and presents an infinite input impedance — a rigid lid — at the even ones, so a partial's fate is decided by the parity of the ratio the arch put it on. The bar's own partials are marked: 1 at 0.0 decibels, 4.00 at -38.5 decibels, 9.03 at -13.1 decibels. — Fig. 1 The power radiated from the mouth of a 311 millimetre stopped tube of 56 millimetre bore, tuned so that its first resonance is a marimba bar’s fundamental at 262 hertz, referred to what it does at that fundamental. The bar’s own partials are marked. The fundamental sits on the peak. The second partial — which the arch spent 58 per cent of the bar’s thickness putting on exactly 4 — sits 38.5 decibels down, in a trough the tube cannot be persuaded out of. The third at 9.03 fares much better, and the reason is the whole of this essay.

Odd resonances and even lids

A cylinder closed at one end and open at the other is the object the clarinet essay took apart, and its resonances are the odd multiples of its fundamental: 1, 3, 5, 7. That much is standard, and it is the half of the result everyone quotes.

The other half is not usually said out loud. The input impedance of a stopped tube driven at its open end is

Z_{\text{in}} = -i\,Z_c \cot(kL)

and the resonances are where the cotangent is zero — where $kL$ is an odd multiple of $\pi/2$ , which gives the odd multiples of the fundamental. But a cotangent also has poles, at $kL = n\pi$ , and at those frequencies the input impedance is infinite. An infinite input impedance means no volume flow enters the tube at all. Driven there, the mouth of the tube behaves as a rigid wall.

Those frequencies are $kL = \pi, 2\pi, 3\pi$ , which for a tube tuned so that $kL = \pi/2$ at the fundamental are the frequencies $2f_1, 4f_1, 6f_1$ — the even multiples, exactly. A stopped tube is not merely a comb of resonances with nothing in between. It is a comb of resonances on the odd numbers and a comb of closed lids on the even ones, and there is nowhere in it that is neither.

This is a statement about a cotangent and it does not depend on how lossy the tube is, how wide it is, or what is above it. Losses turn the infinite impedance into a very large one and the perfect lid into a 38-decibel one, but they cannot move it.

Which is why the two instruments come out opposite

A xylophone bar’s second partial is tuned to 3, which is odd. A marimba bar’s is tuned to 4, which is even. The same tube, hung under either, does opposite things to them.

What a stopped tube tuned to the fundamental does to a xylophone bar's partials. Power radiated from the mouth of a 311 millimetre stopped tube of 56 millimetre bore, tuned so that its first resonance is the bar's fundamental at 262 hertz, plotted against multiples of that fundamental and referred to what it does at the fundamental itself. Its quality factor there is 80. A stopped tube resonates at the odd multiples and presents an infinite input impedance — a rigid lid — at the even ones, so a partial's fate is decided by the parity of the ratio the arch put it on. The bar's own partials are marked: 1 at 0.0 decibels, 3.00 at -4.7 decibels, 6.00 at -32.2 decibels. — Fig. 2 The same tube under a xylophone bar. Its fundamental is on the first peak and its second partial, at exactly 3, is on the second — 4.7 decibels down, which is within a whisker of full reinforcement. The third at 6.000 is even and falls in a trough at 32.2 decibels down. So a xylophone’s tube passes two of the three partials the arch produced, and it is the same tube, at the same length, that shuts on a marimba’s second.

That is the whole finding, and it is worth stating in the plainest form available before the numbers accumulate. A xylophone’s arch and a xylophone’s tube agree, and a marimba’s arch and a marimba’s tube do not. The arch is the same operation on both instruments, executed with the same file to the same standard; the tube is the same object at the same length; and the only thing that differs is the integer the maker aimed the second partial at. One of the two integers is a resonance of the resonator and the other is a lid.

Nothing in the design process would have surfaced that. A maker tuning a bar is holding a bar, and a maker cutting a tube is holding a tube, and the two operations are separated by the workshop as well as by the theory: the tuning target comes from what a bar sounds like when struck in the hand, and the tube length comes from a formula about a quarter wavelength. The interaction between them only appears when the two calculations are done in the same units, which is what the next figure does.

What each tube does to each instrument's tuned second partial. The gain a resonator gives each bar's second partial, relative to what the same tube gives that bar's fundamental. The stopped tube is 311 millimetres long and the open one 639, and both are tuned to the same 262 hertz. A plain bar, second partial at 2.757: -24.8 decibels through the stopped tube, -29.6 through the open one. A xylophone bar, second partial at 3.000: -4.7 decibels through the stopped tube, -2.3 through the open one. A marimba bar, second partial at 4.000: -38.5 decibels through the stopped tube, -3.9 through the open one. A stopped tube's resonances are the odd multiples, so a ratio of 3 is passed and a ratio of 4 is met by a rigid lid. The open tube that would pass both is twice as long, and no marimba is built with one. — Fig. 3 What each of two tubes does to each of three bars’ second partials, relative to what that tube does to the same bar’s fundamental. The stopped tube passes the xylophone’s ratio of 3 at 4.7 decibels down — nearly full reinforcement, since 3 is one of its own resonances — and takes the marimba’s ratio of 4 down by 38.5. The open tube of the same fundamental, twice as long, passes both. The plain bar’s untuned 2.756 sits between the tube’s peaks and comes out 24.8 decibels down.

The third number in that figure is the one worth pausing on. A plain bar’s second partial, at 2.756, is 24.8 decibels down through the tube. Tuning it to 4 takes it to 38.5. The arch does not merely fail to help the partial through the resonator; it moves that partial thirteen and a half decibels further from anything the tube can do for it. The two operations a maker performs on a marimba bar are working against each other, and the second one wins.

The open tube that nobody builds

An obvious repair suggests itself. An open tube — open at both ends — resonates at every whole multiple of its fundamental rather than only the odd ones, so a partial at 4 falls on a peak instead of into a trough. The last figure carries the number: through an open tube the marimba’s second partial comes out 3.9 decibels down instead of 38.5, and its third at 9.03 comes out 9.5 instead of 13.1.

An open tube tuned to the same note is twice as long. For a middle-C bar the stopped tube is 311 millimetres and the open one is 639. Across the four octaves of a concert marimba, whose lowest bars need tubes already approaching a metre and a half, doubling every one of them is not a design that exists.

What an open tube tuned to the fundamental does to a marimba bar's partials. Power radiated from the mouth of a 639 millimetre open tube of 56 millimetre bore, tuned so that its first resonance is the bar's fundamental at 262 hertz, plotted against multiples of that fundamental and referred to what it does at the fundamental itself. Its quality factor there is 102. An open tube resonates at every whole multiple, so nothing a maker can tune a bar to falls between its peaks. The bar's own partials are marked: 1 at 0.0 decibels, 4.00 at -3.9 decibels, 9.03 at -9.5 decibels. — Fig. 4 The repair, drawn. An open tube of the same fundamental resonates on every whole multiple, so the marimba’s second partial at 4 lands on a peak and comes through 3.9 decibels down instead of 38.5, and the third at 9.03 comes through at 9.5. Nothing about the bar has changed. The only difference between this drawing and the one at the head of this essay is which end of the tube is stopped — and the price of the difference is 639 millimetres of tube against 311.

So the resonator’s parity is a consequence of a length constraint and nothing else. Nobody decided that a marimba’s second partial should be suppressed. Somebody decided that the instrument should fit in a room, which forces a stopped tube, which puts resonances on the odd numbers, which cancels every partial a maker tunes to an even one. That is the same shape of argument as a clarinet’s fingering system, where a choice about which end is closed decides something apparently unrelated four steps downstream.

Whether the result survives a real tube

The calculation above uses a lossless-walled tube with viscothermal damping and an unflanged radiation load, which gives a quality factor of 80 at the first resonance. Published measurements of real marimba resonators put the figure much lower — the tubes have tuning stoppers, sharp edges, and a bar sitting partly across the mouth, none of which is in the model. A result that depended on 80 rather than 25 would be a result about an idealisation.

Whether the tube's parity rule survives a lossier tube. The gain a stopped tube gives three tuned partials, against the tube's own quality factor, swept from 80 down to 17 — which spans the range published measurements of real resonators fall in and the value a lossless calculation gives. A xylophone's second partial at three times the fundamental never falls below -5 decibels; a marimba's at four times never rises above -25. The gap between them stays above eighteen decibels everywhere, so the finding is a consequence of parity rather than of a damping figure nobody has measured on this particular tube. — Fig. 5 The gain three tuned partials receive from a stopped tube, against the tube’s own quality factor, swept from 80 down to 17. Broadening the resonances helps the marimba’s partial at 4 — from 38.5 decibels down to 24.6 — because a broader peak spills further into the trough between two of them. It helps the xylophone’s at 3 as well, from 4.7 down to 1.5 decibels up. The gap between them never closes below eighteen decibels anywhere in the sweep, so the finding belongs to the parity rather than to the damping.

At the value a real resonator probably has — a quality factor near 30 — the numbers are that the xylophone’s tuned partial comes through at 0.2 decibels below the fundamental and the marimba’s at 30 decibels below. Thirty decibels is a factor of a thousand in power. Whatever the arch did to place that partial, it is not in the sound the tube radiates.

The one partial the marimba’s tube does help

There is a result here that nothing above predicted, and it is an argument for something the cutting of the arch left open.

The stopped tube’s resonances are not at exactly 3, 5, 7, 9, because a tube’s end correction is frequency-dependent and its higher modes are slightly stretched. Computed, they fall at 1.00, 3.00, 5.01, 7.03 and 9.06 times the fundamental.

The previous rung computed that a single arch holding the second partial at 4 leaves the third at 9.029. That is 0.03 away from the tube’s ninth resonance at 9.06, and the tube passes it at 13.1 decibels down — better than any other partial in the bar apart from the fundamental itself, and eight decibels better than any nearby ratio.

Now put the published figure of 10 into the same tube. Ten is even. It is an antiresonance, and the tube takes it 25.4 decibels down — twelve decibels worse than 9.03.

Where a marimba bar's partials fall against the whole numbers. A marimba bar's first 4 partials on a logarithmic axis, with the whole numbers drawn as dashed lines so that "near" and "not near" can be read rather than assumed. The ratios are 1, 4.000, 9.029, 14.969, of which the second has been put on a whole number by cutting an arch into the underside of the bar. Every ratio is derived from the shape rather than measured on an instrument. — Fig. 6 The marimba bar’s four partials against the whole numbers, with the nearest interval named under each. Two of them land on integers and two do not: 4.000 by design, 9.029 by arithmetic, 14.969 by nothing at all. The two buttons play the set and a harmonic set on the same fundamental — and the difference between the two, which was the whole distance at the outset, has become small. A partial at 4 and a partial at 9 are both members of a harmonic series; what a listener is missing is 2, 3, 5, 6, 7 and 8, which is a thinner sound rather than a different kind of object.

The ratios 1, 4 and 9 have a further property worth noticing here, since the tube has just voted for them. They are the partials a bar pinned at both ends would have — and they are also members of the harmonic series, which the plain bar’s 2.756 and 5.404 are not. A set of partials at 1, 4 and 9 supports a fundamental: it is harmonics one, four and nine of the note the bar is already sounding, missing six of the intervening members. A set at 1, 4 and 10 supports the same fundamental equally well. The choice between them is not about whether a pitch can be inferred; it is about which one the instrument’s own tube will let out.

So the resonator has an opinion about a question the arch could not settle. If a marimba’s third partial is at 9, the instrument’s own tube reinforces it; if it is at 10, the tube shuts on it exactly as it shuts on the second. That is an entirely independent line of evidence for the value the geometry gave, arriving from acoustics rather than from mechanics, and the two agree. It does not settle the discrepancy — only a measured bar profile can do that — but a coincidence that survives a second, unrelated calculation is worth recording as a finding rather than as a curiosity.

What the tube costs

The last thing to say about the resonator is that its amplification is not free, and the accounting is a conservation law rather than a rule of thumb.

The strike puts a fixed amount of energy into the bar’s fundamental mode. The tube can only radiate energy the bar has, so every joule that leaves as sound is a joule the bar no longer has to ring with. Loudness and duration are two halves of one quantity, and a resonator moves along the trade rather than escaping it.

Except in one direction, and it is the direction that matters. A bare bar radiates almost nothing: most of the energy the strike put in is lost as internal friction inside the wood, becoming heat. A rosewood bar with a quality factor of 200 at 262 hertz radiates about one and a half per cent of its energy as sound and turns the other ninety-eight into warmth. The tube does not only redistribute the energy; it recovers energy that was on its way to becoming heat.

What a resonator costs in ring time for what it buys in level. Level against ring time for a bar with a resonator under it, as the resonator is coupled harder. The strike puts a fixed energy into the bar's fundamental and the tube can only radiate energy the bar has, so every decibel is paid for in seconds — except for the part that comes from energy which would otherwise have become heat inside the bar, and on a bar at these frequencies that is nearly all of it. A rosewood bar of quality factor 200 radiates about 1.6 per cent of its energy bare; a tube that raises the figure to 68 per cent makes the note 21 decibels louder while cutting its ring time to 33 per cent. An aluminium bar has ten times the ring time to spend, which is why a vibraphone can afford a tube and a motor that opens and closes it. — Fig. 7 Level against ring time as the resonator is coupled harder, for three bar materials. A rosewood bar radiates 1.6 per cent of its energy as sound when it is bare and turns the other 98 into heat. A tube that raises the radiated share to 68 per cent makes the note 21 decibels louder while cutting its ring time to a third — which is a bargain only because the energy the tube is taking was on its way to warming the bar rather than on its way out of it. An aluminium bar with ten times the quality factor has ten times the ring time to spend, which is why a vibraphone can afford a tube and a motor that opens and closes it.

There is a corollary about the compass. The tube’s length is set by the bar’s fundamental, so it halves every octave: 311 millimetres at middle C, 622 an octave below, and something approaching a metre and a half at the bottom of a five-octave instrument, which is why the bass resonators of a concert marimba are folded or bent and why the frame under them is the size it is. The bore does not scale the same way — a tube’s radiation resistance goes as the square of its mouth’s size in wavelengths, so a bass tube of a treble tube’s proportions would radiate far too little — and the practical consequence is that the low tubes are relatively fatter as well as longer, which lowers their quality factor and broadens their peaks. Reading that off the sweep above: the bass bars’ tubes reinforce their fundamentals less sharply and leak more of everything else, so the parity effect is weakest exactly where the instrument is loudest and most characteristic.

A second consequence is that the whole arrangement is temperature-dependent in a way the bar is not. The speed of sound rises about 0.6 metres per second per degree, so a tube tuned in a cold hall is flat by roughly three cents by the time the hall is warm — while the bar’s own pitch, set by stiffness and density, barely moves. A resonator and its bar therefore drift apart, and the drift shows up as a loss of level rather than as a change of pitch, because the bar is what the listener hears the pitch of. That is the opposite of what temperature does to a wind instrument, where the air column is the pitch and the drift is audible directly.

The vibraphone’s rotating discs are worth naming as what they are in this accounting. A motor turns a butterfly valve at the mouth of every tube, so the coupling between bar and resonator is switched on and off several times a second. That is a duty cycle imposed on the loudness-and-duration trade: while the valve is open the note is loud and draining, while it is closed the note is quiet and preserved, and the audible result is a tremolo whose depth is the difference between the two states.

A marimba cannot do that, and not because nobody thought of it. Its bars ring for under two seconds and there is nothing to spread out.

What the picture cannot show

The tube is modelled as a cylinder with a plane rigid termination, and real resonators are neither: they have adjustable stoppers, sometimes a taper, and on the bass bars of some instruments a folded path. Each of those moves the higher resonances relative to the odd multiples, and the third partial’s near-coincidence with the ninth resonance at 9.06 is exactly the kind of number those details could destroy.

The coupling between the bar and the tube is not modelled at all. The figures ask what the tube does to a frequency presented at its mouth, which is one step short of asking what a bar and a tube do together. A strongly coupled pair splits into two modes about the tuned frequency — the same arithmetic a wolf note is — and a marimba’s tube is deliberately coupled hard enough that the effect is real. The consequence is that the fundamental’s peak in these drawings is narrower than the instrument’s, and the trough at the second partial is not affected either way.

And no part of this says how much of a marimba’s sound comes out of the tube rather than off the bar. The bar radiates directly as well, poorly and in all directions, and a listener hears the sum. The tube’s rejection of the second partial is a statement about the tube’s contribution, and what survives is whatever the bar itself put into the room — which is small, and is what the next rung’s arithmetic is about.

That last limitation is the one a sceptic should press hardest, so it is worth saying what would decide it. The bar’s own radiation is a quadrupole and falls away steeply as the source gets small relative to a wavelength; the second partial is four times the frequency, which makes the bar four times better at radiating it in exactly the terms that matter. So the direct path favours the very partial the tube rejects, and the honest statement is that the second partial reaches a listener almost entirely off the bar rather than out of the tube. That is a testable prediction with an easy test: the sound of a marimba changes character when the resonators are removed, and the direction of the change should be toward more second partial rather than less. Nothing in this collection has measured it.

Where this goes

Two of a marimba’s three design operations are now accounted for, and they disagree. The arch places the second partial on a ratio of 4; the stopped tube meets a ratio of 4 with a rigid lid. The xylophone, with the same two operations and an odd ratio, gets agreement instead — which is the control this argument needs, since it rules out the tube and rules out the arch and leaves only the parity.

Two operations remain, and they are the player’s rather than the maker’s: where the bar is struck, and where it is held. Both turn out to point the same way as the tube. The top of this ladder puts all four on one axis and asks what is left of the partial by the time the four have finished with it, in the four ways a marimba loses what its arch placed.