Instruments and their design

The four ways a marimba loses what its arch placed

The centre of a bar is a node of every even mode by symmetry, so a stroke there gives the second partial exactly nothing and a mallet's width cannot recover it; five millimetres off centre returns 14 per cent. Add the yarn mallet's 3-millisecond contact, the tube's rigid lid and a decay four times faster than the fundamental's, and four independent mechanisms take the tuned partial 80 decibels below it. On a xylophone the same four come to 24.

17 min read 7 figures Where the ends are The design was forced

Assumes: The tube shuts on the partial the arch placed

Two of a marimba’s design decisions are now on record and they disagree. The arch puts the second partial exactly two octaves above the fundamental, at a cost of an octave and a fifth of pitch; the stopped tube under the bar meets a ratio of four with a rigid lid and takes it thirty decibels down.

Two decisions remain and they belong to the player rather than the maker: where the bar is struck, and where it is held. This collection has a nine-rung account of what a strike point does to a string, and the natural thing to do is carry it across. Three of its four terms carry, one of them does not, and the term that carries hardest points the same way as the tube.

What each strike point on a marimba bar excites. The amplitude each of the first 4 partials receives from a strike at each point along the bar, each partial drawn to its own maximum. On a string this comb is |sin nπx| and is periodic, so one strike point silences a whole family; on a bar it is the mode shape itself, whose zeroes are not at rational fractions, so no strike point silences a family. The exception is the middle: every antisymmetric mode is zero there by symmetry, so a centre strike drives partial 1 at 100 per cent of its maximum and partial 2 at nothing at all. The cord's hole at 0.2031 is the mirror case — the fundamental is the partial a strike there cannot reach. — Fig. 1 How hard a strike at each point along a marimba bar drives each of its first four partials, each drawn to its own maximum. On a string this comb is the sine of $n\pi x$ and is periodic, so a strike at one eighth silences partials 8, 16 and 24. On a bar it is the mode shape itself, whose zeroes are not at rational fractions, and no strike point silences a family. The exception is the middle, where the even-numbered modes are zero by symmetry: at the playing spot the fundamental is at its own maximum and the second partial is at nothing at all.

A comb that is not a comb

The excitation-point ladder’s first rung computes what a strike at a fraction $x$ of a string’s length does: partial $n$ is scaled by the absolute value of $\sin(n\pi x)$ , so a hammer at an eighth removes the eighth, sixteenth and twenty-fourth partials and thins the ones near them. The comb is periodic, because a string’s mode shapes are sines and the $n$ -th sine has zeroes at every $k/n$ .

A bar’s mode shapes are not sines. They are combinations of trigonometric and hyperbolic functions and their zeroes sit at 0.2242, 0.1321, 0.0944 and so on — a set with no arithmetic relationship. The comb is therefore aperiodic: no strike point kills a family of partials, because the families do not line up. A player who wants to remove one partial can remove that one and no others, which on a string is impossible.

This is a real difference and it is not the one that matters. What matters is the symmetry.

The centre is a node of every even mode

A bar free at both ends is symmetric about its middle, so its modes alternate: odd-numbered modes symmetric about the centre, even-numbered modes antisymmetric. An antisymmetric mode has zero displacement at the centre, exactly, by symmetry rather than by arithmetic.

The first 4 mode shapes of a marimba bar. Displacement along the bar for its first 4 flexural modes, each drawn to its own maximum, with the nodes marked. Mode n has n+1 of them and they are not at rational fractions of the length: the fundamental's sit at 0.2031 and 0.7969, which is where a cord is threaded through the bar. The odd-numbered modes are symmetric about the centre and the even-numbered ones are antisymmetric, so every even mode has a node exactly at the middle — by symmetry rather than by arithmetic. Partial ratios 1 : 4.000 : 9.029 : 14.969. — Fig. 2 The first four mode shapes of a marimba bar, each to its own maximum, with the centre marked. Modes 1 and 3 have antinodes there and modes 2 and 4 pass through zero. That is not an arithmetic coincidence to be checked at three decimal places: the bar is symmetric about its middle and the arch cut into it is symmetric too, so every even mode must be odd about the centre and every odd function must vanish where it changes sign. The zero survives any depth of arch, any material and any length.
The nodes of the fundamental have moved, though, and by more than they look. On a plain bar they sit at 0.2242 and 0.7758; on this one at 0.2031 and 0.7969, because the arch has softened the middle and the mode has redistributed itself accordingly. That is the number the cord is threaded through, and the last two figures of this essay are about it.

The standard marimba stroke lands in the middle of the bar. It is the spot every method book names and the spot that gives the fullest tone; striking near the ends gives the dry, thin sound used deliberately as a colour. And the middle of the bar is the one point at which the partial the arch was cut to place receives nothing at all.

The symmetry has a consequence that is easy to get wrong. A mallet head is not a point — a marimba mallet is thirty millimetres or so across on a four-hundred-millimetre bar, which is seven and a half per cent of the length, far wider in proportion than a piano hammer’s two per cent. It might seem that so wide a contact would straddle the null and pick up some of the mode either side of it. It does not, and the reason is exact: a symmetric contact patch centred on the middle integrates an odd function, and an odd function integrated symmetrically about its zero gives nothing. Widening the mallet does not recover the partial; only moving it does.

How near the centre a stroke has to be to lose the second partial. The second partial's amplitude against how far the mallet lands from the middle of the bar, in millimetres on a 400 millimetre bar, for three bars with the same arch length and three different arch depths. Exactly at the centre the amplitude is zero for all three, because the mode is antisymmetric and a symmetric contact of any width integrates it to nothing — so a wider mallet does not recover it. Five millimetres off, a marimba bar's second partial is back at 14 per cent of its maximum and twenty millimetres off at 53 per cent. The deeper the arch, the steeper the recovery: the partial a marimba is cut to place is also the partial its player is least able to suppress. — Fig. 3 The second partial’s return against how far the mallet lands from the middle, in millimetres on a 400 millimetre bar. Exactly at the centre it is zero for all three bars. One millimetre off, a marimba bar’s is back at 2.8 per cent of its maximum; five millimetres off, 14 per cent; twenty millimetres off, 53. The deeper the arch, the steeper the recovery — the marimba’s curve is roughly twice as steep as a plain bar’s — so the partial a marimba is cut to place is also the one its player is least able to suppress.

That figure is the honest version of the claim. A player does not hit the exact centre and could not: a five-millimetre error is a good stroke and returns the partial at fourteen per cent of full amplitude, which is seventeen decibels down. So the second partial is not absent from a marimba’s sound. It is attenuated by a mechanism the player cannot switch off and cannot fully engage either, and its level varies stroke to stroke with an accuracy nobody controls to the millimetre.

Which term of the excitation ladder binds

The excitation ladder’s merger expresses each of four terms as a corner partial — the number above which that term is the one taking the energy away — and reads off which is lowest at each pitch. Doing the same for a bar rearranges the answer completely.

The comb survives, and it is a symmetry rather than a corner. It does not remove a family and it does not have a corner partial at all; it removes one partial almost entirely and leaves the rest.

The width does nothing extra. On a piano string the contact’s extent multiplies the comb by a sinc factor whose first null is at partial one hundred, which never binds. On a bar it cannot help even where it might: the null it would fill is a symmetry, and a symmetric contact of any width leaves it a null.

The contact time binds everywhere, and it binds hard. A soft yarn marimba mallet is in contact for about three milliseconds, and the spectrum of a half-sine force pulse of that duration has its first null at 500 hertz.

What each mallet leaves of the partials the arch placed. The spectrum of the force pulse each mallet delivers, normalised to what it gives the fundamental at 262 hertz. A contact lasting τ seconds low-passes the strike with its first null at 1.5/τ, so a soft yarn mallet at 3.0 milliseconds has its corner at 500 hertz and a hard plastic one at 0.4 milliseconds has its corner at 3750. The vertical lines are the two instruments' tuned second partials. The marimba's, two octaves up at 1047 hertz, is -27 decibels down under yarn and -1 under plastic — and yarn is what a marimba is played with. — Fig. 4 The force pulse each of four mallets delivers, referred to what it gives a 262-hertz fundamental. A soft yarn mallet’s corner is at 500 hertz; a hard plastic one’s is at 3,750. The marimba’s tuned second partial is at 1,047 hertz, well above the soft mallet’s corner and well below the hard one’s, so it comes out 27 decibels down under yarn and 1.4 down under plastic. The three buttons play a marimba bar under each mallet and a xylophone bar under the hard one.

And the dispersion term has no analogue. The corner does not come back a corner computes how much the phase error accumulated by a stiff string’s partials degrades the re-excitation of a corner returning to a hammer still in contact. A bar has enormous dispersion — its fifth partial travels 3.65 times as fast as its fundamental — and nothing returns as anything. What the strike does is set each mode’s amplitude once, from the mode shape at the contact and the force pulse’s spectrum, and after that the modes go their separate ways. The term is not small on a bar; it does not exist, because the mechanism it describes requires a wave to come back.

That is a genuinely different verdict from the piano’s. There, three of four terms had corners and the crossing between two of them fell at E3. Here one term is a symmetry, one is inert, one is missing, and the contact time decides everything else.

The mallet is the tone control, and it is aimed at the tuned partial

The numbers in the last figure are worth restating because they say something about how percussion is played that the instruments’ own literature says a different way.

At middle C, a marimba’s tuned second partial sits at 1,047 hertz. A soft yarn mallet takes it 27 decibels down; a medium yarn takes it 22 down; a hard rubber, 10; a hard plastic, 1.4. The mallet’s low pass sweeps straight across the partial the arch placed, and the whole range of mallets a player owns is a range of settings for how much of that partial reaches the room.

That is exactly what players describe, in different words. Soft mallets give the round, fundamental-dominated sound a marimba is bought for; hard mallets give a bright, xylophone-like sound that most marimba writing avoids. What this arithmetic adds is that the axis is not vague — it is one partial, at a known frequency, moving through twenty-six decibels.

There is a second thing the mallet does that this arithmetic makes visible and the vocabulary hides. A harder mallet is louder as well as brighter, and the two are usually described as separate consequences of hitting harder with a harder stick. They are not separate: the pulse’s total impulse is what it is, and shortening the contact redistributes the same impulse over a wider band rather than adding to it. What a hard mallet gains at the second partial it takes from the fundamental, by a small amount — 0.1 decibels against 5.5 for the soft one at 262 hertz, which is the difference between a fundamental delivered whole and one already rolling off. The mallet is a tone control and not a volume control, and the loudness a player associates with hard sticks comes from the attack transient rather than from the tuned partials.

And it explains the pairing of instrument and mallet, which otherwise looks like taste. A marimba is played with yarn and a xylophone with plastic, and the two instruments’ tuned partials are at 4 and 3 times their fundamentals with the xylophone’s bars pitched higher besides. A yarn mallet on a xylophone would remove the twelfth that gives the instrument its character; a plastic mallet on a marimba would let through the two-octave partial and the tube’s lid would be the only thing left standing between it and a listener. The mallets are chosen to agree with the tuning, and on the marimba they agree with it by suppressing it.

Where the cord goes, and the number that has moved

The remaining decision is the suspension. A marimba bar hangs on a cord threaded through holes drilled crosswise near each end, and the holes are at the nodes of the fundamental so that the cord does not damp the note.

The textbook figure is 0.2242 of the length, which is where a plain bar’s fundamental has its node. The arch moves it.

Where the cord's hole should be, as the arch deepens. The position of the fundamental's node against the depth of the arch. A plain bar's is at 0.2242 of the length, which is the number every account of a marimba quotes; a bar arched deeply enough to put its second partial two octaves up has moved it to 0.2031, a shift of 2.1 per cent of the length or 8.4 millimetres on a bar 400 millimetres long. A hole drilled at the quoted figure leaves the fundamental with 10 per cent of its amplitude at the cord, which is the one partial the suspension exists to protect. The node is a property of the bar that was cut, not of bars. — Fig. 5 Where the fundamental’s node sits as the arch deepens. A plain bar’s is at 0.2242; a bar arched to put its second partial two octaves up has moved it to 0.2031 — a shift of 2.1 per cent of the length, or 8.4 millimetres on a 400 millimetre bar. A hole drilled at the quoted figure leaves the fundamental with nearly ten per cent of its amplitude at the cord, on the one partial the suspension exists to protect.

Eight millimetres is not a subtlety on a bar four hundred long, and it is a systematic error rather than a scatter: the node always moves inward, and it moves further the deeper the arch, so the bass bars of an instrument are worst affected. Real makers drill by ear rather than by the quoted number, which is the right procedure and is also why the quoted number has survived so long unexamined. The node is a property of the bar that was cut, not of bars.

The suspension damps everything the arch tuned

The more interesting half of the suspension is what it does to the partials that are not the fundamental. A cord at the fundamental’s node is transparent to the fundamental by construction. It is placed with no regard whatever to the other modes, and the other modes have no reason to have a node there.

What the cord takes from each of a marimba bar's partials. The energy density each partial has at the suspension point, as a fraction of its own maximum, for a marimba bar whose cord passes through the fundamental's node at 0.2031. The fundamental has nothing there, by construction — that is what the position is chosen for. Every other partial has between 18 and 29 per cent of its own, so a suspension chosen to spare the fundamental damps everything else the bar does. The lighter bars are the same quantity at the plain bar's quoted node of 0.2242, where the fundamental is no longer spared either. — Fig. 6 The energy density each partial has at the cord, as a fraction of its own maximum, for a marimba bar whose cord passes through its fundamental’s node at 0.2031. The fundamental has nothing there. The second partial has 22 per cent, the third 18, the fourth 29. The pale bars are the same quantity at the plain bar’s quoted node of 0.2242, where the fundamental is no longer spared either.

There is no position that would spare more than one of them. A cord has to pass through two points, and a bar has two nodes for its fundamental, three for its second partial and four for its third — sets that share no member, because the mode shapes’ zeroes have no arithmetic relationship. Sparing two partials at once would require a pair of positions belonging to both sets, and no such pair exists. So the suspension is not a compromise a better maker could improve on; it is a choice between partials, and the fundamental is the only defensible choice.

So a suspension chosen to spare the fundamental damps everything else the bar does, by roughly a fifth to a third of what a maximally bad position would. That is a slow loss rather than an instantaneous one — it shortens the upper partials’ decay rather than removing them from the strike — and it compounds with something the material was doing anyway.

Internal friction in wood is roughly frequency-independent as a loss factor, which means the time a partial takes to decay is inversely proportional to its frequency. A rosewood bar with a quality factor near 200 rings for about 1.7 seconds at 262 hertz and about 0.42 seconds at 1,047 — the second partial decays four times as fast as the fundamental purely by being four times as high. A listener asked to name a marimba’s pitch is listening mostly to the second half of the note, by which time the tuned partial has gone.

Four mechanisms, one direction

Everything above can be put on one axis, and the picture is the argument this ladder has been building toward.

Everything that happens to the second partial after it is placed. The four things an instrument does to the partial its arch was cut to position, stacked in decibels, for two bars played as their instruments are played. On the marimba the strike lands near the centre where the mode is antisymmetric, the yarn mallet's 3.0 millisecond contact low-passes the strike below the partial, the stopped tube meets it with a rigid lid, and being four times the fundamental it decays four times as fast and so delivers a quarter of the energy: -80 decibels in all. On the xylophone the same four come to -24, because the partial is on an odd ratio the tube passes and the mallet is hard enough to deliver it. Each term is the second partial measured against the fundamental at the same stage, so the four add, and the total is the energy that partial contributes to the note relative to the fundamental's. The arch is the same operation on both instruments and only one of them keeps what it places. — Fig. 7 The four things an instrument does to the partial its arch was cut to position, stacked in decibels, for two bars played as their instruments are played. On the marimba: a stroke five millimetres from a centre where the mode is antisymmetric, a yarn mallet whose corner is below the partial, a stopped tube that meets an even ratio with a lid, and a decay four times faster than the fundamental’s — which delivers a quarter of the energy, six decibels, since a duration is not an amplitude. The four come to 80 decibels below the fundamental. On the xylophone they come to 24, because the partial sits on an odd ratio the tube passes and the mallet is hard enough to deliver it.

Every term in that stack is the second partial measured against the fundamental at the same stage, which is what makes them addable: how much less of it the strike puts in, how much less of it the mallet delivers, how much less of it the tube radiates, and how much less of its energy reaches a listener before it has gone. Fifty-six decibels separate the two instruments’ totals.

Four mechanisms, in four different parts of the instrument, decided by four different people at four different times — a maker cutting an arch, a maker cutting a tube, a manufacturer winding yarn round a core, a player learning where to hit. On a marimba all four take the second partial down. On a xylophone none of them does much.

That is not a coincidence and it is not a design failure. It is the strongest available evidence for the reading the second rung proposed: the marimba’s arch does not place a partial somewhere useful, it removes one from somewhere harmful. A plain bar’s second partial is an octave and a tritone above the fundamental, which is the worst interval available for roughness. Moving it to exactly two octaves makes it harmless. The instrument then discards it four separate ways, which is what would be built if the partial were a nuisance and not what would be built if it were an asset.

The xylophone is the control that makes the argument, rather than a second example of it. Same object, same equation, same kind of cut, same kind of tube — and a second partial tuned to an odd ratio instead of an even one. Every downstream decision comes out the other way: the tube passes it, the mallet delivers it, and the instrument sounds like an instrument that wants its second partial. If the four mechanisms on the marimba were accidents of percussion in general, the xylophone would share them. It shares none.

What this account does not settle

The four terms are added as though they were independent, and they are not quite. A harder mallet is also used with a different stroke; the tube’s rejection and the bar’s direct radiation are two paths to one listener rather than two filters in series. The stack is an accounting of magnitudes, not a synthesis of the note, and the total in decibels should be read as an order of magnitude rather than as a level.

The decay term is the weakest of the four. It compares two ring times as though a single number described each partial’s loss, and a real bar’s damping has contributions from the wood, from the cord, from the air and from the rail the cord runs over, none of which scales the same way with frequency. The direction is secure and the size is not.

And nobody has listened. Every number here is computed, and the claim they add up to — that a marimba’s tuned second partial is nearly inaudible in normal playing — is a claim about what reaches an ear. The experiment is straightforward: record a marimba bar struck at the centre and struck twenty millimetres off, with and without its resonator, under yarn and under plastic, and read the second partial’s level off each. That is four numbers this collection asserts and has not measured.

Where this ladder goes next

A bar is not a string because its equation is fourth order in space, so its frequencies are the squares of an evenly spaced ladder of wavenumbers rather than the ladder itself: 1, 2.756, 5.404, 8.933, which are the odd numbers squared over nine and are uniformly 12.9 cents from them. One cut cannot place two partials, because an arch is a single degree of freedom traced through a two-dimensional plane, and the path it traces passes through the xylophone’s targets of 3 and 6 and under the marimba’s of 4 and 10 by at least 149 cents. The tube shuts on what the arch placed, because a stopped tube resonates on the odd multiples and presents a rigid lid on the even ones, and the two instruments tuned to opposite parities. And the strike and the suspension agree with the tube, because the centre of a bar is a node of every even mode and a cord at the fundamental’s node is at nobody else’s.

What the ladder owes next is a bar it did not compute. Three of its four rungs rest on a thickness law that was stated rather than traced, and the one place where the computation and the published practice disagree — the third partial at 9.03 against a quoted 10 — is exactly where a traced profile would decide the matter. The debt is one measurement: the underside of one marimba bar, as a thickness against position, from which everything here follows without a further assumption. That is not a corpus and not a listener; it is a set of calipers and an afternoon, and it is the cheapest outstanding debt in this field.

Two arguments are also available without it. The first is the coupled pair: this ladder has treated the tube as a filter downstream of the bar, and a marimba’s coupling is strong enough that the two are one system with two modes — the arithmetic a wolf note already uses, applied to a bar and a pipe rather than a string and a body, and computable today. The second is the tabla, the mridangam and the tuned gong, which solve the same problem — an inharmonic object made to carry a pitch — by loading rather than by cutting. The drum ladder reached the same convergence from the membrane’s side and closed there; a rung that put the bar’s cut beside the drum’s paste and the bell’s profile would be about the listener’s inference rather than about any of the three objects, and that is the argument this field is short of.