The second row of resonators keeps the neighbours out of the attack

A bar and its pipe are one object found that a marimba bar and the resonator hanging under it are a single system with two modes rather than a source and a filter, 1.67 times past the point at which that distinction becomes exact. It then turned from the pair to the instrument and named the coupling it had left out. A tube’s mouth sits a couple of centimetres under its own bar and a few centimetres from the next one, and the next one is tuned a semitone away. Every bar is therefore a candidate driver for every nearby tube, and a tube driven by the wrong bar sounds its own pitch into a note that is not its own.

The question that essay put was whether this is large enough to matter, and it attached a practical edge to it. Every concert marimba lays its bars out like a piano keyboard: naturals in a front row, accidentals in a raised row behind, each row with its own line of resonators. If the neighbours’ tubes are doing acoustic work, the second bank would be doing some too, and it ought to be possible to say how much and at which notes.

The answer comes in three parts and each is a number. A bar reaches the next tube in its own row at 0.21 of the pressure it puts on its own mouth, and reaches the nearest tube in the other row at 0.017. In a single chromatic row that is enough for the neighbour’s pitch to be audible as a beat in every attack, for between ten and two hundred and forty milliseconds depending on register — and never for as long as two beats. A second bank halves that time for eight pitch classes of the twelve and cannot touch the other four, because the naturals’ row has two semitones of its own.

A bar is a sheet that pushes both ways at once #

The first number is geometry, and at these distances the bar’s shape is most of it.

A bar in its first mode is not a piston. The first essay on the struck bar gives its mode shape: the centre moves one way, the two ends move the other, and there are two nodal lines a little under a quarter of the length in from each end. So the underside of a bar is a long strip of air-pushers whose sign changes along its length. The top face is the same strip with every sign reversed, since when the underside pushes down the top face pulls away. Seen from a few centimetres below, a vibrating bar is a sheet of dipoles laid out along the mode shape.

What reaches a tube’s mouth is the sum of all of them, each falling off with distance and each arriving with its own phase, averaged over the disc of the mouth. The bar’s own tube sits under its centre, twenty millimetres down, where the underside’s central antinode is closest and dominates. Every other mouth is further from that antinode and relatively closer to the ends, which have the opposite sign, and relatively closer to the top face, which has the opposite sign again. The field does not fall off as one over distance; two sets of cancellations steepen it.

Taking a middle-C bar at 370 by 57 by 20 millimetres, a ten-millimetre gap between bars and a mouth 28 millimetres in radius, the next tube in the row gets 0.211 of the bar’s own-tube pressure, or 13.5 decibels down. The tube after that, 134 millimetres away, gets 0.028 — 31 decibels down. Doubling the distance costs a factor of seven and a half.

The other bank is further away in a direction the bar is long in. An accidental’s resonator sits about a bar-length behind the naturals’ line and half a spacing across, and its bar is raised by about thirty millimetres, so the nearest cross-row mouth is 316 millimetres away and gets 0.017 of the bar’s own. The neighbour in a bar’s own row is reached twelve times harder than the nearest one in the other row.

The dial on the figure is the one dimension a maker can move freely: how far under the bars the mouths sit. It moves the numbers in an unexpected direction. Lowering the tubes from twenty to forty millimetres raises the next tube’s share from 0.21 to 0.36. The bar’s grip on its own mouth weakens with distance faster than its grip on a mouth that was already far away, so the ratio rises. A maker who hangs the resonators low to keep them clear of the mallets is, on this arithmetic, making the neighbours louder relative to the note.

One row of coupled pairs, and what a strike puts into it #

The second number carries the coupling through to sound, reusing the earlier essay’s pair almost unchanged. There, a bar and its tube were two damped oscillators coupled at a rate that was not assumed. It was derived from how much louder the tube makes the note: a bare rosewood bar radiates at a Q of twelve thousand, a tubed one at 320, and the coupling that carries the energy out is the coupling that splits the pair. At middle C it came to 0.62 per cent of the tuned frequency.

A row is that pair repeated, with one addition. Each bar is also coupled to each neighbouring tube, at its own-tube coupling multiplied by the reach above. By reciprocity the neighbouring tube couples back to the bar at the same rate. A strike puts all the energy into one bar; the system’s normal modes are found, the strike is decomposed onto them, and each mode is assigned to the bar-and-tube pair it grew out of. That splits the sound into two parts: the struck note, and whatever is ringing at each neighbour’s pitch.

One detail in that bookkeeping matters more than it looks. Near the exceptional point a pair’s two modes are almost parallel, so a strike puts large and nearly opposite amplitudes into both, and most of each cancels the other. Adding up the energy mode by mode would count the cancelled parts as real and inflate the note several times over. The energy here is computed from the actual time signal of each pair, cross terms included.

What radiates is the tubes. A bare bar’s radiation is so poor that it can be ignored everywhere except the first few milliseconds, before the tube has filled, and it is kept for exactly that reason. Each tube radiates the share of its losses that leave by the mouth rather than into the wall — a little under half, at this size and pitch.

The neighbour is loudest before the note is #

The result for middle C with a semitone neighbour beside it is in the figure. Averaged over the whole note, the energy at C♯ is 33.8 decibels below the C’s, which sounds like nothing. The average is the wrong summary, though, because the C♯ does not arrive with the C. It arrives first.

At the instant of the strike the C’s own tube is empty. The bar has all the energy, and the bar radiates almost none of it. The tube fills over the next several tens of milliseconds, which is why the C curve starts seven decibels under its own peak and climbs. The C♯ tube is driven by the same bar through a weaker coupling, but it is driven off its own resonance, and a resonator driven off resonance answers immediately and at its own frequency before it settles into following the drive. So the C♯'s pitch is there from the first instant. For a few tens of milliseconds the neighbour sits only twenty-odd decibels under a note that has not yet reached full strength.

Twenty decibels down is well inside what an ear can use when the two components are a semitone apart. Two tones at nearby frequencies are heard as one tone that beats at their difference, and the beat a tuner can actually use gives the criterion: a second component at amplitude ratio r modulates the sum by a depth of $m = 2r/(1+r^2)$, and a modulation is detectable when $m$ exceeds about three per cent at slow rates, rising gently above fifty beats a second. At middle C a semitone is 15.6 hertz, the threshold there is 3.1 per cent, and a neighbour becomes audible as a flutter once it is within about 36 decibels of the note.

The C♯ clears that at the strike and stays over it for 90 milliseconds — 1.4 beats at 15.6 a second. Then it falls away far faster than the C does, down to more than fifty decibels under it by two hundred milliseconds. After that it creeps back, at a time when the C itself has fallen far enough that nothing about it counts as the note any more.

The word beat here is being used about something shorter than beats are usually used for. The detection thresholds come from sustained modulations lasting seconds; 1.4 cycles is a flutter in the attack, not a beat a tuner would count. What the model can say is that the modulation depth in those ninety milliseconds exceeds the depth a listener detects when given longer, and that the energy at the wrong pitch is concentrated exactly in the attack. Whether 1.4 cycles of it is heard as a colour of the attack, as roughness or not at all is a question about short stimuli that the criterion was not measured for.

The neighbour’s tube hands it to a bar that cannot speak #

Why the neighbour stops is the most interesting part of the mechanism, and it is the earlier essay’s result showing up in a new place.

The C♯ tube is not a free resonator either. It is half of the C♯ pair, coupled to the C♯ bar exactly as the C tube is coupled to the C bar — 1.61 times past the exceptional point at that pitch. So whatever the C bar puts into the C♯ tube is at once passed across to the C♯ bar, and energy exchanges between the halves of a coupled pair at the rate of the pair’s split.

A lossless exchange at a split of 1.36 hertz would empty the tube half a beat of that split later, at 368 milliseconds. This pair is not lossless, and it is uneven: the tube has a Q of about eighty and the bar about two hundred, so the tube is losing energy both to radiation and to the transfer. Solved for a pair that starts all in its tube, the tube’s amplitude is

$$a(t) \propto e^{-\bar\gamma t}\left[\cos\Omega t - \frac{\Delta\gamma}{2\Omega}\sin\Omega t\right]$$

with $\Omega$ half the split in radians and $\Delta\gamma$ the difference of the two damping rates. It is zero at $\tan\Omega t = 2\Omega/\Delta\gamma$: 211 milliseconds at C♯4, exactly where the full solve finds the tube empty. The agreement is a check that the row has been assembled correctly, since two routes to one number have to meet.

Once the C♯'s pitch is in the C♯ bar it is silent, for the reason that makes a marimba need tubes at all: a bare bar radiates at a Q of twelve thousand. The neighbouring note is excited and carries energy, and it is inaudible because it is sitting in the one part of the instrument that cannot radiate. Its energy then drains through the bar’s internal loss and back through the tube, which is the faint return in the ghost figure after two hundred milliseconds.

This also bears on why the neighbour never manages two beats anywhere on the keyboard. Every rate in the problem — the tube’s loss, the bar’s, the coupling, the handover, and the semitone the neighbour beats at — is a frequency, and if every Q were the same at every pitch, each of them would scale with the note. The whole row would then be the same system at every register, run faster or slower, and the neighbour would last the same number of beats everywhere. The number is a property of the Qs and of the reach, not of the pitch. That is the same kind of ceiling the bar’s own pair met when it tried to beat against itself — 1.24 beats and the note was over — reached from a different direction.

A tone neighbour clears the threshold too #

Whether a second bank does acoustic work depends on what it changes, and what it changes first is which bar sits next to which. With the accidentals moved out, a natural’s neighbours in its own row are a tone away rather than a semitone. That changes two things at once, and they push in opposite directions.

A tone neighbour is driven twice as far from its own resonance, which roughly halves the amplitude that reaches it and costs 5.8 decibels over the note. Against that, a tone beats twice as fast, and the detection threshold rises with beat rate — only slightly at 32 beats a second, 3.6 per cent against 3.1. Both effects shorten the neighbour’s audible time, and together they bring it from 90 milliseconds to 50. That is 1.6 beats, since the beats are now faster. A tone neighbour is still audible in the attack, at every pitch up to the top of the compass.

This is the result that makes the other two cases interpretable. A second bank does not remove a neighbour from the attack of a natural; it swaps a semitone for a tone and roughly halves the time. The large change is somewhere else.

Move the semitone neighbour to the other bank and the reach falls from 0.21 to 0.017. The energy at C♯ drops by 22 decibels to 55.7 below the C, and the modulation never exceeds half the threshold. So the second bank’s real effect is on semitones: it moves every semitone neighbour the accidentals would have had in a single row out of range, and leaves a tone neighbour in range.

Up the keyboard, in beats #

In milliseconds, the bass is where neighbours matter. A semitone neighbour at A2 is audible for 240 milliseconds; at C6, for ten. The level of the neighbour against the note is nearly flat across the compass, 35.1 decibels down at A2 and 30.1 at C6, so the difference is not that the bass neighbour is louder. It is that everything in the bass happens slower. A semitone is 6.5 hertz at A2 and 62 at C6, and the tube’s losses and the coupling slow down in the same proportion, so a bass bar’s neighbour lasts longer in time and not much longer in cycles.

In beats, nothing changes much, and nothing reaches two. The semitone neighbour completes 1.6 beats at A2, 1.4 at middle C, 1.0 at E5 and 0.6 at C6. If every Q were constant the count would be constant, for the reason the previous section gave. It falls toward the treble because the tubes’ Q does: a treble resonator is relatively thinner and lossier, 26 at C6 against 103 at A2, and a lossier tube empties itself, by radiation and by the handover, in fewer of the neighbour’s beats.

The cross-bank semitone is inaudible until the top octave, and at E5 and above it clears the threshold for five milliseconds — a fraction of one cycle, which is as close to nothing as the model can report. The treble tubes are thinner and lossier, which raises the neighbour’s level there by a few decibels, and a few decibels is all the other bank had to spare.

Four notes the second bank cannot help #

The per-note accounting is where a layout argument becomes checkable against what a player hears, and it also contains the result the geometry could not have predicted without counting.

In a hypothetical single chromatic row, every bar has a semitone on each side, and its worst neighbour is audible for between 40 and 95 milliseconds depending on where in the octave it sits.

In two banks the accidentals do best of all. C♯'s neighbours in its own row are D♯ a tone up and A♯ a minor third down across the gap, so its worst is a tone. D♯ has C♯ a tone below and F♯ a minor third above, because there is no E♯, so a third of the accidental row’s neighbours are a minor third away and weaker still. Every semitone neighbour of an accidental is a natural, and every natural is in the other bank. The accidentals lose every semitone from their attack. The naturals D, G and A do almost as well: in the front row their neighbours are a tone away on both sides.

C, E, F and B are unchanged, to the millisecond. The naturals’ row is a diatonic scale, and a diatonic scale has two semitones in it. E and F are adjacent bars in the front row a semitone apart, and so are B and C, and a second bank cannot move them apart. Each of those four notes keeps one semitone neighbour in its own row at full reach, and its attack is exactly what it would be on a single-row instrument.

So “does the second bank do acoustic work” gets a note-by-note answer, not a yes or no. It roughly halves the audible neighbour in the attack of eight notes out of twelve, and does nothing for the other four. The four it cannot help are fixed by the scale, not by the instrument. A maker who wanted to help them would have to space E–F and B–C wider than the other naturals, which would put the front row’s bars at uneven intervals under a player’s mallets. As far as the model shows, no instrument does, and the reason is not acoustic.

What the design was for, and what it happens to do #

None of this means the second bank was put there for its acoustics. The layout mirrors the piano’s, and every argument for it that makers and players give — the mallet player’s reading of the keyboard, the reach, the frame — is independent of anything computed here. The accurate statement is weaker and more interesting. A layout chosen for the hands happens to keep every accidental’s semitones out of every attack. It would have done that whether anyone intended it or not, because it moves half the bars a bar-length away, and the reach falls twelvefold over that distance.

The comparison worth making is with the body as a filter on a bowed instrument, where a violin’s body is a set of resonances that every string drives, tuned to nothing in particular. A marimba’s resonators are each tuned to one bar, and they sit close enough that each also listens to its neighbours. On a violin the sympathetic response is spread over the whole body. On a marimba it is spread over the tubes in a bar’s own row, and the second bank is what confines it to those.

The numbers the argument stands on #

Four quantities in this essay are stated, not derived, and each is a place where the numbers could move.

The bar’s shape and size. The mode shape is a uniform bar’s. A real marimba bar is arched underneath, which one cut cannot place two partials showed concentrates the flexing toward the centre. That makes the central antinode dominate more and narrows the field, so the direction of the error is known: an arched bar reaches its neighbours slightly less than a uniform one. The middle-C dimensions, 370 by 57 by 20 millimetres, are those of a typical concert bar. The model scales the tubes across the compass as the previous essay did but holds the bar’s footprint fixed, which overstates the gap between bars in the bass, where bars are wider, and understates it in the treble.

The gap and the row positions. Ten millimetres between bars, mouths twenty millimetres down, the back row a bar-length behind and thirty millimetres up: typical of the instruments in question, not measured off one of them. The dial on the first figure shows how the in-row reach moves with the mouth depth, and it is the largest sensitivity in the problem. A mouth forty millimetres down reaches its neighbour at 0.36 rather than 0.21, which would lengthen every same-row time here.

The field. The pressure is the free-space field of a sheet of dipoles, with no frame and no neighbouring bars in the way. The bars on either side of the struck one are rigid-ish plates between it and the neighbouring mouth. They would shadow part of the field, again in the direction of reaching the neighbour less.

The criterion. The beat threshold is the three-per-cent figure used across these essays on beating, measured on sustained modulations and applied here to one or two cycles. For stimuli that short it is almost certainly too generous to the listener. That is the most important caveat of the four, and it is the reason the headline is stated in beats as well as in milliseconds.

Where the row stops being a row #

Everything here is the first mode of each bar. The second partial of a marimba bar is placed on four times the fundamental, and the tube shuts on it: its own tube presents a rigid lid there, at the even multiple. A neighbour’s tube a semitone higher meets that partial at 3.78 times its own fundamental, and one a semitone lower at 4.24 — both off the lid and off every resonance, between the third harmonic and the lid in one case and just past the lid in the other. So the second partial drives the neighbours off resonance, as the fundamental does, but with a different mode shape: the second mode has a node at the bar’s centre, so it barely drives its own tube’s mouth and reaches outward more strongly. Whether that is enough to matter is a question of the same size as this one, and it needs the second mode’s field, which the bar’s mode table already has.

The strike is also still a delta. A real mallet delivers its force over three milliseconds or so, and the neighbour’s first instant — the part of this essay’s argument where the neighbour is loudest relative to the note — is inside that contact. A mallet’s pulse is smooth on the scale of a semitone’s beat, so it should change little, but the attack is exactly where the change would land.

Still open: whether a flutter of a beat and a half is heard at all #

The model gives a clean answer to a question that cannot be settled by modelling: whether a listener hears a neighbour’s pitch that modulates an attack by several per cent for a beat and a half.

The existing threshold is for modulation that goes on. What would settle this is a listening test built from the numbers already on the page. Take a synthesised marimba attack, add a second component a semitone or a tone away at the level and with the decay computed here, and ask listeners to tell it from the same attack without the neighbour. The two buttons under the second figure are the stimuli, one at the computed level and one twenty decibels louder. If listeners can separate the computed version, the second bank is doing acoustic work at eight notes out of twelve. If they cannot, the neighbour is real, measurable with a microphone, and below hearing everywhere on the instrument, which is the same kind of verdict the bar’s own pair received.

The same test run with the tone neighbour would decide whether D, G and A are clean on a two-bank instrument or only cleaner. On a real instrument, C, E, F and B are where a player would have to listen for it.