A bar and its pipe are one object

The tube shuts on the partial the arch placed states its own central limitation in one sentence and then names what would fix it:

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.

Every claim in that is correct, which is unusual for a debt and is worth saying before anything else. The pair does split, the arithmetic is a wolf note’s, and the coupling is strong enough — by a factor of 1.67, which is not a comfortable margin. What the paragraph could not say, because nothing on this ladder had run the comparison, is how large the splitting is. It is smaller than the width of the peaks it would have to be seen between, at every note the instrument has.

The boundary is a place rather than a matter of degree #

“Coupled hard enough” sounds like a phrase without a definition and it has one, which is the first thing this rung is for.

Two oscillators with damping rates $\gamma_b$ and $\gamma_t$, coupled at rate $\kappa$, have normal modes at the eigenvalues of a two-by-two matrix whose diagonal is $\mathrm{i}\omega - \gamma$ for each and whose off-diagonal is $\mathrm{i}\kappa$. With both tuned to the same frequency those come out as

$$\lambda_{\pm} = \frac{\lambda_b + \lambda_t}{2} \pm \sqrt{\left(\frac{\gamma_t - \gamma_b}{2}\right)^{2} - \kappa^{2}}$$

and the whole question is the sign under the root. If $\kappa$ is smaller than half the difference of the two damping rates, the root is real: the two modes sit at the same frequency and decay at different rates, which is what “a filter downstream” means. If $\kappa$ is larger, the root is imaginary and the two frequencies separate.

That crossing has a name — an exceptional point — and it is a place rather than a tendency. There is no gradual emergence of a split; below the threshold there is exactly one frequency and above it there are exactly two. So the question the previous rung could not decide is a comparison of two numbers, and the numbers are already on this ladder.

The coupling is not a free parameter #

The obstacle to running that comparison is that $\kappa$ looks like something that would have to be assumed, and an assumed coupling would make the whole rung an argument with itself.

It does not have to be assumed, because the same coupling that splits the pair is the coupling that carries energy into the tube for the tube to radiate — and the previous rung priced that as the whole of its loudness argument. A bare rosewood bar radiates about one and a half per cent of its energy and warms itself with the rest; a tube raises the radiated share to two-thirds and makes the note some twenty decibels louder while cutting its ring time.

In the weak-coupling limit the extra damping a lossy partner adds is $\kappa^2/\gamma_t$, so the Q the tube subtracts from the bar and the coupling that produced it are the same statement twice:

$$\frac{1}{Q_{\text{added}}} = \alpha^{2} Q_{\text{tube}}, \qquad \alpha = \frac{2\kappa}{\omega} = \frac{1}{\sqrt{Q_{\text{added}}\,Q_{\text{tube}}}}$$

At middle C, a bar whose radiation Q goes from 12,000 bare to 320 tubed has an added Q of 329, and the tube’s own Q — computed from its wall losses and its radiation with the bar taken away entirely — is 80.3. That gives a coupling of 0.62 per cent of the tuned frequency.

The threshold is $\alpha_{\text{crit}} = (1/Q_{\text{tube}} - 1/Q_{\text{bar}})/2$, which for a tube of Q 80 against a bar of Q 197 is 0.37 per cent.

So the answer is yes, and it is 1.67 times over. A marimba’s bar and its resonator are one system with two modes, and the ladder’s three previous rungs have been modelling them as two systems in series. Everything computed on this anchor about what the tube does to which partial is the weak-coupling limit of something the instrument is not quite in.

And every consequence of that is under the floor #

Which would be the end of the rung, except that the size of the split is also computable and it is the number that decides whether any of it matters.

At a coupling of 1.67 times the threshold the two modes come out 1.29 hertz apart, both with quality factor 114. A peak of Q 114 at 262 hertz is 2.29 hertz wide at half power.

The split is 0.56 of a linewidth. Two peaks that close are one peak, in any spectrum anybody could measure and in any sense a listener could use. Nothing about the instrument would look like two modes; it would look like one mode of slightly the wrong width, which is precisely what the previous rung’s own caveat predicted — the fundamental’s peak in these drawings is narrower than the instrument’s — and which is therefore a debt that guessed its own answer correctly and could not have known by how much.

The other way to hear a pair is as a beat, and that fails on a different quantity. A split of 1.29 hertz beats 1.29 times a second; a marimba’s note at Q 114 falls sixty decibels in 0.96 seconds.

The pair beats 1.24 times and the note is over. A beat that completes once is not a warble; it is a slight swell in the middle of a decay, indistinguishable from the ordinary two-stage decay any struck object has. And it gets worse toward the treble, where the notes are shorter: at C5 the pair manages 0.43 of a beat.

That is the shape of the result. The debt’s premise is right in every part and its audible consequence is nil, and the two halves have to be reported separately because they point in opposite directions: the ladder’s model of the resonator is wrong, and its error changes no number in this collection by an amount anyone could measure. A debt worth paying does not have to turn out to be a correction, and this one is the other useful kind — a bound where there had been an unknown.

What the coupling does that is visible #

There is one thing it does that a filter model gets wrong by more than a linewidth, and it appears when the tube is not in tune with its bar.

Sweep the tube’s tuning through the bar’s frequency and a filter model predicts a straightforward crossing: the bar’s pitch stays put, the tube’s peak slides through it, and the level rises and falls as the two coincide. A coupled pair does something else. The two branches repel — far from the bar each is one object and near it neither is, and the closest they come is the 1.29 hertz above.

The Qs swap along with the frequencies. Two hundred cents flat, the upper branch has the bar’s own Q of 197 and the lower has 71, which is the tube’s 80 shifted by its own detuning; at exact tuning both branches have 114, which is exactly halfway in the only sense that counts, because each mode carries half of each object’s loss. That is what being one object rather than two means, stated as an arithmetic identity rather than as a metaphor.

This is the same picture a wolf note makes on a cello, and it is worth putting the two side by side, because the mechanism is identical and the outcome is opposite.

A cello’s coupling is 2.8 per cent against this one’s 0.62 — a factor of four and a half — and its body resonance is broader and its string far narrower than either object here. The split comes out at thirteen hertz, on a note that can be held indefinitely, which is thirteen beats a second for as long as the bow moves. A wolf is unplayable and a marimba’s split is inaudible, and the whole of the difference is the ratio of the split to the linewidth and the length of the note. Neither is a difference of mechanism, and a rung that reported only “the pair splits” would have said the same thing about both.

That comparison is also the check on the reasoning. The body is the filter treats a violin body exactly as the third rung of this ladder treats a resonator tube — as something a string’s output passes through — and it is the right model everywhere except at the wolf. Two ladders in this collection have made the same modelling choice for the same good reason and both have exactly one place where it fails; the difference is that on a violin the failure is famous and on a marimba it is 0.56 of a linewidth.

And it says something about the temperature drift the previous rung raised. A tube tuned in a cold hall goes flat by about three cents as the hall warms, while the bar’s own pitch barely moves. Three cents is 0.45 hertz here, which is a third of the split — so a warm hall moves the pair along the flattest part of the avoided crossing, where the two branches are still repelling and neither has yet recovered its own identity. The loss of level the previous rung predicted is real; what a filter model would also predict, a shift of the peak’s frequency, does not happen, because the upper branch is pinned near the bar for the whole of that excursion.

The boundary is inside the instrument #

The last thing the arithmetic says is the one a maker might act on, and it comes from a detail the previous rung recorded and did not use.

A concert marimba’s resonators are not scaled copies of each other. A tube’s radiation resistance goes as the square of its mouth in wavelengths, so a bass tube of a treble tube’s proportions would radiate far too little, and the low resonators are relatively fatter as well as longer. A fatter tube has a lower Q — 103 at A2 against 26 at C6 — and the tube’s Q is on both sides of the threshold comparison.

Run the comparison at every pitch and the ratio falls from 2.34 at A2 to 0.65 at C6, crossing one between C5 and E5.

So the answer to whether a marimba’s bar and its pipe are one system is not yes or no. The bottom four octaves are a coupled pair and the top few notes are a bar with a filter under it, and the instrument crosses its own exceptional point somewhere in the middle of its highest octave. Nothing changes audibly there either — the split at C5 is 0.20 of a linewidth on its way to nothing — but it is the place where the previous rung’s model becomes exactly right after being slightly wrong everywhere below it.

Which computation produced the numbers #

The tube is the third rung’s, unchanged: a stopped cylinder 28 millimetres in radius, its length solved so that its first resonance sits on the bar’s fundamental, with visco-thermal wall losses and a radiation load at the mouth. Its Q is the half-power width of that resonance, measured on the same curve.

The bar’s own numbers are that rung’s too — an internal Q of 200 for rosewood and a bare radiation Q of 12,000, which is what makes a bare bar an extremely poor radiator and a tube worth having.

The coupling is derived rather than chosen, from the identity above, and that is the whole reason this rung is arithmetic rather than a measurement: it is a consequence of how much louder the tube makes the note, which the previous rung already priced. The eigenvalue solve is the two-by-two matrix in full complex arithmetic, so the detuned case comes out with both a frequency and a Q for each branch rather than only a frequency.

The compass sweep scales the tube radius as $f^{-0.35}$, which reproduces the practice the previous rung described in words: a bass resonator relatively fatter than a treble one. That exponent is chosen to match the description rather than measured off an instrument, and it is the one number here that is neither derived nor quoted.

Where the model stops #

Two oscillators is one mode of the bar and one of the tube. A real marimba bar has four partials the arch and the tube place and a stopped tube has a whole comb of resonances; the pair computed here is the fundamental against the tube’s first. That is the right pair to compute, because it is the only place the two are tuned to each other, and it is not the whole system. The bar’s own ladder is fourth-order in space and its partials are nowhere near the tube’s, which is what makes the fundamental the only coupled pair on the instrument — and is also what makes the parity argument of the third rung a statement about everything except the note.

The exponent on the tube radius is the weakest number in the essay. The compass crossing between C5 and E5 moves if the bass tubes are relatively fatter or thinner than $f^{-0.35}$ makes them, and a set of measured resonator diameters from one instrument would replace it in an afternoon. What does not move is that a crossing exists somewhere in the compass, because the tube’s Q falls with pitch faster than the bar’s does and the threshold depends on the difference.

And the weak-coupling identity is used at a coupling that is not weak. Deriving $\alpha$ from the added damping is a first-order result, and the pair here is 1.67 times past the exceptional point, which is where first order starts to fray. The direction of the error is knowable: the true coupling for a given loudness gain is slightly larger than the identity gives, so the pair is at least as far past the threshold as this says. The split would grow with it, and it would have to grow by a factor of 1.8 before it reached a linewidth.

Where this ladder goes next #

Five rungs. A bar’s partials are the odd numbers squared; one cut cannot place two of them; the tube shuts on whichever one the arch placed; four separate mechanisms take away what the arch placed; and now the bar and the tube turn out to be one object rather than two, by a margin of two-thirds, with no audible consequence whatever.

There is one more thing this rung says about the field it sits in, and it is a caution rather than a result. Three of the four instruments this collection has followed for making an inharmonic object carry a pitch — the tuned bar, the bell, and the drum — have a resonator or a body somewhere in them, and every one of the three is modelled here as a filter. The marimba is the only one whose coupling has now been priced, and it came out on the wrong side of the boundary by a margin small enough that neither answer would have been guessable. Nothing licenses assuming the other two are on either side.

Two debts stand and one of them is unchanged. The cheapest is still the fourth rung’s: three of its arguments rest on a thickness law that was stated rather than traced, and the underside of one marimba bar measured as a thickness against position would settle the one place where this collection’s arithmetic and the published practice disagree. That is a set of calipers and an afternoon, and no amount of computation replaces it.

The new one comes out of this rung. Everything above is the tuned pair, and a marimba is a chromatic instrument whose neighbouring bars sit forty millimetres apart over a shared frame. The coupling computed here is between a bar and the tube directly beneath it, and the same tube’s mouth is a few centimetres from the next bar’s — which is a second coupling, between a bar and a resonator tuned a semitone away, at a detuning of a hundred cents where the avoided crossing above says the branches have separated and each object is nearly itself again. Whether that cross-coupling is large enough to matter is the same eigenvalue solve with a three-by-three matrix and one new geometric quantity: how much of a neighbouring tube’s mouth a bar actually sees. It is arithmetic, it is the natural next question, and it would decide whether the practice of laying resonators out in two banks — which every concert instrument does, and which is usually explained as a matter of where the frame can go — is doing acoustic work as well.