Acoustics & Music Theory
why it sounds like that
Series

Struck bar — the series

5 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Second order and fourth order, from the same sequence of wavenumbers. Frequency against wavenumber for a string and for a bar, with each object's own wavenumbers marked on the axis. A string's equation is second order in space, so frequency goes as wavenumber and the straight line carries its modes at 1, 2, 3, 4. A bar's is fourth order, so frequency goes as the SQUARE of wavenumber and the parabola carries the same kind of sequence to 1, 2.757, 5.404, 8.933. Both sequences of wavenumbers are arithmetic — a string's are the even multiples of π/2L and a bar's the odd ones, except that the first is 3.011 rather than 3 — and squaring an arithmetic sequence is the whole of why one object has a fundamental and the other has none. Every number is derived from the boundary condition; none is measured.

    A bar's partials are the odd numbers, squared

    A string's wave equation is second order in space and a bar's is fourth, so a string's frequencies go as its wavenumbers and a bar's go as their squares. Both objects fit a nearly arithmetic sequence of wavenumbers between their ends; squaring one is the whole of why a marimba key has no note in it. The ratios are 1 : 2.756 : 5.404 : 8.933, which are the odd squares over nine, uniformly 12.9 cents flat, and the 12.9 cents is one root that refused to be 3π/2.

    part 1 · instruments
  2. Where one arch can put a bar's second and third partials. The plane of the second and third partials, with the path a single arch traces through it as it is cut deeper, for 4 arch lengths from 50 to 100 per cent of the bar. Every path starts at the plain bar's 2.756 and 5.404 and runs up and to the right. The xylophone's target of 3 and 6 lies on them: reach the second partial and the third is there. The marimba's target of 4 and 10 lies above every one of them — the best any of these arches manages while holding the second partial at 4 is 9.176, which is 149 cents short. A marimba's third partial is not something one cut can place, and that is a statement about the shape of the cut rather than about the maker.

    One cut cannot place two partials

    An arch cut into a bar's underside is one knob, and a maker aims it at two targets. Swept across every arch length, the path it traces through the plane of the second and third partials passes through the xylophone's 3 and 6 — dead through it, within a tenth of a cent at the natural arch length — and under the marimba's 4 and 10 by between 149 and 326 cents. Hitting one target means missing the other, and that is why a marimba bar's underside is not a simple arch.

    part 2 · instruments
  3. 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.

    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.

    part 3 · instruments
  4. 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.

    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.

    part 4 · instruments
  5. Where a bar and its pipe stop being two things. The two normal modes of a bar and a resonator tuned to it, against how strongly they are coupled, at 262 hertz. Below a threshold the pair has one frequency and two different decay rates — the pale curves, which are the damping splitting rather than the pitch — and above it the frequencies separate. The threshold is exact and it is not a matter of degree: it is where the coupling rate equals half the difference between the two damping rates, which for a bar of Q 197 against a tube of Q 80 is a coupling of 0.37 per cent. A marimba's own coupling is 0.62 per cent — 1.67 times the threshold, and not free: it is fixed by how much louder the tube makes the note, since the coupling that splits the pair is the coupling that carries the energy out. So the resonator model's assumption that the tube is a filter downstream of the bar is wrong at middle C, and it is wrong by less than a factor of two.

    A bar and its pipe are one object

    Three earlier essays treat a marimba's resonator as a filter the bar's output passes through, and both of them said in their own caveats that the coupling was not modelled. It is here, and the debt was right: the coupling is 1.67 times the threshold at which the pair acquires two frequencies instead of two decay rates, so the tube is not downstream of anything. Every consequence of that is smaller than the peaks it would have to be seen between.

    part 5 · instruments

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