Generator

The arch cut into a marimba bar, and where the bar bends

A marimba bar in section, with the underside cut away by a thickness law rather than traced from an instrument: at its deepest the arch removes 58 per cent of the thickness and it reaches 40 per cent of the length either side of the middle. Over it are the bending moments of the first two modes, each to its own maximum. Bending stiffness goes as the cube of thickness and mass as its first power, so the cut lowers a mode in proportion to how hard that mode bends where the material has gone — which is why the fundamental, which bends hardest at the centre, falls furthest. The partials come out at 1 : 4.000 : 9.029 : 14.969.

Drawn above with its standard settings, which is almost never how an essay draws it: an essay states the numbers it is arguing about, so the figure a reader meets there is about that argument rather than about the drawing in general. Every option a placement passes is checked against the ones this function actually reads, because an option it does not read is silently ignored and the figure quietly draws what is above instead.

It makes a noise. 37 of its 37 placements carry sound, built from the same numbers as the drawing, offering these buttons: 6 openings a second, A marimba bar on 262 hertz: its partials at 4.00, 9.03, 14.97 times the first, A plain bar on 262 hertz: its partials at 2.76, 5.40, 8.93 times the first, C4 struck, the C♯4 at its computed level for its first 90 ms, C4 struck, the C♯4 at its computed level for its first few ms, C4 struck, the D4 at its computed level for its first 50 ms, C4 struck, with the C♯ tube's own pitch at the level a 20 mm gap gives it, C5 struck with the valve open and 20 more.

Called by 6 essays

the blast radius of changing it

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.

instruments · Struck bar

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.

instruments · Struck bar $What each strike point on a marimba bar excites. The amplitude each of the first 4 partials receives from a strike at each point along the bar, each partial drawn to its own maximum. On a string this comb is |sin nπx| and is periodic, so one strike point silences a whole family; on a bar it is the mode shape itself, whose zeroes are not at rational fractions, so no strike point silences a family. The exception is the middle: every antisymmetric mode is zero there by symmetry, so a centre strike drives partial 1 at 100 per cent of its maximum and partial 2 at nothing at all. The cord's hole at 0.2031 is the mirror case — the fundamental is the partial a strike there cannot reach.$

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.

instruments · Struck bar

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.

instruments · Struck bar

The C♯4 tube speaks in C4's attack. A marimba bar at C4, 262 hertz, struck, with the bar a semitone above it next to it in the same row. One curve is the struck note as the tubes radiate it; the other is what radiates at the neighbour's own pitch, C♯4; both in decibels against the struck note's peak. The dashed curve is where the neighbour would have to be to modulate the note by the beat-detection threshold at 15.6 beats a second, 3.1 per cent. The neighbour is above it for the first 90 milliseconds — 1.4 beats — and then drops away, while the note itself is near its loudest. Over the whole note the energy at the neighbour's pitch is 33.8 decibels below the struck note's. It comes out of the neighbour's tube, which the bar reaches at -13.5 decibels, and it leaves because that tube hands it to its own bar, which cannot radiate it.

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

A marimba bar reaches the mouth of the tube under the next bar at a fifth of what it puts on its own, and the next bar is tuned a semitone away. Solved as a row of coupled pairs, the neighbour's pitch speaks in every attack for about a beat and a half and is then handed to its own bar, which cannot radiate it. Laying the accidentals out in a second bank halves that for eight notes of the twelve, and leaves C, E, F and B exactly where they were.

instruments · Struck bar

A vibraphone's motor makes a note quieter and far longer, with a tremolo that is shallower than the valve's two states at middle C. The power a struck bar and its tube radiate at middle C, in decibels, with the coupling switched at 6 openings a second. A tube's filling time here is 98 ms. With the bar holding the same energy, the open valve radiates 13.8 dB more than the shut one; with the motor running the tremolo's depth is 10.0 dB.

A vibraphone's valve is a door on the sound and a drain on the bar

A vibraphone's motor is usually described as a duty cycle: loud while the valve is open and quiet while it is shut, with a tremolo as deep as the gap between them. Put the tube's own time back and the description comes apart. The door and the drain obey different laws. A low note's tremolo is a third shallower than a high note's at the same speed, the note rings almost three times longer than with the valve left open and well beyond any mean of the two states, and the same stroke can land eight decibels apart according to where the motor happens to be.

instruments · Struck bar

All figures · What can be heard