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

Almost every struck instrument gets its loudness by spending its length. The tube shuts on the partial the arch placed priced that trade for a marimba: a tube makes a rosewood bar twenty-one decibels louder and takes two thirds of its ring time away to pay for it. The vibraphone is the one common instrument that can change its mind during a note. A motor turns a pair of blades in the mouth of every tube, and the tube’s connection to its bar opens and closes several times a second.

That earlier essay disposed of the mechanism in a paragraph, in the terms everyone uses. It called the motor a duty cycle on the trade: loud and draining while the valve is open, quiet and preserved while it is shut, and a tremolo as deep as the difference between the two states. It is a natural account, and it makes three claims that can each be tested with the same arithmetic. The depth of the tremolo is the gap between two states. The ring is something like an average of two rings. And nothing else about the note should change.

None of the three holds at any speed anybody plays. The reason in each case is one quantity that the description leaves out, which is that a tube takes time.

A tube fills in a tenth of a second at middle C #

A bar and its pipe are one object showed that a marimba’s resonator is not a filter the sound passes through but the second member of a coupled pair, exchanging energy with the bar at a rate set by how much louder the tube makes the note. The same pair applies here with one addition. The coupling is multiplied by how open the valve is at each moment, a sine-squared curve for a pair of blades that pass open twice a revolution, so the number that matters is openings a second and it is twice the motor’s speed.

The bar is aluminium, with an internal quality factor of 2,000, which is the ten times rosewood the earlier essay used. The tubed quality factor of 500 is a round figure chosen to put middle C’s ring near a few seconds with the valve open. Neither is a measured instrument, and only their ratio to the bare bar’s radiation matters to most of what follows. What is not a free choice is the tube. Its own damping sets the time it takes to fill or to empty, and that runs from 171 milliseconds at F3, through 98 at middle C and 30 at C5, to about 4 at F6. A valve opening six times a second has a period of 167 milliseconds, and it spends about half of that substantially open. So at the bottom of the instrument the tube is being asked to fill in a third of the time it takes to do so, and at the top it fills and empties several times over within each opening.

That figure sets the note against the valve left open and left shut. The dashed lines are the two states the description names. With the valve left open, the note jumps up as the tube fills, then falls steadily through more than thirty decibels in two seconds. With it shut, the bar radiates feebly on its own and hardly declines. The running motor cuts between them and belongs to neither: below the open line at first and above it after about a second, and above the shut line and heading for it. Everything below is a reading of that curve.

The door works at any speed and the drain does not #

Hold the bar’s stored energy fixed and the two states are a definite distance apart. The open tube radiates $2\kappa^2/\gamma$ of it, where $\kappa$ is the coupling and $\gamma$ the tube’s damping, and the bare bar radiates its own tiny share. That ratio comes out as the bare bar’s quality factor over the tubed one, about 14 decibels on every note. If the tremolo were the difference between two states, it would be 14 decibels deep everywhere.

At an ordinary speed the tremolo is as deep as the two states only at the top of the instrument. At six openings a second, C6 is 14.8 decibels deep, F3 is 9.5, a third shallower in decibels and about two thirds less as a ratio of power. That much was expected from the filling time. What was not expected is the floor. The prediction, from the tube’s filling time alone, was that a fast enough motor would average the coupling and leave a tube too slow to respond, with the tremolo collapsing towards nothing in the low register. It does not. It stops falling at about nine decibels, and it stays there at any speed.

The reason is that a valve does two things and the word duty cycle mixes them up. It is a door on the sound: a tube with a shut mouth does not radiate into the room, however much is stored in it. And it is a drain on the bar: only an open tube takes energy from the bar. The two work on different clocks. The door works instantly and at any speed, because the sound leaving the mouth follows the mouth. The drain works only as fast as the tube can respond. So a fast motor on a low note keeps its door and loses its drain: the tube sits nearly full at a steady level, and what the listener gets is that steady stock being let out and shut in. That accounts for the nine decibels, and the difference between that and the deep tremolo a slow motor gives is what a tube’s filling time is worth, one to five decibels depending on the note.

A slow motor can exceed 14 decibels for the opposite reason, door and drain acting together: at one opening a second, a C6 reads 18.6. The motor has time to show the listener a tube that is both shut and emptied, and then one that is both open and full.

This is a claim that depends on how the door has been modelled. The tube here keeps its damping when shut, as though what it held were lost at the same rate behind the door. A real shut tube loses less, and on opening it would return what it kept, and so the low-register floor may be somewhat different from nine. But the distinction between a door and a drain does not depend on that, since a stopped mouth cannot radiate at any rate of rotation.

Why the motor rings longer than either story #

The second claim was that the ring is some kind of average of two rings. At middle C the valve left open gives a T60 of 2.7 seconds and left shut, 14.4. If the running motor simply spent half its time in each, and draining means adding up rates, the bar would lose energy at the mean of the two rates, for a ring of 4.5 seconds.

The motor rings for 7.5 seconds, almost three times as long as an open valve and two thirds longer than the mean. The reason is a square law, and it is already written down in the coupling’s own algebra: the loss a tube adds to a bar goes as the square of the coupling, $\kappa^2/\gamma$. Half the coupling takes a quarter of the drain, not half. A tube that is too slow to follow the blades responds to their average opening of one half, so at six or more openings a second it takes a quarter of what the open valve takes, plus the bar’s own internal loss. That is why the running motor has the same ring as a valve left half open and held there: 7.5 against 7.6 seconds. To the bar, a fast motor and a half-open valve are one object.

A slow motor is different, because then the tube follows the blades moment by moment, and the drain is the average of the squares rather than the square of the average. Sine-squared gives three eighths and not one quarter, so the motor drains half as fast again. That is what the bottom of the rising curve in the figure shows: 6.0 seconds at one opening a second, 7.5 from about six. And it leads to a counter-intuitive prediction anyone with a vibraphone and a stopwatch could check. Turning the motor up should lengthen the note, slightly, at the bottom of the speed range, and then stop lengthening it.

Read across the compass, the same fact takes a different shape. The ratio between the motor’s ring and the mean of the two drains is largest in the bass, 11.3 seconds against 6.6 at F3, and smallest in the treble, 1.2 against 1.0 at F6, because six openings a second is fast for a 171-millisecond tube and slow for a 4-millisecond one. So the same motor speed is two different mechanisms on one instrument, a half-open valve at the bottom and a real on-off switch at the top. A player hears one motor. The bar on the left of the instrument has been handed a much better deal than the bar on the right.

What the long ring is paid for with #

An instrument that rings almost three times as long might seem to have got something for nothing, and it has not. The earlier essay’s trade rested on the fact that a tube recovers energy that would otherwise have become heat in the bar. A valve that lets the bar keep its energy longer is giving that energy more time to become heat.

With the valve left open, 77 per cent of the strike leaves as sound. With the motor running at F3, 30 per cent does. The rest is warming the aluminium. The cost in level during the note’s first second is 5.6 decibels at F3, 4.7 at middle C and about 2 at the top. So the ring gained, a factor of 2.9 at F3 and 2 at F6, is bought with a real loss of energy to heat, and the deal is struck on the same notes: the bass gains the most ring and loses the most sound. In a sense the running motor makes the vibraphone a poorer radiator in exchange for being a longer one. That is the marimba’s trade run backwards, with the same exchange rate, and it is why the motor is a choice and not an improvement.

It also bears on the observation that closed the earlier account, that a marimba cannot do this because its bars ring for under two seconds and there is nothing to spread out. That is true but the argument is more specific now. A rosewood bar’s internal loss dominates, so a marimba keeping its energy away from the tube would be sending it into the wood almost at once: the share that a valve saved would be heat almost before it could be spent. Aluminium’s quality factor is what gives the bar somewhere to hold energy while the valve is shut. The door is cheap on any instrument; the drain is worth controlling only where the bar has a stock to keep.

The stroke the player does not control #

The third claim, that nothing else about the note changes, fails in the place a player might feel it most. A bar is struck at some moment in the motor’s cycle, and nobody can see which.

At six openings a second, the same stroke on C5 can arrive 7.7 decibels louder or softer according to where the motor happens to be. On C6 the figure is 7.3, on middle C 4.2 and on F3 2.3. The reason it is largest high up is again the tube’s time: a quick tube’s output follows the door during the first tenth of a second, where a slow tube has hardly started to fill whichever phase the door is in. At a slow motor the spread reaches over ten decibels on the upper notes. It falls almost to nothing near ten openings a second, not for any acoustical reason but because the tenth of a second in which the level is read now contains a whole cycle of the valve. That window is an assumption, a stated figure for something like how an attack is heard. Using a shorter one would move the speed at which the spread collapses and leave the rest alone.

This is the surprising connection, and it is to a rhythm rather than to a timbre. The essays on ensemble timing treat a performer’s control over dynamics and placement as limited by a body and a listener; here the same limit is imposed by a machine under the keyboard. A vibraphonist who keeps a motor at a moderate speed is playing an instrument whose loudness on a given stroke carries a random component of several decibels on the upper notes, set by a rotation that bears no relation to the tempo. On the upper notes that is about the distance between two written dynamic markings, taking that distance as a round figure of six decibels, and the crescendo inside a fast run is the same kind of thing: a change of level that belongs to the instrument’s time constants and not to anything the player decided. A player might well choose a speed that is a whole multiple of the music’s pulse so that the valve lands the same way every time, which would remove that randomness and put a regular accent in its place. It is an interesting prediction: on this arithmetic, a motor locked to the pulse has a rhythmic effect at any speed below about eight openings a second. Nobody here has asked a vibraphonist whether that is something they listen for.

What this pair leaves out #

One partial. The whole calculation is the bar’s fundamental in a tube tuned to it. A stopped tube favours odd multiples, and a vibraphone’s bar has its second tuned partial placed near the fourth harmonic, so the tube shuts on that partial and it hardly takes part in any of this. That means the fundamental pulsates and the fourth partial does not: the tremolo is a tremolo of brightness as well as level, and the note is at its brightest in proportion while the valve is shut. That may be a large part of what a listener hears as the sound of the motor, and none of it is here. The bar’s upper partials also fall away at their own rates, as every struck note’s do, and the second partial’s own damping is part of how a bar loses what it was given, so the brightness of a tremolo would change through the note.

The valve is one number. A real blade is a rotating disc in a flow, shading the mouth with its own geometry, and its opening curve is not exactly sine-squared. It presumably affects the tube’s end correction as it turns, which would detune the tube a little as the valve closes, and so move the pair along the avoided crossing that the earlier essay drew and put a pitch or level component into the tremolo that has been left out. The tube is tuned exactly here and held there. The neighbours are not here either, though a row of resonators hands an attack to the bars beside it and with a valve in every mouth driven by one shaft they would all move together.

Hearing is not computed. Nothing here says what a fluctuation of nine or fifteen decibels at three or six cycles a second sounds like. Fluctuation is a well-studied percept with a broad peak around four cycles a second, a rate that vibrato also occupies, and all of the speeds in common use lie close to it, so the numbers may understate how much the speed matters by reading level only. A tenth of a second as the window for an attack is a round figure. The stated T60s are predictions from stated quality factors, and the real instrument’s bars, felted suspension cords and pedal damper all contribute, so the ratios should be trusted before the seconds.

Repertoire. The instrument in view is the motorised vibraphone of the twentieth-century pedal-bar design, as played in concert and jazz with the motor either running or stopped open. Many players leave it off altogether, and a great deal of its literature asks for a motor rarely or never, so all of this is a claim about a sound some of its repertoire uses and much of it does not. Whether the sense of a long, shimmering note that players ascribe to the motor is the long ring computed here, or the tremolo, or a brightness that is moving, cannot be decided from these sums.

Still open: stopping the blades on the way #

The figures consider a valve that runs or is parked wide open. A player can do something in between, and the arithmetic has a clear reason to care. Since a bar responds to the square of the coupling, a valve stopped a third of the way open takes about a ninth of the drain and keeps much of the door, which suggests that parked positions are not on the same curve as running speeds. And a player who starts or stops the motor during a held note can spend a stock accumulated while the valve was shut, releasing a bar that has lost far less than a bar with an open tube would have, a kind of swell a struck instrument should not be able to make. The computation is this same pair with the valve’s opening set by a player, which would say how large that swell can get, at what moment it is largest, and whether there is a pattern of opening that gives the most sound over the longest ring.