The loudspeaker that goes round

The note that changes on approach began the Doppler effect with an ambulance going past: a pitch held high while it approaches, dropping as it passes, held low as it recedes. One pass, one change. Make the source go round a small circle instead of along a road, and the change repeats every revolution — a pitch that rises and falls at the rate of rotation. That is a vibrato, and since the 1940s it has been built into a piece of furniture.

The Leslie speaker, invented by Donald Leslie to give the Hammond electric organ something of the life of a pipe organ heard in a large hall, is a cabinet in which the loudspeakers do not move but their sound is thrown out through rotating parts: a horn, turning on a vertical axis, for the treble, and a rotating drum with a scooped opening below a fixed woofer for the bass. Switched to its fast setting, the horn turns about six or seven times a second, and the sound that comes out is not a steady note but a swirling, shimmering one that became the signature of a great deal of jazz, gospel and rock organ. The physics of the swirl is a Doppler shift, and following it carefully shows that a Doppler shift repeated periodically is not a shift at all.

A source going round #

A source moving in a straight line crowds its wavefronts ahead of it and spreads them behind. A source moving round a circle does the same at every moment, but the direction of “ahead” keeps turning.

The figure exaggerates the speed, to six-tenths of the wave speed, so that the effect is visible: each crest is a circle centred where the source was when it left, and the crowding sweeps round with the motion, leaving a pattern that is lopsided and slowly spiralling rather than the egg shape of a straight-moving source. A listener anywhere outside receives crests crowded together and then spread apart, alternately, once per revolution. A Leslie horn’s mouth moves at about six metres a second, under two per cent of the speed of sound, and its pattern is far too slight to draw. The effect on the pitch is not slight at all.

The Doppler formula for a moving source and a stationary listener, $f' = f/(1 - v_r/c)$, depends only on the component of the velocity towards the listener, $v_r$. For a mouth going round a circle of radius $r$ at $f_{rot}$ revolutions a second, that component swings sinusoidally between $\pm 2\pi r f_{rot}$ for a listener in the plane of rotation, and the heard frequency swings between $f(1 \pm v/c)$, very nearly.

The traces above are computed without that approximation. The sound heard at each instant left the horn’s mouth at an earlier instant, when the mouth was somewhere else on its circle, and the computation finds that earlier instant exactly — the retarded time — and differentiates. For a mouth fifteen centimetres from its axis turning 6.7 times a second, the speed is 6.3 metres a second and a listener level with the horn hears every note swing up and down by 32 cents, a third of a semitone. A listener 45 degrees above the plane of rotation sees only the cosine of that motion and hears 23 cents. A listener directly above the axis sees the mouth go round without ever coming nearer or going farther, and hears no vibrato at all: the same geometry as the shift that survives at right angles, where for sound a source passing at right angles produces no first-order shift.

Two features of this vibrato matter for music. Its depth is the same fraction of every note, so every partial of every note in a chord is moved by the same number of cents and the chord wavers together, its intervals intact. And its rate is the rotation rate, here 6.7 per second, which happens to be close to the natural vibrato rate of a singer’s voice or a violinist’s finger, five to seven per second. On its slow setting, under one revolution a second, the cabinet’s motion is heard as a slow sweep rather than a vibrato at all.

A vibrato is a comb #

A frequency that swings sinusoidally is called frequency modulation, and the spectrum it produces is not what the word “vibrato” suggests. A note whose pitch wanders between two limits might be expected to spread its power over the band between them, most of it near the limits where the pitch lingers. For slow wandering that is roughly true. When the wandering is periodic, the signal itself is periodic in a particular way, and the Fourier analysis that the even wave train that cannot stay even applies to a modulated wave gives an exact answer: the spectrum is a comb of discrete lines at the original frequency plus and minus every multiple of the modulation rate, with no power at all between the lines.

The comb’s lines are spaced at 6.7 hertz, the rotation rate. Their strengths are set by a single number, the modulation index $\beta = 2\pi f r/c$, which is the peak swing in frequency divided by the modulation rate: the power in the $n$-th line is $J_n(\beta)^2$, the square of the Bessel function of order $n$. The 1 kHz comb computed from the retarded-time signal matches these powers line for line. For a 250-hertz tone the index is 0.69 and the comb is narrow — 78 per cent of the power stays at the original frequency, and most of the rest is in the first line on either side. For a 4 kHz tone the index is 11 and the power is spread across some two dozen lines, about eighty hertz either side, with almost none left at the original frequency.

That proportionality is the heart of the sound. Because the index is proportional to the frequency, the turning horn leaves a note’s fundamental nearly clean while smearing its upper partials into a dense cluster of lines. The ear hears the clean low part as a pitch that wavers, and the smeared high part as a shimmer — a quality that a vibrato produced by wobbling an oscillator’s frequency electronically does not reproduce in the same way, because that applies a vibrato to the note without the cabinet’s other effects. The same Bessel-function spectrum describes FM radio, where a programme signal swings a carrier’s frequency, and the rule that the occupied bandwidth is about twice the peak swing plus the modulation rate — Carson’s rule, from 1922 — predicts the 4 kHz comb’s ±80 hertz. It also underlies FM sound synthesis, introduced by John Chowning in the 1970s, which deliberately uses large modulation indices to manufacture rich spectra from two oscillators.

A comb heard as a wobble #

If the spectrum is a comb of lines with nothing between them, the listener might be expected to hear a cluster of steady tones a few hertz apart, rather than one tone whose pitch moves. The reason the ear hears motion is that it cannot resolve lines that close together in the time it allows itself. Sharpness has to be paid for gives the trade-off: telling two frequencies 6.7 hertz apart takes a listening time of about 1/6.7 of a second or longer, during which nothing must change. The ear’s frequency analysis works over a few tens of milliseconds, so it sees the comb’s lines as a single band whose centre moves, and the moving centre is the vibrato. An instrument that listened for seconds at a time — a spectrum analyser set to fine resolution — would show the comb, and indeed it does; the comb in the figure is computed over exactly one revolution, the shortest stretch over which the lines are defined.

The two descriptions, a moving pitch and a fixed comb, are the same signal described at two resolutions, and neither is more correct. Which one is useful depends on the question. For the musician the moving pitch is the right picture, because the ear works at short times; for the radio engineer allocating channels the comb is the right one, because a neighbouring station’s receiver works at long times and will hear every line that strays into its band. The point at which the two pictures cross over is set by the modulation rate: a vibrato much slower than the ear’s integration time is heard as a moving pitch, and one much faster — hundreds of hertz, as in FM synthesis — is heard as a fixed timbre made of the comb’s lines.

The same comb appears in radar. A helicopter’s rotating blades return a radar signal with a periodically varying Doppler shift, and the shift a mirror gives twice notes the sidebands that result, spaced at the blade rate; identifying aircraft by the spacing and spread of those sidebands is the radar version of hearing that an organ is playing through a Leslie. And it is the classical cousin of the Doppler shift with nothing coming closer, where a spinning surface shifts light carrying orbital angular momentum by a fixed amount; there the rotation produces a single new line because the beam has a definite twist, here a comb because the source has a definite position going round.

The room joins in #

A Leslie speaker is almost never heard in open air. In a room, the listener hears the direct sound and also the sound reflected from every wall, and each reflection behaves as though it came from an image of the horn behind the wall — the same mirror construction that turns a reflector into a second source.

The image of the horn behind each wall turns in the opposite sense, as a mirror image must, and it is seen from a different direction, so the reflected sound’s pitch swings with the same depth and a different timing. At any instant the five copies in the figure — direct and four first reflections — arrive at five different pitches, up to two-thirds of a semitone apart. They therefore slip in and out of step with each other, and the loudness of their sum, below, swells and collapses several times a turn, falling at its deepest to six per cent of its peak. The pattern repeats each revolution, but its shape depends on the room and on exactly where the listener stands: move a step and the timings, and so the pattern of swells, change.

Real rooms add dozens of later reflections, each a further copy with its own timing, and the result is a sound that seems to move about the room and to have a depth that the cabinet’s own output does not. Recordings of Leslie cabinets are often made with two microphones on opposite sides of the horn, which hear its vibratos in opposite phase, to recover some of that spatial quality. The amplitude of the direct sound varies too, as the horn’s mouth, which beams its treble forward, turns towards and away from the listener; the combination of this tremolo, the Doppler vibrato and the room’s copies is the whole of the effect, and none of them alone sounds like it.

Slow, fast and the swirl between #

The cabinet has two speeds, and players switch between them as part of the music. On the slow “chorale” setting the horn turns under once a second; on the fast “tremolo” setting six or seven times. The switch is not instantaneous, because the rotors have inertia: the light horn reaches its new speed in about a second, the heavier bass drum in several.

The vibrato’s depth is proportional to the mouth’s speed and its rate to the rotation, so as a rotor accelerates its vibrato grows both deeper and faster together — the horn’s from four cents at the slow speed to thirty-two at the fast. The bass drum’s mouth is closer to its axis and it turns a little more slowly, so its vibrato is shallower, and because it lags the horn by seconds, the treble and bass sections of every chord waver at different rates while the cabinet changes speed. That rising, unsynchronised swirl is one of the most recognisable sounds of the instrument, and organists use it deliberately, switching speeds in the middle of a phrase to make the sound accelerate.

The bass drum’s construction reflects another piece of physics. Low frequencies have wavelengths of several metres, far larger than any rotating part that would fit in a cabinet, and a loudspeaker radiates them almost equally in every direction whichever way it faces. Spinning the woofer itself would therefore achieve nothing but the Doppler vibrato — the beaming that produces the tremolo would be absent. The drum instead rotates a baffle with a single opening around a downward-facing woofer, which forces the bass out through a moving aperture. At the frequencies where the opening is comparable to the wavelength the bass acquires some directivity; below them only the small Doppler shift of the moving aperture survives.

Why this is a Doppler effect and not a trick #

It might be objected that what reaches the listener from a rotating horn is simply a modulated signal, and that calling it a Doppler effect adds nothing. The retarded-time computation answers the objection: nothing in it refers to modulation. It asks only when the sound arriving at each instant left the mouth, given the mouth’s motion and the speed of sound, and the vibrato, the comb, the Bessel weights and the dependence on the listener’s elevation all follow. A modulation imposed electrically would be the same for every listener; the Doppler vibrato is not, because it depends on the component of the mouth’s motion along each listener’s line of sight. A listener above the cabinet hears none of it, and two listeners on opposite sides hear it in opposite phase, which is how the stereo recordings work.

The same distinction appeared in everything from an exchange of pulses, where a Doppler factor emerged from nothing but the timing of signals between moving observers. The rotating horn is the case where the source’s motion is periodic, so the timing of its signals is periodically compressed and stretched; a periodic compression of time is a frequency modulation, and frequency modulation is a comb. Each step is a statement about when sound left and when it arrived.

A horn is not a point, and a room has more than four walls #

The computations treat the horn’s mouth as a point source radiating equally in all directions, which isolates the Doppler effect; a real horn beams its higher frequencies within a cone of some tens of degrees, so the loudness swings along with the pitch, and the two are partly coupled. The room model uses only the four first reflections from walls of a rectangular room, with a fixed loss at each, and ignores the floor, the ceiling and every later reflection; it shows the mechanism, not the sound of any real room. The spin-up uses a simple exponential approach to the new speed with representative time constants, whereas real rotors are driven by small motors through belts and accelerate in a way that depends on the cabinet’s age and state of maintenance. The radius, speeds and crossover of the cabinet are those of common models, approximately; Leslie cabinets were built in many versions over several decades.

Still open: how much of the sound is the room #

How much of the Leslie’s character comes from the Doppler vibrato, how much from the horn’s amplitude modulation and how much from the room is a question with a practical edge, because it decides whether the effect can be reproduced electronically. Digital simulations of rotating speakers are now built into most electronic organs, and they model all three to varying degrees; listening tests disagree about how close they come, and the answer appears to depend on the playing style and on whether the comparison is made in a room or through headphones, where the real cabinet’s spatial character is partly lost. Whether a listener can distinguish a real cabinet from a good simulation with the room reflections included, and which of the physical effects carries the distinction, has not been settled by controlled experiment.

The habit worth carrying away is to ask whether a shift is steady or periodic. A source going round a circle shifts every frequency by the instantaneous v/c towards each listener, ±32 cents for a horn mouth moving at 6.3 metres a second — but a periodically swinging frequency is a comb of lines spaced by the rotation rate with Bessel weights $J_n(2\pi f r/c)$, so the same rotation that barely touches a 250-hertz fundamental spreads a 4-kilohertz partial over ±80 hertz. The swirl of a Hammond organ is the Doppler effect turned into a spectrum, and then multiplied by every wall in the room.