The Leslie

The Leslie, named after its inventor, Don Leslie,6.9 is a popular audio processor used with electronic organs and other instruments [59,189]. It employs a rotating horn and rotating speaker port to ``choralize'' the sound. Since the horn rotates within a cabinet, the listener hears multiple reflections at different Doppler shifts, giving a kind of chorus effect. Additionally, the Leslie amplifier distorts at high volumes, producing a pleasing ``growl'' highly prized by keyboard players. At the time of this writing, there is a nice Leslie Wikipedia page, including a stereo sound-example link under the first picture (that is best heard in headphones). Papers on computational audio models of the Leslie include [468,191].

The Leslie consists primarily of a rotating horn and a rotating speaker port inside a wooden cabinet enclosure [189].

Rotating Horn Simulation

The heart of the Leslie effect is a rotating horn loudspeaker. The rotating horn from a Model 600 Leslie can be seen mounted on a microphone stand in Fig.5.7. Two horns are apparent, but one is a dummy, serving mainly to cancel the centrifugal force of the other during rotation. The Model 44W horn is identical to that of the Model 600, and evidently standard across all Leslie models [189]. For a circularly rotating horn, the source position can be approximated as

$\displaystyle \underline{x}_s(t) = \left[\begin{array}{c} r_s\cos(\omega_m t) \\ [2pt] r_s\sin(\omega_m t) \end{array}\right] \protect$ (6.8)

where $ r_s$ is the circular radius and $ \omega_m $ is angular velocity. This expression ignores any directionality of the horn radiation, and approximates the horn as an omnidirectional radiator located at the same radius for all frequencies. In the Leslie, a conical diffuser is inserted into the end of the horn in order to make the radiation pattern closer to uniform [189], so the omnidirectional assumption is reasonably accurate.6.10

Figure 5.7: Rotating horn recording set up (from [468]).
By Eq.$ \,$(5.3), the source velocity for the circularly rotating horn is

$\displaystyle \underline{v}_s(t) = \frac{d}{dt}\underline{x}_s(t) = \left[\begi...\omega_m t) \\ [2pt] r_s\omega_m\cos(\omega_m t) \end{array}\right] \protect$ (6.9)

Note that the source velocity vector is always orthogonal to the source position vector, as indicated in Fig.5.8.

Figure 5.8: Relevant geometry for a rotating horn (from [468]).

Since $ \underline{v}_s$ and $ \underline{x}_s$ are orthogonal, the projected source velocity Eq.$ \,$(5.4) simplifies to

$\displaystyle \underline{v}_{sl}= {\cal P}_{\underline{x}_{sl}}(\underline{v}_s...
...line{x}_s\,\right\Vert^2}\left(\underline{x}_l-\underline{x}_s\right). \protect$ (6.10)

Arbitrarily choosing $ \underline{x}_l=(r_l,0)$ (see Fig.5.8), and substituting Eq.$ \,$(5.8) and Eq.$ \,$(5.9) into Eq.$ \,$(5.10) yields

$\displaystyle \underline{v}_{sl}= \frac{-r_l r_s\omega_m\sin(\omega_m t)}{r_l^2...
...l-r_s\cos(\omega_m t) \\ [2pt] -r_s\sin(\omega_m)t \end{array}\right]. \protect$ (6.11)

In the far field, this reduces simply to

$\displaystyle \underline{v}_{sl}\approx -r_s\omega_m\sin(\omega_m t) \left[\begin{array}{c} 1 \\ [2pt] 0 \end{array}\right]. \protect$ (6.12)

Substituting into the Doppler expression Eq.$ \,$(5.2) with the listener velocity $ v_l$ set to zero yields

$\displaystyle \omega_l = \frac{\omega_s }{1+r_s\omega_m\sin(\omega_m t)/c} \approx \omega_s \left[1-\frac{r_s\omega_m}{c}\sin(\omega_m t)\right], \protect$ (6.13)

where the approximation is valid for small Doppler shifts. Thus, in the far field, a rotating horn causes an approximately sinusoidal multiplicative frequency shift, with the amplitude given by horn length $ r_s$ times horn angular velocity $ \omega_m $ divided by sound speed $ c$. Note that $ r_s\omega_m $ is the tangential speed of the assumed point of horn radiation.

Rotating Woofer-Port and Cabinet

It is straightforward to extend the above computational model to include the rotating woofer port (``baffle'') and wooden cabinet enclosure as follows:

  • In [189], it is mentioned that an AM ``throb'' is the main effect of the rotating woofer port. A modulated lowpass-filter cut-off frequency has been used for this purpose by others. Measured data can be used to construct angle-dependent filtering in a manner analogous to that of the rotating horn, and this ``woofer filter'' runs in parallel with the rotating horn model.

  • The Leslie cabinet multiply-reflects the sound emanating from the rotating horn. The first few early reflections are simply handled as additional sources in Fig.5.6.

  • To qualitatively simulate later, more reverberant reflections in the Leslie cabinet, we may feed a portion of the rotating-horn and speaker-port signals to separate states of an artificial reverberator (see Chapter 3). This reverberator may be configured as a ``very small room'' corresponding to the dimensions and scattering characteristics of the Leslie cabinet, and details of the response may be calibrated using measurements of the impulse response of the Leslie cabinet. Finally, in order to emulate the natural spatial diversity of a radiating Leslie cabinet in a room, ``virtual cabinet vent outputs'' can be extracted from the model and fed into separate states of a room reverberator. An alternative time-varying FIR filtering approach based on cabinet impulse-response measurements over a range of horn angles is described in [191].

In summary, we may use multiple interpolating write-pointers to individually simulate the early cabinet reflections, and a ``Leslie cabinet'' reverberator for handling later reflections more statistically.

Recent Research Modeling the Leslie

As mentioned above, modeling the Leslie via interpolating delay-line writes and cabinet image-sources was described in [468].6.11More recently, Leslie simulation via time-varying FIR filtering has been developed [191]. See these papers and their cited references for further details.

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Ideal Vibrating String
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Chorus Effect