## 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

where is the circular radius and 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}

By Eq.(5.3), the source velocity for the circularly rotating horn is

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

Since and are orthogonal, the projected source velocity Eq.(5.4) simplifies to

Arbitrarily choosing (see Fig.5.8), and substituting Eq.(5.8) and Eq.(5.9) into Eq.(5.10) yields

In the far field, this reduces simply to

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

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 times horn angular velocity divided by sound speed . Note that 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].^{6.11}More 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