The Leslie effect heard on the Leslie Speaker, has spatial motion properties while listening to a sound source. In particular Hammond B-3 organs add expressive cues to a tone which is static on its duration envelope. Great organ performers like Jimmy Smith, Lonnie Smith and the Deep Blue Organ Trio, make clever use of this effect. Because of frequency shifts due to moving speakers, this effect effect gives the illusion of not only one sound source but several. At slow rotating speeds, there is the effect of a moving source, while at fast rotating rates, there is the effect of a chorale with multiple sound sources. Time varying delay lines are used for various effects and simulation of moving sound sources. Below is ChucK code for modeling the Leslie speaker cabinet as an example for applications of dynamics from moving sound sources.
The "Leslie" speaker cabinet, nowadays an intrinsic part of the Hammond organ system, is an example of a moving sound source. Perception of moving sources and localization cues depend on static variables such as amplitude or intensity of a sound, further function of distance from source to the listener. Timbre or spectral cues, pitch and reverberation of the environment are also static. Dynamic variables consist of changing distances from sound sources to the listener and velocities of this motion [Kronland-Martinet et al, 2008]. These parameters in addition to velocity of sound dispersion are the factors for Doppler shifts. More about the Leslie cabinet: [here].
Doppler effect models have been used to add realism of simulated moving sources on composition contexts in a variety of computer music pieces and electroacoustic composition [Chowning, 1971], [Lopez-Lezcano, 2008]. Immersive environments with invisible objects use sound motion to delineate possible images and locations for perception of sources. Bill Fontana's [Sonic Shadows] is an example of real and physical dynamic sound sources inside an acoustical environment.
A Doppler simulator can be used to model sound source motion. For a reasonable application such as the Leslie effect, a simulator with delay lines as building-blocks has been proposed on [Smith et al., 2002]. It is a well known fact that a time varying delay line results in a frequency shift see [here]. Since Doppler effects give the illusion of a pitched sound rising or falling, given the motion of the source or the listener in relation to each other, a delay line mimics this effect[ibid]. In the case of the Leslie speaker delay lines cast a Doppler simulator by modeling motion of the rotating horn. In this, the listener hears multiple reflections at different Doppler shifts, giving the illusion of a multiple voice chorale. More on time-varying delay effects [here].
Delay lines are part of major software synthesis packages. STK and therfore ChucK implement [DelayL] providing continous and linearly interpolated variable delay lengths. For reference, a technical despcrition on modeling Doppler and the Leslie is found [here]. Operation of the Leslie is described on [Wikipedia]. Note that to implement a continuously varying delay, a "delay growth parameter" 'g' needs to added to the delay line. When g is 0, there is a fixed delay line. When g>0, delay increments g samples per sample, and thus seconds per second.
To calculate a growing factor for the rotating horn in the Leslie speaker we need its geometrical factors. Therefore, using Doppler shift equation (14) on [Smith et al., 2002], a growing factor for frequency shift is obtained by the product of the trace of the circumference of the horn, its angular velocity and cosine (or sine) of the angle of the horn at time 't', over the speed of sound. See equation (16) on [ibid]. In modeling the cabinet four delay lines are needed for the system to get shifts from the four sides of the cabinet. Similarly, motion of lower baffle speaker is achieved by a set of four low pass filters. Their center frequency is also obtained by a growing factor, this time describing the circular motion of the lower Leslie bass rotor (see code listing).