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Spatial Modulation Synthesis ICMC 2015 – Sept. 25 - Oct. 1, 2015 – CEMI, University of North Texas Spatial Modulation Synthesis Ryan McGee Media Arts and Technology University of California, Santa Barbara [email protected] ABSTRACT Sound Source In 1962 Karlheinz Stockhausen’s “Concept of Unity in Elec- Bounded Periodic tronic Music” introduced a connection between the param- Motion eters of intensity, duration, pitch, and timbre using an ac- celerating pulse train. In 1973 John Chowning discovered Source that complex audio spectra could be synthesized by increas- Frequency ing vibrato rates past 20Hz. In both cases the notion of in- creased speed to produce timbre was critical to discovery. Although both composers also utilized sound spatialization Source Source in their works, spatial parameters were not unified with their Speed Position synthesis techniques. Spatial Modulation Synthesis is intro- duced as a novel, physically-based control paradigm for audio- Spatialization Doppler Gain Amplitude visual synthesis, providing unified control of spatialization, Algorithm Shift Attenuation Panning timbre, and visual form using high-speed sound trajectories. High Speed 1. INTRODUCTION Spatial Spatial modulation synthesis (SM) is a paradigm through which AM and AM and Modulation Doppler FM modulation synthesis and granulation effects can be produced Synthesis Granulation Spread via high-speed sound spatialization within periodic trajectory orbits. Hence the name, spatial parameters are modulated to result in frequency and amplitude modulation of an input Figure 1. Spatial Modulation Synthesis signal rather than controlling FM and AM directly. Spatial- ization algorithms include a means for distributing the out- put to each loudspeaker, such as VBAP [1] or Ambisonics over distance. Using existing spatialization, Doppler, and at- [2], and are typically paired with multiple distance cues in- tenuation techniques, spatial modulation synthesis provides a cluding Doppler shift and a distance-based gain attenuation. new control and visualization paradigm through which space Additional distance cues may include air absorption filtering, and timbre are unified. presence filtering, and reverberation. For now, the concept of spatial modulation synthesis focuses only on Doppler shift 1.1 Background and gain attenuation, leaving other possible cues as future re- search. Beginning with Karlheinz Stockhausen’s monumental Gesang While the spatialization of sounds along calculated trajec- der Junglinge in 1956, several 20th century works of elec- tories and FM have been used together in many monumental tronic and computer music have utilized sound spatialization compositions, spatialization has not been used to synthesize as an independent parameter of composition choreographed tones through precise control of sound source trajectories. alongside timbre, pitch, intensity and duration. While early Similarly, while the spatialization of sound grains has been choreography of spatial trajectories relied simply on ampli- implemented by various composers, spatialization itself has tude panning between speakers, John Chowning’s 1971 pa- not been used to create granular streams through rapid motion per, The Simulation of Moving Sound Sources [3], detailed various techniques to implement localization cues. There- after, many spatial compositions have utilized more realistic, Copyright: © 2015 Ryan McGee. This is an open-access article distributed physically based distance simulation including Doppler shift under the terms of the Creative Commons Attribution License 3.0 Unported, and gain attenuation with distance. which permits unrestricted use, distribution, and reproduction in any Both FM and granular synthesis techniques have allowed medium, provided the original author and source are credited. composers to fuse perceptibly discrete sound events into new – 246 – ICMC 2015 – Sept. 25 - Oct. 1, 2015 – CEMI, University of North Texas timbres. Chowing accomplished this by increasing vibrato Most digital audio Doppler simulations use an approxima- rates into the audio domain (above 20 Hz) to discover FM tion for Doppler shift that is relatively accurate for vs < synthesis [4], and Stockhausen morphed a series of impulses 100m=s and gives an equal shift of frequency both towards into tone by altering their duty cycle. Stockhausen’s Concept and away from the listener [6]. This symmetric depth Doppler of Unity in Electronic Music [5] states the composition and approximation is given by (Equation 2) and the resulting shift decomposition of timbres as a fundamental principle of elec- is shown in figure 3. tronic music and expresses the idea of unity between timbre, vs pitch, intensity, and duration. To achieve this unity, Stock- f = 1 ± fs (2) hausen had to take pulse trains to extremely short durations c while Chowning took vibrato rates to extreme frequencies. Likewise, spatial modulation synthesis is based on extremes of sound source velocity to produce new timbres. 2. SPATIAL MODULATION SYNTHESIS 2.1 Doppler FM Figure 3. Symmetric Doppler Appoximation, vs c Doppler shift is the physical phenomena of increased pitch for sound sources moving toward a listener and decreased pitch Although Equation 2 is not physically accurate for high val- for sound sources moving away from a listener. Through the ues of vs, the approximation conveniently provides a sym- simulation of precise trajectories and velocities of moving metrical modulator than can used used to replicate exact FM sound sources it is possible to sculpt the resulting Doppler timbres from SM. However, an additional implementation of curve for use as a frequency modulator. The term Doppler the true Doppler shift (Equation 1) has also been implemented FM is used to describe audio rate frequency modulation re- since its use is necessary for SM to achieve higher modula- sulting from bounded, high-velocity sound source movement tion indices in addition to keeping the paradigm as physically oscillating around a listener. First, the simplest case involving accurate as possible. Though not symmetrical in depth, us- the motion of a sound source with a constant magnitude of ve- ing true Doppler modulation still produces more partials in locity moving back and forth along one dimension is demon- the resulting spectra for greater timbral variation. For sim- strated. Then, the acceleration required to produce classic si- plicity, the following equations will consider the symmetrical nusoidal FM is described. Finally, the technique is extended Doppler approximation that shows similar relationships be- to 3-dimensional motion resulting in parallel multiple modu- tween velocity, bounds, modulation depth, and frequency. lator FM. The Doppler shift for a stationary listener and moving sound 2.1.1 SM Square Modulation source is given by Equation (1). Consider the scenario of a single sound source with frequency f moving on the x-axis at a constant velocity, v , past a lis- c s s f = fs (1) tener at x = 0. If the motion of the sound source oscillates c ± vs such that the direction of its velocity is inverted at a bounds at distance B in either direction from the listener then the result- f is the observed frequency, fs is the frequency of the source, ing Doppler shift becomes periodic. For this back and forth c is the speed of sound in air (≈ 340m=s) and vs is the speed of the moving sound source, positive when moving motion at constant velocity, the resulting Doppler curve will away from the listener and negative when moving towards resemble a square wave. the listener. For vs c the magnitude of the shift is roughly equal as the source moves towards and away from the listener at a constant vs. However, for higher velocities, the change in frequency, ∆f = jf − fsj, will be uneven and greater when the source is approaching as shown in figure 2. For vs = c the observed frequency goes to 1 and, in reality, a sonic boom occurs. Figure 4. Oscillating Doppler Shift for Constant Velocity . Now, considering the Doppler curve as a frequency modu- lator for our source with a rate of modulation determined by the time it takes the source to move a distance of 2B and a modulation depth directly related to the sound source’s ve- Figure 2. Physically Accurate, Asymmetrical Depth Doppler locity, the modulation depth (∆f), frequency (fm), and index . (I) are given by – 247 – ICMC 2015 – Sept. 25 - Oct. 1, 2015 – CEMI, University of North Texas v and sawtooth modulators. Also, asymmetrical bounds may ∆f = s f (3) be used to further shape the modulation. c s v f = s (4) 2.2 Multi-dimensional Motion m 2B Expanding source velocity to a 3-dimensional vector, ∆f 2Bf hv ; v ; v i, produces complex modulation similar to par- I = = s (5) sx sy sz fm c allel multiple modulator frequency modulation (MMFM) [8]. Not only are the timbres more complex, but the resulting 3D v The speed of the moving sound source, s, is directly pro- orbits can produce beautiful Lissajous trajectories that modu- portional to both the frequency and depth of modulation. The late in correlation with the changing FM parameters. size of the bounds, B is inversely proportional to the fre- quency of modulation, but directly proportional to the index of modulation. The frequency of the moving sound, or car- rier, is directly related to the depth and index of modulation. This is a departure from classic FM in which the carrier fre- quency is a separate control from the depth and index. Sim- ilarly, this unique relationship between carrier frequency and index of modulation was observed by Lazzarini et. al [7] in their delay-line phase modulation implementation. A fundamental concept of FM synthesis is that integer ratios of the carrier to modulator frequencies will produce harmonic spectra. In the case of SM square wave modulation the ratio appears as Figure 6. 3D Asymmetrical SM Orbit f 2Bf s = s (6) fm vs 2.3 Morphing Tones to Spatial Grains Thus, proportions of bounding space and velocity become Sounds resulting from SM can replicate classic FM timbres, critical to shape the harmonic content of a sound. but have the ability to morph into completely different modu- 2.1.2 SM Sinusoidal Modulation lating or granular timbres by changing only a two spatial con- trols, namely velocity and bounds.
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