2. Large-scale Physical Modeling Synthesis Stefan Bilbao Acoustics and Fluid Dynamics Group / Music University of Edinburgh NUI Maynooth, 2008 1.Abstract sound synthesis and physical modeling 2.Physical modeling techniques 3.Direct simulation 4.Implementation Issues

3. Digital Sound Synthesis: Motivations Various goals: Achieving complete parity with sound produced by existing instruments… Creating new sounds and instruments… The goal determines the particular methodology and choice of technique…and the computational complexity as well as implementation details! Main groups of techniques: Sampling synthesis Abstract techniques Physical Modeling

4. Abstract Digital Sound Synthesis Origins…early computer hardware/software design, speech (Bell Labs, Stanford) Basic operations: delay lines, FFTs, low-order filters Some difficulties: Sounds produced are often difficult to control… betray their origins, i.e., they sound synthetic. BUT: may be very efficient! AmplitudeFrequency Sinusoid FM output Carrier ModulatorVariable rate read Table of data Wavetable Synthesis (1950s) Oscillators/ Additive Synthesis (1960s) FM Synthesis (1970s) Others: Subtractive Waveshaping Granular…

5. Physical Modeling Physical models: based of physical descriptions of musical “objects” can be computationally demanding… potentially very realistic sound control parameters: few in number, and perceptually meaningful digital waveguides, modal synthesis, finite difference methods, etc.

6. Linearity and Nonlinearity A nonlinear system is best defined as a system which is not linear (!) A linear system, crudely speaking: a scaling in amplitude of the excitation results in an identical scaling in amplitude of the response Many interesting and useful corollaries… Many physical modeling techniques are based on this simplifying assumption… Resonator Excitation Gestural data (control rate) Sound output (audio rate) Strongly nonlinearLinear (to a first approximation)

7. Digital Waveguides (J. O. Smith, CCRMA, Stanford, 1980s--present)  A delay-line interpretation of 1D wave motion:  Useful for: strings/acoustic tubes  Waves pass by one another without interaction  Extremely efficient…almost no arithmetic!  See (Smith, 2004) for much more on waveguides… Rightward traveling wave Leftward traveling wave Add waves at listening point for output

8. Products Using Waveguide Synthesis Physical modeling synthesizers Yamaha VL-1 & VL-7, 1994 Korg Prophecy, 1995 Sound cards Creative Sound Blaster AWE64 Creative Sound Blaster Live Technology patented by Stanford University and Yamaha Yamaha VL-1 (Sound examples from: http://www-ccrma.stanford.edu/~jos/waveguide/Sound_Examples.html)http://www-ccrma.stanford.edu/~jos/waveguide/Sound_Examples.html (This slide courtesy of Vesa Valimaki, Helsinki University of Technology, 2008.)

9. Waveguide Stringed Instruments (Helsinki University of Technology, Department of Acoustics and Signal Processing) Sound examples: Full harpsichord synthesis ( Valimaki et al., 2004) Guitar modeling ( Valimaki et al., 1996) See http://www.acoustics.hut.fi/~vpv/ for many other sound examples/related publications http://www.acoustics.hut.fi/~vpv/

10. Modal Synthesis (Adrien et al., IRCAM, 1980s-- present)  Vibration is decomposed into contributions from various modes, which oscillate independently, at separate frequencies  Basis for Modalys/MOSAIC synthesis system (IRCAM) Sound output

11. Limitations  Digital waveguides: work well in 1D, but do not extend well to problems in higher dimensions Cannot handle nonlinearities: Linear StringNonlinear String Cannot extract efficient delay-line structures…  BUT: when waveguides may be employed, they are far more efficient than any other technique!

12. Limitations  Modal synthesis Not computationally efficient Irregular geometries  huge memory costs (storage of modes) Also cannot handle distributed nonlinearities: Linear Plate Nonlinear (von Karman) Plate  These methods are extremely useful, as first approximations…

13. Observations These methods can be efficient, but: They are really “physical interpretations” of abstract methods: Wavetable synthesis  waveguides Additive synthesis  modal synthesis Can deal with some simple physical models this way, but not many.

14. Physical Modelling Synthesis: Time- domain Methods System of equations Numerical method (recursion) Musical instrument Output waveform Finite Difference Methods Finite Element Methods Spectral/Pseudospectral Methods Methods are completely general— no assumptions about behaviour Vast mainstream literature, 1920s to present.

15. FD schemes as recursions All time domain methods operate as recursions over values on a grid Recursion updated at a given sample rate f s Typical audio sample rates: 32000 Hz 44100 Hz 48000 Hz 96000 Hz

16. FD schemes as recursions  Solution evolves over time  Output waveform is read from a point on the grid  Entire state of object is computed at every clock tick

17. Sound Examples: Nonlinear Plate Under struck conditions, a wide variety of possible timbres: Under driven conditions, very much unlike plate reverberation…  Nonlinear plate vibration the basis for many percussion instruments: cymbals, gongs, tamtams  Audible nonlinear phenomena: subharmonic generation, buildup of high-frequency energy, pitch glides

18. FD Cymbal Modeling Cymbals: an interesting synthesis problem: Simple PDE description Regular geometry Highly nonlinear Time-domain methods are a very good match… A great example of a system which is highly nonlinear…linear models do not do justice to the sound! Linear modelNonlinear model Difference methods really the only viable option here…

19. FPGA percussion instrument (R. Woods/K. Chuchasz, Sonic Arts Research Centre/ECIT, Queen’s University Belfast)

20. FD Wind Instruments … Wind instrument models: Also very easily approached using FD methods… Clarinet Saxophone Squeaks! BUT: for simple tube profiles (cylindrical, conical), digital waveguides are far more efficient!

21. FD Modularized Synthesis: Coupled Strings/Plates/Preparation Elements String/soundboard connection Prepared plateBowed plate Spring networks A complex nonlinear modular interconnection of plates, strings, and lumped elements…

22. FD Plate Reverberation  Physical modeling…but not for synthesis!  Drive a physical model with an input waveform  In the linear case: classic plate reverberation (moving input, pickups)

23. RenderAIR: FD Room Acoustics Simulation RenderAIR: FD Room Acoustics Simulation (D. Murphy, S. Shelley, M. Beeson, A. Moore, A. Southern, University of York, UK)  Audio bandwidth 3D models = High Memory/High Computation load. Possible Solutions?  Uses Collada (Google Earth/Sketchup) format geometry files.  “Grows” a mesh to fit the user defined geometry.  Mesh topology/FDTD-Scheme plug-ins for speed of development.  Contact and related publications: Damian Murphy, University of York, UK  dtm3@ohm.york.ac.uk dtm3@ohm.york.ac.uk  http://www-users.york.ac.uk/~dtm3/research.html http://www-users.york.ac.uk/~dtm3/research.html

24. Musical Example Untitled (2008)---Gordon Delap stereo, realized in Matlab(!)

25. A general family of systems in musical acoustics A useful (but oversimplified) model problem: Parameters : d: dimension (1,2, or 3) p: stiffness (1 or 2) c: ‘speed’ V : d-dim. ‘volume’ p\d123 1 2  strings  acoustic tubes  membranes  room acoustics  bars  plates

26. Computational Cost: A Rule of Thumb Result: bounds on both memory requirements, and the operation count: # memory locations # arithmetic ops/sec Some points to note here: As c decreases, or as V becomes larger, the “pitch” decreases and computation increases: low-pitched sounds cost more… Complexity increases with dimension (strongly!) Complexity decreases with stiffness(!) The bound on memory is fundamental, regardless of the method employed…

27. Computational Costs 10 6 10 7 10 8 10 9 10 10 11 10 12 10 13 10 14 10 15 10 16 10 17 Arithmetic operations/second, at 48 kHz: Large acoustic spaces Plate reverberation Wind instruments Single string Bass drum Small-medium acoustic spaces Great variation in costs… Full piano Approx. limit of present realtime performance on commercially available desktop machines

28. Difficulties: Numerical Stability  For nonlinear systems, even in isolation, stability is a real problem.  Solution can become unstable very unpredictably…  Problems for composers, and, especially: live performers!  Even trickier in fixed- point arithmetic.

29. Parallelizability: Modal Synthesis Each mode evolves independently of the others: Result: independent computation for each mode (zero connectivity) Obviously an excellent property for hardware realizations. Each mode behaves as a “two-pole” filter:

30. Parallelizability: Explicit finite difference methods A useful type of scheme: explicit Each unknown value calculated directly from previously computed values at neighboring nodes “local” connectivity… Useful for linear problems… Unknown Known Update point

31. Parallelizability: Implicit finite difference methods Other schemes are implicit… Unknowns coupled to one another (locally) Useful for nonlinear problems… (stability!) Unknown Known Update group of points

32. Parallelizability: Sparse matrix representations Can always rewrite explicit updates as (sparse) matrix multiplications: = State transition matrix Last state Next state Sparse, often structured (banded, near Toeplitz) Size N by N, where N is the number of FD grid locations. NNZ entries: O(N)

33. Parallelizability: Sparse matrix representations Can sometimes write implicit updates as (sparse) linear system solutions: = Last state Next state Many fast methods available: Iterative… Thomas-type for banded matrices FFT-based for near-Toeplitz Different implications regarding parallelizability!

34. I/O: Modal Methods Modal representations are non-local: input/output at a given location requires reading/writing to all modes: Excitation point Readout point Location-dependent expansion coefficients Input Output Location-dependent expansion coefficients Expansion coefficients calculated offline! Must be recalculated for each separate I/O location Multiple outputs: need structures running in parallel…

35. I/O: Finite Difference Methods Finite difference schemes are essentially local: Input/output is very straightforward: insert/read values directly from computed grid… O(1) ops/time step Connect excitation element/insert sample Read value

36. I/O: Finite Difference Methods Multichannel I/O is very simple… No more costly than single channel! Connect excitation element/insert sample Read values Read/write over trajectory Moving I/O also rather simple Interpolation (local) required…

37. Boundary conditions Updating over interior is straightforward… Need spcialized updates at boundary locations… …as well as at coordinate boundaries

38. Modularized synthesis Idea: allow instrument designer (user) to connect together components at will: Basic object types: Strings Bars Plates Membranes Acoustic tubes Various excitation mechanisms Need to supply connection details (locations, etc.) Object 1 Object 2 Object 3

39. Challenges: Modular Stability Easy enough to design stable simulations for synthesis for isolated objects… Even for rudimentary systems, problems arise upon interconnection: Stable ConnectionUnstable Connection Mass/spring system Ideal String For more complex systems, instability can become very unpredictable…

40. Energy based Modular Stability Key property underlying all physical models is energy. For a system of lossless interconnected objects, each has an associated stored energy H: Each energy term is non-negative, and a function only of local state variables---can bound solution size: Numerical methods: assure same property in recursion in discrete time, i.e., H1H1 H2H2 H3H3 H4H4 Need to ensure positivity in discrete time…

41. Energy: Coupled Strings/Soundboard/Lumped Elements System Soundboard Energy Energy of Prepared Elements Energy of Strings Total Energy Can develop modular numerical methods which are exactly numerically conservative… A guarantee of stability… A useful debugging feature! Returning to the plate/string/prepared elements system, time

42. Concluding remarks Digital waveguides: Ideal for 1D linear uniform problems: ideal strings, acoustic tubes Extreme efficiency advantage… Modal synthesis: Apply mainly to linear problems Zero connectivity I/O difficulties (non-local excitation/readout) Possibly heavy precomputation Good for static (i.e., non-modular) configurations FD schemes Apply generally to nonlinear problems Local connectivity Stability difficulties I/O greatly simplified Minimal precomputation Flexible modular environments possible

43. References General Digital Sound Synthesis: C. Roads, The Computer Music Tutorial, MIT Press, Cambridge, Massachusetts,1996. R. Moore, Elements of Computer Music, Prentice Hall, Englewood Cliffs, New Jersey, 1990. C. Dodge and T. Jerse, Computer Music: Synthesis, Composition and Performance, Schirmer Books, New York, New York, 1985. Physical Modeling (general) V. Valimaki and J. Pakarinen and C. Erkut and M. Karjalainen, Discrete time Modeling of Musical Instruments, Reports on Progress in Physics, 69, 1—78, 2005. Special Issue on Digital Sound Synthesis, IEEE Signal Processing Magazine, 24(2), 2007. Digital Waveguides J. O. Smith III, Physical Audio Signal Procesing, draft version, Stanford, CA, 2004. Available online at http://ccrma.stanford.edu/~jos/pasp04/http://ccrma.stanford.edu/~jos/pasp04/ V. Välimäki, J. Huopaniemi, M. Karjalainen, and Z. Jánosy, “Physical modeling of plucked string instruments with application to real-time sound synthesis,” J. Audio Eng. Soc., vol. 44, no. 5, pp. 331–353, May 1996. V. Välimäki, H. Penttinen, J. Knif, M. Laurson, and C. Erkut, “Sound synthesis of the harpsichord using a computationally efficient physical model,” EURASIP Journal on Applied Signal Processing, vol. 2004, no. 7, pp. 934–948, June 2004. Modal Synthesis D. Morrison and J.-M. Adrien, MOSAIC: A Framework for Modal Synthesis, Computer Music Journal, 17(1):45—56, 1993. Finite Difference Methods S. Bilbao, Numerical Sound Synthesis, John Wiley and Sons, Chichester, UK, 2009 (under contract).