Citation
David Yeh. "Numerical solution of nonlinear differential equations in musical synthesis". Talk or presentation, 11, March, 2008; From the CHESS Seminar.

Abstract
In creating algorithms for musical synthesis based upon physical modeling, one often assumes a linear system to take advantage of transformations that simplify the computation. In many cases, nonlinearities are responsible for the musically interesting variations over time that characterize an instrument, and such assumptions reduce the realism of the algorithm. Often one seeks efficient methods to emulate the nonlinear effect; however, the full dynamics are not captured unless the nonlinear differential equations are being solved.

The talk will begin with a summary of results from the presenter's work investigating the performance of various simple numerical integration methods on a particularly difficult circuit example involving a saturating nonlinearity. This talk then reviews two numerically based methods in the musical acoustics literature to directly solve nonlinear ordinary differential equations in real-time for musical synthesis. These methods can generate explicit computations to process incoming signals robustly in real-time even though the nonlinearities in the system may be implicitly defined. This explicitness provides opportunities to improve run-time performance on parallel hardware. An example found in the literature to simulate the partial differential equations of nonlinear plates using an FPGA will demonstrate the need for parallel hardware. Suggestions are sought from the audience regarding the parallelizability of these and other algorithms to solve differential equations numerically.

Electronic downloads

• chesssem20080311.pdf · application/pdf · 2467 kbytes

• HTML

  David Yeh. <a
  href="http://chess.eecs.berkeley.edu/pubs/403.html"><i>Numerical
  solution of nonlinear differential equations in musical
  synthesis</i></a>, Talk or presentation,  11,
  March, 2008; From the <a
  href="http://chess.eecs.berkeley.edu/seminar.htm">CHESS
    Seminar</a>.

• Plain text

  David Yeh. "Numerical solution of nonlinear differential
  equations in musical synthesis". Talk or presentation,  11,
  March, 2008; From the CHESS  
  Seminar.

• BibTeX

  @presentation{Yeh08_NumericalSolutionOfNonlinearDifferentialEquationsInMusical,
      author = {David Yeh},
      title = {Numerical solution of nonlinear differential
                equations in musical synthesis},
      day = {11},
      month = {March},
      year = {2008},
      note = {From the CHESS
                  Seminar},
      abstract = {In creating algorithms for musical synthesis based
                upon physical modeling, one often assumes a linear
                system to take advantage of transformations that
                simplify the computation. In many cases,
                nonlinearities are responsible for the musically
                interesting variations over time that characterize
                an instrument, and such assumptions reduce the
                realism of the algorithm. Often one seeks
                efficient methods to emulate the nonlinear effect;
                however, the full dynamics are not captured unless
                the nonlinear differential equations are being
                solved. 
  The talk will begin with a summary of
                results from the presenter's work investigating
                the performance of various simple numerical
                integration methods on a particularly difficult
                circuit example involving a saturating
                nonlinearity. This talk then reviews two
                numerically based methods in the musical acoustics
                literature to directly solve nonlinear ordinary
                differential equations in real-time for musical
                synthesis. These methods can generate explicit
                computations to process incoming signals robustly
                in real-time even though the nonlinearities in the
                system may be implicitly defined. This
                explicitness provides opportunities to improve
                run-time performance on parallel hardware. An
                example found in the literature to simulate the
                partial differential equations of nonlinear plates
                using an FPGA will demonstrate the need for
                parallel hardware. Suggestions are sought from the
                audience regarding the parallelizability of these
                and other algorithms to solve differential
                equations numerically.},
      URL = {http://chess.eecs.berkeley.edu/pubs/403.html}
  }
  

Posted by Christopher Brooks on 14 Mar 2008.
For additional information, see the Publications FAQ or contact webmaster at chess eecs berkeley edu..

Notice: This material is presented to ensure timely dissemination of scholarly and technical work. Copyright and all rights therein are retained by authors or by other copyright holders. All persons copying this information are expected to adhere to the terms and constraints invoked by each author's copyright.