*banner
 

Numerical solution of nonlinear differential equations in musical synthesis
David Yeh

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

Citation formats  
  • 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 <a
    href="http://chess.eecs.berkeley.edu/seminar.htm"
    >CHESS   Seminar</a>.
  • 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 <a
                  href="http://chess.eecs.berkeley.edu/seminar.htm"
                  >CHESS   Seminar</a>},
        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. <p>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.
Groups: chess
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.

You are not logged in 
©2002-2014 Chess