This page contains the bibliography for the AMS Notices article:  Music: Broken Symmetry, Geometry, and Complexity.  Many of the references are available for downloading by clicking on the appropriate link.

[1]  Acoustical Research Institute of the Austrian Academy of Sciences. STx software.    Download 

[2]  J. Adams.  Hallelujah Junction: Composing an American Life.  Farrar, Straus and Giroux, New York, 2008.

[3]  J.B. Allen and L.R. Rabiner.  A unified approach to short-time Fourier analysis and synthesis.  Proc. IEEE, Vol. 65 (1977), 1558-1564.

[4]  J.F. Alm and J.S. Walker. Time-frequency analysis of musical instruments.  SIAM Review, Vol. 44 (2002), 457-476.    Download

[5]   Appalachian Spring, Rodeo, Fanfare for the Common Man.  Atlanta Symphony Orchestra, cond. Louis Lane.  Telarc 1982.  Track 6:  Appalachian Spring (Suite).

[6]  R.B. Ash.  Information Theory.  Dover, New York, 1990.

[7]  G. Assayag, H. Feichtinger, J.-F. Rodrigues, (Eds.), 2002, Mathematics and music: a Diderot Mathematical Forum. Springer, New York.

[8]  Ave Verum Corpus--Motets And Anthems Of William Byrd, by Cambridge Singers. Recorded at Great Hall of University College School, London. Collegium Records, 2003.  Track~16:  Non Vos Relinquam.

[9]  P. Balazs.  Regular and Irregular Gabor Multiplier With Application To Psychoacoustic Masking.   PhD thesis, Univ. of Vienna, 2005.                                                                                                                                                                                                          
[10]  M. Barnsley.  Fractals Everywhere, Second Edition .  Academic Press, Cambridge, MA, 1993.

[11] The Beatles (The White Album).  Capitol/EMI, 1968. Disc 1, Track 11:Blackbird.   Converted to mono by Audacity sound editor.

[12]  J.W. Beauchamp (Ed.).  Analysis, Synthesis, and Perception of Musical Sounds:  The Sound of Music.  Springer, 2007.

[13]  T.C. Bell, J.G. Cleary, and I.H. Witten.  Text Compression.  Prentice-Hall, Englewood Cliffs, NJ, 1990.

[14]   Benny Goodman Orchestra, Jazz Collector Edition.  Laser Light Digital, 1991. Track 13:  Sing, Sing, Sing.  Circa 1943 live broadcast of Benny Goodman Orchestra.  

[15]  D. Benson.  Music: A Mathematical Offering.  Cambridge, 2006.

[16]  L. Bernstein.  The Joy of Music.  Amadeus Press, Pompton Plains, NJ, 2004.

[17]  Carreras, Domingo, Pavarotti in concert.  Orchestra del Maggio musicale fiorentino and Orchestra del Teatro dell'Opera di Roma, conducted by Zubin Mehta. Decca, 1990.  Track 12: Nessun Dorma.

[18]  P. Cassaza. The art of frame theory.  Taiwanese J. of Math.  Vol.4 (2000), 129--201.  Download

[19]  D. Chen and S. Qian.  Joint Time-Frequency Analysis: Methods and Applications.  Prentice Hall, Englewood Cliffs, NJ, 1996.              

[20]  X. Cheng, J.V. Hart, J.S. Walker.  Time-frequency analysis of musical rhythm.  Notices of the American Mathematical Society,
Vol. 56 (2009), 344--360.  Download

[21]  E. Chew. Math & Music -- The Perfect Match.  OR/MS Today, Vol. 35 (2008), 26--31.  Download

[22]  N. Chomsky.  Syntactic Structures.  Mouton de Gruyter, New York, 2002.

[23]  O. Christensen. Pairs of dual Gabor frame generators with compact support and desired frequency localization.  Appl. Comput. Harmon. Anal., Vol. 20 (2006), 403--410.   Download

[24]  O. Christensen.  An Introduction to Frames and Riesz Bases.  Birkauser, Boston, 2003.

[25]  O. Christensen and W. Sun.  Explicitly given pairs of dual frames with compactly supported generators and applications to irregular B-splines.  J. Approx. Theory, Vol. 151 (2008), 155--163.                                                                                                

[26]  C.K. Chui.  Wavelets: A Mathematical Tool for Signal Analysis.  SIAM, Philadelphia, PA, 1997.

[27]  R. Cogan.  New Images of Musical Sound.   Harvard University Press, Cambridge, MA, 1984.

[28]  Computer Music Journal.  Webpage:

[29]  T.M. Cover, J.A. Thomas.  Elements of Information Theory.  Wiley, New York 1991.

[30]   C-Sound is available at

[31]  I. Daubechies.  Ten Lectures on Wavelets.  SIAM, Philadelphia, PA, 1992.

[32]  I. Daubechies and S. Maes.  A nonlinear squeezing of the continuous wavelet transform based on auditory nerve models. In
Wavelets in Medicine and Biology,   pp. 527--546, CRC Press, Boca Raton, FL, 1996.

[33]  I. Daubechies, A. Grossman, and Y. Meyer. Painless nonorthogonal expansions.  J. Math. Phys., Vol. 27 (1986), 1271--1283.

[34]  G.W. Don.  Brilliant colors provocatively mixed: Overtone structures in the music of Debussy.  Music Theory Spectrum, Vol. 23 (2001), 61--73.

[35]  G.W. Don, Fern.  Unpublished recording, available at

[36]  G.W. Don and J.S. Walker, Sierpinski Round.  Unpublished recording, available at                                                                                                    

[37]  M. Dörfler.  Gabor Analysis for a Class of Signals called Music.  Dissertation, University of Vienna, 2002.  Download

[38]  M.Dörfler.  Time-Frequency Analysis for Music Signals---a Mathematical Approach.  Journal of New Music Research, Vol. 30, No. 1, March 2001.  Download

[39]  M. Dörfler and H. Feichtinger. (2004). Quilted Gabor families I: reduced multi-Gabor frames.  Appl. Comput. Harmon. Anal., Vol. 356, 2001--2023.   Download

[40]  M. Dörfler and H. G. Feichtinger.  Quantitative Description of Expression in Performance of Music, Using Gabor Representations.  Proceedings of the Diderot Forum on Mathematics and Music, 1999, Vienna.

[41]  M. Dörfler and B. Torresani.  Representation of operators in the time-frequency domain and generalized Gabor multipliers. To appear in J. Fourier Anal. Appl.   Download

[42]  J.S. Downie. Music Information Retrieval online bibliography:

[43]  J. Du, M.W. Wong, and H. Zhu.  Continuous and discrete inversion formulas for the Stockwell transform.  Integral Transforms Spec. Funct., Vol.  18 (2007), 537--543.

[44]  J. Duesenberry.  Convolution reverb and beyond.  Electronic Musician, April 1, 2005.  Download

[45]  R.J. Duffin and A.C. Schaeffer.  A class of nonharmonic Fourier series.  Trans. A.M.S., Vol. 72 (1952), 341--366.

[46]  Electric Ladyland, Jimi Hendrix Experience.  Reprise, 1968. Track~15:  All Along the Watchtower.  Converted to mono by Audacity sound editor.

[47]  J. Fauvel, R. Flood, and R. Wilson, Eds.  Music and Mathematics: From Pythagoras to Fractals.  Oxford, New York, 2003.

[48]  H.G. Feichtinger, F. Luef, and T. Werther. A Guided Tour from Linear Algebra to the Foundations of Gabor Analysis. In Gabor and Wavelet Frames, IMS Lecture Notes Series, Vol. 10 (2007), 1--49.   Download

[49]  H.G. Feichtinger and K. Nowak, 2003, A First Survey of Gabor Multipliers. In ibid.  Download

[50]  H.G. Feichtinger and T. Strohmer, Eds.  Gabor Analysis and Algorithms.  Birkhauser, Boston, MA, 1998.

[51]  H.G. Feichtinger and T. Strohmer, Eds.  Advances in Gabor Analysis.  Birkhauser, Boston, MA, 2002.

[52]  S. Flinn, Shepard's Tones webpage:

[53]  D. Gabor.  Theory of communication.  Journal of the Institute for Electrical Engineers, Vol. 93 (1946), 873--880.

[54]  P.C. Gibson, M.P. Lamoureux, and G.F. Margrave.  Letter to the editor: Stockwell and wavelet transforms.  J. Fourier Anal. Appl., Vol. 12 (2006), 713--721.

[55]  K. Gröchenig.  Foundations of Time-Frequency Analysis.  Birkhauser, Boston, MA, 2001.

[56]  L. Harkleroad.  The Math Behind the Music.  Cambridge University Press, Cambridge, UK, 2006.

[57]  D.M. Healy Jr. and S. Li.  A parametric class of discrete Gabor expansions.  IEEE Trans. Signal Process., Vol. 44 (1996), 201--211.

[58]  C.E. Heil and D.F. Walnut. Continuous and discrete wavelet transforms.  SIAM Review,Vol. 34 (2001), 628--666.  Download

[59]  H. Helmholtz.  On the Sensations of Tone.  Dover, NY, 1954.

[60]  D.R. Hofstadter.  Gödel, Escher, Bach: an Eternal Golden Braid.  Vintage, New York, 1980.

[61]  D. Kroodsma.  The Singing Life of Birds.  Houghton-Mifflin, NY, 2005.

[62]  J. Latartara.  Pedagogic Applications of Fourier Analysis and Spectrographs in the Music Theory Classroom.  J. of Music Theory Pedagogy, Vol. 22 (2008), 61--90.

[63]  Layla and Other Assorted Love Songs, Derek and the Dominoes.  Original recording remastered.  Polydor, 1970. Track 13:  Layla. Converted to mono by Audacity sound editor.

[64]  Louis Armstrong: Gold.  Hip-O Records, 1996.  Original June 26, 1950 recording remastered.  Disc 2, Track 3:  La Vie En Rose.

[65]  G. Loy.  Musimathics: The Mathematical Foundations of Music, Vol. 2.  MIT Press, Cambridge, MA, 2007.

[66]  Ludwig van Beethoven, Piano Sonata in E (Opus 109), Movements 1 and 2.  Performed by Daniel Anez Garcia on January 25, 2007 at Jeneusses Musicales du Canada. Available at

[67]  F.B. Mache.  Music, Myth and Nature. Contemporary Music Studies, Vol.6.  Taylor & Francis, London, 1993.

[68]  S. Mallat.  A Wavelet Tour of Signal Processing. Second Edition.  Academic Press, San Diego, CA, 1999.

[69]  MatLab Time-frequency toolboxes:

[70]  G. Mazzola. (2002).  The Topos of Music.  Birkhauser, Basel.

[71]  MetaSynth music software:

[72]  L.B. Meyer.  Music the Arts and Ideas: Patterns and Predictions in Twentieth-Century Culture.  University of Chicago Press, 1967.

[73]  S. Mithen.  The Singing Neanderthals:  The Origins of Music, Language, Mind, and Body.  Harvard University Press, 2006.

[74]  Journal of New Music.  Website:

[75]  W. O'Grady, M. Dobrovolsky, and M. Arnoff.  Contemporary Linguistics, An Introduction.  St. Martins Press, New York, 1993.

[76]  L.R. Rabiner and J.B. Allen.  Short-time Fourier analysis techniques for FIR system identication and power spectrum estimation. IEEE Trans. Acoust. Speech Signal Process., Vol. 27 (1979), 182--192.

[77]  Perspectives of New Music.  Website:

[78]  Rahn, J., webpage:

[79]  A. Ross.  The Rest is Noise:  Listening to the Twentieth Century.  Farrar, Straus and Giroux, New York, 2007.

[80]  D. Rothenberg.  Why Birds Sing: A Journey into the Mystery of Bird Song.  Basic Books, NY, 2005.

[81]  Satie: L'Oeuvre pour piano, vol. 1.  Aldo Ciccolini, piano, recorded in 1983. EMI, 1987.  Track 7: Gymnopedie I.

[82]  K. Schnass.  Gabor Multipliers: A self-contained survey.  Master's thesis, Univ. of Vienna, 2004.

[83]  W. Sethares.  Rhythm and Transforms.  Springer, New York, NY, 2007.

[84]  W. Sethares.  Tuning, Timbre, Spectrum, Scale.  Springer, London, 2005.

[85]  R.N. Shepard. Circularity in judgements of relative pitch.  J. of the Acoustical Soc. of America, Vol. 36 (1964), 2346--53.

[86]  L.M. Smith. Modelling rhythm perception by continuous time-frequency analysis.  Proceedings of the 1996 International Computer Music Conference, Hong Kong, 392--395.  Download

[87]  L.M. Smith. A multiresolution time-frequency analysis and interpretation of musical rhythm.  Thesis, University of Western Australia, 2000.   Download

[88]  L.M. Smith and H. Honing. Time-frequency representation of musical rhythm by continuous wavelets.  J.of Mathematics and Music, Vol. 2 (2008), 81--97.   Download

[89]  E. Stern, quoted from the Pandolous website.  Click here.

[90]  R.G. Stockwell.  A basis for efficient representation of the S-transform.  Digital Signal Processing, Vol. 17 (2007), 371--393.

[91]  R.G. Stockwell, L. Mansinha, and R.P. Lowe.  Localization of the complex spectrum: the S transform.  IEEE Trans. Signal Processing, Vol. 44 (1996), 998--1001.   Download

[92]  T. Strohmer.  Numerical algorithms for discrete Gabor expansions.  See [50, pp. 267--294].

[93]  Suite: The Firebird, Igor Stravinsky.  Boston Symphony Orchestra, 1964 recording, cond. Erich Leinsdorf.  BMG Classics, 1991.

[94]  Time Further Out, Dave Brubeck Quartet.  SONY, 1961.  Track 7: Unsquare Dance.  Recorded on June 8, 1961.

[95]  D. Tymoczko, The geometry of musical chords.  Science, Vol. 313 (2006), 72--74.  Download

[96]  J.S. Walker.  A Primer on Wavelets and their Scientific Applications, Second Edition.  Chapman & Hall/CRC Press, Boca Raton, FL, 2008.  Material referenced from Chapters 5 and 6 is available at

[97]  T. Werther et al.  CPR Artifact Removal in Ventricular Fibrillation ECG Signals Using Gabor Multipliers.  IEEE Trans. Biomedical Engineering, Vol. 56 (2009), 320--327.

[98]   Why Birds Sing website:

[99]  R.M. Young.  An Introduction to Nonharmonic Fourier Series.  Academic Press, New York, 1980.

[100]  M. Zibulski and Y. Zeevi.  Discrete multiwindow Gabor-type transforms.  IEEE Trans. Signal Processing, Vol. 45 (1997), 1428--1442.