Musical Instruments as Synchronized Systems

  • Rolf Bader
Part of the Springer Handbooks book series (SHB)


Most musical instrument families have nearly perfect harmonic overtone series, for example plucked, bowed, or wind instruments. However, when considering the complex geometry and nonlinear driving mechanisms many of these instruments have we would expect them to have very inharmonic overtone series. So to make musical instruments play notes that we accept as harmonic sounds, synchronization needs to occur to arrive at the perfect harmonic overtone series the instruments actually produce. The reasons for this synchronization are different in the singing voice, organs, saxophones or clarinets, violin bowing or in plucked stringed instruments. However, when examining the mechanisms of synchronization further, we find general rules and suitable algorithms to understand the basic behavior of these instruments.


finite-difference time domain


finite element method


impulse pattern formulation


  1. 10.1
    H. Haken: Synergetics (Springer, Berlin, Heidelberg 1990)zbMATHGoogle Scholar
  2. 10.2
    J. Argyris, G. Faust, M. Haase, R. Friedrich: An Exploration of Dynamical Systems and Chaos (Springer, Berlin, Heidelberg 2015)CrossRefGoogle Scholar
  3. 10.3
    V. Aschoff: Experimentelle Untersuchungen an einer Klarinette. [Experimental investigations of a clarinet], Akust. Z. 1, 77–93 (1936)Google Scholar
  4. 10.4
    J. Sundberg: The Science of the Singing Voice (Nothern Illinois University Press, DeKalb 1988)Google Scholar
  5. 10.5
    I.R. Titze: The physics of small-amplitude oscillation of the vocal folds, J. Acoust. Soc. Am. 83, 1536–1552 (1988)CrossRefGoogle Scholar
  6. 10.6
    T. Fitch, J. Neubauer, H. Herzel: Calls out of chaos: The adaptive significance of nonlinear phenomena in mammalian vocal production, Animal Behav. 63(3), 407–418 (2002)CrossRefGoogle Scholar
  7. 10.7
    P. Mergell, H. Herzel, T. Wittenberg, M. Tigges, U. Eysholdt: Phonation onset: Vocal fold modeling and high-speed glottography, J. Acoust. Soc. Am. 104(1), 464–470 (1998)CrossRefGoogle Scholar
  8. 10.8
    P. Mergell, H. Herzel, I.R. Tietze: Irregular vocal-fold vibration – High-speed observation and modeling, J. Acoust. Soc. Am. 108(6), 2996–2300 (2000)CrossRefGoogle Scholar
  9. 10.9
    A. Behrmann, R.J. Baken: Correlation dimension of electroglottographic data from healthy and pathologic subjects, J. Acoust. Soc. Am. 102(4), 2371–2379 (1997)CrossRefGoogle Scholar
  10. 10.10
    J. Neubauer, M. Edgerton, H. Herzel: Nonlinear phenomena in contemporary vocal music, J. Voice 18(1), 1–12 (2004)CrossRefGoogle Scholar
  11. 10.11
    K. Ishizaka: Equivalent lumped-mass models of vocal fold vibration. In: Vocal Fold Physiology (1981) pp. 231–244Google Scholar
  12. 10.12
    I.R. Titze, Sh S. Schmidt, M.R. Titze: Phonation threshold pressure in a physical model of the vocal fold mucosa, J. Acoust. Soc. Am. 97(5), 3080–3084 (1995)CrossRefGoogle Scholar
  13. 10.13
    J.J. Jiang, Y. Zhang: Modeling of chaotic vibrations in symmetric vocal folds, J. Acoust. Soc. Am. 110(4), 2120–2128 (2001)CrossRefGoogle Scholar
  14. 10.14
    J.J. Jiang, Y. Zhang: Chaotic vibration induced by turbulent noise in a two-mass model of vocal folds, J. Acoust. Soc. Am. 112(5), 2127–2133 (2002)CrossRefGoogle Scholar
  15. 10.15
    J.G. Švec, H.K. Schutte, D.G. Miller: On pitch jumps between chest and falsetto registers in voice: Data from living and excised human larynges, J. Acoust. Soc. Am. 106(3), 1523–1531 (1999)CrossRefGoogle Scholar
  16. 10.16
    I.T. Tokuda, M. Zemke, M. Kob, H. Herzel: Biomechanical modeling of register transition and the role of vocal tract resonators, J. Acoust. Soc. Am. 127(3), 1528–1536 (2010)CrossRefGoogle Scholar
  17. 10.17
    J.C. Lucero: Dynamics of the two-mass model of the vocal folds: Equilibria, bifurcations, and oscillation region, J. Acoust. Soc. Am. 94(6), 3104–3111 (1993)CrossRefGoogle Scholar
  18. 10.18
    J.C. Lucero: A theoretical study of the hysteresis phenomenon at vocal fold oscillation onset-offset, J. Acoust. Soc. Am. 105(1), 423–431 (1999)CrossRefGoogle Scholar
  19. 10.19
    J.C. Lucero, L.L. Koenig, K.G. Lourenço, N. Ruty, X. Pelorson: A lumped mucosal wave model of the vocal folds revisited: Recent extensions and oscillation hysteresis, J. Acoust. Soc. Am. 129(3), 1568–1579 (2011)CrossRefGoogle Scholar
  20. 10.20
    P.Å. Lindestad, M. Södersten, B. Merker, S. Granqvist: Voice source characteristics in Mongolian ‘throat singing’ studied with high-speed imaging technique, acoustic spectra, and inverse filtering, J. Voice 15(1), 78–85 (2001)CrossRefGoogle Scholar
  21. 10.21
    I. Steinecke, H. Herzel: Birfurcations in an asymmetric vocal fold model, J. Acoust. Soc. Am. 97, 1874–1884 (1995)CrossRefGoogle Scholar
  22. 10.22
    M.-H. Lee, J.N. Lee, K.-S. Soh: Chaos in segments from Korean traditional singing and Western singing, J. Acoust. Soc. Am. 103(2), 1175–1182 (1998)CrossRefGoogle Scholar
  23. 10.23
    D.A. Berry, H. Herzel, I.R. Titze, K. Krischer: Interpretation of biomechanical simulations of normal and chaotic vocal fold oscillations with empirical eigenfunctions, J. Acoust. Soc. Am. 95(6), 3595–3604 (1994)CrossRefGoogle Scholar
  24. 10.24
    Q. Xue, R. Mittal, X. Zhang: A computational study of the effect of vocal-fold asymmetry on phonation, J. Acoust. Soc. Am. 128(2), 181–187 (2010)CrossRefGoogle Scholar
  25. 10.25
    F.A. Berry, H. Herzel, I.R. Tieze, B.H. Story: Bifurcations in excised larynx experiments, J. Voice 10, 129–138 (1996)CrossRefGoogle Scholar
  26. 10.26
    T. Lerch: Vergleichende Untersuchung von Bohrungsprofilen historischer Blockflöten des Barock (Comparative investigation of bore profiles of historical Barock recorder flutes) (Staatliches Institut für Musikforschung. Preussischer Kulturbesitz Musikinstrumentenmuseum, Berlin 1996)Google Scholar
  27. 10.27
    C.J. Nederveen: Acoustical Aspects of Musical Instruments (Northern Illinois University Press, DeKalb 1998)Google Scholar
  28. 10.28
    A.H. Benade: Fundamentals of Musical Acoustics (Oxford Univ. Press, New York 1976)Google Scholar
  29. 10.29
    G. Krassnitzer: Multiphonics für Klarinette mit deutschem System und andere zeitgenössische Spielarten. (Multiphonics for clarinet with german system and other contemporary styles) (edition ebenos, Aachen 2002)Google Scholar
  30. 10.30
    P.A. Durbin, R. Pettersson: Statistical Theory and Modeling for Turbulent Flows (Wiley, Chichester 2001)zbMATHGoogle Scholar
  31. 10.31
    B. Fabre, A. Hirschberg, A.P.J. Wijnands: Vortex shedding in steady oscillation of a flue organ pipe, Acta Acust. United Acust. 82, 863–877 (1996)Google Scholar
  32. 10.32
    J.-P. Dalmont, J. Gilbert, J. Kergomard, S. Ollivier: An analytical prediction of the oscillation and extinction thresholds of a clarinet, J. Acoust. Soc. Am. 118(5), 3294–3305 (2005)CrossRefGoogle Scholar
  33. 10.33
    R. Kaykayoglu, D. Rockwell: Unstable jet-edge interaction. Part 1. Instantaneous pressure fields at a single frequency, J. Fluid Mech. 169, 125–149 (1986)CrossRefGoogle Scholar
  34. 10.34
    R. Kaykayoglu, D. Rockwell: Unstable jet-edge interaction. Part 2: Multiple frequency pressure fields, J. Fluid Mech. 169, 151–172 (1986)CrossRefGoogle Scholar
  35. 10.35
    A. Richter, R. Grundmann: Numerical investigations of the bassoons aeroacoustic, J. Acoust. Soc. Am. 123, 3448 (2008)CrossRefGoogle Scholar
  36. 10.36
    R. Bader: Nonlinearities and Synchronization in Musical Acoustics and Music Psychology, Springer Series Current Research in Systematic Musicology, Vol. 2 (Springer, Heidelberg 2013)CrossRefGoogle Scholar
  37. 10.37
    J.W. Coltman: Sounding mechanism of the flute and organ pipe, J. Acoust. Soc. Am. 44(4), 983–992 (1968)CrossRefGoogle Scholar
  38. 10.38
    M. Abel, S. Bergweiler, R. Gerhard-Multhaupt: Synchronization of organ pipes: Experimental observations and modeling, J. Acoust. Soc. Am. 119, 2467 (2006)CrossRefGoogle Scholar
  39. 10.39
    W. Lottermoser: Orgeln, Kirchen und Akustik (Organs, Churches, and Acoustics) (Erwin Bochinsky, Frankfurt a.M. 1983)Google Scholar
  40. 10.40
    C. Koehn: A bowed bamboo tube zither from Southeast Asia. In: ISMA, Le Mans 2014 (2014) pp. 499–502Google Scholar
  41. 10.41
    G. Müller, W. Lauterborn: The bowed string as a nonlinear dynamical system, Acustica 82, 657–664 (1996)Google Scholar
  42. 10.42
    C.V. Raman: On the mechanical theory of the vibrations of bowed strings and of musical instruments of the violin family, with experimental verification of the results, Bull. Indian Assoc. Cultivat. Sci. 15, 1–158 (1918)Google Scholar
  43. 10.43
    L. Cremer: The Physics of the Violin (MIT Press, Cambridge 1985)Google Scholar
  44. 10.44
    A. Askenfeld: Measurements of bow motion and bow force in violin playing, J. Acoust. Soc. Am. 80, 1007–1015 (1986)CrossRefGoogle Scholar
  45. 10.45
    P. Duffour, J. Woodhouse: Instability of systems with a frictional point contact: Part 1, Basic modelling, J. Sound Vib. 271, 365–390 (2004)CrossRefGoogle Scholar
  46. 10.46
    P. Duffour, J. Woodhouse: Instability of systems with a frictional point contact: Part 2, Model extensions, J. Sound Vib. 271, 391–410 (2004)CrossRefGoogle Scholar
  47. 10.47
    W. Güth: A comparison of the Raman and the oscillator models of string excitation by bowing, Acustica 82, 169–174 (1996)zbMATHGoogle Scholar
  48. 10.48
    M.E. McIntyre, J. Woodhouse: Fundamentals of bowed-string dynamics, Acustica 43, 93–108 (1979)zbMATHGoogle Scholar
  49. 10.49
    M.E. McIntyre, J. Woodhouse: On the oscillations of musical instruments, J. Acoust. Soc. Am. 74(5), 1325–1345 (1983)CrossRefGoogle Scholar
  50. 10.50
    R. Bader: Whole geometry finite-difference modeling of the violin. In: Proc. Forum Acusticum 2005 (2005) pp. 629–634Google Scholar
  51. 10.51
    R.J. Hanson, A.J. Schneider, F.W. Halgedahl: Anomalous low-pitched tones from a bowed violin string, J. Catgut Acoust. Soc. 2, 1–7 (1994)Google Scholar
  52. 10.52
    M. Kimura: How to produce subharmonics on the violin, New Music Res. 28, 178–184 (1999)CrossRefGoogle Scholar
  53. 10.53
    J. Angster, J. Angster, A. Miklós: Coupling between simultaneously sounded organ pipes, AES E-Library 94, 1–8 (1993)Google Scholar
  54. 10.54
    D.H. Keefe, B. Laden: Correlation dimension of woodwind multiphonic tones, J. Acoust. Soc. Am. 90(4), 1754–1765 (1991)CrossRefGoogle Scholar
  55. 10.55
    D. Borgo: Sync or Swarm. Improvising Music in a Complex Age (Bloomsbury Academic, New York, London 2005)Google Scholar
  56. 10.56
    V. Gibiat: Phase space representations of acoustical musical signals, J. Sound Vib. 123(3), 529–536 (1988)MathSciNetCrossRefGoogle Scholar
  57. 10.57
    R.V. Velazques: Ancient aerophones with mirliton. In: Proceedings ISGMA (2004) pp. 363–373Google Scholar
  58. 10.58
    N.H. Fletcher: Mode locking in nonlinearly excited inharmonic musical oscillators, J. Acoust. Soc. Am. 64, 1566–1569 (1978)CrossRefGoogle Scholar
  59. 10.59
    S. Dubnov, X. Rodet: Investigation of phase coupling phenomena in sustained portion of musical instruments sound, J. Acoust. Soc. Am. 113, 348–359 (2003)CrossRefGoogle Scholar
  60. 10.60
    K.A. Legge, N.H. Fletcher: Nonlinear generation of missing modes on a vibrating string, J. Acoust. Soc. Am. 76(1), 5–12 (1984)CrossRefGoogle Scholar
  61. 10.61
    R. Bader: Theoretical framework for initial transient and steady-state frequency amplitudes of musical instruments as coupled subsystems. In: Proc. 20th Int. Symp. Music Acoust. (ISMA) (2010) pp. 1–8Google Scholar
  62. 10.62
    P. Cariani: Temporal codes, timing nets, and music perception, J. New Music Res. 30(2), 107135 (2001)Google Scholar
  63. 10.63
    F. Messner: Friction blocks of New Ireland. In: Australia and the Pacific Islands, Garland Encyclopedia of World Music, Vol. 9, ed. by A.L. Kaeppler, J.W. Love (Routledge, London 1998) pp. 380–382Google Scholar
  64. 10.64
    N.J. Conrad, M. Malina, S.C. Münzel: New flutes document the earliest musical tradition in southwestern Germany, Nature 460, 737–740 (2009)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2018

Authors and Affiliations

  • Rolf Bader
    • 1
  1. 1.Institute of Systematic MusicologyUniversity of HamburgHamburgGermany

Personalised recommendations