Physics of Music

Physics of Music

Science Day lecture at The Institute of Mathematical Sciences

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Ronojoy Adhikari

February 28, 2010
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Transcript

  1. The sound of music and the physics of sound Ronojoy

    Adhikari
  2. The sound of music

  3. None
  4. Nad Brahma Sound is Divine!

  5. Can we understand the mundane parts of what goes into

    making music ? The physics of sound.
  6. Some aspects of the physics of sound • Classification of

    musical instruments. • The normal modes a string : lutes. • The normal modes of an uniform membrane : drums. • The normal modes of a non-uniform membrane : Indian drums. • Can one hear the shape of a drum ? The isospectral problem. • The numerical synthesis of sound.
  7. Classification : Mahillon’s fourfold way • Proposed by Victor Mahillon

    (1878) in Brussells. • Idiophones : the body of the instrument vibrates. • Aerophones : the air enclosed by the instrument vibrates. • Chordophones : the string of the instrument vibrates. • Membranophones : the membrane of the instrument vibrates.
  8. Natyashashtra classification • Natyashastra (200 BCE - 200 CE) is

    a comprehensive treatise on the performing arts. • The Natyashastra classification of instruments is ... • Ghana (solid) : ideophones • Sushira (hollow) : aerophones • Tata (stretched) : chordopones • Avanadda (covered) : membranophones
  9. Did Mahillon follow the Natyashastra classification ? Yes! Interesting route

    of transmission, ask later.
  10. Idiophone : ghana

  11. Aerophone : sushira

  12. Chordophone : tata

  13. Membranophone : avanaddha

  14. String vibrations Overtones are integer multiples of the fundamental. This

    is what makes string vibrations well-defined in pitch.
  15. (Uniform) membrane vibrations frequencies given by zeros of Bessel functions

    Overtones are not integer multiples of the fundamental. This gives drums a very diffuse sense of pitch.
  16. Tabla, mridangam, many other Indian drums have a very sharp

    sense of pitch. They are tuned to the tonic of the musician. ?
  17. Raman : first systematic study Raman made a study of

    classic simplicity to obtain the normal modes of the mridangam. He found that the first five overtones were integer multiples of the fundamental. We will reproduce Raman’s study using modern methods in the afternoon demos.
  18. What about a mathematical model ? mass density tension Ramakrishna

    and Sondhi : step function loading, analytical solution possible for concentric loading. Uncontrolled approximation for eccentric loading.
  19. a b Tabla Mathematical model G. Sathej and RA :

    more realistic loading, spectrally accurate solution, easy to handle both concentric and eccentric loadings.
  20. 1 2 3 4 5 1 1.5 2 2.5 3

    3.5 4 4.5 5 5.5 m t t 1 t 2 t 3 t 4 t 5 t 6 t 7 t 9 t 8 Variation of eigenvalues with loading
  21. At what parameters is the drum most harmonic ?

  22. 1 2 3 4 5 6 7 8 9 10

    11 12 13 14 15 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 N−th overtone Frequency ratio Experimental Numerical Analytical Experiment, theory, numerics
  23. h = 1.0995 h = 2.0000 h = 2.0000 h

    = 2.9582 h = 2.9582 h = 3.0535 h = 3.9541 h = 3.9541 h = 4.0204 h = 4.0204 h = 4.9041 h = 4.9687 h = 4.9687 h = 4.9727 h = 4.9727 h = 5.7441 h = 5.7441 h = 5.9409 h = 5.9409 h = 5.9897 Nodal lines
  24. Mode 1 Mode 2 Mode 3 Mode 4 h =

    1.0345 h = 2.0000 h = 3.0394 h = 3.0534 Shapes of the eigenmodes
  25. Eccentric loading e

  26. 0 0.1 0.2 0.3 0.4 0.5 0.5 1 1.5 2

    2.5 3 3.5 4 4.5 ¡ t t 1 t 2 t 3 t 4 t 5 t 6 t 7 t 8 t 9 Eccentric drum : variation with eccentricity
  27. h = 0.4846 h = 0.9960 h = 1.0000 h

    = 1.3843 h = 1.5617 h = 1.5628 h = 1.7341 h = 1.7752 h = 2.0903 h = 2.1012 h = 2.1017 h = 2.2158 h = 2.2166 h = 2.4850 h = 2.4934 h = 2.5593 h = 2.5613 h = 2.7926 h = 2.7928 h = 2.8490 Nodal contours : eccentric
  28. h = 1.0995 h = 2.0000 h = 2.0000 h

    = 2.9582 h = 2.9582 h = 3.0535 h = 3.9541 h = 3.9541 h = 4.0204 h = 4.0204 h = 4.9041 h = 4.9687 h = 4.9687 h = 4.9727 h = 4.9727 h = 5.7441 h = 5.7441 h = 5.9409 h = 5.9409 h = 5.9897 Nodal contours : concentric
  29. Mode 1 h = 0.4846 Mode 2 Mode 3 Mode

    4 h = 0.9960 h = 1.0000 h = 1.3843 Shapes of the eigenmodes
  30. Can one hear the shape of a drum ? (Kac,

    1960) If the eigenvalues of a membrane bounded by an arbitrary region is given, can one infer the shape of the region from the eigenvalues ? Answer is no! Gordon, Webb, Wolpert, 1991
  31. None
  32. None
  33. None
  34. Acknowledgement Most of this work was done by G. Sathej

    and Ganesh Saraswat, both from IIT-M.
  35. Thank you for your attention!