Physics of Music

Science Day lecture at The Institute of Mathematical Sciences

Transcript

1. The sound of music and the physics of sound Ronojoy Adhikari
2. The sound of music
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
34. Acknowledgement Most of this work was done by G. Sathej and Ganesh Saraswat, both from IIT-M.
35. Thank you for your attention!