The sound of music and the physics of sound
Ronojoy Adhikari
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The sound of music
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Nad Brahma
Sound is Divine!
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Can we understand the
mundane parts of what goes into
making music ?
The physics of sound.
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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.
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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.
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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
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Did Mahillon follow the Natyashastra classification ?
Yes!
Interesting route of transmission, ask later.
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Idiophone : ghana
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Aerophone : sushira
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Chordophone : tata
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Membranophone : avanaddha
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String vibrations
Overtones are integer
multiples of the fundamental.
This is what makes string vibrations
well-defined in pitch.
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(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.
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Tabla, mridangam, many other Indian drums have a very sharp sense of pitch.
They are tuned to the tonic of the musician.
?
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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.
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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.
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a
b
Tabla
Mathematical model
G. Sathej and RA : more realistic loading, spectrally accurate solution, easy to handle
both concentric and eccentric loadings.
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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
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
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Mode 1 Mode 2
Mode 3 Mode 4
h = 1.0345
h = 2.0000
h = 3.0394
h = 3.0534
Shapes of the eigenmodes
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Eccentric loading
e
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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
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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
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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
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Mode 1
h = 0.4846
Mode 2
Mode 3 Mode 4
h = 0.9960
h = 1.0000 h = 1.3843
Shapes of the eigenmodes
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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
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Acknowledgement
Most of this work was done
by G. Sathej and Ganesh
Saraswat, both from IIT-M.