HOW DO YOU MAKE A DRUM HARMONIC?
James James HaguesHagues
33rdrd March March 20102010
Harmonicity
Harmonics are produced when any note played
Perfectly harmonic = Harmonics with frequencies of integer multiples
Harmonic instruments – violin, piano Inharmonic instruments – ordinary drum,
cymbal
The Tabla
Tabla – indian set of drums Harmonicity achieved by loaded region
Membrane of right-handed tabla
Solving the ordinary drum
Wave Equation - Description of displacement over time
Solution for the ordinary drum
Two-Density Drum
Use 2-density drum to model tabla Solve t first to get
Helmholtz equations for different regions
ab
y1
y2
Make substitutions
Permitted frequencies given by equation
Equations for different regions
Modes of Vibration
Comparison of Harmonic Error
Table of relative frequencies of vibration modes
Graph of the Harmonic Error
6
2
5
4
3
1
Ordinary Drum
2 Density Drum
Conclusion
2-density model a good approximation
Doesn’t achieve perfect harmonicity
Better model could be produced
References Ramakrishna, B. S. and Sondhi, M. M.; Vibrations of Indian Musical Drums
Regarded as Composite Membranes, Journal of the Acoustical Society of America, 26, 2, 523-529, (1954).
Gaudet, S., et al.; The evolution of harmonic Indian musical Drums: A mathematical perspective, Journal of Sound and Vibration, 291, 388-394, (2006).
Benson, D. J.; Music: A Mathematical Offering, Cambridge University Press, (2007).
Any Questions?