HOW DO YOU MAKE A DRUM HARMONIC? James Hagues 3 rd March 2010

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 a b y1y1 y2y2

 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 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, , (1954).  Gaudet, S., et al.; The evolution of harmonic Indian musical Drums: A mathematical perspective, Journal of Sound and Vibration, 291, , (2006).  Benson, D. J.; Music: A Mathematical Offering, Cambridge University Press, (2007). Any Questions?