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

2. 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

3. The Tabla  Tabla – indian set of drums  Harmonicity achieved by loaded region Membrane of right-handed tabla

5. Solving the ordinary drum  Wave Equation - Description of displacement over time  Solution for the ordinary drum

6. Two-Density Drum  Use 2-density drum to model tabla  Solve t first to get  Helmholtz equations for different regions a b y1y1 y2y2

7.  Make substitutions  Permitted frequencies given by equation  Equations for different regions

8. Modes of Vibration

9. Comparison of Harmonic Error  Table of relative frequencies of vibration modes

10. Graph of the Harmonic Error 6 2 5 4 3 1 Ordinary Drum 2 Density Drum

11. Conclusion  2-density model a good approximation  Doesn’t achieve perfect harmonicity  Better model could be produced

12. 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?