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?