What makes Tabla a harmonic drum?

How drums make sound?

When taught membranes are struck they vibrate and those vibrations creates a compression wave in air which is nothing but sound. Striking these membranes in a certain fashion makes membrane to vibrate at different frequencies. In general, lowest frequency, also called as fundamental frequency, the drums can produce need not have any relation to their higher frequencies. Which brings us to the tabla.

Tabla

Tabla is an Indian percussion instrument. It holds a special place in Indian classical music.

Unlike other percussion instruments which are anharmonic, meaning their higher frequencies are not positive integer multiple of fundamental frequency, the tabla is harmonic, thus its higher modes of vibration are positive integer multiple of fundamental frequency. Not just tabla, Mridangam also have this property. To be precise only dayan, literal translation right side, smaller drum of the tabla exhibit this property.

Say, the lowest frequency that the dayan can produce is 10Hz, then all other higher frequencies it can produce can only be an integer multiple of 10Hz, i.e., 20Hz, 30Hz, 40Hz, etc. Where as this is not true in general for all drums.

In a classical paper published in 1934, Sir C.V. Raman studied this property and attributed this interesting behaviour to the central black patch called Syahi. Syahi's literal translation is darkness. Syahi is a paste made from water, flour, and iron. Crafts-person coats the tabla membrane in precise fashion to change membrane mass density, which converts dayan into a harmonic instrument.

Modeling

In this project I modeled the tabla as two dimensional finite membrane. In absence of any empirical data the membrane mass density was reasonable assumed to have a parabolic profile, plotted below. Then the 2-D wave equation was solved using modal analysis. For this crude membrane profile the first 9 modes were approximately the integer multiple of fundamental frequency. The vibrating frequencies are tabulated in the slide 14 below.