acoustics of stretched circular membrane (studies on the mridangam-A south indian drum)
Mathematical Methods course project JNCASR, Supervisor: Prof Rama Govindarajan
Background
The Mridangam and Tabla are standard percussive accompaniments for Carnatic and Hindustani music and display some remarkable acoustical properties. The most important of these are their implementation of a practical solution to transform the in-harmonic overtones of a stretched circular membrane to harmonic ones. It is a well-known mathematical result that the Eigen-modes of a stretched circular membrane are roots of the Bessel's equation and are hence not integer multiples of the fundamental mode. Since the harmonic content of a sound is directly tied to its musicality, a circular membrane in general does not produce a musical sound. However, the right hand drum of the Mridangam clearly does produce a musical sound and has a well-defined pitch which is tuned to the tonic note of the main performer. It is indeed the case that a sizable number of overtones of the Mridangam and Tabla are harmonic, a result first shown by the physicist C V Raman. This is achieved mainly due to the black patch seen on the right-hand side drum (see picture) of the Mridangam and Tabla which is an axisymmetric loading on the membrane.
There are still some open questions with respect to acoustics of the Mridangam. For instance, one of the simplest sounds the Mridangam can produce even in the hands of a beginner is the 'open' sound which is the excitation of the fundamental mode of the membrane. This can be excited by simply striking the black patch without much thought to technique. Interestingly this produces a note which is exactly a semitone higher than the expected frequency of the fundamental. This is true of any Mridangam anywhere. Though some theories exist linking this raising of frequency with the coupled vibrations of the air column within the Mridangam shell, none have settled the question completely.
Ongoing Work
As a Mridangam player I have always wanted to explore this topic in depth. I got an opportunity to do so in the Mathematical Methods course at JNCASR and was supervised by Prof Rama Govindarajan. I began by writing down the solution for the simple case of an un-damped circular membrane (Bessel's equation -see above). I then solved the equation with damping terms added to model the effects of the annular ring present in the Mridangam. I also made careful recordings of the various tones a Mridangam can produce and analyzed them using a standard FFT tool. I confirmed that the frequency spectrum indeed was harmonic for the first five overtones. The observed shift up in the frequency of the fundamental is also apparent from the frequency spectrum.
We now plan to build a simple setup to investigate the reason for the upward shift of the fundamental mode and efforts are underway towards this.
The Mridangam and Tabla are standard percussive accompaniments for Carnatic and Hindustani music and display some remarkable acoustical properties. The most important of these are their implementation of a practical solution to transform the in-harmonic overtones of a stretched circular membrane to harmonic ones. It is a well-known mathematical result that the Eigen-modes of a stretched circular membrane are roots of the Bessel's equation and are hence not integer multiples of the fundamental mode. Since the harmonic content of a sound is directly tied to its musicality, a circular membrane in general does not produce a musical sound. However, the right hand drum of the Mridangam clearly does produce a musical sound and has a well-defined pitch which is tuned to the tonic note of the main performer. It is indeed the case that a sizable number of overtones of the Mridangam and Tabla are harmonic, a result first shown by the physicist C V Raman. This is achieved mainly due to the black patch seen on the right-hand side drum (see picture) of the Mridangam and Tabla which is an axisymmetric loading on the membrane.
There are still some open questions with respect to acoustics of the Mridangam. For instance, one of the simplest sounds the Mridangam can produce even in the hands of a beginner is the 'open' sound which is the excitation of the fundamental mode of the membrane. This can be excited by simply striking the black patch without much thought to technique. Interestingly this produces a note which is exactly a semitone higher than the expected frequency of the fundamental. This is true of any Mridangam anywhere. Though some theories exist linking this raising of frequency with the coupled vibrations of the air column within the Mridangam shell, none have settled the question completely.
Ongoing Work
As a Mridangam player I have always wanted to explore this topic in depth. I got an opportunity to do so in the Mathematical Methods course at JNCASR and was supervised by Prof Rama Govindarajan. I began by writing down the solution for the simple case of an un-damped circular membrane (Bessel's equation -see above). I then solved the equation with damping terms added to model the effects of the annular ring present in the Mridangam. I also made careful recordings of the various tones a Mridangam can produce and analyzed them using a standard FFT tool. I confirmed that the frequency spectrum indeed was harmonic for the first five overtones. The observed shift up in the frequency of the fundamental is also apparent from the frequency spectrum.
We now plan to build a simple setup to investigate the reason for the upward shift of the fundamental mode and efforts are underway towards this.
A Mridangam (showing the right-side drum head or "valanthalai") source: wikipedia
A dismantled right-side drum head showing the central loading (black patch)
A edge-on view of the right-side drum head showing the axisymmetric thickness variation of the black patch
