# The Indian musical drums

There is a well-known paper by the famous scientist and Nobel laureate C. V. Raman about the harmonic drums of India – the mridangam and the tabla. While the paper was written in the 1930s, it is quite detailed and refreshing in its clear description of how these instruments work. This post attempts to popularize his description.

The idea of a tuned drum is quite alien to music outside India. Drums are percussive instruments, tuning is something performed on instruments that are musical in nature, is the general feeling. But it is clear that the principle of all musical instruments is some kind of stretched “thing”, which is forced to vibrate with a fundamental and several harmonics – the various harmonics being excited by the design of the instrument. Hence, for instance, note the excitement over expensive violins and cellos – they are expensive precisely because they have a particularly pleasing combination of harmonics when the air columns in the instrument vibrate sympathetically with a bow plied over the stretched string.

A stretched membrane, as found in a regular (timpani) drum, is not particularly harmonious sounding except for its percussive, repetitive property. The modes of the uniform, circular membrane can be solved – the solution is an exercise in solving the equation for a harmonic oscillator in cylindrical coordinates. The modes are described in the paper referred to above – a modern, color picture is below.

The dashed lines represent nodes of the membrane, where the membrane isn’t moving. The adjacent segments, across the nodes, move in opposite directions. The fundamental note is the mode named ${\bf 01}$ in the above – the whole membrane vibrates as a whole – to excite this mode, you would bang the drum right in the center. The next mode is the mode named ${\bf 11}$; it has a frequency $1.59$ times the frequency of the fundamental. This is a little more difficult to create. You’d have to find a way to limit the vibration down a diameter, then bang the drum a quarter of the way away from the diametrical line to excite that mode. However, its frequency is $1.59$ times the fundamental – is that any good?

Now, if you know anything about how musical notes are organized, you will understand that there is a whole lot of personal choice involved in what combination of notes in a scale sound “pleasing”. The only uniformity is the central, organizing parts of the scale.

The central note in the scale could be chosen as, say the middle “C”, in Western music. In Indian music, any note could be chosen as the “reference” note – the center of “your” octave.

Then, the note whose frequency is double the frequency of your centered note is the upper end of this “middle” octave. It should really be called a “duplex”, but it is (by Western tradition at least), the eighth note (inclusive) from the middle “C”, so the name is not totally inappropriate.

The note with $50 \%$ higher frequency is the “Pa” in Indian music (“so” in Western music) is a note that is particular pleasing sounding, when played with the central (or “reference”) note.

This principle, as described, is very common in all forms of Indo-European musical traditions, which Indian and Western styles belong to (as do Iranian and Middle-Eastern music). In fact, in the Indian musical drone (the tanpura), the four strings play the “reference” note, its “Pa” (the note of $50\%$ higher frequency) and the upper octave “reference” (double the frequency of the “reference” note).

The other notes are between the “reference”, the “pa” and the upper octave “reference” (of double frequency). The complexity of Indian music (and indeed other kinds of music in the Middle East, for instance) is buried in the larger number of notes “in-between” compared to Western music. Western music has three flat notes between the “reference” and the “so”, then two more between “so” and the upper octave’s start. South Indian music has closer to twenty one; I have never bothered to count. In addition, notes sound different because the “attack” (how the note is approached) and “gamaka ” (how the note is shaken) is different for different ragas.

The intermediate notes are picked in different ways – the “equal tempered” scale (with equal ratios between successive notes as the twelfth root of 2) favored by Western orchestral instruments is a “medium” to allow different instruments to play together, The “harmonic” scale, with simple fractional ratios between various notes sounds better (and has sounded better since the days of Pythagoras) and is the basis of most non-Western musical traditions.

Now, let’s look at how the mridangam produces sound. Look at its vibrational modes, first (as detailed in Raman’s paper),

The numbers below each mode are the frequencies of each mode, as multiples of the fundamental frequency. Note that they are all “pleasing” multiples of the fundamental. There are nine such modes. There are higher modes, but since they involve vibrations that are have shorter distance between the “nodes”, they are easy to suppress by weighting the stretched membrane near its anchors at the ends of the cylindrical case.

The purpose of the black iron-oxide/gum paste at the center of the mridangam and tabla face, as well as the width of the border of the membrane, is to ensure the frequencies are as above and not as is usual for a plain stretched membrane. Note also, that to excite the various modes, you need to place some fingers on the membrane at some places, while striking it (“smartly”, to quote Raman) at other specific spots. In addition, if you have noticed tuning a mridangam, they rotate it on their thighs while testing the tension in the sixteen strings holding the membranes in place. This is because if the tension is not uniform, you might not get the appropriate harmonic unless you hit the “right” diametric spots. That becomes difficult to adjust to, especially when you are playing in situations where the temperature varies from place to place (from outside to inside an auditorium for instance).

Learning to play involves understanding how to strike the membrane, as well as producing adequate volume from the places you do strike. In addition, there are many variables that can be played with – the material used for the face, the diameter of the instrument, the attachment of the face to the body, the exact composition of the material used for the central black patch. All these serve to change the frequency of the fundamental, to suit the pitch of the singer that these instruments are supposed to accompany.

There are some videos that describe the techniques in detail on YouTube, linked to here and here.

${\bf \: Note \: added \: post \: publication:}$

A couple of Indian physicists (then graduate students) studied the frequencies produced by a real mridangam using discrete Fourier transform techniques (sampling the sound at $200 \mu$second intervals) as also a computer simulation of a “loaded” stretched membrane. They report that they notice the general pattern of a fundamental that is $7-10 \%$ higher in frequency than expected, but with the harmonics related by integer fractions to each other, i.e., the frequencies are 1.075, 2, 3, 4.025… in one experiment. This result appears to be borne out in the numerical simulation they perform and they see similar results for the ratio between the fundamental and the harmonics, as well as between harmonics. They speculate that audiences are simply not able to discern the difference between the expected fundamental and the “real” fundamental.

First, the numerical simulation is of a simple membrane with a denser central region that mimics the iron-oxide spot in the center of the mridangam’s face. However, the mridangam also has a third stretched membrane under the basic membrane, separated by short wooden sticks. This is a rather complex setup that doesn’t precisely match the simulated system.

Also, given the extreme sophistication of audiences, as well as performers (and especially critics!), in discerning “sruti” lapses in performances, it is frankly hard to believe that a $10 \%$ error in the tuned fundamental would not be noticed (this would imply the “reference” note is a sharp (second) “ri” that is played with the upper octave’s “sa” – this is discernibly dissonant!). Tuning a mridangam is difficult and keeping it tuned is hard and it is not clear from the paper what methods were used to keep the instrument (whose sound was sampled) tuned after it was initially set up. This needs more research, maybe another such study – stay tuned.

Pics. courtesy:

Stretched membrane: http://hyperphysics.phy-astr.gsu.edu/hbase/Music/cirmem.html