June 2018 – Simply Curious

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

Tanpura: Ravi Maharjan http://www.youtube.com/123koolabhi

The Rule of 72 – and what does the Swiss National Bank have to do with it

I was listening to an academic talk and someone mentioned the “Rule of 72”. Apparently invented by Einstein, it is a simple numerical approximation that helps you understand the power of compound interest. This, according to legend, became popular when interest rates offered on deposits by the Swiss National Bank dropped to $2-3 \%$ in the 1930s. Only the Germans appeared to be suffering hyperinflation, the Swiss clearly weren’t (though that was before the advent of modern monetary policy, which made the connection between interest rates and inflationary expectations).

Einstein is also touted as the source of a quote on compound interest – “Compound interest is the eighth wonder of the world. He who understands it, earns it…he who doesn’t …pays it”. By the way, I have seen several of the physical wonders of the world and have learnt several of the wonders of theoretical physics. While, for instance, the power of the exponential is to be seen to be believed (read Perelman’s book for how to hold a ship with a few loops of rope around a post, as in the image above), one can see it also in the ability of tiny humans to combine forces to build buildings as big as the Pyramid of Khufu, the Madurai Meenakshi temple, or the Burj Khalifa building in Dubai (below)

I would point to those things, rather than the mere accumulation of interest, as a more picturesque depiction of the power of the exponential. Also, as the website snopes.com seems to indicate, the attribution of the quote to Einstein might be an urban legend.

For what it is worth, the Rule of 72 is as follows. If you want to know how long it will take to double your money at an interest rate of $X \%$ , the number of years is $\frac{72}{X}$ . Obviously, this rule doesn’t apply if the interest rate is negative, as has been the case in some European countries in the last several years.

A quick check on Excel tells you that a better approximation is to use $69$ or $70$ in the numerator. Using $72$ harkens back to a predilection for multiples of 12, something that dates back to Babylonian times.

You can also use this formula to deduce the ruinous effects of inflation.

A complete description, that I found after preparing this article, is to be found in this page. The Rule of 72 actually appears in articles aimed at investors in the NASDAQ market, as well as in bank advertisements.

Credits:

The photograph of the rope at dock: Pratik Panda at Dreamstime.com

The photograph of the pyramid : By Nina – Own work, CC BY 2.5, https://commons.wikimedia.org/w/index.php?curid=282496

The photograph of the Meenakshi temple: Wikipedia

The photograph of the Burj Khalifa building: Wikipedia

Bucking down to the Bakhshali manuscript

The Bakhshali manuscript is an artifact discovered in 1881, near the town of Peshawar (in then British India, but now in present-day Pakistan). It is beautifully described in an article in the online magazine of the American Mathematical Society and I spent a few hours fascinated by the description in the article (written excellently by Bill Casselman of the University of British Columbia). It is not clear how old the 70 page manuscript fragment is – its pieces date (using radiocarbon dating) to between 300-1200 AD. However, I can’t imagine this is simply only as old as that – the mathematical tradition it appears to represent, simply by its evident maturity, is much older.

Anyway, read the article to get a full picture, but I want to focus on generalizing the approximation technique described in the article. One of the brilliant sentences in the article referred to above is “Historians of science often seem to write about their subject as if scientific progress were a necessary sociological development. But as far as I can see it is largely driven by enthusiasm, and characterized largely by randomness”. My essay below is in this spirit!

The technique is as follows (and it is described in detail in the article)

Suppose you want to find the square root of an integer $N$ , which is not a perfect square. Let’s say you find an integer $p_1$ whose square is close to $N$ – it doesn’t even have to be the exact one “sandwiching” the number $N$ . Then define an error term $\mathcal E_1$ in the following way,

$N = p_1^2 + \mathcal E_1$

Next, in the manuscript, you are supposed to define, successively

$p_2 = p_1 + \frac{\mathcal E_1}{2 p_1}$

and compute the $2^{nd}$ level error term using simple algebra

$\mathcal E_2 = - \frac{\mathcal E_1^2}{4 p_1^2}$

This can go on, according to the algorithm detailed in the manuscript and write

$p_3 = p_2 + \frac{\mathcal E_2}{2 p_2}$

and compute the error term with the same error formula above, with $p_2$ replacing $p_1$ . The series converges extremely fast (the $p_n$ approaches the true answer quickly).

Here comes the generalization (this is not in the Bakhshali manuscript, I don’t know if it is a known technique) – you can use this technique to find higher roots.

Say you want to find the cube root of $N$ . Start with a number $p_1$ whose cube is close to $N$ . Then we define the error term $\mathcal E_1$ using

$N = p_1^3 + \mathcal E_1$

Next, define $p_2 = p_1 + \frac{\mathcal E_1}{3 p_1^2}$

and compute the $2^{nd}$ level error term using simple algebra

$\mathcal E_2 = -3 \frac{\mathcal E_1^2}{9 p_1^3} - \frac{\mathcal E_1^3}{27 p_6}$

and carry on to define $p_3 = p_2 + \frac{\mathcal E_2}{3 p_2^2}$ and so on.

The series can be computed in Excel – it converges extremely fast. I computed the cube root of $897373742731$ to Excel’s numerical accuracy in six iterations.

Carrying on, we can compute the fourth root of $N$ , as should be obvious now, by finding a potential fourth root $p_1$ that is a close candidate and then

$p_2 = p_1 + \frac{\mathcal E_1}{4 p_1^3}$

and so on.

You can generalize this to a general $q^{th}$ root by choosing $p_2=p_1 + \frac{\mathcal E_1}{q p_1^{q-1}}$ and carrying on as before. It is a little complex to write down the error term, but it just needs you to brush up the binomial theorem expansion a little. I used this to compute the $19^{th}$ root of $198456$ , starting from the convenient $2$ , rather than $1$ (see below!). It is $1.900309911$ . If I started from the intial number $3$ , convergence is slower but it still gets there quite fast (in twelve iterations).

Here’s a bit of the Excel calculation with a starting point of 2

	p	E
p_1	2	-325832
p_2	1.93458156	-80254.8113
p_3	1.90526245	-10060.9486
p_4	1.90042408	-226.670119
p_5	1.90030997	-0.12245308
p_6	1.90030991	-3.6118E-08
p_7	1.90030991	0

and here’s a starting point of 3. Notice how much larger the initial error is (under the column E).

	p	E
p_1	3	-1162063011
p_2	2.84213222	-415943298
p_3	2.69261765	-148847120
p_4	2.55108963	-53231970.3
p_5	2.41732047	-19003715.9
p_6	2.29140798	-6750923.06
p_7	2.17425159	-2365355.74
p_8	2.06867527	-797302.733
p_9	1.98149708	-240959.414
p_10	1.92430861	-53439.1151
p_11	1.90282236	-5045.06319
p_12	1.90033955	-58.8097322
p_13	1.90030991	-0.00825194
p_14	1.90030991	0 : CONVERGED!

The neat thing about this method is, unlike what Bill Casselman states, you don’t need to start from an initial point that sandwiches the number $N$ . If you start reasonably close (though never at 1, for obvious reasons!), you do pretty well. The reason why these iterations converge is that the error term $\mathcal E_m$ is always smaller than $p_q^{m-1}$ .

The actual writer of the Bakhshali manuscript did not use decimal notation, but used rational numbers (fractions) to represent decimals, which required an order of magnitude more work to get the arithmetic right! It is also interesting how the numerals used in the manuscript are so close to the numerals I grew up using in Hindi-speaking Mumbai!

Note added: Bill Casselman wrote to me that the use of rational numbers with a large number of digits instead of decimals represents “several orders of magnitude” more difficulty. I have no difficulty in agreeing with that sentiment – if you want the full analysis, read the article.

There is also a scholarly analysis by M N Channabasappa (1974) that predates the AMS article that analyzes the manuscript very similarly.

A wonderful compendium of original articles on math history is to be found at Math10.