Registered with the Registrar of Newspapers for India under R.N.I 53640/91

Vol. XXV No. 19, January 16-31, 2016

**A memorable speech at the Music Academy in December 2015 by Manjul Bhargava on Music and Mathematics.**

first came to concerts at the Music Academy 10 years ago, and sat mesmerised and inspired by so many breathtaking performances of great artists. I certainly never imagined at that time I’d be up here on stage one day felicitating some of those very artists who I had been admiring. Of course, my journey in music goes back much further, my mother being a Hindustani vocalist and tabla player. I grew up hearing her sing and play. But it was the tabla (and percussion in general) that always attracted me the most, as my instrument of choice, and as a possible profession.

Tabla, and of course, mathematics. As you know, I did definitely also consider pursuing mathematics. I actually never considered these subjects of music and mathematics that different. Much of my interest in mathematics arose through music, and vice versa. Since my grandfather was a scholar of ancient Indian history, I had the opportunity of, while growing up, acquainting myself with various ancient books of India, including many of the classics on mathematics and music. As everyone knows, Bharata’s Natyashastra and Sharngadeva’s Sangita-ratnaakara are two of the most groundbreaking works ever published on music, and they laid the foundations of modern day Carnatic and Hindustani music. What many people don’t know though is that they were at the time two of the most groundbreaking works on mathematics ever published.

Some of the beautiful mathematical problems in music that they discussed lie at the root of some of my interests in number theory, which is the area of mathematics that I specialize in. Number theory is the branch of mathematics that studies the whole numbers: 1, 2, 3,… and 0, -1, -2, etc. Number theorists aim to understand special sequences of whole numbers, like the square numbers and the prime numbers, and they aim to understand how to solve equations with solutions in the whole numbers.

Both the Natyashastra and the Sangita-ratnaakara considered problems in number theory which remain very important in modern mathematics, and which for them had fundamental applications to music. They considered some of the remarkable mathematical problems in rhythm, understanding, for example, in how many ways you can break a 17-beat cycle into partitions of length 1,2,4, and 6. As shown in the SR, the answer is 6236, in case you want to try your hand at it!

I thought I’d say a little more today about how mathematics also plays a fundamental role in melody, particularly in Carnatic and Hindustani melody. Truly fundamental problems in number theory immediately come up which were already considered in great detail by Bharata in his Natyashastra, and these problems lie at the foundation of Carnatic and Hindustani music, and indeed all music.

The basic problem of melody is which notes to use. Although we don’t think about it much anymore, the question is a truly fundamental one – what notes or frequencies should we use in our music when we compose melodies? In other words, which notes will sound good together? And which ones will not? There are infinitely many, a continuous range of, notes and frequencies out there, yet, in the end, why do we use 7 notes (saptaswara) in our melakarta ragas? Why are there 12 notes in each saptak on the modern veena, and why are there 22 shruti-s?

It is a question that has faced all musical cultures of the world. Which notes should we use?

The answer to the question is entirely a mathematical one, and different solutions were obtained across the world. The solution obtained in Carnatic and Hindustani music, as documented in Bharata’s Natyashastra back in 200 B.C.E., is one of the truly remarkable solutions, and that mathematical solution is the seed that lies behind what eventually led to classical Indian music developing to the incredible form and the oldest style of classical music in the world today, known the world over for its rich and expressive melodies.

So what do the choices of notes have to do with mathematics? The choices of frequencies that sound good in music are governed by mathematical principles; two notes sound good together (i.e., are resonant) if the ratio of their frequencies is a simple whole number ratio, like 2:1 (which is the distance of one saptak, from sa to sa) or 3:2 (which is the distance from sa to pa). Resonance from simple whole number ratios of frequencies sounds beautiful for reasons of physics – you feel the concurrence of vibrations of overtones! That’s why two people singing together in the same pitch sounds really nice – their fundamental frequencies and their overtones are all lining up.

But it also sounds very nice when two people sing one saptak apart (i.e., one person sings at twice the frequency of the other person), for example at a birthday, antakshari, etc. Two notes one saptak apart sounds so good that they are given the same name in essentially all cultures.

The goal of ancient Indian musicians as described by Bharata was maximising resonance. The idea was to fix a tonic note (sa), and the scales of notes in Indian music are then taken from a set of 22 whole number ratio frequencies from that tonic (called shruti-s). 2/1 gives the higher sa, 3/2 gives pa, 4/3 gives shuddha ma, 5/4 gives shuddha ga, and so on. There are 22 simple whole number ratios specified by Bharata that sound nice together, yet are far apart from each other so that musicians can feel clearly that they are different notes.

These choices of notes documented by Bharata in 200 B.C.E., although they probably go back even earlier than him, allowed for very rich and resonant melodies in Hindustani and Carnatic music. Shruti-s were chosen in raga-s in ancient times so that certain intervals in the raga would be maximally resonant, and that’s why listening to melodies this music season has sounded so good!

Of course, musicians are not thinking about these ratios any more when they sing; the frequencies have become innate through practice, and passed on from generation to generation. Musicians also don’t necessarily follow these 22 recommended shruti frequencies exactly all the time; sometimes variations, even irrational numbers for frequencies, are introduced for artistic effect. For example, the komal re that is used in some raga-s is flattened, moved really close to the sa, so that it creates a feeling of longing. There are also sets of notes that might not have been recommended by older texts, such as what are called vivadi or dissonant raga-s, of which there are 40. These raga-s are being performed more often than ever before.

So Carnatic music continues to move forward, into new territory, but a basic and beautiful mathematical framework is what lies under it all.

I was inspired to think more about mathematics and number theory in particular when I was growing up and thinking about these musical problems. But even when I wasn’t thinking about mathematics of the music I was listening to, I found it inspirational. It would take me to a place where I was doing. This is not unique to me. Ramanujan, one of my major mathematical heroes and also from this part of the world – his mother was an accomplished Carnatic vocalist and he grew up listening to Carnatic music as he did his mathematics. He worked in the lobby of his temple listening to Carnatic drumming. Prof. Seshadri, one of the great mathematicians, who is also from Chennai and who started the Chennai MI, is also an avid classical Indian vocalist.

There is something about classical music that makes a person more creative even in other endeavours. Justice Mudgal, former Chief Justice of the Punjab and Haryana High Court, who is also a supporter of the Music Academy and was here all week at concerts, has been saying that in his experience, judges with training in fine arts tend to show greater judgement and humanity in their decisions and judgments than those who don’t. Steve Jobs was famous for his ideas for products that married top-notch aesthetics with top-notch engineering. When asked about why the Macintosh computer revolutionized computing, he remarked it was because he loved to hire computer scientists who were also trained in the fine arts, who showed far greater innovative ability.

Art is what separates humans from other species, from artificially intelligent computers – it’s what makes humans human. For that reason, as we all support the Music Academy and the artists here, let’s also all try to make sure that our young people are educated in the fine arts.

Classical Indian music is a mathematically deep and aesthetically complex and extremely expressive form of art that is truly beautiful, which unifies India, and which can help the country produce not just better artists, but better scientists, judges, innovators, and in general better human beings. Let’s try to make Classical Indian music part of the school curriculum, to help make sure this amazing form of art continues, and so that we continue to evoke the very best from our young people.

Manjul Bhargava

Manjul Bhargava is a Canadian-American mathematician. He is the R. Brandon Fradd Professor of Mathematics at Princeton University, the Stieltjes Professor of Number Theory at Leiden University, and also holds Adjunct Professorships at the Tata Institute of Fundamental Research, the Indian Institute of Technology, Bombay, and the University of Hyderabad. He is known primarily for his contributions to number theory.

His mother, a mathematician at Hofstra University, New York, was his first mathematics teacher. He completed all of his high school maths and computer science courses by age 14. He graduated from high school in 1992 as the class valedictorian and obtained his B.A. from Harvard University in 1996.

When he received tenure as Full Professor in 2003. He was 29 and the third youngest Full Professor in Princeton University’s history. Bhargava has won several awards for his research, the most prestigious being the Fields Medal, the highest award in the field of mathematics, which he won in 2014.

Bhargava is also an accomplished tabla player, having studied under gurus such as Zakir Hussain. He also studied Sanskrit from his grandfather Purushottam Lal Bhargava, a well-known scholar of Sanskrit and ancient Indian history. He is an admirer of Sanskrit poetry.

In 2013, Bhargava was elected to the National Academy of Sciences (U.S.).