**Rhythmatics**

**Fibonacci – Hemachandra Sequence**

Some of my readers will m remember, one Krishnagiri Kittappa, the official percussionist of Oho Productions in the great Tamil romantic comedy of 1960s, Kathalikaa Neramillai (No time for romance). He was initially a self-taught mridangam (Drum) player. He wanted to learn to play Tabla also. He went to a Tabla player to learn the same. He was started on his first lesson, of course, in Teen Tal (or Triputa Tal in Carnatic music) of 8 beats.

*Na Din Dinnah – Na Din Dinnah*

*Na Din Dinnah – Na Din Dinnah*

*Na Din Dinnah – Na Din Dinnah . . . . . . .*

This went on for quite some time. Our man got bored of playing the same rhythm. It is the same 1,1,2 – 1,1,2 all the time for the 8-beat cycle. Why not 1,2,1, – 1,1,2, he thought.

*Din Dinnah Din – Na Dhin Dhinna*

*Din Dinnah Din – Na Dhin Dhinna*

Then, why not 2,1,1-1,2,1

*Dinnah Din Din – Din Dinnah Din*

Good. Now he further thought about how many such combinations of 1 and 2 (Din and Dinnah), he can make in a cycle of 8 beats. Ancient Indians have already thought about this and so, I gave him the answer as 34 different combinations. He was surprised. So many? How did they calculate?

Ancient Indians always depended on recursive technique in solving such problems. They started from 1-beat, then to 2-beats, 3-beats etc.

No. of Beats (n) | Syllables – 1 & 2 | Combinations | Total Combinations K_{n} |

1 | Din | 1 | 1 |

2 | Din, Din
Dinnah |
2 |
2 |

3 | Din Din Din
Dinnah Din Din Dinnah |
3 |
3 |

4 | Dinnah (+ 2-beats)
Din (+ 3-beats) |
2
3 |
5 |

Now they generalized: |
|||

(n+1) Beats | Dinnah + (n-1) beats
Din + (n) beats |
K_{(n-1)}
K |
K_{(n+1)} = K_{(n-1) }+ K_{(n)} |

Therefore,

K_{4} = K_{2} + K_{3} = 2 + 3 = 5

K_{5} = K_{3} + K_{4} = 3+ 5 = 8

K_{6} = K_{4} + K_{5} = 5 + 8 = 13

K_{7} = K_{5} + K_{6} = 8 + 13 = 21

K_{8} = K_{6} + K_{7} = 13 + 21 = 34

Hence we have 34 combinations of 1-2 in an 8-beat cycle.

Now look at the series K1, K2, ….. Kn:

1, 2, 3, 5, 8, 13, 21, 34, 55 ……

This is the famous Fibonacci series ‘invented’ by Fibonacci (alias Leonardo Pisano Bogollo) in 13th Century AD. Ancient Indians knew about this, at least, a thousand years before him. Fibonacci himself acknowledges this fact. Fibonacci also helped spread Hindu- Arabic Numerals (like our present numbers 0,1,2,3,4,5,6,7,8,9) through Europe in place of Roman Numerals (I, II, III, IV, V, etc). The olden day knowledge route was from India to Alexandria (Egypt) to Europe.

Susantha Goonatilake (Ref-2) writes that the development of the Fibonacci sequence ” is attributed in part to Pingala (200 BC), later being associated with Virahanka (c. 700 AD), Gopāla (c. 1135), and Hemachandra (c. 1150). Parmanand Singh cites Pingala’s cryptic formula *misrau cha* (“the two are made together”) and cites scholars who interpret it in the context as saying that the cases for ‘*n’* beats (*K _{n+1}*) is obtained by adding [Short or 1] to

*K*cases and [Long or 2] to the K

_{n }

_{n}_{−1}cases. He dates Pingala before 450 BC ”.

“However, the clearest exposition of the sequence arises in the work of Virahanka (c. 700 AD), whose own work is lost, but is available in a quotation by Gopala (c. 1135). The sequence is also discussed by Gopala (before 1135 AD) and by the Jain scholar Hemachandra (c. 1150) “. Fibonacci was born only in 1170 AD.

Prof. Manjul Bhargava (R Brandon Fradd Professor of Mathematics, Princeton University, USA) gave a Lec-Dem on Music & Mathematics at the Music Academy, Madras during their annual conference 2015, on 31^{st} December 2015. Being a Tabla player himself, he dealt with the above aspect of rhythm variations in detail. His talk was the inspiration for me to write this blog.

References:

- https://www.mathsisfun.com/numbers/fibonacci-sequence.html
- https://en.wikipedia.org/wiki/Fibonacci_number
- Manjul Bhargava’s Lec-Dem at the Music Acedemy, Madras (2015)

L V Nagarajan

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