Saturday, September 20, 2008

Fibonicci and Pascal - Hemachandra and Pingla

Background Posting

On Calculus - Did Newton Lie


The Mathematics of Poetry
Multicultural Mathematics
Dr. Rachel Hall

The mathematical field of Combinatorics includes the mathematics of combinations and
permutations. Combinatorics starts with questions of how many ways there are to do or
make something. Some early studies in Combinatorics were done by Ancient Indian
scholars studying poetry!

Hemacandra.

The Jain writer ¹Ac¹arya Hemacandra (c. 1150 AD) studied the rhythms
of Sanskrit poetry. Syllables in Sanskrit are either long or short. Long syllables have twice
the length of short syllables. The question he asked is How many rhythm patterns with a
given total length can be formed from short and long syllables?

For example, how many patterns have the length of five short syllables (i.e. five \beats")?

There are eight:

SSSSS, SSSL, SSLS, SLSS, LSSS, SLL, LSL, LLS

As rhythm patterns, these are (corresponding to above)

xxxxx, xxxx., xxx.x, xx.xx, x.xxx, xx.x., x.xx., x.x.x

To investigate the general rule, we can make a chart:

Patterns Length Number
S 1 1
SS, L 2 2
3
4
5 8
6

It turns out that the sequence of numbers of patterns is called the Fibonacci sequence,
after the Italian mathematician Fibonacci, whose work was published 70 years after
Hemacandra's. The numbers in the sequence are called Fibonacci numbers. In your own
words, give the rule for finding the next number in the Fibonacci sequence.

Returning to our musical question, the answer is that the number of rhythm patterns with length n is the sum of the number of patterns of length n - 1 and the patterns of length n - 2.

Why? Write out the eight patterns of length five beats in a special way. First list the ones
that start with a L|there are three of these:

Now list the five patterns that start with a S:

Write out the three patterns of length 3:

and the five of length 4:

Explain how the patterns of length 3, 4, and 5 are related:


Pingala.

Another question inspired by the study of rhythm is: how many patterns can be formed from a given number of syllables? For example, how many patterns consist of three syllables? The syllables can be all L's, all S's, or some mixture of L's and S's. The answer is eight; see if you can discover how I grouped them.

SSS SSL SLL LLL
SLS LSL
LSS LLS

Pingala's Chandahsutra (c. 200 B.C.) enumerated the possible poetic meters of a fixed
number of syllables. Syllables are short (1 beat) or long (2 beats).

He classified 16 different meters of four syllables in the following way:

1 meter of four short syllables
4 meters of three shorts and a long
6 meters of two shorts and two longs
4 meters of one short and three longs
1 meter of four longs

Now nd the number of patterns consisting of two syllables, grouped by length:

Let's make a chart of the results and look for a pattern:

Number of Syllables Number of patterns Grouping of patterns

1
2 4
3 8 1 + 3 + 3 + 1
4 16
5 1 + 5 + 10 + 10 + 5 + 1
6 1 + 6 + 15 + 20 + 15 + 6 + 1

We see that the answer to our original question is that the number of patterns composed of
n syllables is 2n. In addition, there's something really interesting going on with the
groupings of patterns. Fill in the following triangle with your values for the groupings:

1 note 1 + 1 = 2 patterns
2 notes 1 + 2 + 1 = 4 patterns
3 notes 1 + 3 + 3 + 1 = 8 patterns
4 notes 1 + 4 + 6 + 4+ 1 = 16 patterns
5 notes 1 + 5 + 10 + 10 + 5 + 1 = 32 patterns
6 notes 1 + 6 + 15 + 20 + 15 + 6 + 1 = 64 patterns
7 notes

This is called Pascal's Triangle!

This observation was first made by the Indian writer Pingala (c. 200 B.C.), who lived
eighteen centuries before Pascal! It also appears in the commentary on this work by
Halayudha (c. 10th century).