Numbers: Their Tales, Types, and Treasures.
Chapter 5: Counting for Poets
5.7.THE ART OF COUNTING SYLLABLES
Let's reconsider the problem of classifying verse meters. This was the original topic where the old Indian scholars, more than two thousand years ago, formulated for the first time typical questions of a scientific discipline nowadays called mathematical combinatorics.
In this section, we consider again the theoretical number of verse meters, but we slightly shift our point of view. Instead of asking for the number of meters with a given duration, we ask for the number of verse meters with a given length in syllables. Among the old Indian verse meters, it is the group of so-called Aksarachandas that are characterized by a fixed number of syllables. Again, every syllable has a fixed duration and is either long or short.
Pingala's second problem:
How many different verse meters exist that have a total length of n syllables?
In order to realize one of the meters with a given number of n syllables, we have to distribute the long and short syllables in the verse. For the first syllable we can choose either a long or a short syllable. For each of these two beginnings, we have another two possibilities for the second syllable. This gives a total of four different possibilities for the first two syllables (namely ¯ ¯ , ¯ ˘, ˘ ¯, and ˘ ˘). For each one of these four forms, we have two possibilities to add a third syllable. Therefore, with every syllable we add, the number of possibilities is multiplied by two. We have
For 3 syllables: 2 × 2 × 2 = 23 = 8 different meters
For 4 syllables: 2 × 2 × 2 × 2 = 24 = 16 different meters
For 5 syllables: 2 × 2 × 2 × 2 × 2 = 25 = 32 different meters
For n syllables: 2 × 2 ×…× 2 (n factors) = 2n different meters
In this way we obtain, for example, in case of 24 syllables, more than 16 million possible verse meters—exactly 16,777,216. Very often, however, these verses consist of four similar parts (padas), and the first part alone determines the structure of the whole verse. But since Indian poetry has meters up to a length of more than one hundred syllables, we still obtain an enormous number of possible verse meters, of which only a few (actually several hundred) occur in practice.
It is interesting that old Indian scholars engaged in this kind of number game, which at first sight had little practical relevance. This could happen only because mathematics had already reached a fairly high level. Scholars had a deep knowledge about how to deal with numbers, and they were obviously proud to handle exceedingly large numbers. And they had already cultivated the ability to prove facts on the basis of logical arguments. In their examination of verse meters, we find a mathematical way of thinking that is visible in the ambition to understand all theoretically possible variants of a problem, even if not all the variants occur in reality.