Metrics and combinatorics: prastāraḥ or “spreading out”

There were “six combinatorial methods” (ṣaṭ pratyayāḥ, lit. ‘six notions’) that South Asian thinkers applied to metrical forms (as well as other combinatorial problems), and Hēmacandra’s discussion of them, in the seventh chapter of his Chandōnuśāsanam, was nicely explained by Ludwig Alsdorf (“Die Pratyayas: Ein Beitrag zur indischen Mathematik” in Zeitschrift für Indologie und Iranistik 9 [1933]: 97–157, downloadable here).

The general question that these methods were devised to answer is: given a certain number of syllabic positions (k), in which each syllable can either be light or heavy (n = 2), how many distinct combinations of syllables are possible? Having figured out that it is nk, Indian scholars started to ask other questions about these combinations.

The first method is called prastāraḥ or “spreading out.” This is a brute-force method that involves writing out each combination, but it has two important features: (1) each combination is assigned a serial number, and (2) each combination is derived from the previous one in a consistent, algorithmic manner. It is:

  1. On the first line (serial number 1), write k heavy syllables.
  2. For each subsequent line:
    1. take the first heavy syllable of the previous line and change it into a light syllable;
    2. for all of the syllables that follow, copy them exactly as they were in the previous line;
    3. for all of the syllables that preceded, restore them to their original (i.e., heavy) form.
  3. Proceed until you are left with all light syllables.

When k = 4 (the class of meters called pratiṣṭhā), the following prastāraḥ is made (resulting in 16 = 24 possibilities), using the symbols ऽ for “heavy” and । for “light”:

1. ऽऽऽऽ
2. ।ऽऽऽ
3. ऽ।ऽऽ
4. ।।ऽऽ
5. ऽऽ।ऽ
6. ।ऽ।ऽ
7. ऽ।।ऽ
8. ।।।ऽ
9. ऽऽऽ।
10. ।ऽऽ।
11. ऽ।ऽ।
12. ।।ऽ।
13. ऽऽ।।
14. ।ऽ।।
15. ऽ।।।
16. ।।।।

As Alsdorf explains, the same procedure can be applied to mora-counting meters like the gāthā. In such cases, however, the number of moras in a certain position must be kept constant, so when a heavy syllable is replaced in the prastāraḥ with a light syllable, another light syllable must be added (so a heavy syllable is effectively replaced by two light syllables). These “facultative” light syllables are marked in gold in the table below. Generally they do not count as syllables in the prastāraḥ. Hence he gives the following prastāraḥ for the first group of moras (gaṇa-) in the gāthā:

1. ऽऽ
2. ।ऽ
3. ऽ।
4. ।।।

Note that the third combination is ruled out by the rules of the gāthā, which state that the pattern ।ऽ। must not occur in even-numbered positions. Hence from combination no. 2 we change the first heavy syllable (i.e., the last syllable) into a light syllable, revert the initial (non-facultative) light syllable into a heavy, and add a facultative light syllable between them, yielding ऽ।।.

You might have noticed a pattern, which Hēmacandra also describes. In the first column, representing the first syllabic position, you can write ऽ and । alternating with each other for the odd and even serial numbers. Then, in the second column, write ऽ and ऽ on top of each other (for serial numbers 1 and 2) and then । and । on top of each other (for serial numbers 3 and 4). Then, in the third, write ऽ, ऽ, ऽ and ऽ on top of each other (serial no. 1–4) and ।, ।, ।, and । on top of each other (serial numbers 5–8). And so on and so forth. Thus write 2r-1 heavy syllables on top of 2r-1 light syllables in position r of k total positions.