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01-17-2025, 08:46 PM
Chandas and mathematics
The attempt to identify the most pleasing sounds and perfect compositions led ancient Indian scholars to study permutations and combinatorial methods of enumerating musical metres. The Pingala Sutras includes a discussion of binary system rules to calculate permutations of Vedic metres. Pingala, and more particularly the classical Sanskrit prosody period scholars, developed the art of Matrameru, which is the field of counting sequences such as 0, 1, 1, 2, 3, 5, 8 and so on (Fibonacci numbers), in their prosody studies.
The first five rows of the Pascal's triangle, also called the Halayudha's triangle. Halayudha discusses this and more in his Sanskrit prosody bhashya on Pingala.
The 10th-century Halāyudha's commentary on Pingala Sutras, developed meruprastāra, which mirrors the Pascal's triangle in the west, and now also called as the Halayudha's triangle in books on mathematics. The 11th-century Ratnakarashanti's Chandoratnakara describes algorithms to enumerate binomial combinations of metres through pratyaya. For a given class (length), the six pratyaya were:
prastāra, the "table of arrangement": a procedure for enumerating (arranging in a table) all metres of the given length,
naṣṭa: a procedure for finding a metre given its position in the table (without constructing the whole table),
uddiṣṭa: a procedure for finding the position in the table of a given metre (without constructing the whole table),
laghukriyā or lagakriyā: calculation of the number of metres in the table containing a given number of laghu (or guru) syllables,
saṃkhyā: calculation of the total number of metres in the table,
adhvan: calculation of the space needed to write down the prastāra table of a given class (length).
Some authors also considered, for a given metre, (A) the number of guru syllables, (B) the number of laghu syllables, © the total number of syllables, and (D) the total number of mātras, giving expressions for each of these in terms of any two of the other three. (The basic relations being that C=A+B and D=2A+B.)
The attempt to identify the most pleasing sounds and perfect compositions led ancient Indian scholars to study permutations and combinatorial methods of enumerating musical metres. The Pingala Sutras includes a discussion of binary system rules to calculate permutations of Vedic metres. Pingala, and more particularly the classical Sanskrit prosody period scholars, developed the art of Matrameru, which is the field of counting sequences such as 0, 1, 1, 2, 3, 5, 8 and so on (Fibonacci numbers), in their prosody studies.
The first five rows of the Pascal's triangle, also called the Halayudha's triangle. Halayudha discusses this and more in his Sanskrit prosody bhashya on Pingala.
The 10th-century Halāyudha's commentary on Pingala Sutras, developed meruprastāra, which mirrors the Pascal's triangle in the west, and now also called as the Halayudha's triangle in books on mathematics. The 11th-century Ratnakarashanti's Chandoratnakara describes algorithms to enumerate binomial combinations of metres through pratyaya. For a given class (length), the six pratyaya were:
prastāra, the "table of arrangement": a procedure for enumerating (arranging in a table) all metres of the given length,
naṣṭa: a procedure for finding a metre given its position in the table (without constructing the whole table),
uddiṣṭa: a procedure for finding the position in the table of a given metre (without constructing the whole table),
laghukriyā or lagakriyā: calculation of the number of metres in the table containing a given number of laghu (or guru) syllables,
saṃkhyā: calculation of the total number of metres in the table,
adhvan: calculation of the space needed to write down the prastāra table of a given class (length).
Some authors also considered, for a given metre, (A) the number of guru syllables, (B) the number of laghu syllables, © the total number of syllables, and (D) the total number of mātras, giving expressions for each of these in terms of any two of the other three. (The basic relations being that C=A+B and D=2A+B.)