This is in continuation of my earlier articles on Pingala’s Chandahshastra – Binary Number System, Algorithm to Calculate the Value of an Exponential.
Till now, we have dealt with poetic meters where all the four quarters of a verse had the same pattern of Laghus and Gurus. Such meters are called Samavrutta (समवृत्त) (even meters).
For a quarter having ‘n’ syllables, we know that the total number of possible even meters is 2n.
What if all the four quarters of a verse are not the same?
If the first and the third quarters of a verse are similar and the second and the fourth quarters are similar, the meters are known as Ardhasamavrutta (अर्धसमवृत्त) (semi-even meters). In semi-even meters the halves have the same pattern.
If the numbers of syllables in all the four quarters are the same but the patterns are different it is called Visamavrutta (विषमवृत्त) (odd meters).
In Chandahshastra, Pingala has given sutras to calculate the sankhya (संख्या) total number of possible semi-even and odd meters for quarters having ‘n’ syllables each.
समं तावत्कृत्वः कृतमर्धसमम्। विषमं च। राश्यूनम्।
(छन्द: शास्त्रम् ५.३-५)
There are 2^n x 2^n possibilities in the first two quarters (as well as the third and the fourth). To get the number of semi-even meters we have to deduct the number of even-meters from it.
Therefore, the total number of possible semi-even meters for a quarter having ‘n’ syllables is (2^n)^2 – 2^n = 2^2n – 2^n
Now, let us take up the case of odd meters. For a verse having ‘n’ syllables in each quarter, we have 2^n x 2^n x 2^n x 2^n = (2^n)^4 = 2^4n possibilities. From this we have to deduct the number of even and semi-even meters.
2^4n – 2^n – (2^2n – 2^n) = 2^4n – 2^2n
Hence, the total number of possible odd meters for each quarters having ‘n’ syllables is 2^4n – 2^2n.
In the 11th century AD, Bhaskaracharya too had given the sutra to calculate the sankhyas of even, semi-even, and odd meters in his book Lilavati.
पादाक्षरमितगच्छे गुणवर्गफलं चये द्विगुणे ।
समवृत्तानां संख्या तव्दर्गो वर्गवर्गश्च ॥
स्वस्वपदोनी स्यातां अर्धसमानं च् विषमाणाम् ॥१३९॥
Bhaskaracharya had also given a problem for finding the total number of possible even, semi-even and odd meters for Anustupa (अनुष्टुप) meter. Anustupa meter has 32 syllables in a verse and 8 syllables in each of the four quarters.
समानामर्धतुल्यानं विषमाणां पृथक् पृथक् ।
वृत्तानां वद मे संख्यामनुष्टुप्छंदसि द्रुतम् ॥१४०॥
Find the number of even, semi-even, and odd meters in Anustupa meter?
Using the above formula, we get:
Number of even meters = 2^8 = 256
Number of semi-even meters = (2^8)^2 – 2^8 = 65,536 – 256 = 65280
Number of odd meters = (2^8)^4 – (2^8)^2 = 4294901760
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