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Saṅkhyā Series

Adapted from Chapter 5 of 'Saṅgītaratnākara - A Study' by R.Rangaramanuja Ayyangar, 2nd Ed.

Prastāra dissects a tāla or a phrase into its smallest units through graded steps. Saṅkhyā tabulates the number of such steps in a series of horizontal blocks, runs a commentary.

From the figures in Appendix 4 of the book that lists details of the various prastāra-s, the total number of permutation steps in various prastāra-s may be noted and tabulated as follows:

 Prastāra d l dl g dg p dp lp dlp gp dgp pp dpp Saṅkhyā 1 2 3 6 10 19 33 60 106 191 340 610 1089 Druta-s 1 2 3 4 5 6 7 8 9 10 11 12 13
Note: d - Druta; l - Laghu; g - Guru; p - Pluta

The first row in the table lists the principal form of thirteen prastāra-s in sequence; the second row, the number of permutation steps in each of them; and the third the number of druta-s in the final step of the prastāra-s. The second row is the Saṅkhyā series.

The tablulation gives not just the total number of permutation steps but also the number of druta, laghu, guru and pluta ending steps in each prastāra. To illustrate, consider the druta-pluta (d-p) prastāra with seven druta-s in the final step. As seen from the above table, this prastāra has 33 permutation steps. Write down the series till this value, assigning a serial number from right to left to each block:
 Serial 6 5 4 3 2 1 0 Saṅkhyā 1 2 3 6 10 19 =33

Consider the Saṅkhyā values at serial numbers 0, 1, 2, 4 and 6. At these positions, the values are 33, 19, 10, 3 and 1 respectively. The value at position 0, i.e. 33, represents the total number of permutation steps in the prastāra. The value at position 1, i.e. 19, reveals the number of druta ending steps in the prastāra. At position 2 is the number of steps in the prastāra that end in a laghu element, i.e. 10. The next two values 3 and 1 at positions 4 and 6 represent the number of steps that end in the guru and pluta elements respectively.

Add up the Saṅkhyā values at positions 1, 2, 4 and 6:

19 + 10 + 3 + 1 = 33

This is the total number of permutation steps in the prastāra. The steps in this prastāra may be seen in Appendix 4 of the book.

The relative positioning of the blocks holding the number of druta, laghu, guru and pluta ending steps - i.e. 1, 2, 4 and 6 - is of significance in determining the numbers in the Saṅkhyā series as well as in the naṣṭam and uddiṣṭam methods that are elaborated in the next section.

The elements in the Saṅkhyā series may be represented by the following formula where Sn represents the value at position 'n'.

Sn = Sn-1 + Sn-2 + Sn-4 + Sn-6

Note: for the initial elements, values at Sn-1 through Sn-6 may not be found. A work-around requires starting the Saṅkhyā series with a dummy value of 1. For computational reference, the series may be tabulated as follows:
 Prastāra d l dl g dg p dp lp dlp gp dgp pp dpp Saṅkhyā 1 1 2 3 6 10 19 33 60 106 191 340 610 1089 Druta-s 1 2 3 4 5 6 7 8 9 10 11 12 13

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