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Naṣṭam Method

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

Naṣṭam (that which is lost, hidden) was a method to determine the rūpam (form) of a specified step in a prastāra. Both Naṣṭam and uddiṣṭam methods used saṅkhyā series. The number of permutation steps that ended in each of the four tāla elements and their relative positioning in the series assumed significance.

The naṣṭam method is illustrated through an example to determine the tāla rūpam for the seventh step in the laghu-pluta prastāra.

Note: The derivation process is interspersed with a tabular interpretation represented through steps 'a' to 'm'.

1. Write down the saṅkhyā series for this prastāra (step a in table below). The series for the laghu-pluta prastāra extends till the value 60. This value represents the total number of permutation steps in the prastāra. The number of druta, laghu, guru and pluta ending permutation steps are at one, two, four and six blocks to the left of this position. Tag these positions with a set of five markers, say, m0, m1, m2, m3 and m4 (step b).

2. At position m0, deduct the desired step number from the total number of permutation steps in the prastāra and obtain the remainder. 60 - 7 = 53 (step c). Only this first reduction is carried out at position m0. All subsequent reductions are at positions starting from m1 onwards.

a. 1 2 3 6 10 19 33 =60
b.   m4   m3   m2 m1 m0
c.               60-7=53

3. Attempt to reduce this remainder (53) by the saṅkhyā values at the position of one or more markers m1 through m4. A reduction is possible if the value at the marker position is greater than or equal to the remainder at hand. If a reduction can be made over the values at two consecutive marker positions, collect a guru for the rūpam. If a reduction is possible over one position, collect a laghu. If no reduction is possible, collect a druta.

4. At the initial setting of markers, the saṅkhyā values at marker positions m1, m2, m3 and m4 are 33, 19, 6 and 2 respectively. The remainder, 53, may be reduced by values at marker positions m1 and m2. 53-33-19 = 1 (step d). Reduction cannot be carried down further. As this is a reduction over two marker positions, collect a guru for the rūpam (step e).

5. Relocate the markers to start from the position where the reduction stopped. Thus, the new starting marker, m0, is now set at the position of earlier m3. The other markers, m1 through m4 are now positioned 1, 2, 4 and 6 positions to the left of the new m0. Note that some of these markers are off the grid (step f).
a. 1 2 3 6 10 19 33 =60
b.   m6   m3   m2 m1 m0
c.               60-7=53
d.           53-33-19=1  
e.           g  
f.   m2 m1 m0        

6. Attempt to reduce the current remainder, 1, with saṅkhyā values at new marker positions m1, m2, m3 and m4. Value at m1 is now 3. As this is greater than the remainder no reduction is possible; so collect another druta and reset the markers (steps g, h, i).
a. 1 2 3 6 10 19 33 =60
..                
f.   m2 m1 m0        
g.     3 > 1          
h.     d          
i. m2 m1 m0          

7. With the markers reset, the value at position m1 is now 2. Again, this is greater than the remainder of 1. Collect another druta and reposition the markers once more (steps j, k, l).
a. 1 2 3 6 10 19 33 =60
..                
i. m2 m1 m0          
j.   2 > 1 m0          
k.   d            
l. m1 m0            

8. Now the value at position m1 is 1. This may be reduced by the remainder that is also 1. As reduction over one position is possible, collect a laghu per 3 above (steps m, n).
a. 1 2 3 6 10 19 33 =60
..                
m 1-1=0              
n. l              

9. Assemble all the elements gathered to form the rūpam of the seventh step of the prastāra as 'l-d-d-g'.



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