Extending the logic of Shruti-Nirman-Chakras (Topic 8) ,it may appear that if we complete the cycles S:P and S:M respectively by adding 50 % and 33.333333 % to the frequencies of the earlier notes, we can get the frequency values of 24 Shrutis.
To our horror, if we complete such cycles, we will have a problem in the end.
Because, if the starting point is taken as 100, S:P cycle which should end at 200, actually ends at 202.7286526 ! This constitutes the famous 'Pythagorean Error'.
No one knows why this happens but this is termed as 'Pythagorean Comma' (Comma means Error in Greek), meaning, not an error made by Pythagoras, but an error or natural flaw in musical numbers as found by Pythagoras. (Refer to following table)
The accompanying table shows Pythagorean Comma (Error) : Sa-Pa Cycle
Remember : Multiply every note by 3/2 = 1.5 to get the frequency of the next note (Pancham of previous note).
(# Under the decimal column, when the value goes beyond 2 in the next Saptak,
it is brought back to the original Saptak by dividing the value by 2, or it's multiples)
Very interestingly, it can be seen that the upper S' which should be 2, comes actually somewhat more as 2.027286529. This difference is known as 'Pythagorean Comma', the mathematical value of which is 2.027286529 divided by 2.0 = 1.013643264. This cycle is therefore 'Chadhi' or giving us 'higher frequency notes'.
All the Swaras are therefore suffixed by 2 indicating a 'higher' Shruti.
Therefore, if we start from a Shadja with say frequency of 100 Hz, the upper Shadja will come at 202.7286529, and not at 200. (Refer to following table.)
Table shows that S:P cycle goes beyond S' (Pythagorean Comma) (Take exact values from above table)
Even the Sa:Ma cycle ends at 197.3080737 and not 200. (Refer to following table)
Accompanying Table shows Pythagorean Comma (Error) : Sa-Ma Cycle
Remember : Multiply every note by 4/3 = 1.333333333 to get the frequency of the next note (Madhyam of previous note)
(# Under the decimal column, when the value goes beyond 2 in the next Saptak,
it is brought back to the original Saptak by dividing the value by 2, or it's multiples.)
This cycle is therefore 'Utari' or giving us 'lower frequency notes'. All the Swaras are therefore suffixed by 1 indicating a 'lower' Shruti.
Therefore, if we start from a Shadja with say frequency of 100 Hz, the upper Shadja will come at 197.3080737, and not at 200. (Refer to following table)
Table shows S:M cycle falls short of S' (Take exact values from above tables)
This means that the Shrutis thus obtained are not accurate as S:P cycle 'overshoots' and S:M cycle 'falls short'!