The other important number we must understand before we can move on to the mathematics of 22 Shrutis, is called as 'Syntonic Comma' or 'Ptolemic Comma' because it is credited to the famous scientist Ptolemi (90-168 AD). This number is the difference in the value of Gandhar in the 3rd Saptak as found by 2 different methods.
Method 1: If we take 5 consecutive Panchams from S (100 Hz) at a distance of 50%, we come to the value of Gandhar in 3rd Saptak as 506.25 as shown below.
Method 2: However, if we double the S twice, i.e., from 100 to 200 (Upper or Tara S'), and then to 400 (Still upper S'' or Ati-tara S''), and thereafter add 25% (the natural distance of G from S), we arrive at the value of Gandhar in 3rd Saptak as 400 + 25% = 500.
Thus, we got 2 values of Gandhar, 506.25 and 500.
This mathematical discrepancy by arriving at Gandhar of the 3rd Saptak by the 2 above methods should not have arisen. This natural error was demonstrated by Ptolemi and this is known as 'Syntonic or Ptolemic Comma'. (Comma means Error as we have seen earlier).
506.25 divided by 500 = 1.0125
This measurement of 1.0125 is also fundamental in the creation of 22 Shrutis.