Carnatic Guitar

The computed values below are surprisingly close to the ones computed using cycle of fifths. I've used a string length of 9*13*4 = 468 to have integer units for the vibrating string lengths. Sruti - Sruti Number VSL - Vibrating sring length. Ratio = String Length / VSL SA Grama - Ratios and corresponding Swara positions in Bharata's SA Grama MA Grama - Ratios and corresponding Swara positions in Bharata's MA Grama Sruti VSL Ratio SA Grama MA Grama 0 468 1.0000 1.7778 N 1.1852 G 1 455 1.0286 1.8286 1.2190 2 442 1.0588 1.8824 1.2549 3 429 1.0909 1.9394 1.2929 4 416 1.1250 1.0000 S 1.3333 M 5 403 1.1613 1.0323 1.3763 6 390 1.2000 1.0667 1.4222 7 377 1.2414 1.1034 R 1.4713 P 8 364 1.2857 1.1429 1.5238 9 351 1.3333 1.1852 G 1.5802 10 342 1.3684 1.2164 1.62...

In Indian music the tonic (1/1) is called Sa (S) and the perfect fifth (3/2) is called Pa (P). Indian music being monophonic, S and P are the only notes that are allowed to be heard along with any other note. So strings are tuned in S - P relationship only. That way even if the strings are accidentally struck not much harm is done. In my view, 22 srutis appear as the solution to a problem of identifying the optimal number of fret positions in a fixed fret ( straight frets perpendicular to strings ) - stringed instrument, tuned to S (1/1 ) - P (3/2) , while satisfying the following constraints. - The notes should be consonant with both strings S - P - It should be possible to play the scale assuming either string as the tonic, in which case the string relationship is S (1/1) - M ( 4/3 ) - Exploit octave complements, since we have to live with them - example S - P, P - 2 S This places the following mathematical constraints on the solution set which is a set of d...