Derivation of 22 srutis of indian music as a fret spacing problem

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 decimal fractions in the range 1.0 to 2.0.

The ratios are symmetric within an octave, i.e - for every R, 2/R (octave complement ) must be present in the set

Since this is a fixed fret instrument, placing a fret on the S string would also introduce a note on the P string, (3/2)*R

Also if P string is taken to be the S string, then S string becomes M in which case the ratio R*4/3 is also part of the set.

So, for every ratio R in the set, the ratios 2.0/R, R*(4/3), 2.0/(R*(4/3))and R*(3/2), 2.0/(R*(3/2)) must also be present.

These numbers were computed by starting with a set of just one ratio, the number 1. It takes about 14 iterations to converge to point where new ratios generated are only different from the existing ratios in the set by 0.001 cents. Here is the ruby code using which I generated these numbers.

Here is the solution set.

This is nothing but cycle of fifths and fourths with additional constraints requiring symmetry.

These additional constraints help in not running into the problems that necessitated the 12 step equal tempered compromise of Western music.

While the results are already well known, it is the problem definition that is interesting.

Also looking at the pattern of the ratios that are clustered, although there are 22 possible frets, it should be sufficient if we provide frets for the flatter note in each closely spaced pair, and achieve the adjacent sharper note in the pair by string bending, which is the way veena and sitar are played.

My next step is to verify this by building a custom fretboard with these fret positions.

It is also interesting that the major third is just flat by only 2 cents, unlike the 12 tone tempered scale. I hear beats when I tried a major chord using scala. Perhaps a guitar player can overcome this by uneven finger pressure on the strings.

If you create a scala scale file with the following lines, you can hear these ratios and verify.

!

! mani-22-sruti.scl

!

symmetric scale with ratios satisfying - R, 2/R, (3/2)R, (4/3)R

22

!

90.225

113.685

180.450

203.910

294.135

317.595

384.360

407.820

498.045

521.505

588.270

611.730

701.955

792.180

815.640

882.405

905.865

996.090

1019.550

1086.315

1109.775

1200.000

Picking from the nearest 53-edo we get the following scale. Seems like a good one to try building a fretboard for a string instrument tuned in fifths and octaves.

!

! mani-22-sruti.scl

!

symmetric 22 scale from 53 et

22

!

90.566

113.208

181.132

203.774

294.340

316.981

384.906

407.547

498.113

520.755

588.679

611.321

701.887

792.453

815.094

883.019

905.660

996.226

1018.868

1086.792

1109.434

1200.000