Geometric derivation of 22 srutis

This is a continuation of my earlier post showing the derivation of 22 srutis, as a geometric construction in a log-log plot. This was created using LibreCAD.

Figure.1, shows a cycle of pure fourths and fifths.
Figure.2, shows a cycle of 53EDO tempered fourths and fifths.
The numbers shown are frequency ratios in cents.
An octave (2:1) is 1200 cents wide. Standard Western tuning divides this in 12 equal semitones ( 2^(1/12) : 1 ) 100 cents wide.

How to read this construction.
The X and Y axes are 1200 cents long, and they represent frequency ratios between 1 and 2.
zero cents represents the ratio 1:1
1200 cents represents the ratio 2:1
Because of octave equivalence 1:1 and 2:1 refer to the same musical note.
We are interested in finding the optimal number of frequency ratios that we can fit inside this space.
We introduce a new ratio by adding a fret.
If our instrument has two strings tuned an octave apart, each new ratio that we introduce ( say 'a' ), also introduces its octave inverse ( 2/a ), as the ratio between the fretted note on the tonic string to the open note on the octave string.

This computation is obtained by the hypotenuse (H1) of the largest triangle in the figure. This connects (x=0,y=1200) to (x=1200, y=0).

When we divide the string into two equal parts by placing a fret at mid-length and stop one half, we get the octave.
When we further divide each half into halves ( quarters ), we get octave of octave, which is equivalent, and a new ratio (4/3), which is called fourth ( western music ) or madhyamam ( indian music ). madhyamam means middle or center in sanskrit, which could have got this name because the fret is placed midpoint between the nut and the octave fret. fourth could have been named this way because it is the ratio when a fourth of the string is stopped.

We start with this ratio (4/3). In cents this is 498.045.
So we draw a vertical line at x=498.045. This line intersects the hypotenuse (H1), At this point y=701.955 cents, which is the ratio (3/2). This geometric operation has computed the octave inverse.
for x = 4/3, y = 2/x = 2/(4/3) = 3/2.
In cents, x = 498.045, y = 1200 - 498.045 = 701.955
Division of frequency ratios becomes subtraction in cents because cents is logarithmic scale.

If our instrument has strings tuned in octaves and fifths, it can also be considered as tuned in octaves and fourths, if the fifth is considered the tonic.

So we add two more hypotenuses through (x = y= 498.045, and x = y = 701.955 ), though these are not directly used in the figure.

When a square of side 498.045 is placed at the lower left corner of the octave (1200) triangle, we
get two large triangles (701.955) atop and aside the square overlapping to form a small triangle (203.910) at the top right corner of the square.

When a square of side 203.910 is placed inside a triangle of (498.045) we get two large triangles of ( 294.135 ) overlapping to form a small triangle (90.225).

90.225 square inside 294.135, gives two large triangles (203.910), overlapping to form a smaller triangle (113.685).

90.225 square inside 203.910, gives two large triangles (113.685), overlapping to form a small triangle (23.46).

Up to this point the size of small triangular overlap kept reducing at each step.

If we fit 23.46 square inside 90.225 triangle, the triangular overlap is larger than the current smallest triangle 23.46. This means our computation will start to diverge.

If we create a symmetric figure by packing triangles inside each other, we can see the vertical lines are spaced at the points of the srutis.

Figure .1 Cycle of pure fourths and fifths

(download dxf file )
click on image to take a closer look


Since we always have S and P in the tambura providing the constant background drone sound, choosing notes that are too close to these which would be at a distance (23.46) would be very dissonant.

So we drop (0 + 23.46 ) and (701.955 - 23.46 = 678.495 ). We can see from the figure that 498.045 ( madhyamam, fourth) is the ninth sruti from from the shadja ( tonic ), and 701.955 ( panchamam, fifth ) is the thirteenth sruti from the shadja ( tonic ). Since a fourth remains above the fifth within an octave, the pattern of srutis from shadja ( tonic ) to madhyamam ( fourth ) is repeated starting from the panchamam ( fifth ), to form the upper tetrachord.


These sruti values are already provided in my earlier post.

 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


23.46 is the pramaana sruti, which is the difference between the two semitones 113.685 and 90.225.

If we can find a pramaana sruti such that we can define both the semitones as integral multiples of pramaana sruti, we can work in simple integral numbers.

Fortunately such a value exists, (498.045/22 = 22.6384) which is almost the same as (1200/53 = 22.6415) the 53rd root of 2. This is called 53 EDO ( 53 equal divisions of the octave ), or 53 equal tempered ratios (53 ET).

This value 22.6415 divides madhyamam (498.045) into almost 22 (21.9969875 ) equal parts.

Here is an interesting observation, which leads to a question.

Although there are 22 sruti values in our list, should the octave (1200) be considered a sruti, since it is the equivalent of shadja (octave equivalence) ? Are there only 21 srutis, then ?

Since madhyamam (498.045) was the first ratio that we started with, and since we see no benefit if we further divide madhyamam beyond 22 equal parts, could this be the reason for the name 22 srutis ?

For the sake of comparison, I've also created a similar geometric figure using a madhyamam of 22-pramaana srutis. (1200*22/53 = 498.11321 cents).



Figure.2 Cycle of 53EDO tempered fourths and fifths

(download dxf file )
click on image to take a closer look


The sruti values are pretty close to the ones obtained earlier.



 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


In other words, here is a ruby code snippet that shows the process

def recursive_divide(large, small)
  new = large - 2 * small
  puts "#{'%8.3f' % large} #{'%8.3f' % small} #{'%8.3f' % new}\t#{new < small}"
  if ( new < small ) then 
    recursive_divide(small, new)
  end
end

puts " Large     Small     New\t(is new < small ?)"
recursive_divide(1200.0, 498.045)

Output
-------------------------------------
 Large     Small     New (is new < small ?)
1200.000  498.045  203.910 true
 498.045  203.910   90.225 true
 203.910   90.225   23.460 true
  90.225   23.460   43.305 false

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