You messed up everything. Let's see how equal temperament works.
You are trying to create equal temperament based on having 2 intermediate micro-tones between tones of 12-tone European system. It it also pretty apparent that you use the
equal temperament and
A440 pitch standard as the base. Please see:
http://en.wikipedia.org/wiki/Equal_temperament[
^],
http://en.wikipedia.org/wiki/A440_%28pitch_standard%29[
^].
See also the article on microtones we would need to discuss:
https://en.wikipedia.org/wiki/Microtonal_music[
^].
Now, let's see. The physics of musical hearing and musical cognition is based on the
harmonics and hence, frequency ratios between the tones. The fixed ratios close to simplest
rational numbers are easy to memorize, are clearly recognized, create strong associations in all musical constructs when perceived by hearing.
It doesn't depend on particular culture. This is rooted deeply in physics and physiology. We perceive the sounds of the
octave interval (exactly double frequency) as "logically equivalent" and two sounds with frequency ratio close to 4/3 and 3/2 as "harmonic" — perfect
quarta and
quinta. They are really harmonic.
There are many intonational systems in the world, and modern classical music system is the European 12-tone system. You are trying to calculate the frequencies of the systems with 2 (not 3 as you wrote) intermediate tones in the semitone, so you are trying to describe 36-tone system. There are other systems, notable 24-tone ones (see, for example,
https://en.wikipedia.org/wiki/Arab_tone_system[
^]), and I even heard of 60-tone (like minutes in hour) one.
You should also understand that the desire to make the musical pieces in different keys "logically equivalent", the desire to make all keys "equavalent" which lead to the
equal temperaments forces us into using
irrational numbers for frequency ratios,
which makes equal temperaments non-harmonic where the harmonic frequencies
only approximation of truly harmonic ratios, which are always rational numbers. Please see:
http://en.wikipedia.org/wiki/Rational_number[
^],
http://en.wikipedia.org/wiki/Irrational_number[
^],
just for example:
http://en.wikipedia.org/wiki/Just_intonation[
^].
In modern so-called
authentic music, people often go back to non-equal temperament tone systems, trying music to sound as more harmonic, albeit in some fixed key. It requires instruments to be re-tuned to each specific piece of music, which is actually done, to one or another degree. European baroque equivalents of modern strings instruments (characteristic example: viola de gamba instead of cello) tend to have frets, but each fret, made of the piece of gut, can be tuned individually, which is actually done.
We well see those irrational numbers soon, but make one note:
we need to use floating-point data types for calculation of frequencies, never integer types.
We can easily build the frequencies of any of the tonal system. For 12-tone example, let's see.
A and A(II), A and A in next octave, gives us double frequency. Each semitone gives you fixed ratio S.
A = 440 Hz;
A# = A * S;
B = A * S * S;
...
A(II) = A * (S * S * ... 12 times) = A*2.
Therefore S is the 1/12 power if 2, the root of 12 degree of 2, 2
1/12,
pow(2, 12).
The same calculation will go for your 36-tone system. The interval between microtones will be finer, 2
1/36.
Again, this is the ratio, not fixed number if Hertz!
I got S = 2
1/12 = 1.0594630943592953 and your microtone M = 2
1/36 = 1.0194406437021448.
Let's calculate:
Assuming A = 440,
we go in previous octave with C:
C = A *S*S*S / 2 = 261.6255653005987
C♯ = C * S = 277.18263097687213;
now, micro-intervals:
C1 = C * M = 266.71173469897985
C2 = C1 * M = 271.8967825044437
As you can see, my figures are close to yours, by apparent reasons, but you are using totally wrong idea.
[EDIT]
You did not answer my questions on the notation you've shown in [pa,ma,ni,sa] and [pa-SA-SA-sa] examples. But I found that those are the notes in
Carnatic rāgas, which I kinds know how they sound, somewhat similar to European 7/12-note system.
But this is a different system, not the same as one you are trying to build based on two intermediate microtones between European semitons. It is based on 7 main notes and 16 varieties. I would need more information to analyze the rāgas, which I don't presently have. There are different flavors of rāga, and so on. I'm not sure the calculations I demonstrated can be applied,
but I'm sure you can apply the general principle I tried to explain: musical intervals are defined by the ratio between tones.
Indian music is very interesting to me, I would gladly learn a lot more and would be very grateful if someone gives me interesting and informative links on this diverse topic.
I would also like to encourage Indian people to take more interest in their own culture, no less then I, a total foreigner to India, do. :-)
—SA
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