Pi and Sanathana Dharma
The following was an interesting aspect in an interview which I read after one of my brother took the pains to tell me to read it , otherwise I have an healthy contempt not to touch Chennai Times as it is full of filmy gossip and glossy photos of actresses with perky breasts and piercing thighs, not necessarily the ectomorphic variety of the west but the bulky bosomed one of the subcontinent. However, rarely such interviews do pop up as the one from Yann Martel (Canadian Novelist), a great admirer of Hinduism and India. He said the following in his interview, “In mathematics, Pi is an irrational number which means it is number that goes on forever. Yet it's constant in science, so we use this irrational number to get to the rational understanding. It's that contradiction that I liked. To me religion is like that - an irrational number, it doesn't make sense on its own, but it helps make sense of the world.”
After I read this I was not only savoring his comparison but I was also thinking that Indians, especially the Sanathana Dharmic practioners, who have always had an unrestrained foray into many domains , especially in their highly logical, scientific , ritualistic [probably due to the inherent contradiction of life and inevitable necessity to be highly demographically inclusive], mathematical and musical aspects of spiritual and cultural expressions must have had something definitely, I was convinced more so because recently I read a wonderful book called ‘THE ADVENTURES IN NUMBERLAND ‘ by a French mathematician wherein he after doing extensive research into the evolution of numbers and all aspects of mathematics writes in that book a lot about the advanced concepts in mathematics and their multiple usages that Indian had in practical application far ahead of Westerners even coming to decipher them, including Fibonacci numbers and Pi . I am not mentioning these as the outpouring of an ethnocentric fanatic soul but as statement of facts.
I also realized that in our over enthusiasm we must not "mathmaticize" Religion, in case of Hindus the Sanatan Dharma. I am also aware that there is this big difference between Pi and Religion in the "progression". In case of Pi it is fixed repeated numbers, whereas Religion, especially Dharma as we know, does not have any "dimension" or can be "quantified". Another difference, from Mr. Martel's philosophy if I am allowed to digress, is that Dharma includes "unknowns" beyond this world (also known as Earth) and our solar Universe, or to be more precise Universes.
Not everything is unreasonable as far as similarity between these two [ 2] topics. For example, mathematics provide a path to expressions & queries from creative and ever-active Human minds, whereas such mental engines are designed & developed by super-Power that also gave us all the Dharma. Not only highly evolved concepts of Mathematics did exist since or even before the time of Ramayana, and surely indications are there in Holy Gita, but there were many practical and superb applications of mathematical concepts in many of the spiritual and cultural expressions of practioners of Sanathana Dharma.
Well as serendipity has this positive prejudice of popping up from unexpected areas to encourage the enquiring mind, enhance and enrich the understanding and enlighten further, it did pop up when I was explaining to someone the melakartha ragas system and then I was lead to these information taken from many sources but vindicating that we Indians had far greater applications of Pi in most of spiritual enquiries and cultural arena.
The "ka-Ta-pa-ya" scheme
and its application to
mELAkarta raagas of Carnatic music
and its application to
mELAkarta raagas of Carnatic music
The ka-Ta-pa-ya scheme:
The "ka-Ta-pa-ya" rule used by ancient Indian mathematicians and grammarians
is a tool to map names to numbers. Writing the consonants of the Sanskrit
alphabet as four groups with "ka, Ta, pa, ya" as the beginning letters of
the groups we get
1 2 3 4 5 6 7 8 9 0
ka kha ga gha ~ma cha Cha ja jha ~na
Ta Tha Da Dha Na ta tha da dha na
pa pha ba bha ma
ya ra la va Sa sha sa ha
Now, each letter of the group is numbered from 1 through 9 and 0 for the tenth
letter. Thus, ka is 1, sa is 7, ma is 5, na is 0 and so on. So to indicate
the number 356 for example one would try and come up with a word involving
the third, fifth and sixth letters of the groups like "gaNitam" or "lESaca".
However, in the Indian tradition, the digits of a number are written left to
right in the increasing order of their place value - exactly opposite the way
we are used to writing in the western way. Therefore 356 would be indicated
using letters in the 6th, 5th, and 3rd positions of the group e.g. "triSUlaM".
There apparently were upto 4 flavors of this scheme in use in ancient
India. These differ in how to interpret the conjoint consonant. The popular
scheme was to use only the last consonant. And any consonant not attached
to a vowel is to be disregarded. These rules should be used while decoding
a phrase in "katapayadi" scheme.
The following phrase found in "sadratnamAla" a treatise on astronomy,
bhadram budhi siddha janma gaNita Sraddha@h mayadbhUpagi@h
when decoded yields
4 2 3 9 7 8 5 3 5 6 2 9 5 1 4 1 3
which when reversed gives
3 1 4 1 5 9 2 6 5 3 5 8 7 9 3 2 4
which is readily recognised as the digits in "pi" (except that the 17th
digit is wrong - it should be 3) :-)!
(source: The article "The Katapayadi Formula and Modern Hashing Technique"
by Anand V Raman, appearing in "Computing Science in Ancient India", edited
by T.R.N.Rao and Subhash Kak, published by the Center for Advanced Computer
Studies, University of South Western Louisiana, Louisiana LA 70504)
The raagas of Carnatic music are said to be derived from a definite set of
72 ragas known as mELakarta or janaka or sampoorNa raagas.
Before we go any further let me remind you that in an octave there are 12
tones each separated by a half note. These 12 tones are named as the 7 notes
and variations on some of those 7 notes. These names and their western music
equivalents are as follows:
1. shaDjamam S Doh C
2. Suddha rishabham R1 C# or Db
(read as C-sharp or D-flat)
3. chatuSruti rishabham R2
Suddha gAndhAram G1 Re D
4. shaTSruti rishabham R3
sAdhAraNa gAndhAram G2 D# or Eb
5. antara gAndhAram G3 Me E
6. Suddha madhyamam M1 Fa F
7. prati madhyamam M2 F# or Gb
8. pancamam P Sol G
9. Suddha dhaivatam D1 G# or Ab
10. chatuSruti dhaivatam D2
Suddha nishaadam N1 La A
11. shaTSruti dhaivatam D3
kaiSika nishaadam N2 A# or Bb
12. kAkali nishaadam N3 Ti B
The properties of the janaka raagas are :
a) they contain all the 7 notes of an octave (hence the name saMpoorNa)
exactly once in the scale.
b) the tones of the notes must all be in ascending order in the
aarOhaNa. i.e You cannot pick S, R3, G1 ... because the tone of R3
is higher than the tone of G1. Also the ArOhaNa/avarOhaNa cannot have
jumps back and forth like S, G3, R1, ... or S, N2, P, D1 ... etc..
c) the avarOhaNa should contain the same notes as ArOhaNa in the reverse
Given these properties/rules, we can easily surmise that there cannot be
more than 72 sampoorNa raagas. Because of the need for ascending order of
tones the permissible combinations of R,G and D,N are limited to 6 each, viz.
R1G1, R1G2, R1G3, R2G2, R2G3, R3G3 and D1N1, D1N2, D1N3, D2N2, D2N3, D3N3.
There are two varieties of M viz. M1, M2. So the number of possible different
sampoorNa raagas is 6 x 6 x 2 = 72. So if these 72 are arranged in a regular
order, we can figure out the scale of a janaka raaga if its number in the
list is given. Now, from our "kaTapayaadi" scheme if we can name the raaga
in such a way that the name yields the number, then we have further reduced
Application of "kaTapayaadi" to mELakarta raagas
That is exactly what venkaTamakhi of the 18th century is purported to have done.
He applied the "kaTapaya" scheme to name the janaka raagas to fit their place
in the mELakarta list. Some of these already had suitable names and some had
unsuitable names that were in common use. He changed those names a little to
fit the naming scheme. Thus "kalyANi" becomes "mEcha kalyANi","SankarAbharaNam"
becomes "dheera SankarAbharaNam" etc.
Under this naming scheme, the number of a janaka raaga is obtained by decoding
the first two letters using the "kaTapaya" scheme. For the naming scheme used
for the mELakarta raagas, apart from the decoding rules mentioned above for,
conjoint consonants, in case one of the consonants is from the 'ya' group,
the first consonant is to be considered instead of the last. And finally,
to get back to our familiar western notation, reverse the decoded digits.
"kharaharapriya" : kha = 2 and ra = 2 i.e 22 reversing the digits : 22
"shaNmukhapriya" : sha = 6 and mu = 5 i.e 65 reversing the digits : 56
"naThabhairavi" : na = 0 and Tha = 2 i.e 02 reversing the digits : 20
"divyamaNi" : di = 8 and va = 4 i.e 84 reversing the digits : 48
Once you get the number, figuring out the notes is easy. The 72 raagas are
arranged such that the first 36 raagas contain M1 and the next 36 contain M2.
In each half, the various possible combinations of R,G and D,N
occur cyclically with the R,Gs varying slower than the D,Ns.
i.e: for the first six raagas
R1G1 occurs with each of D1N1, D1N2, D1N3, D2N2, D2N3, D3N3
for the next six raagas
R1G2 occurs with each of D1N1, D1N2, D1N3, D2N2, D2N3, D3N3
and so on.
So given a janaya raaga number you perform the following calculation:
1. if NUM is from 1-36, raaga has M1, from 37-72 raaga has M2.
2. if NUM is greater than 36 subtract 36 from it.
3. divide NUM by 6;
a. if remainder=0
i. the sixth D,N combination occurs.
ii. the quotient gives which of the R,G combinations occurs.
b. if remainder is not zero
i. the remainder gives which of the D,N combinations occurs.
ii. the quotient+1 gives which of the R,G combinations occurs.
Taking the example of "shaNmukhapriya":
From the "kaTapaya" rule its number is 56.
56 is greater than 36. So M2 occurs.
20 divided by 6 : quotient=3, remainder=2
so 3+1=4th RG combination : R2G2 occurs.
and 2nd DN combination : D1N2 occurs.
So shaNmukhapriya has the notes:
S R2 G2 M2 P D1 N2 S
Another example : "varuNapriya"
From the "kaTapaya" rule its number is 24.
24 is less than 36. So M1 occurs.
24 divided by 6 : quotient=4, remainder=0
so 4th R,G combination: R2G2 occurs.
and 6th D,N combination: D3N3 occurs.
thus varuNapriya has the notes:
S R2 G2 M1 P D3 N3 S