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.
It is based on something called as Katapayadi system. Those of
you who are interested in reading further about its meaning, history ,
geographical spread etc can read this link http://en.wikipedia.org/wiki/Katapayadi_system
I have the book by Ramon Campayo Maximize
your Memory wherein he uses something similar
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)
mELakarta raagas:
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
order.
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
the memorising!
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.
For example:
"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.
56-36=20.
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
|
