**PRAMANA SRUTI**

*(i.e. the lower and the upper boundaries of an octave)*. Western musicologists could, therefore, modify their ‘Notes’ mathematically, in order to keep pace with changing times. But how did they discover at all that ‘mathematical fractions’ and music were deeply related? The core idea seems to have descended from the age-old Sumerian tradition (4000 B.C.) of worshipping gods in the form of ‘fractions’. It is incredible to realize that a ‘primitive’ civilization such as the Sumerians could ever conceptualize an advanced mathematical concept such as ‘fractions’ in their culture. As I followed this thread further, I was in for bigger surprises! Their pantheon of twelve major gods in effect, symbolized a “family of 22 simple fractions” which appears to be the basic building blocks for an advanced class of music that existed during a pre-historic era! Interesting?

*For more details, please read my ‘Blog’ and ‘Presentation Slides’ placed on the internet:*

*Gods of Sumeria symbolize our Ancient Music of 22 Srutis (Blog):*http://sumeriangods.blogspot.com/

*Gods of Sumeria symbolize our Ancient Music of 22 Srutis (Presentation):*

As we turn towards
India, we are in for even greater surprises! Vedic Indians

*(3500 B.C.)*conceptualized their musical octave in a ‘mystical and extraordinary’ manner. They calibrated the octave in an abstract ‘tonal domain’, i.e. in terms of degree of ‘shrillness’. The limits for this ‘shrillness’ were set between ‘0’ and ‘22’. Within this octave, the ‘rising index of shrillness’ was calibrated in 22 equal steps. Each step was known as a ‘sruti’*(‘Shruti’ as per Shastriya Sangeet traditions)*or more specifically, the*‘Pramana Sruti’**(i.e. the most elemental sruti).*In other words,**Such a fact, however, raises several eyebrows within the mathematical community. ‘Tonal calibration’ of the octave, as a concept, presumes that the ancient Indian musicians knew how to***the Indian octave comprised of 22 ‘Pramana srutis’.**determine the 22*^{nd}root of Number ‘2’!*(Incredible indeed! Our civilization had learnt to determine even the 12*^{th}root of number ‘2’, only a few centuries back; in this backdrop, granting our primitive ancestors a capability to determine the 22^{nd}root of number ‘2’ would be an absurd presumption)!
Bharata Muni

*(200 B.C; the earliest scribe who had documented the erstwhile oral traditions of Indian musicology)*had described that a set of seven ‘Divine’ tones of ‘Sama Veda’*(Sama Veda is a musical mode of chanting the Holy Scripture that dates back to an antiquity of 3500 B.C.)*formed the foundation for music, in the name of ‘Sadja grama’. These ‘seven tones’*(known as ‘swaras’ in Indian musicology)*were designed in three different bunches of Pramana srutis (i.e. bunches of ‘2’ or ‘3’ or ‘4’): The swaras ‘Sadja’, ‘Madhyama’ and ‘Panchama’ measured ‘4’ Pramana srutis; ‘Rishabha’ and ‘Dhaivata’ measured ‘3’ Pramana srutis and ‘Gandhara’ and ‘Nishada’ measured ‘2’ Pramana srutis; the aggregate of these seven swaras accounted for the 22 Pramana srutis within the octave. ‘Seats’*(known as ‘swara-sthaanas’)*of these seven swaras were quantified and assigned in the octave: The Tonic ‘Sadja’ = at the ‘0’^{th}Pramana sruti position, ‘Rishabha’ (the ‘Second’) at the 3^{rd}Pramana sruti position, ‘Gandhara’ (the ‘Third’) at the 5^{th}Pramana sruti position, ‘Madhyama’ (the ‘Fourth’) at the 9^{th}Pramana sruti position,**‘Dhaivata’ (the ‘Sixth’) at the 16***‘Panchama’ (the ‘Fifth’) at the 13*^{th}Pramana sruti position,^{th}Pramana sruti position, ‘Nishada’ (the ‘Seventh’) at the 18^{th}Pramana sruti position and the ‘Octave Sadja’ at the 22^{nd}Pramana sruti position (i.e. the ‘0’^{th}Pramana sruti position of the Higher Octave).
During this era, musicological
knowledge was confined to these ‘seven tones’ only. The underlying mathematics
for the design of this 22-sruti octave as well as the seven Sama Vedic tones was

**unknown**to the musicologists. They were also**not aware**of any process that would resolve and distil out a single pramana sruti from the bunches of Pramana srutis, to enable further vertical development of music. These seven tones were described as ‘Suddha’ (i.e. ‘pure’) as they were derived directly from the Sama Veda. Even musicologists of high stature such as Bharata Muni and Dattila Muni would not dare to enlarge the swara base of seven ‘Suddha’ tones through experimentations, due to the apprehension that it might lead to ‘distortions’ (i.e. creation of vikrta swaras) and would amount to breaching Vedic Holiness.
Bharata Muni also
describes another ‘group of swaras’, known as ‘Madhyama grama’. This was required
to be derived from ‘Sadja grama’.

*(These two ‘gramas’ acted as equal partners in the subsequent vertical development of music in India, by way of ‘murchanas’ and ‘Tanas’. Therefore, understanding the design of ‘Madhyama grama’ is quite important).*Bharata Muni scribes an age-old musicological Rule: “Madhyama grama will get evolved when the Panchama swara of Sadja grama is reduced by one Pramana sruti”. There can be two interpretations for implementing this age-old musicological Rule. However, all Indian musicologists since the medieval era, have trekked the ‘beaten track’ of only one of the methods, to arrive at the following configuration for ‘Madhyama grama’: The Tonic ‘Sadja’ = at the ‘0’^{th}Pramana sruti position, ‘Rishabha’ at the 3^{rd}Pramana sruti position, ‘Gandhara’ at the 5^{th}Pramana sruti position, ‘Madhyama’ at the 9^{th}Pramana sruti position,**‘Dhaivata’ at the 16***‘Panchama’ at the 12*^{th}Pramana sruti position,^{th}Pramana sruti position, ‘Nishada’ at the 18^{th}Pramana sruti position and the ‘Octave Sadja’ at the 22^{nd}Pramana sruti position.**in the configuration of the new entity is noticeable; i.e. the swara-sthaana of ‘Panchama’ has been lowered by one position (from the 13***Only one change*^{th}to the 12^{th}). By observing this ‘triviality’ of the difference between the two ‘gramas’, some modern critics state that our understanding of the concept of derivation of ‘Madhyama-grama’ had been quite inadequate.
I decided to study the hitherto ‘unexplored’ “alternative option” for
the derivation of Madhyama grama. I was amazed
myself to discover that a

**had remained ‘hidden’ from our view, all these days! This option brought about a major change in the profile of Madhyama grama, as given: On reducing one Pramana sruti from ‘Panchama’, the ‘Madhyama’ swara-sthaana moved from position ‘9.00’ to ‘10.00’ srutis with respect to the tonic. Similarly, ‘Gandhara swara sthaana’ moved from position ‘5.00’ to ‘6.00’; ‘Rishabha swara sthaana’ moved from ‘3.00’ to ‘4.00’; ‘Nishada swara sthaana’ moved from ‘18.00’ to ‘19.00’ and ‘Dhaivata swara sthaana’ moved from ‘16.00’ to ‘17.00’. However, the ‘Panchama swara sthaana’***new phenomenon***. The viewers may please appreciate that this is an important deduction. ‘Panchama’ is a very important Note that stands tall in an octave and is next in importance to the Tonic Sadja. The ‘Panchama swara sthaana’ had been fixed at 13.00 srutis by the Sama Veda itself and hence its sanctified seat should not be altered in both ‘gramas’! It is also common knowledge that the position assigned to ‘Panchama’ at 13.00 srutis is a universal practice followed in all other traditions of music also.**__continued to remain at 13.00 srutis__
Having studied the basic features of the
ancient musical culture of Sumeria and India, I decided to draw some comparison
between them. However, direct comparison was difficult as the Sumerian
tradition was founded on ‘fractions’ and the Indian tradition was based on
‘tones’. I therefore, mathematically transformed the family of 22 simple
fractions (i.e. the Sumerian gods) into the tonal domain of a 22-srutis octave
and tabulated the results. I was amazed to realize that the

**As I compared the swaras of the newly evolved ‘Madhyama grama’, again I found that they were the replicas of some more members of the Sumerian god family. I extended this observation by experimenting on the age-old Indian concept of ‘murchanas’***Indian Sama Vedic tones (i.e. the swaras of Sadja grama) were nothing but the replicas of seven Sumerian gods!**{‘murchans’ were the fountain-head for the vertical development of music in medieval India. Seven ‘murchanas’ each were derived from each ‘grama’. These are somewhat similar to the seven ‘modes’ of ancient Greek music.}*and found that the remaining gods of the Sumerian Pantheon were embedded there in the form of tones!**For details, please read my ‘Blogs’ and ‘Presentation Slides’ on the Internet:***In other words, the god-fractionss worshipped by the Sumerians and the tones advocated in ancient Indian musicology were the same!*
Sadja-grama
(Blog): http://sadjagrama-nambirajan.blogspot.com

Sadja-grama--(Presentation): https://docs.google.com/open?id=0B6Qw6H3PDIHNM1kwRi1TWVlIUEk

Murchanas (Presentation): https://docs.google.com/open?id=0B6Qw6H3PDIHNdEp1Q1cwOHA3dzg

Having
achieved some spectacular breakthrough with my earlier experiments, I decided
to extend the same towards validating the contention of Bharata Muni with
regard to the “Pramana Sruti”. Bharata Muni narrates in ‘Natyasastra’ verses 27
and 28 that the phenomenon of Pramana Sruti is ‘realizable’ at the ‘Panchama’
boundary when Sadja grama is transformed into Madhyama grama. I, therefore,
compared the positions of ‘Panchama’ swara in both the ‘grama groupings’. I
found that the upper boundary of ‘Panchama’ (i.e. the swara-sthaana of ‘Panchama’)
remained unaltered in both the gramas. However, the lower boundary of ‘Panchama’
was found https://docs.google.com/open?id=0B6Qw6H3PDIHNbnh1aWdXNHVBN0E

**from its erstwhile position of ‘9.13’ srutis (in Sadja grama) to ‘10.11’ srutis (in Madhyama grama), i.e. a difference of ‘one Pramana-sruti’ (‘0.98’ srutis to be mathematically exact!). For better comprehension, please see my ‘Presentation Slides’ placed on the internet:***‘shifted’*
Let
us now take a re-look at this ‘phenomenon’ from the angle of ‘Sumerian- Fractions’! When ‘Panchama’ is compressed at its lower
boundary, the ‘Madhyama swara sthaana’ moves from position ‘3/4’ to position
‘8/11’. For quantifying the ‘shift’ that had occurred, we have to divide ‘3/4’
by ‘8/11’; and we get the result as ‘33/32’. This may be mathematically
re-written as 1.03125. This value very favourably compares with the modern accuracy
of 1.032000828, which has been obtained by finding the 22

^{nd}root of number ‘2’.
There is another
method for determining the value of Pramana sruti too! The octave extends
between ‘1’ and ‘1/2’, i.e. a length of ‘1/2’. If we apportion this segment
between 22 sruti entities, each sruti segment will measure ‘1/44’ length; i.e.
Pramana Sruti should measure a segmental length of ‘1/44’. With this
background, let us examine the gramas once again. We have seen in the earlier paragraph
that the ‘Madhyama swara sthaana’ had moved from position ‘3/4’ to position
‘8/11’. For quantifying this ‘shift’, let us subtract ‘8/11’ from ‘3/4’; we get
‘1/44’!

**This method of validation of Indian Sastric statements with the help of Sumerian Divine Fractions, establishes that there was**

*These accuracies are quite amazing indeed!***between the ancient Indians and the Sumerians in the field of musicology and religion. The sophistication of mathematical knowledge displayed in these two traditions indicates that the ‘pre-historic music’ inherited by them in ‘coded forms’ must be a priceless cultural gift passed on to our civilization by our pre-historic ancestors!**

*“Total Covergence”*
For
more details, contact me on Teles: 91 20 26729256, 9890266845, 98501 21834.
E-mail: snnambirajan@rediffmail.com.
Please also visit my web-site which provides links to access my other ‘Blogs’
and ‘Presentation Slides’ pertaining to ‘Our Ancient Music of 22 srutis’: http://www.22sruti.com . I would also
recommend the viewers to peruse my Book:

**(available at my postal address: Srinivasan Nambirajan, A-7/ 103, Florida Estate, Keshav Nagar, Mundhwa, Pune-411036).***“The Mystic Citadel of 22 Srutis Music”*