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Mathematical concepts in Veda

A technical article explaining the mathematical concepts of the different chanting of the Vedas namely Vakkya and Pada, Krama , Jata and Gana Paathas and thereby the Vedas were preserved in their original forms without interpolation.

FOUR VEDAS – Oral tradition that lives! – Vedic Chanting

Siva Subramanian and Chandra Mouli


‘Hindu’ is a terminology given to the people living around river Sindhu by Persians and hence the religion practiced by them came to be called  ‘Hinduism’. Our fore fathers called it as Sanaatana Dharma (Eternal Doctrine or Religion) since its origin is not known and is still part of a living culture. Vedas, the prime scripture of the Hindus is verily considered the breadth of one transcendent Lord Brahman and handed to humanity through the vision of the Rishis (the wise men). The root word of “Veda” is “Vid” which in Samskritam means ‘to know’.  The Vedas thus carry the epithets “Anaadi” (without a beginning) and “Apouresheyam” (without an author). Sanaatana Dharma could also be called Vedic Religion. The great Rishi Sri Veda Vyasa codified the undifferentiated huge body of knowledge contained in the Vedas into four for our learning, understanding and comprehension. They are Rik, Yajur, Sama and Atharva Vedas. Vyasa after codifying the Vedas, instructed them one each to his disciple Rishis Paila, Vaisampayana, Jaimini and Sumantu respectively to be passed on to posterity.  A division appeared later in Yajur Veda called the Sukla (white) Yajur Veda and the original Yajur Veda was named Krishna (Black) Yajur Veda. “The Vedas reveal the one truth to us in the form of many deities. The Vedas contain lofty truths and is storehouse of knowledge. The mantras of the Vedas bring blessings to the world in the form of sound”.

Each of the four Vedas in turn has many Saakhas (branches). It is now known that originally there were 1180 Saakhas in total. Rik Veda had 21 Saakhas, Krishna Yajur Veda 94, Sukla Yajur Veda 15, Sama Veda 1000 and Atharva Veda had 50 Saakhas. Depending upon the type of knowledge that they impart, four major divisions have been identified in each Shakha. They are: 1. Samhitas 2. Brahmanas 3. Aranyakas and 4. Upanishads. However, due to diminishing number of sadhakaas (students who get initiated, learn and practice), many Saakhas have become extinct. Thus we have a situation where there are only 2 Saakhas in Rik, 4 in Yajur (Sukla and Krishna Yajur combined), 3 in Sama and 3 in Atharva Veda and in them too not all of the 4 divisions (Brahmanas etc) are available.

The Vedas were given to the world by the great Rishis (by observing them in the universe through their divine powers – mantra drishtah) for the benefit of all lives in the universe. In order to achieve this effect, the Vedas must be chanted in a way that preserves their pristine purity. Hence rules were codified for: pronouncing each word with appropriate upward or downward drift in the note, time taken to utter word using the concept of maatras or meter (the time taken to pronounce a short vowel).Etc.. These rules, together with knowledge required for using them in rituals, are found in a set of texts called Vedangas (literally meaning the body parts of the Veda). There are six such Vedangas and they are: 1. Sikshaa or phonetics 2. Chandas or prosody 3. Vyaakarana or grammar 4. Nirukta or etymology 5. Jyotisha or astronomy/astrology 6. Kalpa or manual of rituals.

“The sounds of some mantras have greater value than their meaning! Their syllables chanted in a particular manner create a special energy”. Hence the Rishis devised ways to maintain the purity, clarity and originality of the Vedas by clearly instructing the appropriate chanting techniques. Each of these techniques has a precise mathematical pattern.  As Sri Chandrasekharendra Sarasvati, the 68th Pontiff of Sri Kanchi Kamakoti Peetham had observed:  “our forefathers put down specific rules to make sure that not a syllable was changed in chanting, not a svaraa (note) was altered”.

 “In order to make sure that the words (and hence their meanings) do not get corrupted or mutated, it was laid down that each Vedic mantra should be chanted in various patterns and combinations known as “Vaakya, Pada, Krama, Jata. Maalaa. Sikha, Rekha, Dhwaja, Danda, Ratha and Ghana” 

 The description of each of the patterns and combinations using mathematical notations and symbols given below, are based in part from the book on ‘Sri Rudra Ghanam’ by Sri K.Suresh written in Tamil language. Any mistakes that might have crept in translation and elaboration are entirely ours.

 Vaakya Paatha (also called Samhita Paatha):   In this first pattern, a sentence in a mantra is chanted as is. The only rule observed is that the words in the sentence that are conjoined (sandhi) in chanting are done so appropriately.

 Pada Paatha: In this, the sentence is broken into words or “pada” and is recited word by word instead of joining the words and stringing them together.

 These two methods are called “Prakriti Paatha” or natural way of chanting as the words of the mantra occur in normal sequence. If these were the only prescribed methods of chanting, then it would have been impossible to prevent interpolations and alterations and changes over thousands of years. So our forefathers devised additional recitation patterns to protect the integrity of the mantras.

 Example Sentence:

“Gajaananam Bootha Ganaathi Sevitham”

 Krama Paatha – this is a sentence structure that combines the first and second words for the first turn, then the second and third words for the second turn, then third and fourth words for the third turn and so on. There are certain other rules to preserve the ending and beginning of sentences as well. 

 1-2; 2-3; 3-4; and so on

 Example Sentence:

“Gajaananam Bootha, Bootha Ganaathi, Ganaathi Sevitham”

 Mathematical Sequence for Krama Paatha:

 n, m are integers. n>1, m>0

S = original sentence (i.e. sequence of words/Padas)

P(n) = nth word/Pada of a sentence

K = full Krama Paatha of original sentence

K(m) = mth turn of Krama Paatha


S = P(1), P(2), P(3), … , P(n-2), P(n-1), P(n)


Turn 1: K(1) = P(1), P(2) ;

Turn 2: K(2) = P(2), P(3);




Turn (m-1): K(m-1) = P(n-2), P(n-1)

Last Turn:    K(m) = P(n-1), P(n);


 So general combination for Krama Paatha is:

K(m) = P (n-1), P(n);

K = K(1), K(2), K(3), … , K(m-1), K(m)

Where, n >1, and m < n

 The following are considered as "Vikriti paathas" or artificial way of chanting because it involves reverting the words like first word after second, second after three etc.

 Jata Paathathis is a more complicated sentence structure where each Jata turn itself contains: (Krama Turn) + (reverse Krama Turn) + (Krama Turn).

 1-2-2-1-1-2;  2-3-3-2-2-3; 3-4-4-3-3-4; and so on

 Example Sentence:

“Gajaananam Bootha Bootha Gajaananam Gajaananam Bootha;

Bootha Ganaathi Ganaathi Bootha Bootha Ganaathi;

Ganaathi Sevitham Sevitham Ganaathi Ganaathi Sevitham”

 Mathematical Sequence for Jata Paatha:

Leaving some of the complexities about Veshtanas…


n, r are integers. n>1, r>0

S = original sentence (i.e. sequence of words/Padas)

P(n) = nth Pada in a sentence

J = full Jata Paatha of original sentence

J(r) = rth turn of Jata Paatha


S = P(1), P(2), P(3), … , P(n-2), P(n-1), P(n)


Turn 1: J(1) = P(1), P(2), P(2), P(1), P(1), P(2);

Turn 2: J(2) = P(2), P(3), P(3), P(2), P(2), P(3);



Turn (r-1): J(r-1) =  P(n-2), P(n-1), P(n-1), P(n-2), P(n-2), P(n-1);

Last Turn:  J(r)  =  P(n-1), Pn, Pn, P(n-1), P(n-1), Pn;


Hence General Combination for Jata Paatha is

J(r) = P(n-1), Pn, Pn, P(n-1), P(n-1), Pn;

J = J(1), J(2), J(3), … , J(r-1), J(r)

Where, n >1, and r < n (without any Veshtana)


Ghana Paatha – this is the most complicated sentence structure where each Ghana turn itself contains: (Jatai Turn) + (3rd Pada) + (reverse 3 Paathas) + (straight 3 Paathas)


1-2-2-1-1-2-3-3-2-1-1-2-3; 2-3-3-2-2-3-4-4-3-2-2-3-4;

3-4-4-3-3-4-5-5-4-3-3-4-5;  and so on


Example Sentence:

“Gajaananam Bootha Bootha Gajaananam Gajaananam Bootha Ganaathi Ganaathi Bootha Gajaananam Gajaananam Bootha Ganaathi;

Bootha Ganaathi Ganaathi Bootha Bootha Ganaathi Sevitham Sevitham Ganaathi Bootha Bootha Ganaathi Sevitham;”


Mathematical Sequence for Ghana Paatha:


Leaving some of the complexities about Veshtanas…


n, s are integers. n>2, s>1

S = original sentence (i.e. sequence of words/Padas)

P(n) = nth Pada in a sentence

G = full Ghana Paatha of original sentence

G(s) = sth turn of Ghana Paatha


S = P(1), P(2), P(3), … , P(n-2), P(n-1), P(n)


Turn 1: G(1) = P(1), P(2), P(2), P(1), P(1), P(2), P(3), P(3), P(2), P(1), P(1), P(2), P(3);

Turn 2: G(2) = P(2), P(3), P(3), P(2), P(2), P(3), P(4), P(4), P(3), P(2), P(2), P(3), P(4);



Turn (s-1): G(s-1) =  P(n-2), P(n-1), P(n-1), P(n-2), P(n-2), P(n-1), Pn, Pn, P(n-1), P(n-2), P(n-2), P(n-1),  P(n);

Last Turn:   G(s)  =  P(n-1), Pn, Pn, P(n-1), P(n-1), Pn; = J(r)


Hence General Combination for Ghana Paatha is

G(s-1) =  P(n-2), P(n-1), P(n-1), P(n-2), P(n-2), P(n-1), Pn, Pn, P(n-1), P(n-2), P(n-2), P(n-1),  Pn;

G = G(1), G(2), G(3), … , G(s-1), G(s)

Where, n > 2, s > 1 and s < n (without any Veshtana)


It is said that chanting using Pada Paatha is twice as beneficial as chanting using Samhita Paatha. The benefit ratio multiplies with each subsequent paatha as Krama paatha gives 4 times and Jata paatha 1000 times the benefits with the final Ghana paatha showering infinite blessings.

From the descriptions of some of the chanting patterns, it is clear that our Rishis have devised a foolproof system to preserve the purity of word, sound, intonation, pronunciation, accent and sound combinations for thousands of years and hopefully for all ages if the Vedic learning tradition continues. No wonder that the Vedas carry the appellation 'Sruthi' (meaning that which is heard) as the oral tradition can transmit the phonetics much more effectively than written symbols.

Sri Siva Vishnu Temple is bringing more than 190 priests who are experts in the currently existing Veda saakhas of all four Vedas for chanting some important Vedic mantras using multiple paathas for the benefit of our congregation. This is a rare opportunity indeed even in India. Thirty-two of these visiting scholars and pundits can also recite all these different saakhas up to Ghana Paatha and are reverentially called Ghanapattis.  Please come with family and friends to listen to this unique and rare ways of chanting mantras from our glorious Veda on the evening of Sunday, August 3, 2008 and avail of its beneficial vibrations.



1.    Pujyashri Chandrasekharendra Sarasvati Svaamigal; Hindu Dharma – The Universal Way of Life published by Bharathiya Vidya Bhavan, 1996

2.    Pujyashri Chandrasekharendra Sarasvati Svaamigal; Voice of God 

3.    Sri K. Suresh (Venkatrama Sarma); Sri Rudra Ghanam published by Latha, 2004