mathematics in sanskrit poetry


Mathematics in Sanskrit poetry

       It is interesting to observe that in Sanskrit poetry, the concept of binomial coeficients, Fibonacci numbers and binary numeration has been in use right from the days of Pingala who was the first to write a treatise on Chandas-shastra relating to metres in Sanskrit poetry.

    In Sanskrit poetry, we have stanzas with four quarters. Each quarter may have the same number of syllables or the same number of time units, a short one being assigned 1 time unit and a long one 2 time units.  There are metres in which the odd quarters have the same number of syllables or time units, while the even quarters have a different number of equal units.

         A Sanskrit stanza or padya consists of four padas or four quarters, which are regulated by

                      (i) The number of syllables in each quarter, or

                      (ii) The number of syllabic time units or matras, a short

                         sound being assigned one unit of time and the long one                                                                                    two units of time.

   Each stanza or padya can thus be of two broad types, vritta or the one in which metre is regulated by the number and composition of syllables, and the other jati, where the metre is regulated by the number of syllabic instants or time units in each quarter.

   If all the padas are similar, the padya is called samavritta,; if alternate quarters are similar, it is called ardhasamavritta, and if all the quarters are dissimilar it is called vishamavritta.

   A syllable can be short or long; short vowels followed by anuswara or a visarga or a conjunct consonant are considered to be long. Any exceptions to this are generally granted as poetic licence to be exercised, depending upon the exigencies of the case.

    8 sets of triplets of long or short sounds or a mixture of both have been classified by Pingala and called ganas. He has chosen syllable triplets consisting of all short or all long or a mixture of these two sounds.   The interesting thing is that he has used the concept of binary numbers in place of the decimal numbers from 0 to 7. One does not know whether the present symbol for zero was known to him or not.  Assuming that he had used * for 0 and 1 for long sounds, his classification of ganas was as follows, (S standing for short and L for long sound units)  

Decimal       Name of    Binary Pingala's   Syllable

Number         Gana         form    form       Triplet

0                      Na        000      ***      SSS

1                      Sa         001      **1      SSL     

2                      Ja          010      *1*      SLS 

3                      Ya         011      *11      SLL

4                      Bha       100      1**       LSS

5                      Ra         101      1*1       LSL

6                      Ta          110     11*       LLS

7                      Ma         111     111       LLL  

 An easy way to remember this is to use the mnemonic

Sa Na JaYaBhaRaTa Mey (which in Sanskrit would mean: He is  not my Jayabharata.        

        Pingala had considered how many patterns can be formed from a given number of syllables, for instance, how many patterns consist of 3 syllables, how many consist of 4 syllables and so on?  They can be of different types, all short or all long or a mixture of long and short sounds.  (Let S denote short sound and L denote long sound).  In other words, combinatorics or the principle of permutations/combinations and the binomial coefficients had a large role to play in this regard. We examine a few cases with a fixed number of syllables, long or short or a mixture of both.


      Pattern with 1 syllable (Long or short): L, S        No. of patterns = 2  =   21   

      Pattern with 2 syllables: (Long, short or mixture) LL, SS, LS, SL   No. of patterns = 4 = 22

      Pattern with 3 syllables: (Long, short or mixture)

          All short      SSS                                      No.of patterns = 1

          One short and two long:

            SLL, LLS, LSL, LLS,                           No. of patterns = 4

          One long and two short:

            LSS, SLS, SSL                                      No. of patterns = 3

                       Total number of patterns: 23 = 8.


      Pattern with 4 syllables (Long, short or mixture)

          4 short syllables:     SSSS                       No. of patterns = 1 =  4C0

          3 short & 1 long:     SSSL, SSLS,

                                          SLSS, LSSS            No. of patterns = 4 =  4C1

          2 short & 2 long:   SSLL, LLSS.

                                        SLSL, LSSL,

                                        LSLS, SLLS              No. of patterns = 6 =  4C2

          1short & 3 long: SLLL, LSLL,

                                     LLSL, LLLS                No. of patterns = 4 =  4C3


                                   4 long:  LLLL                             No. of patterns = 1 =  4C4


          Pattern with 5 syllables (Long, short or mixture):

          5 short syllables:   SSSSS:                      No. of patterns = 1 = 5C0

          4 short and 1 long: SSSSL, LSSSS, SLSSS,

                                         SSLSS, SSSLS         No. of patterns = 5 =  5C1

              3 short and 2 long: SSSLL, LLSSS,

                                        LSLSS, LSSLS,


                                        SLSLS, SSLLS,


                                        SLLSS, SLSSL         No. of patterns =  10 = 5C2

                                                  2 short and 3 long: LLLSS, LSLLS,

                                   LLSLS, LSLSL,                 

                                                                 LLSSL, LSSLL


                                           SLLSL, SSLLL,



                                            LLSLS      No. of patterns = 10 =  5C3


                   1 short and 4 long: LLLLS, LSLLL,

                                            LLSLL, LLLSL

                                            SLLLL       No. of patterns = 5 =  5C4

                  5 long:                      LLLLL      No. of patterns = 1 =  5C5

                                                    Total umber of patterns = 26 = 32.

         As seen in the analysis, if each quarter contains 6 syllables, (long, short or a mixture of both), theoretically the number of possible metres is 26 = 64.     

     If each quarter is to contain 26 syllables (long, short or a mixture of both), the theoretic combination of syllables is 226 = 67,108,864.  If we consider cases where alternate quarters are similar or all are dissimilar, the number of possible metres becomes very large.  However, poets have used only a very limited number of metres. (Please see Appendix A and B for metres with constant number of syllables in each quarter, and metres with constant time units in each quarter, which poets have actually used).


      It is interesting to see that when the number of syllables is n (long, short or a mixture of both) the number of patterns follows the pattern of binomial coefficients in the expansion of (a + b)n, and the total number of patterns possible is 2n and when n = 4 it becomes = 16, and when n is 5, it becomes 32.  Further, for s short and l long syllables, or l short and s long syllables, the number of patterns available in this case of a total of s + l syllables is  the binomial coefficient s+l Cs =  s+lCl,. It looks strange that binomial coefficients should find some reference in Sanskrit poetry right from the early days.

       Now, we see the pattern for time units 1 to 6, taking 1 unit for short and 2 for long.

       Pattern with 1 short syllable: S                                               1    

       Pattern with 2 short syllables: SS, L                                       2

       Pattern with 3 short syllables: SSS. SL. LS                            3

       Pattern with 4 short syllables: SSSS, SSL. SLS, LSS, LL      5

       Pattern with 5 short syllables: SSSSS, SSSL, SSLS, SLSS.   8

       Pattern with 6 short syllables: SSSSSS, SSSSL, LSSSS,

                                                          SLSSS, SSLSS, SSSLS,

                                                          LLSS, SSLL, LSSL,               13


                                                          SLSL, LSLS, SLLS


    When the number of time units is n, the number of patterns follows the Acharya Hemachandra-Gopala pattern of numbers, now known as the Fibonacci numbers, such that Fib(1) = 1, and (Fib(2) =2, and Fib(n) = Fib(n-1) + Fib (n-2), for all n > 2.  We thus see for n = 3, Fib(3) = 1 + 2 = 3; Fib (4) = 3+ 2 = 5; Fib(5) = 5 +3 = 8;  Fib (6) = 5 + 8 = 13.

    Acharya Hemadeva had noticed this and called this set of numbers as Hemadeva-Gopala numbers which are the forerunners for the Fibonacci numbers which came nearly 70 years later. Perhaps, Fibonacci drew his inspiration from this aspect of metres in Sanskrit poetry.  


    Pingala’s Chandas-shastra as well as Lilavati and Vrittaratnakara give directions for computing the number of possible varieties and finding their places or that of any single one in a regular enumeration of them.

    For instance, the number of padas and syllables in a few of the metres is as follows:

          Gayatri(tripada):     3 padas and 8 syllables in each

          Anushtup: 4 padas and 8 syllables in each

          Viraja:       4 padas and 10 syllables in each

          Trishtup:   4 padas and 11 syllables in each

          Jagati:        4 padas and 12 syllables in each.

    A well-known quantitative scheme in the traditional literature classifies the common metres according to the syllable count of a stanza, as multiples of 4, as follows:  

        Dwipadaviraja        20 syllables

        Gayatri:                  24 syllables

        Ushnik:                   28 syllables

        Anushtup:               32 syllables

        Brihati:                    36 syllables

        Pankti:                     40 syllables

        Trishtup:                  44 syllables

        Jagati:                       48 syllables.

There are many comprehensive schemes in the traditional literature, each distinct type of stanza carrying its own name.


                               (Please see Appendix A and B which follow)


                                                                                APPENDIX A





4 syllables: Kanya: L LLL (Gu Ma)


5 syllables: Pankti LSS LL   (Bha Gu Gu)


6 syllables: Gayatri:--

    (1) Tanumadhyama: LLS SLL (Tha, Ya)

    (2) Vidyullekha (Vani, Sashivadana): LSS LLL   (Bha Ma)

    (3) Somarati: SLL SLL  (Ya Ya)


7 Syllables: Ushnik:--

     (1) Kumaralalita: SLS SSL L  ( Jha, Sa,Gu)

     (2) Madalekha: LLL SSL L    (Ma Sa Gu)

     (3) Madhumati: SSS SSS L     (Na, Na, Gu)


8 syllables: (1) Anushtup: (Most popular metre: The fifth syllable in each quarter is short and the 6th syllable in each quarter is long, while in the 1st and 3rd quarters, the 7th syllable is long and in the 2nd and 4th quarters the 7th  syllable is short):--

    (2) Gajagati: SSS LSS S L  (Na, Bha, La, Gu)

    (3) Pramanika: SLS LSL S L  (Ja, Ra, La, Gu)

                (4) Manavaka: LSS SSS S L   (Bha, Tha, La, Gu)

                (5) Vidyunmala: LLL LLL L L  (Ma, Ma, Gu, Gu)

                (6) Samanika: LSL SLS L S   (Ra, Ja, Gu, La)


9 syllables: Brihati:--

    (1) Bhujagasishubhruta: SSS SSS LLL (Na, Na, Ma)

    (2) Bhujangasangata: SSL SLS LSL   (Sa, Ja,  Ra)

    (3) Manimadhya: LSS LLL SSL     (Bha, Ma, Sa)


10 syllables: Pankti:--

     (1) Tvaritagati: SSS SLS SLS L  (Na, Ja, Na, Gu)

     (2) Matta: LLL LSS SSL L  (Ma, Bha, Sa, Gu)

     (3) Rukmavati (Champakamala): LSS LLL SSL L (Bha, Ma, Sa, Gu)


11 syllables: Trishtup:--

    (1) Indravajra: LLS LLS SLS L L  (Tha, Tha, ja, Gu, Gu)

    (2) Upendravajra: SLS LLS SLS L L  (Ja, Tha, Ja, Gu, Gu)

    (3) Upajati: (This is a mixture of (1) and (2) above, and there are 14 varieties. Even when other metres are mixed in one stanza, it is still called upajati): (The first and third quarters are as in Indravajra while the 2nd and fourth quarters are as in Upendravajra).

 (4) Dodhaka: LSS LSS LSS L L  (Bha, Bha, Bha, Gu, Gu)

 (5) Bhramaravilasitam: LLL LLL SSS S L  (Ma, Bha, Na, La, Gu)

 (6) Rathoddhata: LSL SSS LSL S L (Ra, Na, Ra, La, Gu)

 (7) Vatormi: LLL LLL LLS LL  (Ma, Ma, Tha, Gu, Gu)

 (8) Shalini: LLL LLS LLS LL  (Ma, Tha, Tha, Gu, Gu)

 (9) Swagata: LSL SSS LSS LL  (Ra, Na, Bha, Gu, gu)


12 syllables: Jagati:--

   (1) Indravamsa: LLS LLS SLS LSL (Tha, Tha, Ja, Ra)

   (2) Chandravartma: LSL SSS LSS SSL (Ra, Na, Bha, Sa)

   (3) Jaladharamala: LLL LSS SSL LLL (Ma, Bha, Sa, Ma)

   (4) Jaloddhatagati: SLS SSL SLS SSL (Ja, Sa, Ja, Sa) 

   (5) Tamarasa: SSS SLS SLS SLL (Na, Ja, Ja, Ya)

   (6) Totaka: SSL SSL SSL SSL (Sa, Sa, Sa, Sa)  

   (7) Drutavilambita: SSS LSS SLS LSL  (Na, Bha, Ya, Ya)

           According to Pingala: SSS LSS LSS LSL (Na, Bha, Bha,Ra)

           According to Kalidasa: LSS LSS LSS LSL (Bha, Bha, Bha, Ra)

                             According to Apte: SSS LSS SLS LSL (Na, Bha, Ja, Ra)

   (8) Prabha: SSS SSS LSL LSL (Sa, Sa, Ja, Ja)

   (9) Pramitakshara: SSL SLS SSS SSS  (Sa, Ja, Sa, Sa)

  (10) Bhujangaprayata: SLL SLL SLL SLL  (Ya, ya, Ya Ya)

  (11) Manimala: LLS SLL LLS SLL (Ta, ya, Tha,Ya)

  (12) Malati: SSS SLS SLS LSL  (Na, Ja, Ja, Ra)

  (13) Vamsasthavila: SLS LLS SLS LSL (Ja, Tha, Ja, Ra)

  (14) Vaishwadevi: LLL LLL SLL SLL  (Ma, Ma, Ya, Ya)

  (15) Sragini: LSL LSL LSL LSL (Ra, Ra, Ra, Ra)


13 syllables: Atijagati:

   (1) Kalahamsa: SSL SLS SSL SSL L  (Sa, Ja, Sa, Sa, Gu)

   (2) Kshama: SSS SSS LLS LLS L  (Na, Na, Tha, Tha, Gu)

   (3) Manjubhahsini (Sunandini, Prabodhita): SSL SLS SSL SLS L

         (Sa, Ja, Sa, Ja,Gu)

               (4) Mattamayuri:L LLL LLS SLL SSL L (Ma, Tha, Ya, Sa, Gu)

   (5) Ruchira: SLS LSS SSL SLS L (Ja, Bha, Sa, Ja Gu)


14 syllables: Shakvari:--

   (1) Aparajita: SSS SSS LSL SSL S L (Na, Na, Ra, Sa, La Gu)

   (2) Asambhadha: LLL LLS SSS SSL LL (Ma, Tha, Na, Sa, Gu, Gu)

   (3) Pathya: SSL SLS SSL SLL S L  (Sa, Ja, Sa, Ya, La, Gu)

   (4) Pramada: SSS SLS LLL SLS S L (Na, Ja, Ma, Ja, La Gu)

   (5) Praharanakalika: SSS SSS LSS SSS S L  (Na, Na, Bha, Na, La, Gu)

   (6) Madhyakshama: LLL LSS SSS SLL L L (Ma, Bha, Na, Ya, Gu, Gu)

   (7) Vasantatilaka: (Uddhaarshani, Simhonnata): LLS LSS SLS SLS L L

           (Tha, Bha, Bha, Ja, Ja, Gu, Gu)

   (8) Vaasanti: LLL LLS SSS LLL LL  (Ma,Tha, Na, Ma, Gu, Gu)


15 syllables: Atishakvari:

    (1) Toonaka: LSL SLS LSL SLS LSL (Ra, Ja, Ra, Ja, Ra, Gu)

    (2) Malini: SSS SSSLLL SLL SLL (Ja, Ra, Ja, Ra, Ja,Gu)

    (3) Leelakhela: LLL LLL LLL LLL LLL (Na, Ja, Bha, Ja, Ra, Gu)

    (4) Shashikala: SSS SSS SSS SSS SSL (Na, Na, Na, Na, Sa)


16 syllables: Ahi:

   (1) Chitra: LSL SLS LSL SLS LSL L (Ra, Ja, Ra, Ja, Ra, Gu)

   (2) Panchachamara: SLS LSL SLS LSL LSL L (Ja, Ra, Ja, Ra, Ja, Gu)

   (3) Vanini: SSS  SLS LSS SLS LSL L  (Na, Ja, Bha, Ja, Ra, Gu)


17 syllables: Ashti:--

   (1) Chitralekha: SSL SSL SLS LSS SLS LL  (Sa, Sa, Ja,  Bha, Ja, Gu, Gu) 

   (2) Nardataka (Kokikala): SSS SLS LSS SLS SLS S L (Na, Jha, Bha,    Ja, Ja, La, Gu)

   (3) Prithvi: SLS SSL SLS SSL SLL S L  (Ja, Sa, Ja, Sa, Ya, La, gu)

   (4) Mandakranta: LLL LSS SSS LLS LLS LL  (Ma, Bha, Na, Tha,Tha Gu, Gu)


 18 syllables: Dhruti:--

    (1) Kusumalatovellita: LLL LLS SSS SLL SLL SLL (Ma, Tha, Na, Ya, Ya, Ya)

    (2) Chitrakala: LLS LSS SSS SLL SLL SLL (Tha, Bha, Na, Ya, Ya Ya)

    (3) Nandana: SSS SLS LSS SLS LSL LSL  (Na, Na, Bha, Ja, Ra Ra)

    (4) Naaracha (Mahamalika,Mahamalini)): SSS SSS LSL LSL LSL  LSL (Na, Na, Ra, Ra, Ra, Ra)

    (5) Shardoolavilasita: LLL LLS SLS SSL LLS SSL (Ma, Tha, Ja, Sa, Tha, Sa)


19 syllables: Atidhruti:--

    (1) Meghavispoorjita: SLL LLL SSS SSL LSL LSL L (Ya, Ma, Na, Sa, Ra,  Ra, Gu)

    (2) Shardoolavikreedita: LLL SSLSLS SSL LLS SSS L (Ma, Sa, Ja, Sa, Tha, Tha, Gu)

    (3) Mumadhura: LLL LSL LSS SSS LLL SSS S (Ma, Ra, Ma, Na, Ma,  Na,Gu)

                        (4) Surasa: LLL LSL LLL SSS SLL SSS L (Ma, Ra, Ma, Na, Ya, Na, Gu)


             20 syllables: Kruti:--

          (1) Geetika: SSL SLS SSL LSS LSL SSL S L  (Sa, Jha, Sa, Bha, Ra,  Sa, La Gu)

      (2) Suvadana: LLL LSL LSS SSS SLL LLL S L (Ma, Ra,  Bha, Na, Ya, La, Gu)


               21 syllables: Prakruti:--

(1) Panchakavali: SSS LSL LSS SLS SLS SLS LSL  (Naa, Ja, Bha, Ja, Ja, Ja, Ra)

                           (2) Sragdhara:LLL LSL LSS SSS SLL SLL SLL (Ma, Ra, Bha, Na, Ya,

                       Ya, Ya)

              22 syllables: Akruti:--

                 (1) Hamsi: LLL LLL LLS SSS SSS SSS LLS L (Ma, Ma, Tha, Na, Na,  

                     Na, Na, Tha, Gu)

              23 syllables: Vikruti:--

                 (1) Adritanaya: SSS SLS LSS SLS LSS SLS LSS S L (Na, Ja, Bha, Ja,

                    Bha, Ja, Bha, La, Gu)

              24 syllables: Sanskruti:--

                 (1) Tanvi: LSS LLS SSS SSL LSS LSS SSS SLL (Bha, Tha, Na, Sa,

                       (Bha, Bha, Na, Ya)


              25 syllables: Atikruti:--

                 (1) Kraunchapada: :LSS LLL SSL LSS SSS SSS SSS L  (Bha, Tha,

                         Na, Sa, Bha, Bha, Na, Ya)


              26 syllables: Utkruti:--

                 (1) Bhujangavijrumbhita: LLL LLL LLS SSS SSS SSS LSL SSL S L

                          (Ma, Ma, Tha, Na, Na, Na, Ra, Sa, La, Gu)

  27 syllables or more ( up to a maximum of  999): Dandaka:

   In each quarter SSS SSS (Na, Na) must occur and the remaining may be either LSL or SLL (Ra or Ya) or all may be SSL (Ya)  .

   Some of the famous Dandakas are Chandavrushtiprayata, Prachitaka Mattamatangaleelakara, Simhavikranta, Kusumasthabaka, Anangasekhara, and Sangrama.


      Now, we consider the syllable patterns in metres whose quarters are half equal, that is, the 1st and 3rd quarters have equal syllables, while the 2nd and 4th have a different number of equal syllables.


1.Aparavaktra (Vaitaleeya): 1 & 3: SSS SSS  LSL  S L  (11)

                                                          (Na, Na, Ra, La, Gu)

                                              2 & 4: SSS SLS SLS SLS   (12)

                                                           (Na, Ja, Ja, Ra)

2.*Upachitra: 1 & 3: SSL SSL SSL S L (11)** (Sa, Sa, Sa, La, Gu)

                       2 & 4: LSS LSS LSS L L (11)  (Bha, Bha, Bha,Gu, Gu)

3.Pushpitagra (Aupachandasika): 1 & 3: SSS SSS LSL SLL  (12)

                                                                 (Na, Na, Ra, Ya)

                                                       2 & 4: SSS SLS SLS LSL L (13)

                                                                 (Na, Ja, Ja, Ra, Gu)

4.*Viyogini: (Vaitaleeya or Sundarii: 1 & 3: SSL SSL SLS L (10)

                                                                  (Sa, Sa, Ja, Gu)

                                                          2 & 4: SSL LSS LSL S L (11)

                                                                   (Sa, Bha, Ra, La, Gu)

5.Vegavati: 1& 3: SSL SSL SSL L (10)** (Sa, Sa, Sa, Gu)

                    2 & 4: LSS LSS LSS L(10)  (Bha, Bha, Bha, Gu)    

         ** Although the number of syllables is the same in all quarters, the pattern is different in the odd and even number quarters.


6.Harinipluta: 1 & 3: SSL SSL SSL S L (11) (Sa, Sa, Sa, La, Gu)

                        2 & 4: SSS LSS LSS LSL (12) (Sa, Bha, Bha, Ra)

    * These are treated as Jati metres but sometimes they are defined with triplets of syllables called ganas.


        Metres with unequal number of syllables in different quarters


Udgata: 1st: SSL SLS SSL S (10) Sa, Ja, Sa, La)

             2nd: SSS SSL SLS L (10) (Na, Sa, Ja, Gu)

             3rd: LSS SSS SLS S L (11) (Bha, Na, Ja, La, Gu)

             4th: SSL SLS SSL SLS L (13) (Sa, Ja, Sa, Ja,gu)

                      In another variant: 3rd: LSS SSS LSS  L (10) (Bha, Na, Bha, Gu)


  There are metres where the number of syllables may vary in each quarter, and these are generally called Gathas, and this includes stanzas where the number of quarters may also be different from 4, or two or more quarters of a regular metre may be combined to form half-equal or unequal quarters.


Metres regulated by the number of syllabic instants or time units

or matra-chandas

The most common variety is:

1.Arya: 1st and 3rd quarters: 12 matras or time units

               2nd quarter: 18 time units

               4th quarter: 15 time units.

   This metre is said to have nine sub–divisions, the most common among them being the following: 


2.Geeti:  1st and 3rd quarters: 12 time units

               2nd and 4th quarters: 18 time units

3.Upageeti: 1st and 3rd quarters; 12 time units

                     2nd quarter: 15 time units

                     4th quarter: 18 time units

4.Udgeeti: 1st and 3rd quarters: 12 time units

                     2nd and 4th quarters: 20 time units

5.Aryageeti: 1st and 3rd quarters: 12 time units

                      2nd quarter: 15 time units

                      4th quarter: 18 time units

6.Vaitaleeya:1st and 3rd  quarters: 14 time units (6 syllabic instants followed by Ra and La and Gu or SLS and SL

     2nd and 4th quarters: 16 time units (8 syllabic instants followed by Ra, La, Gu or SLS SL

7.Aupachandasika: Here, the pattern is the same as above, except that at the end we have Ra and Ya (SLS and SLL); 1st and 3rd quarters: syllabic instants are not all to be short or long syllables. In each quarter, 2nd, 4th and 6th one should not be formed jointly with the next ones.  






































                                                                                     APPENDIX B


          Metres with constant number of time instants in each quarter


There are metres in which each quarter consists of 16 syllabic instants or time units, with the 9th one being short and the last being long.


(1)   Veenaakamsika: 9th and 12th instants: S

                           15th and 16th instants: L

       And the rest of the units are optional, short or long.


(2)   Chitra: 5th, 8th, 9th and 10th instants: S

                                          15 and 16th instants: L


(3)   Upachitra: 5th, 8th, 9th and 10th  instants: S

                                                  15th and 16th  instants: L; (The rest indeterminate)


(4)   Vishloka: 5th, 8th and 12th  instants : S

                                               15th and 16th  instants:  L (The rest indeterminate)

 If two or more varieties are contained in the same stanza, they are called Padakulaka, and there is no restriction except that each quarter should have 16 time units.