Pascal’s triangle in Karnatic music and Bharata Natyam

 According to Hindu mythology, Mount Meru is the sacred and the centre of the metaphysical universe. The word Meru roughly translates to the great or highest. The texts of Hindu mythology provide the description of mount Meru. It’s conical in nature at the outer appearance similar to the triangle. It's tapered in nature.

Among many mathematicians, Acharya Pingala and Bhaskar Acharya were the first in the world to explain the concept of numbers in triangles. It is woeful to see how the discovery is created by the French mathematician Blaise pascal and the term is coined as pascal’s triangle.

Combinatorics is a substantial branch of mathematics that deals with the counting of numbers and finite structures. Computational molecular biology, computer architecture and pattern analysis are the few areas of applications of combinatorics. Analysis of patterns present in images or signals helps scientists understand the structure and behaviour of elements. Dance is not far from all of these; patterns are embedded throughout the sequences. One needs a sense of combinatorics to analyze this. All the students of the academy were compelled to count the number of beats (Thalam) before actually doing a new dance sequence. This helped students in understanding the pattern of choreography and the structure present in it.

 

I was immensely inspired by my Sanskrit teacher in my childhood, I was just nine years old when I composed four Sanskrit verses with correct grammar. I learnt that mathematics was the basis of any composition or dance in India. Any Sanskrit verse will always have an equal number of letters in each quarter. An unequal number of letters in each section is considered grammatically incorrect. Let’s analyze the famous verse in praise of Lord Ganapathi.

वक्रतुण्ड महाकाय सूर्यकोटि समप्रभ ।

निर्विघ्नं कुरु मे देव सर्वकार्येषु सर्वदा ॥

The verse has two halves, which is indicated by the comma (|) and period (||). The number of letters in each half should be equal.

व	क्र	तु	ण्ड	म	हा	का	य	सू	र्य	को	टि	स	म	प्र	भ
1	2	3	4	5	6	7	8	1	2	3	4	5	6	7	8

 

नि	र्वि	घ्नं	कु	रु	मे	दे	व	स	र्व	का	र्ये	षु	स	र्व	दा
1	2	3	4	5	6	7	8	1	2	3	4	5	6	7	8

 

Note: At the fourth position of the first half’ ण्ड’ is not two letters even though it has ‘ण’ it not in its full form, it is just half of ण. Hence all such combinations in Sanskrit are not counted as a letter. 

During my music classes, I was made to sing the basics for a long time. I was bored like any other student as I was not proceeding further in my lessons. Over the years I realized every exercise of the class was helpful in many ways to understand the mathematical structure of music and dance associated with it. The work of all the legends like Pingala, Bhaskaracharya, Bartha Muni and also Blaise pascal all started making sense in terms of Mathematics the day my teacher asked me to sing the following lesson.

 

S

SRS

SRGRS

SRGMGRS

SRGMPMGRS

SRGMPDPMGRS

SRGMPDNDPMGRS

SRGMPDNSNDPMGRS

 

I followed him blindly without realizing why was he making me sing in a particular sequence. I saw a right-angled triangle when tried to write the notations of it. To my surprise instead of writing it from the left corner, I wrote the first musical note ‘S’ in the centre of the page and started arranging the successive notes below it. Was it the same Meru mountain that Halayudha and others were talking about? I had no idea.

 

S

SRS

SRGRS

SRGMGRS

SRGMPMGRS

SRGMPDPMGRS

SRGMPDNDPMGRS

SRGMPDNSNDPMGRS

This was easy to remember and sing as it involves basic ascending and descending of notes by just one note. The very next day my teacher experimented more on my singing skills and asked me to sing a new pattern of notes, it was not easy as the previous one.

S

S S

S R S

S G G S

S M D M S

I came with the mathematical interpretation to remember the sequence as I had to skip a few notes to get the sequence correct. I constructed a table in which I coded all the musical notes, the results which I obtained was astonishing.

Notes	Number
S	1
R	2
G	3
M	4
P	5
D	6
N	7

 

It resulted in a beautiful mathematical pattern that looked like mountain Meru.

Pascal’s triangle is a construction of a triangular pattern using numbers arranged row-wise. It begins with number one in the first row followed by two 1’s in the second row. The numbers of the successive rows are obtained by adding the number above a given number.

 

The sequence of the musical noted decoded exactly follows the same pattern as pascals triangle.

S

S S

S R S

S G G S

S M D M S

 

1

1  1

1  2  1

1  3  3  1

1  4  6  4  1

One can observe patterns similar to this in many dances and songs. 

 

The creator of Bharat Natyam Sri Bharat Muni explains in the three verses of natya shastra about the constructions of syllables to the meters. The order of the letters or syllables can either be in ascending order or descending order.

 

Nāṭyaśāstra:

 

ekādhikāṃ tathā saṃkhyāṃ chandaso viniveśya tu |

yāvat pūrñantu pūrveña pūrayeduttaraṃ gañaṃ ||

 

evaṃ kṛtvā tu sarveṣāṃ pareṣāṃ pūrvapūrañaṃ | kramānnaidhanamekaikaṃ pratilomaṃ visarjayet ||

 

sarveṣāṃ chandasāmevaṃ laghvakṣaraviniścayaṃ |

jānīta samavṛattānāṃ saṃkhyāṃ saṃkṣepatastathā ||

 

Translation: 

Start the sequence by increasing by one till the number of syllables in a meter.
After this, sum the successive number to the previous one till the end. Remove each syllable in reverse sequence till the final in a sequential manner. This is the rule for equal (sama) forms.

 

After watching many Thillana and learning quite a few, I started observing the choreography of the korvai follows the rules established by Pingala, Barthaamuni. The sequence taught by my guru had many such patterns which looked like Mount Meru. One such pattern was in Kedragowlam Thillana set to Adi Thalam.

The sequence:

Thai

Dhi                  Thai

Thai                Thai                tham

Thai                Dhi                  Dhi                  Thai

 

Dhi                  Thai

Thai                Thai                Tham

Thai                Dhi                  Dhi                  Thai

 

Dhi                  Thai

Thai                Thai                Tham

Thai                Dhi                  Dhi                  Thai

 

 

Numbering the sequence will result in a pattern like a mountain.

1

1         2

1         2         3

1         2         3         4

1         2

1         2         3

1         2         3         4

1         2

1         2         3

1         2         3         4

It’s so fascinating to see how our ancestors used the concepts of combinatorics in music and dance. Every time I come across such patterns in a dance sequence many similar patterns appear in mind. Sometimes it's either a tree or mountain or face beak of the bird. What’s more fascinating is the mathematical order implemented in the construction of musical syllables. These patterns are either in arithmetic progression with a common difference of one or in geometric progression.