Understanding Mathematics in Kathak | Shreeyaa Popat | TEDxSIUNashik

Key Concepts

• Le and Tal: Speed/tempo and rhythmic cycle in Indian classical music and Kathak dance.
• Tal: A cycle of a defined number of beats that repeats.
• Sum: The first beat of any Tal, marking the beginning and end of the cycle.
• Thihai: A phrase repeated thrice, ending on "Sum."
• Fermishi Chakradar: A longer composition with a "Mukra" (introductory phrase) and a "Thihai" embedded, repeated thrice, with each repetition of the Thihai ending on "Sum."
• Mukra: An introductory phrase in a Fermishi Chakradar.
• Paltas: Permutations of a main phrase in Kathak.
• Laya: Tempo or speed.
• Lehri: Manipulating the tempo of counts while maintaining a constant clap tempo.
• Sangam Tal: A decimal Tal of 8.75 beats, composed of three parts with varying speeds.
• Hustak: Hand gestures.
• Nohaka: A sequence demonstrating spins.

Understanding the Mathematics in Kathak

Introduction

The lecture demonstration explores the mathematical principles underlying Kathak dance, highlighting the role of numbers, arithmetic, algebra, geometry, permutations, and speed (Laya) in its structure and composition.

Le and Tal

• Le: Speed or tempo.
• Tal: A cycle of a defined number of beats. It's a cycle, meaning it loops.
• Tintal: A 16-beat Tal cycle is demonstrated. The beats are counted from 1 to 16, then back to 1.
• Sum: The first beat of the Tal, marking the beginning and end of each cycle. Every Kathak composition is designed to fit within this Tal cycle.

Arithmetic in Kathak

• The four basic arithmetic operations (addition, subtraction, multiplication, and division) are used in composing Kathak phrases.
• Thihai: A phrase repeated thrice to land on "Sum."
  • If one unit of the Thihai is "x," the entire Thihai is "3x."
  • The formula is 3x = n + 1, where "n" is a multiple of 16 (representing cycles of Tintal).
  • Example: An 11-beat phrase repeated thrice equals 33 beats, fitting into two cycles of 16 beats (16 * 2 = 32, 32 + 1 = 33).
• Fermishi Chakradar: A longer composition with a Mukra and a Thihai.
  • The formula is (m + 3x) * 3 = n + 1, where "m" is the Mukra, "x" is the Thihai, and "n" is a multiple of 16.
  • Example: If n = 81 and x = 5, then m = 12. The Mukra is 12 beats, the Thihai is 15 beats, and the entire phrase is 27 beats. Repeated thrice, it equals 81 beats, fitting into five cycles of 16 beats (16 * 5 = 80, 80 + 1 = 81).

Algebra in Kathak

• Algebraic formulas simplify the composition of Thihai and Fermishi Chakradar.
• The presenter's guru, Pandit Anupam Yogi, has contributed a list of formulas for composing Fermishi Chakradar compositions for various Tal cycles (5 to 32 beats).
• A Fermishi Chakradar in Teental is demonstrated.

Geometry in Kathak

• Spins (Chakras): A dancer makes a circle around themselves. A circle has four quadrants. Spotless Chakras involve facing the first quadrant and completing the remaining three-quarters.
• Stage Rounds: The dancer imagines a circumference on the stage, and the leading foot should follow this imaginary line.
• Hand Gestures (Hustak):
  • A Hustak can resemble a 180-degree line.
  • Another can resemble a V or angle bisector.
  • Another can resemble a perpendicular line.
• Posture (Thai): In a posture, the right hand should be parallel to the right leg. The shoulder to elbow should be parallel to the hip socket and knee, the elbow to wrist parallel to the knee and ankle, and the wrist to fingertips parallel to the ankle and toes. The chin and eyesight should be in line with the right hand.
• A Nohaka is demonstrated to showcase the geometry of spins.

Permutations in Kathak

• Paltas: Permutations of a main phrase.
• Example: A main phrase "1 2 3 1 2 3 1 2 3 1 2 3 1 2" can be permutated by shifting the position of "1 2" to create variations like "5 4 3 2 1."
• The slant in the permutations is heard in the footwork.

Speed (Laya) in Kathak

• Lehri: Manipulating the tempo of counts while maintaining a constant clap tempo.
• Example: Counting 1, 2, 3, 4, 5, 6, 7, 8 while clapping at a constant rate, then increasing the count tempo.
• Basic Lehri involves 1, 2, or 4 counts per clap.
• The presenter's guru contributed a list of all Leharis from 1 to 8 and how to perform them practically.

Kathak and Decimal System

• Sangam Tal: A decimal Tal of 8.75 beats.
  • Part 1: Six beats in speed x.
  • Part 2: Three beats in speed 2x (3 * 2 = 1.5).
  • Part 3: Five beats in speed 4x (5 * 0.25 = 1.25).
  • Total: 6 + 1.5 + 1.25 = 8.75 beats.
• A Fermishi Chakradar in Sangam Tal is recited and demonstrated.

Conclusion

Mathematics plays an integral role in Kathak, providing a framework for its structure and composition. The challenge in dance is to present these mathematical principles in an aesthetically pleasing manner.