Ganita in Kathak: A Modular-Arithmetic Model for Progressive Jathi Construction and Deriving Cyclic Star-Polygon Sequences from Fractional Matra Increments.

Abstract

This paper documents an original technique for constructing jathis (syllable phrases) within teentaal,developed through practical experimentation with progressive syllable addition inside a vibhag. Bysystematically increasing the count at which each successive syllable group is added — in steps of aquarter matra (kaud) — a numerical sequence emerges that describes which beat-position each newaddition lands on. On tabulating these sequences across a full cycle of counts, a consistent underlying rulewas found: each row is generated by a linear recurrence of the form aₙ = 1 + (n−1)k (mod 8), where k itselfprogresses as k = 4c mod 8 for count c. This is structurally identical to the method used to construct {8/k}star polygons, meaning the jathi-building technique traces, in time, the same geometry that star polygonstrace in space. The paper presents the Kathak-theoretical background, the empirical procedure, theresulting data, and the formal derivation of the governing formula.