We develop a mathematical framework for Indian classical music based on a tonic-centred reindexing of cyclic pitch space. The chromatic pitch system is modelled as a cyclic group of order 12, reindexed so that the tonic Sa occupies position 12 (the full-rotation point) rather than position 0. This reindexing, which we call the RS-12 (Root Space of 12) convention, is isomorphic to Z/12Z but provides a musically natural coordinate system in which Sa is the maximal position and Pa (the perfect fifth, 7 semitones above Sa) falls at position 7. We prove that 12 is the smallest positive integer n for which the continued-fraction convergent to log₂ (3/2) has denominator n, making it the natural minimal cycle for Pythagorean fifth-closure. We introduce a binary ASCII notation system for Hindustani music in which komal (flat) degrees are lowercase and shuddha/tivra degrees are uppercase, giving a clean 12-token encoding of all RS-12 positions with no subscripts required. The 10 Bhatkhande thaats are formalized as heptatonic subsets of the RS-12 chromatic cycle satisfying a coverage condition over five variable degree slots. This yields a combinatorial space of 32 candidate thaats isomorphic to the binary hypercube 0, 1⁵. We prove that the thaat adjacency graph (single-degree substitution) is connected, define the Hamming-based thaat metric, and characterize the 10 Bhatkhande thaats as the tritone-bounded elements of this space (verified by exhaustive enumeration). Raga grammar is formalized as a finite-state traversal system over a 7-position swara cycle (RS-7), extending Vijayakrishnan's (2007) constraint-based grammar framework with explicit cycle-position arithmetic. Together these contributions provide a self-contained, computationally tractable symbolic foundation for Indian classical music that is compatible with existing transformational theory (Lewin 1987, Fiore 2007) and geometric music theory (Tymoczko 2011).
Abu Sanjid Shanto (Sun,) studied this question.