Seventy-two parents. Eighteen orbits. One Z-relation. One walk.
The Encoding
A seven-tone Melakarta is a popcount-seven vector in F212, set under a mirrored-endianness convention. Sa and Pa are fixed; the three regions — lower (R-G), madhyama, upper (D-N) — partition the remaining ten bits.
The Klein Quilt
Two involutions — madhyama swap (K1) and Sa-Pa-fixing antipodal complement (K2) — generate a Klein four-group acting freely on the seventy-two set, partitioning it into eighteen orbits of size four.
The Distance Field
The graph whose edges connect pairs at Hamming distance two is nine-regular, with three hundred and twenty-four edges. Its full pairwise distance distribution is (324, 936, 936, 324, 36); the thirty-six pairs at distance ten are exactly the antipodal pairs.
The Gray Walk
Warnsdorff's heuristic, starting at Kanakāṅgī, traces a Hamiltonian path through all seventy-two melas with every consecutive pair at distance two. It crosses the madhyama partition exactly once — at step thirty-five, from Dhīraśaṅkarābharaṇaṃ to Mēcakalyāṇī — which is the minimum.
The Z-Triangle
Among the seventy-two, exactly one Forte Z-pair is realised: 7-Z18 and 7-Z38, carried by Navanītaṃ, Divyamaṇi and Dhavaḷāmbarī. All three share the interval-class vector ⟨4, 3, 4, 4, 4, 2⟩; the system permits exactly this.
The Vivadi Lab
A Melakarta is extremally vivadi iff its R-G or D-N nibble lies at the boundary of its four-bit window — the patterns 0x3 and 0xC. Exactly forty Melakartas satisfy this; the fourteen that do not, but would under a naïve predicate, form a coherent middle-chromatic exception set.
Cross-currents
The Western diatonic family realises the Forte class 7-35. Hungarian minor, Siṃhēndramadhyamaṃ, and Double harmonic major, Māyāmāḷavagauḷa, both sit in 7-22 and are T7-related: the latter is the parent, the former its fourth mode.
Empirical
Forte-class population is a robust positive predictor of per-Melakarta janya count: Spearman ρ = +0.46, p < 10−4 on a sample of nine hundred and two janyas. This page reads only the fifteen counts hardcoded in Appendix C; the remainder, where unknown to this codebase, are marked accordingly.
Colophon
References, the kaṭapayādi key, methods, and a citation.