A pitch-class set is a subset of Z/12 — a "necklace" of 12 beads, each on or off (2 colors). Two sets are the "same" if one maps to the other under the chosen symmetry group. Burnside's lemma counts the orbits (distinct sets) as the average number of colorings each group element leaves fixed. Click a row's count to load an example.
A finite group G acts on the set of colorings. The number of orbits (distinct objects) is |Orbits| = (1/|G|) Σg∈G |Fix(g)|. For necklaces, a coloring is fixed by g iff it is constant on every cycle of g's permutation of the beads. Counting only colorings with exactly k black beads gives Fixk(g) = [xk] ∏cycles c(1 + x|c|). Summing over sizes recovers Polya's cycle index Z(G) with two colors, total Z(G; 2,2,…) = 352 for C12 and 224 for D12.
C12 (order 12) is generated by transposition T1. Rotation by j has gcd(j,12) cycles, each of length 12/gcd(j,12). D12 (order 24) adds inversion TnI: six reflections fix two beads (axis through opposite pitches, cycle type 1225) and six fix none (axis through gaps, type 26).
The normal order is the most compact rotation (smallest span, ties broken from the top). The prime form (Rahn/Straus) compares the set's normal order against its inversion's and picks the one packed farthest left, transposed to 0. Each equivalence class has a Forte number card–index; a Z marks a Z-pair (two classes sharing one interval vector). The interval vector <ic1..ic6> tallies each of the six interval classes; it is invariant under the whole dihedral group.
A set is maximally even if its notes are as evenly spread as possible (floor(i·12/k)) — e.g. whole-tone, diminished-7th, diatonic. It has Myhill's property if every generic interval (2nd, 3rd, …) comes in exactly two chromatic sizes; equivalently it is a well-formed generated scale. The diatonic and pentatonic collections are generated by the perfect fifth (ic5).
Independent explorer built from scratch; math learned from Ian Ring's Study of Musical Scales and Michael Keith's From Polychords to Polya. No external code, fonts, or network.