Pitch-Class Sets, Normal Order, and Transformations
A pitch-class set is an unordered collection of distinct pitch-class integers, written in curly braces — for example {0, 4, 7} for a C major triad. Sets are the basic objects of post-tonal analysis: any chord, motive, or simultaneity can be reduced to a pc set by stripping away octave and ordering information.
Normal order is the most compact arrangement of a pc set, providing a canonical representation for comparison. To find normal order: (1) list the pcs in ascending order within one octave; (2) among all rotations of this list, choose the one with the smallest span from first to last element; (3) if two rotations tie on the outer span, choose the one with the smaller interval from first to second element, and continue inward to break further ties. The result is written as an ordered list in square brackets, e.g. [0, 4, 7].
Transposition (T_n) shifts every element of a set by n semitones mod 12: T_n({x, y, z}) = {(x+n) mod 12, (y+n) mod 12, (z+n) mod 12}. Inversion (I) maps each pc to its mod-12 complement: I({x, y, z}) = {(−x) mod 12, (−y) mod 12, (−z) mod 12} = {12−x, 12−y, 12−z}. The combined operation T_nI first inverts then transposes. Two sets related by any T_n or T_nI operation belong to the same set class.