Set Theory (music) - Set and Set Types

Set and Set Types

The fundamental concept of musical set theory is the (musical) set, which is an unordered collection of pitch classes (Rahn 1980, 27). More exactly, a pitch-class set is a numerical representation consisting of distinct integers (i.e., without duplicates) (Forte 1973, 3). The elements of a set may be manifested in music as simultaneous chords, successive tones (as in a melody), or both. Notational conventions vary from author to author, but sets are typically enclosed in curly braces: {} (Rahn 1980, 28), or square brackets: (Forte 1973, 3). Some theorists use angle brackets to denote ordered sequences (Rahn 1980, 21 & 134), while others distinguish ordered sets by separating the numbers with spaces (Forte 1973, 60–61). Thus one might notate the unordered set of pitch classes 0, 1, and 2 (corresponding in this case to C, C♯, and D) as {0,1,2}. The ordered sequence C-C♯-D would be notated or (0,1,2). Although C is considered to be zero in this example, this is not always the case. For example, a piece (whether tonal or atonal) with a clear pitch center of F might be most usefully analyzed with F set to zero (in which case {0,1,2} would represent F, F♯ and G. (For the use of numbers to represent notes, see pitch class.)

Though set theorists usually consider sets of equal-tempered pitch classes, it is possible to consider sets of pitches, non-equal-tempered pitch classes, rhythmic onsets, or "beat classes" (Warburton 1988, 148; Cohn 1992, 149).

Two-element sets are called dyads, three-element sets trichords (occasionally "triads", though this is easily confused with the traditional meaning of the word triad). Sets of higher cardinalities are called tetrachords (or tetrads), pentachords (or pentads), hexachords (or hexads), heptachords (heptads or, sometimes, mixing Latin and Greek roots, "septachords"—e.g., Rahn 1980, 140), octachords (octads), nonachords (nonads), decachords (decads), undecachords, and, finally, the dodecachord.