Editing Set theory (music) (section)

==Types of sets==
{{Main|Set (music)}}

The fundamental concept of musical set theory is the (musical) set, which is an unordered collection of pitch classes.{{sfn|Rahn|1980|loc=27}} More exactly, a pitch-class set is a numerical representation consisting of distinct integers (i.e., without duplicates).{{sfn|Forte|1973|loc=3}} The elements of a set may be manifested in music as [[simultaneity (music)|simultaneous]] chords, successive tones (as in a melody), or both.{{Citation needed|date=January 2010|reason=Must they be "manifested" at all? If so, why are these the only way they may be manifested? And who says so?}} Notational conventions vary from author to author, but sets are typically enclosed in curly braces: {},{{sfn|Rahn|1980|loc=28}} or square brackets: [].{{sfn|Forte|1973|loc=3}}

Some theorists use angle brackets {{angbr| }} to denote ordered sequences,{{sfn|Rahn|1980|loc=21, 134}} while others distinguish ordered sets by separating the numbers with spaces.{{sfn|Forte|1973|loc=60–61}} Thus one might notate the unordered set of pitch classes 0, 1, and 2 (corresponding in this case to C, C{{Music|#}}, and D) as {0,1,2}. The ordered sequence C-C{{Music|#}}-D would be notated {{angbr|0,1,2}} or (0,1,2). Although C is considered 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{{Music|#}} 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,{{Citation needed|date=September 2010|reason=Warburton and Cohn discuss beat-classes of rhythmic onsets, but not non-equal-tempered pitch classes.}} rhythmic onsets, or "beat classes".{{sfn|Warburton|1988|loc=148}}{{sfn|Cohn|1992|loc=149}}

Two-element sets are called [[dyad (music)|dyad]]s, three-element sets [[trichord]]s (occasionally "triads", though this is easily confused with the traditional meaning of the word [[Triad (music)|triad]]). Sets of higher cardinalities are called [[tetrachord]]s (or tetrads), [[pentachord]]s (or pentads), [[hexachord]]s (or hexads), [[heptachord]]s (heptads or, sometimes, mixing Latin and Greek roots, "septachords"—e.g. [[John Rahn|Rahn]]),{{sfn|Rahn|1980|loc=140}} [[octachord]]s (octads), [[nonachord]]s (nonads), [[decachord]]s (decads), [[undecachord]]s, and, finally, the [[dodecachord]].