List of set classes

Set 3-1 has three possible versions: [01112T], [011TE1], and [0TT1E1], where the subscripts indicate adjacency intervals. The normal form is the smallest "slice of pie" (shaded) or most compact form, in this case: [01112T].

This is a list of set classes, by Forte number.[1] A set class (an abbreviation of pitch-class-set class) in music theory is an ascending collection of pitch classes, transposed to begin at zero. For a list of ordered collections, see: list of tone rows and series.

Sets are listed with links to their complements. The prime form of unsymmetrical sets is marked "A". Inversions are marked "B" (sets not marked "A" or "B" are symmetrical). "T" and "E" are conventionally used in sets to notate ten and eleven, respectively, as single characters. The ordering of sets in the lists is based on the string of numerals in the interval vector treated as an integer, decreasing in value, following the strategy used by Forte in constructing his numbering system. Numbers marked with a "Z" refer to a pair of different set classes with identical interval class content that are not related by inversion, with one of each pair listed at the end of the respective list when they occur. [The "Z" derives from the prefix "zygo"—from the ancient Greek, meaning yoked or paired. Hence: zygosets.]

There are two slightly different methods of obtaining the prime form—an earlier one due to Allen Forte and a later (and generally now more popular) one devised by John Rahn—both often confusingly described as "most packed to the left". However, a more precise description of the Rahn spelling is to select the version that is most dispersed from the right. The precise description of the Forte spelling is to select the version that is most packed to the left within the smallest span. [a] This results in two different prime form sets for the same Forte number in a number of cases. The lists here use the Rahn spelling. The alternative notations for those set classes where the Forte spelling differs are listed in the footnotes.[3][4]

Elliott Carter had earlier (1960–67) produced a numbered listing of pitch class sets, or "chords", as Carter referred to them, for his own use.[5][6] Donald Martino had produced tables of hexachords, tetrachords, trichords, and pentachords for combinatoriality in his article, "The Source Set and its Aggregate Formations" (1961).[7]

The difference between the interval vector of a set and that of its complement is <X, X, X, X, X, X/2>, where (in base-ten) X = 12 – 2C, and C is the cardinality of the smaller set. In nearly all cases, complements of unsymmetrical sets are inversionally related—i.e. the complement of an "A" version of a set of cardinality C is (usually) the "B" version of the respective set of cardinality 12 – C. The most significant exceptions are the sets 4-14/8-14, 5-11/7-11, and 6-14, which are all closely related in terms of subset/superset structure.

  1. ^ Forte, Allen (1973). The Structure of Atonal Music. Yale University Press. ISBN 0-300-02120-8.
  2. ^ Nelson, Paul (2004). "Two Algorithms for Computing the Prime Form", ComposerTools.com.
  3. ^ Rahn, John (1980). Basic Atonal Theory. New York: Longman. ISBN 978-0028731605.
  4. ^ Straus, Joseph N. (1990). Introduction to Post-Tonal Theory. Prentice-Hall. ISBN 9780131898905.
  5. ^ Schiff, David (1983/1998). The Music of Elliott Carter.
  6. ^ Carter, Elliott (2002). The Harmony Book, "Appendix 1". ISBN 9780825845949.
  7. ^ Schuijer, Michael (2008). Analyzing Atonal Music: Pitch-Class Set Theory and Its Contexts, p.97. University of Rochester. ISBN 978-1-58046-270-9.


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