• The primary criticisms of Forte’s nomenclature are: (1) Forte’s labels are arbitrary and difficult to memorize, and it is in practice often easier simply to list an element
  of the set class; (2) Forte’s system assumes equal temperament and cannot easily be extended to include diatonic sets, pitch sets (as opposed to pitch-class sets), multisets or sets in other tuning systems; (3) Forte’s original system considers
  inversionally related sets to belong to the same set-class.

• One branch of musical set theory deals with collections (sets and permutations) of pitches and pitch classes (pitch-class set theory), which may be ordered or unordered, and
  can be related by musical operations such as transposition, melodic inversion, and complementation.

• Furthermore, where musical set theory refers to ordered sets, mathematics would normally refer to tuples or sequences (though mathematics does speak of ordered sets, and although
  these can be seen to include the musical kind in some sense, they are far more involved).

• Sets belonging to the same transpositional set class are very similar-sounding; while sets belonging to the same transpositional/inversional set class could include two chords
  of the same type but in different keys, which would be less similar in sound but obviously still a bounded category.

• [citation needed] Since transpositionally related sets share the same normal form, normal forms can be used to label the Tn set classes.

• Comparison with mathematical set theory Although musical set theory is often thought to involve the application of mathematical set theory to music, there are numerous differences
  between the methods and terminology of the two.

• [9][10] 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).

• The second notational system labels sets in terms of their normal form, which depends on the concept of normal order.

• Since transposition and inversion are isometries of pitch-class space, they preserve the intervallic structure of a set, even if they do not preserve the musical character
  (i.e.

• Its main connection to mathematical set theory is the use of the vocabulary of set theory to talk about finite sets.

• [18] Thus the chromatic trichord {0, 1, 2} belongs to set-class 3–1, indicating that it is the first three-note set class in Forte’s list.

• 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.)

• Moreover, musical set theory is more closely related to group theory and combinatorics than to mathematical set theory, which concerns itself with such matters as, for example,
  various sizes of infinitely large sets.

• [20] Mathematicians and computer scientists most often order combinations using either alphabetical ordering, binary (base two) ordering, or Gray coding, each of which lead
  to differing but logical normal forms.

• [8] 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}.

• In practice, set-theoretic musical analysis often consists in the identification of non-obvious transpositional or inversional relationships between sets found in a piece.

• [4] More exactly, a pitch-class set is a numerical representation consisting of distinct integers (i.e., without duplicates).

• Types of sets The fundamental concept of musical set theory is the (musical) set, which is an unordered collection of pitch classes.

• Symmetries The number of distinct operations in a system that map a set into itself is the set’s degree of symmetry.

• 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]
  rhythmic onsets, or “beat classes”.

• To put a set in normal form, begin by putting it in normal order, and then transpose it so that its first pitch class is 0.

• The resulting set labels the initial set’s Tn/In set class.

• To put a set in normal order, order it as an ascending scale in pitch-class space that spans less than an octave.

• “[17] Transpositional and inversional set classes Two transpositionally related sets are said to belong to the same transpositional set class (Tn).

• Two sets related by transposition or inversion are said to belong to the same transpositional/inversional set class (inversion being written TnI or In).

• Because of this, music theorists often consider set classes basic objects of musical interest.

• Sets related by transposition or inversion are said to be transpositionally related or inversionally related, and to belong to the same set class.

 

Works Cited

[‘1. Schuijer 2008, 99.
2. ^ Hanson 1960.
3. ^ Forte 1973.
4. ^ Rahn 1980, 27.
5. ^ Jump up to:a b Forte 1973, 3.
6. ^ Rahn 1980, 28.
7. ^ Rahn 1980, 21, 134.
8. ^ Forte 1973, 60–61.
9. ^ Warburton 1988, 148.
10. ^ Cohn 1992, 149.
11. ^
Rahn 1980, 140.
12. ^ Forte 1973, 73–74.
13. ^ Forte 1973, 21.
14. ^ Hanson 1960, 22.
15. ^ Cohen 2004, 33.
16. ^ Schuijer 2008, 29–30.
17. ^ Schuijer 2008, 85.
18. ^ Forte 1973, 12.
19. ^ Forte 1973, 179–181.
20. ^ Rahn 1980, 33–38.
21. ^
Rahn 1980, 90.
22. ^ Alegant 2001, 5.
23. ^ Rahn 1980, 91.
24. ^ Jump up to:a b Rahn 1980, 148.
2. Alegant, Brian. 2001. “Cross-Partitions as Harmony and Voice Leading in Twelve-Tone Music”. Music Theory Spectrum 23, no. 1 (Spring): 1–40.
3. Cohen,
Allen Laurence. 2004. Howard Hanson in Theory and Practice. Contributions to the Study of Music and Dance 66. Westport, Conn. and London: Praeger. ISBN 0-313-32135-3.
4. Cohn, Richard. 1992. “Transpositional Combination of Beat-Class Sets in Steve
Reich’s Phase-Shifting Music”. Perspectives of New Music 30, no. 2 (Summer): 146–177.
5. Forte, Allen. 1973. The Structure of Atonal Music. New Haven and London: Yale University Press. ISBN 0-300-01610-7 (cloth) ISBN 0-300-02120-8 (pbk).
6. Hanson,
Howard. 1960. Harmonic Materials of Modern Music: Resources of the Tempered Scale. New York: Appleton-Century-Crofts.
7. Rahn, John. 1980. Basic Atonal Theory. New York: Schirmer Books; London and Toronto: Prentice Hall International. ISBN 0-02-873160-3.
8. Schuijer,
Michiel. 2008. Analyzing Atonal Music: Pitch-Class Set Theory and Its Contexts. ISBN 978-1-58046-270-9.
9. Warburton, Dan. 1988. “A Working Terminology for Minimal Music”. Intégral 2:135–159.
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