 Set theory (music)

Musical set theory provides concepts for categorizing musical objects and describing their relationships. Many of the notions were first elaborated by Howard Hanson (1960) in connection with tonal music, and then mostly developed in connection with atonal music by theorists such as Allen Forte (1973), drawing on the work in twelvetone theory of Milton Babbitt. The concepts of set theory are very general and can be applied to tonal and atonal styles in any equallytempered tuning system, and to some extent more generally than that. One branch of musical set theory deals with collections (sets and permutations) of pitches and pitch classes (pitchclass set theory), which may be ordered or unordered, and which can be related by musical operations such as transposition, inversion, and complementation. The methods of musical set theory are sometimes applied to the analysis of rhythm as well.
Contents
Mathematical set theory versus musical 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. For example, musicians use the terms transposition and inversion where mathematicians would use translation and reflection. 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 these can be seen to include the musical kind in some sense, they are far more involved).
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. In combinatorics, an unordered subset of n objects, such as pitch classes, is called a combination, and an ordered subset a permutation. Musical set theory is best regarded as a field that is not so much related to mathematical set theory, as an application of combinatorics to music theory with its own vocabulary. The main connection to mathematical set theory is the use of the vocabulary of set theory to talk about finite sets.
Set and set types
Main article: Set (music)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 pitchclass 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.^{[citation needed]} 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 CC♯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 equaltempered pitch classes, it is possible to consider sets of pitches, nonequaltempered pitch classes,^{[citation needed]} rhythmic onsets, or "beat classes" (Warburton 1988, 148; Cohn 1992, 149).
Twoelement sets are called dyads, threeelement 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.
Basic operations
The basic operations that may be performed on a set are transposition and inversion. Sets related by transposition or inversion are said to be transpositionally related or inversionally related, and to belong to the same set class. Since transposition and inversion are isometries of pitchclass space, they preserve the intervallic structure of a set, and hence its musical character. This can be considered the central postulate of musical set theory. In practice, settheoretic musical analysis often consists in the identification of nonobvious transpositional or inversional relationships between sets found in a piece.
Some authors consider the operations of complementation and multiplication as well. (The complement of set X is the set consisting of all the pitch classes not contained in X (Forte 1973, 73–74).) However, since complementation and multiplication are not isometries of pitchclass space, they do not necessarily preserve the musical character of the objects they transform. Other writers, such as Allen Forte, have emphasized the Zrelation which obtains between two sets sharing the same total interval content, or interval vector, but which are not transpositionally or inversionally equivalent (Forte 1973, 21). Another name for this relationship, used by Howard Hanson (1960), is "isomeric" (Cohen 2004, 33).
Operations on ordered sequences of pitch classes also include transposition and inversion, as well as retrograde and rotation. Retrograding an ordered sequence reverses the order of its elements. Rotation of an ordered sequence is equivalent to cyclic permutation.
Transposition and inversion can be represented as elementary arithmetic operations. If x is a number representing a pitch class, its transposition by n semitones is written T_{n} = x + n (mod12). Inversion corresponds to reflection around some fixed point in pitch class space. If "x" is a pitch class, the inversion with index number n is written I_{n} = n  x (mod12).
Equivalence relation
"For a relation in set S to be an equivalence relation [in algebra], it has to satisfy three conditions: it has to be reflexive [...], symmetrical [...], and transitive [...]."(Schuijer 2008, p.2930) "Indeed, an informal notion of equivalence has always been part of music theory and analysis. PC set theory, however, has adhered to formal definitions of equivalence" (Schuijer 2008, 85).
Transpositional and inversional set classes
Two transpositionally related sets are said to belong to the same transpositional set class (T_{n}). Two sets related by transposition or inversion are said to belong to the same transpositional/inversional set class (inversion being written T_{n}I or I_{n}). Sets belonging to the same transpositional set class are very similarsounding; while sets belonging to the same transpositional/inversional set class are fairly similar sounding. Because of this, music theorists often consider set classes to be basic objects of musical interest.
There are two main conventions for naming equaltempered set classes. One, known as the Forte number, derives from Allen Forte, whose The Structure of Atonal Music (1973), is one of the first works in musical set theory. Forte provided each set class with a number of the form cd, where c indicates the cardinality of the set and d is the ordinal number (Forte 1973, 12). Thus the chromatic trichord {0, 1, 2} belongs to setclass 31, indicating that it is the first threenote set class in Forte's list (Forte 1973, 179–81). The augmented trichord {0, 4, 8}, receives the label 312, which happens to be the last trichord in Forte's list.
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 pitchclass sets), multisets or sets in other tuning systems; (3) Forte's original system considers inversionally related sets to belong to the same setclass.This means that, for example a major triad and a minor triad are considered the same set. Western tonal music for centuries has regarded major and minor as significantly different. Therefore there is a limitation in Forte's theory.^{[citation needed]} However, the theory was not created to fill a vacuum in which existing theories inadequately explained tonal music. Rather, Forte's theory is used to explain atonal music, where the composer has invented a system where the distinction between {0, 4, 7} (called 'major' in tonal theory) and its inversion {0, 8, 5} (called 'minor' in tonal theory) may not be relevant.
The second notational system labels sets in terms of their normal form, which depends on the concept of normal order. To put a set in normal order, order it as an ascending scale in pitchclass space that spans less than an octave. Then permute it cyclically until its first and last notes are as close together as possible. In the case of ties, minimize the distance between the first and nexttolast note. (In case of ties here, minimize the distance between the first and nexttonexttolast note, and so on.) Thus {0, 7, 4} in normal order is {0, 4, 7}, while {0, 2, 10} in normal order is {10, 0, 2}. 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 (Rahn 1980, 33–38). 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.^{[citation needed]}
Since transpositionally related sets share the same normal form, normal forms can be used to label the T_{n} set classes.
To identify a set's T_{n}/I_{n} set class:
 Identify the set's T_{n} set class.
 Invert the set and find the inversion's T_{n} set class.
 Compare these two normal forms to see which is most "left packed."
The resulting set labels the initial set's T_{n}/I_{n} set class.
Symmetry
The number of distinct operations in a system that map a set into itself is the set's degree of symmetry (Rahn 1980, 90). Every set has at least one symmetry, as it maps onto itself under the identity operation T_{0} (Rahn 1980, 91). Transpositionally symmetric sets map onto themselves for T_{n} where n does not equal 0. Inversionally symmetric sets map onto themselves under T_{n}I. For any given T_{n}/T_{n}I type all sets will have the same degree of symmetry. The number of distinct sets in a type is 24 (the total number of operations, transposition and inversion, for n = 0 through 11) divided by the degree of symmetry of T_{n}/T_{n}I type.
Transpositionally symmetrical sets either divide the octave evenly, or can be written as the union of equallysized sets that themselves divide the octave evenly. Inversionallysymmetrical chords are invariant under reflections in pitch class space. This means that the chords can be ordered cyclically so that the series of intervals between successive notes is the same read forward or backward. For instance, in the cyclical ordering (0, 1, 2, 7), the interval between the first and second note is 1, the interval between the second and third note is 1, the interval between the third and fourth note is 5, and the interval between the fourth note and the first note is 5. One obtains the same sequence if one starts with the third element of the series and moves backward: the interval between the third element of the series and the second is 1; the interval between the second element of the series and the first is 1; the interval between the first element of the series and the fourth is 5; and the interval between the last element of the series and the third element is 5. Symmetry is therefore found between T_{0} and T_{2}I, and there are 12 sets in the T_{n}/T_{n}I equivalence class (Rahn 1980, 148).
See also
 Identity (music)
 Pitch interval
 Tonnetz
 Transformational music theory
References
 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 0313321353.
 Cohn, Richard. 1992. "Transpositional Combination of BeatClass Sets in Steve Reich's PhaseShifting Music". Perspectives of New Music 30, no. 2 (Summer): 146–77.
 Forte, Allen (1973). The Structure of Atonal Music. New Haven and London: Yale University Press. ISBN 0300016107 (cloth) ISBN 0300021208 (pbk).
 Hanson, Howard (1960). Harmonic Materials of Modern Music: Resources of the Tempered Scale. New York: AppletonCenturyCrofts.
 Rahn, John (1980). Basic Atonal Theory. New York: Schirmer Books; London and Toronto: Prentice Hall International. ISBN 0028731603.
 Schuijer, Michael (2008). Analyzing Atonal Music: PitchClass Set Theory and Its Contexts. ISBN 9781580462709.
 Warburton, Dan. 1988. "A Working Terminology for Minimal Music". Intégral 2:135–59.
Further reading
 Carter, Elliott (2002). Harmony Book, edited by Nicholas Hopkins and John F. Link. New York: Carl Fischer. ISBN 0825845947.
 Lewin, David (1993). Musical Form and Transformation: Four Analytic Essays. New Haven: Yale University Press. ISBN 0300056869. Reprinted, with a foreword by Edward Gollin, New York: Oxford University Press, 2007. ISBN 9780195317121
 Lewin, David (1987). Generalized Musical Intervals and Transformations. New Haven: Yale University Press. ISBN 0300034938. Reprinted, New York: Oxford University Press, 2007. ISBN 9780195317138
 Morris, Robert (1987). Composition With PitchClasses: A Theory of Compositional Design. New Haven: Yale University Press. ISBN 0300036841.
 Perle, George (1996). TwelveTone Tonality, second edition, revised and expanded. Berkeley: University of California Press. ISBN 0520201426. (First edition 1977, ISBN 0520033876)
 Straus, Joseph N. (2005). Introduction to PostTonal Theory, 3rd edition. Upper Saddle River, NJ: PrenticeHall. ISBN 0131898906.
External links
 Tucker, Gary (2001) "A Brief Introduction to PitchClass Set Analysis", Mount Allison University Department of Music.
 Nick Collins "Uniqueness of pitch class spaces, minimal bases and Z partners", Sonic Arts.
 "Twentieth Century Pitch Theory: Some Useful Terms and Techniques", Form and Analysis: A Virtual Textbook.
 Solomon, Larry (2005). "Set Theory Primer for Music", SolomonMusic.net.
 Kelley, Robert T (2001). "Introduction to PostFunctional Music Analysis: PostFunctional Theory Terminology", RobertKelleyPhd.com.
 Kelley (2002). "Introduction to PostFunctional Music Analysis: Set Theory, The Matrix, and the TwelveTone Method".
 "SetClass View (SCv)", Flexatone.net. An athenaCL netTool for online, webbased pitch class analysis and reference.
 Tomlin, Jay. "All About {Musical} Set Theory", JayTomlin.com.
 "Java Set Theory Machine" or Calculator
 Helmberger, Andreas (2006). "Projekte: Pitch Class Set Calculator", www.AndreasHelmberger.de. (German)
 "PitchClass Set Theory and Perception", OhioState.edu.^{[dead link]}
 "Colors are sounds: How to See the Music", Creativelab. The method for transformation of music into an image.
 "Software Tools for Composers", ComposerTools.com. Javascript PC Set calculator, twoset relationship calculators, and theory tutorial.
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