View source for Set theory (music)

{{Short description|Branch of music theory}}
[[File:Z-relation Z17 example.png|thumb|upright=1.5|Example of [[Interval vector#Z-relation|Z-relation]] on two pitch sets analyzable as or derivable from Z17,{{sfn|Schuijer|2008|loc=99}} with intervals between pitch classes labeled for ease of comparison between the two sets and their common interval vector, 212320]]

'''Musical set theory''' provides concepts for categorizing [[music]]al objects and describing their relationships. [[Howard Hanson]] first elaborated many of the concepts for analyzing [[tonality|tonal]] music.{{sfn|Hanson|1960}} Other theorists, such as [[Allen Forte]], further developed the theory for analyzing [[atonal]] music,{{sfn|Forte|1973}} drawing on the [[twelve-tone technique|twelve-tone]] theory of [[Milton Babbitt]]. The concepts of musical set theory are very general and can be applied to tonal and atonal styles in any [[equal temperament]] tuning system, and to some extent more generally than that.

One branch of musical set theory deals with collections ([[set (music)|sets]] and [[permutation (music)|permutations]]) of [[pitch (music)|pitches]] and [[pitch class]]es ('''pitch-class set theory'''), which may be [[Order theory|ordered or unordered]], and can be related by musical operations such as [[Transposition (music)|transposition]], [[melodic inversion]], and [[Complement (music)|complementation]]. Some theorists apply the methods of musical set theory to the analysis of [[rhythm]] as well.

==Comparison with mathematical set theory==
{{Unreferenced section|date=November 2023}}
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 (music)|transposition]] and [[Melodic inversion|inversion]] where mathematicians would use [[translation (geometry)|translation]] and [[reflection (mathematics)|reflection]]. Furthermore, where musical set theory refers to ordered sets, mathematics would normally refer to tuples or sequences (though mathematics does speak of [[ordered set]]s, and although 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 {{var|n}} objects, such as [[pitch class]]es, is called a [[combination]], and an ordered subset a ''permutation''. Musical set theory is better regarded as an application of combinatorics to music theory than as a branch of mathematical set theory. Its main connection to mathematical set theory is the use of [[naive set theory|the vocabulary of set theory]] to talk about finite sets.

==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]].

==Basic operations==
{{Main|Transformation (music)}}
[[File:Pitch class 5-6 inversion example.png|upright=0.5|thumb|Pitch class inversion: 234te reflected around 0 to become t9821]]

The basic operations that may be performed on a set are [[transposition (music)|transposition]] and [[Melodic inversion|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 [[isometry|isometries]] of pitch-class space, they preserve the intervallic structure of a set, even if they do not preserve the musical character (i.e. the physical reality) of the elements of the set.{{Citation needed|date=July 2019|reason=This claim recently said just the opposite: that the musical character is in fact preserved.}} This can be considered the central postulate of musical set theory. In practice, set-theoretic musical analysis often consists in the identification of non-obvious transpositional or inversional relationships between sets found in a piece.

Some authors consider the operations of [[complement (music)|complementation]] and [[multiplication (music)|multiplication]] as well. The complement of set X is the set consisting of all the pitch classes not contained in X.{{sfn|Forte|1973|loc=73–74}} The product of two pitch classes is the product of their pitch-class numbers modulo 12. Since complementation and multiplication are not [[isometries]] of pitch-class space, they do not necessarily preserve the musical character of the objects they transform. Other writers, such as Allen Forte, have emphasized the [[Z-relation]], which obtains between two sets that share the same total interval content, or [[interval vector]]—but are not transpositionally or inversionally equivalent.{{sfn|Forte|1973|loc=21}} Another name for this relationship, used by Hanson,{{sfn|Hanson|1960|loc=22}} is "isomeric".{{sfn|Cohen|2004|loc=33}}

Operations on ordered sequences of pitch classes also include transposition and inversion, as well as [[Permutation (music)|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 {{var|x}} is a number representing a pitch class, its transposition by {{var|n}} semitones is written T<sub>{{var|n}}</sub>&nbsp;=&nbsp;{{var|x}}&nbsp;+&nbsp;{{var|n}}&nbsp;mod&nbsp;12. Inversion corresponds to [[Reflection (mathematics)|reflection]] around some fixed point in [[pitch class space]]. If {{var|x}} is a pitch class, the inversion with [[index number]] {{var|n}} is written I<sub>{{var|n}}</sub>&nbsp;=&nbsp;{{var|n}}&nbsp;-&nbsp;{{var|x}}&nbsp;mod&nbsp;12.

==Equivalence relation==
{{See also|Equivalence class (music)}}

"For a relation in set ''S'' to be an [[equivalence relation]] [in [[algebra]]], it has to satisfy three conditions: it has to be [[reflexive relation|reflexive]] ..., [[Symmetry in mathematics|symmetrical]] ..., and [[transitive relation|transitive]] ...".{{sfn|Schuijer|2008|loc=29–30}} "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."{{sfn|Schuijer|2008|loc=85}}

==Transpositional and inversional set classes==
Two transpositionally related sets are said to belong to the same transpositional set class (T<sub>{{var|n}}</sub>). Two sets related by transposition or inversion are said to belong to the same transpositional/inversional set class (inversion being written T<sub>{{var|n}}</sub>I or I<sub>{{var|n}}</sub>). 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. Because of this, music theorists often consider set classes basic objects of musical interest.

There are two main conventions for naming equal-tempered 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 {{var|c}}–{{var|d}}, where {{var|c}} indicates the cardinality of the set and {{var|d}} is the ordinal number.{{sfn|Forte|1973|loc=12}} 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.{{sfn|Forte|1973|loc=179–181}} The augmented trichord {0, 4, 8}, receives the label 3–12, 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 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. 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 well as chord inversions, as significantly different. They generate indeed completely different physical objects. Ignoring the physical reality of sound is an obvious limitation of atonal theory. However, the defense has been made that 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, 3, 7} (called 'minor' in tonal theory) may not be relevant.

The second notational system labels sets in terms of their [[Set (music)#Non-serial|normal form]], which depends on the concept of ''normal order''. To put a set in ''normal order,'' order it as an ascending scale in pitch-class 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 next-to-last note. (In case of ties here, minimize the distance between the first and next-to-next-to-last 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.{{sfn|Rahn|1980|loc=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|date=September 2010}}

Since transpositionally related sets share the same normal form, normal forms can be used to label the T<sub>{{var|n}}</sub> set classes.

To identify a set's T<sub>{{var|n}}</sub>/I<sub>{{var|n}}</sub> set class:
* Identify the set's T<sub>{{var|n}}</sub> set class.
* Invert the set and find the inversion's T<sub>{{var|n}}</sub> set class.
* Compare these two normal forms to see which is most "left packed."
The resulting set labels the initial set's T<sub>{{var|n}}</sub>/I<sub>{{var|n}}</sub> set class.

==Symmetries==
The number of distinct operations in a system that map a set into itself is the set's [[degree of symmetry]].{{sfn|Rahn|1980|loc=90}} The degree of symmetry, "specifies the number of operations that preserve the unordered pcsets of a partition; it tells the extent to which that partition's pitch-class sets map into (or onto) each other under transposition or inversion".{{sfn|Alegant|2001|loc=5}} Every set has at least one symmetry, as it maps onto itself under the identity operation T<sub>{{var|0}}</sub>.{{sfn|Rahn|1980|loc=91}} Transpositionally symmetric sets map onto themselves for T<sub>{{var|n}}</sub> where {{var|n}} does not equal 0 (mod 12). Inversionally symmetric sets map onto themselves under T<sub>{{var|n}}</sub>I. For any given T<sub>{{var|n}}</sub>/T<sub>{{var|n}}</sub>I type all sets 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&nbsp;=&nbsp;0 through 11) divided by the degree of symmetry of T<sub>{{var|n}}</sub>/T<sub>{{var|n}}</sub>I type.

Transpositionally symmetrical sets either divide the octave evenly, or can be written as the union of equally sized sets that themselves divide the octave evenly. Inversionally symmetrical 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.{{sfn|Rahn|1980|loc=148}}

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<sub>0</sub> and T<sub>2</sub>I, and there are 12 sets in the T<sub>{{var|n}}</sub>/T<sub>{{var|n}}</sub>I equivalence class.{{sfn|Rahn|1980|loc=148}}

==See also==
* [[Identity (music)]]
* [[Pitch interval]]
* [[Tonnetz]]
* [[Transformational theory]]

==References==
{{reflist|15em}}

'''Sources'''
{{div col|colwidth=45em}}
* {{wikicite|ref={{harvid|Alegant|2001}}|reference=Alegant, Brian. 2001. "Cross-Partitions as Harmony and Voice Leading in Twelve-Tone Music". ''Music Theory Spectrum'' 23, no. 1 (Spring): 1–40.}}
* {{wikicite|ref={{harvid|Cohen|2004}}|reference=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}}.}}
* {{wikicite|ref={{harvid|Cohn|1992}}|reference=[[Richard Cohn|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.}}
* {{wikicite|ref={{harvid|Forte|1973}}|reference=[[Allen Forte|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).}}
* {{wikicite|ref={{harvid|Hanson|1960}}|reference=[[Howard Hanson|Hanson, Howard]]. 1960. ''[https://archive.org/details/harmonicmaterial00hans Harmonic Materials of Modern Music: Resources of the Tempered Scale]''. New York: Appleton-Century-Crofts.}}
* {{wikicite|ref={{harvid|Rahn|1980}}|reference=[[John Rahn|Rahn, John]]. 1980. ''Basic Atonal Theory''. New York: Schirmer Books; London and Toronto: Prentice Hall International. {{ISBN|0-02-873160-3}}.}}
*{{wikicite|ref={{harvid|Schuijer|2008}}|reference=Schuijer, Michiel. 2008. ''Analyzing Atonal Music: Pitch-Class Set Theory and Its Contexts''. {{ISBN|978-1-58046-270-9}}.}}
* {{wikicite|ref={{harvid|Warburton|1988}}|reference=Warburton, Dan. 1988. "A Working Terminology for Minimal Music". ''[[Intégral: The Journal of Applied Musical Thought|Intégral]]'' 2:135–159.}}
{{div col end}}

==Further reading==
{{div col|colwidth=45em}}
* [[Elliott Carter|Carter, Elliott]]. 2002. ''Harmony Book'', edited by Nicholas Hopkins and John F. Link. New York: Carl Fischer. {{ISBN|0-8258-4594-7}}.
* [[David Lewin|Lewin, David]]. 1993. ''Musical Form and Transformation: Four Analytic Essays''. New Haven: Yale University Press. {{ISBN|0-300-05686-9}}. Reprinted, with a foreword by Edward Gollin, New York: Oxford University Press, 2007. {{ISBN|978-0-19-531712-1}}.
* Lewin, David. 1987. ''Generalized Musical Intervals and Transformations''. New Haven: Yale University Press. {{ISBN|0-300-03493-8}}. Reprinted, New York: Oxford University Press, 2007. {{ISBN|978-0-19-531713-8}}.
* [[Robert Morris (composer)|Morris, Robert]]. 1987. ''Composition With Pitch-Classes: A Theory of Compositional Design''. New Haven: Yale University Press. {{ISBN|0-300-03684-1}}.
* [[George Perle|Perle, George]]. 1996. ''Twelve-Tone Tonality'', second edition, revised and expanded. Berkeley: University of California Press. {{ISBN|0-520-20142-6}}. (First edition 1977, {{ISBN|0-520-03387-6}})
* Starr, Daniel. 1978. "Sets, Invariance and Partitions". ''[[Journal of Music Theory]]'' 22, no. 1 (Spring): 1–42.
* Straus, Joseph N. 2005. ''Introduction to Post-Tonal Theory'', third edition. Upper Saddle River, New Jersey: Prentice-Hall. {{ISBN|0-13-189890-6}}.
{{div col end}}

==External links==
* Tucker, Gary (2001) [http://www.mta.ca/faculty/arts-letters/music/pc-set_project/pc-set_new/ "A Brief Introduction to Pitch-Class Set Analysis"], ''Mount Allison University Department of Music''.
* Nick Collins [https://web.archive.org/web/20120717011213/http://www.sonic.mdx.ac.uk/research/nickpitch.html "Uniqueness of pitch class spaces, minimal bases and Z partners"], ''Sonic Arts''.
* [http://www.lsu.edu/faculty/jperry/virtual_textbook/20th_c_pitch_theory.htm "Twentieth Century Pitch Theory: Some Useful Terms and Techniques"], ''Form and Analysis: A Virtual Textbook''.
* Solomon, Larry (2005). [https://web.archive.org/web/20080115120710/http://solomonsmusic.net/setheory.htm "Set Theory Primer for Music"], ''SolomonMusic.net''.
* Kelley, Robert T (2001). [http://www.robertkelleyphd.com/atnltrms.htm "Introduction to Post-Functional Music Analysis: Post-Functional Theory Terminology"], ''RobertKelleyPhd.com''.
* Kelley, Robert T (2002). [http://www.robertkelleyphd.com/12-tone.htm "Introduction to Post-Functional Music Analysis: Set Theory, The Matrix, and the Twelve-Tone Method"].
* [http://www.flexatone.net/athenaSCv.html "SetClass View (SCv)"], ''Flexatone.net''. An athenaCL netTool for on-line, web-based pitch class analysis and reference.
* Tomlin, Jay. [http://www.jaytomlin.com/music/settheory/help.html "All About Set Theory"]. ''JayTomlin.com''.
* [http://www.jaytomlin.com/music/settheory/ "Java Set Theory Machine"] or Calculator
* Kaiser, Ulrich. [http://musikanalyse.net/entdecken/#k=&p=am "Pitch Class Set Calculator"], ''musikanalyse.net''. {{in lang|de}}
* [https://web.archive.org/web/20080725115152/http://dactyl.som.ohio-state.edu/Gibson/research.summary.html "Pitch-Class Set Theory and Perception"], ''Ohio-State.edu''.
* [http://www.composertools.com/Tools/ "Software Tools for Composers"], ''ComposerTools.com''. Javascript PC Set calculator, two-set relationship calculators, and theory tutorial.
*"[https://www.mta.ca/pc-set/calculator/pc_calculate.html PC Set Calculator]", ''MtA.Ca''.
* Taylor, Stephen Andrew. [http://stephenandrewtaylor.net/setfinder/index.html "SetFinder"], ''stephenandrewtaylor.net''. Pitch class set library and prime form calculator.

{{Music theory}}
{{Set theory (music)}}
{{Twelve-tone technique}}

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[[Category:Musical set theory| ]]