Dictionary## Set_theory |

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## Mathematical set theory versus musical set theoryAlthough 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 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 ## Set and set typesMain 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 pitch-class 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 Though set theorists usually consider sets of equal-tempered pitch classes, it is possible to consider sets of pitches, non-equal-tempered pitch classes, Two-element sets are called dyads, three-element sets trichords (occasionally triads, though this is easily confused with the traditional meaning of the word). Sets of higher cardinalities are called ## Basic operationsThe basic operations that may be performed on a set are 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 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 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 + 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 ## Transpositional and inversional set classesTwo transpositionally related sets are said to belong to the same transpositional set class (T There are two main conventions for naming equal-tempered set classes. One, known as the Forte number, derives from Allen Forte, whose 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 significantly different. Therefore there is a limitation in Forte's theory. The second notational system labels sets in terms of their normal form, which depends on the concept of Since transpositionally related sets share the same normal form, normal forms can be used to label the T To identify a set's T - 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 ## SymmetryThe 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 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. 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 ## See also## 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 Beat-Class Sets in Steve Reich's Phase-Shifting 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 0-300-01610-7 (cloth) ISBN 0-300-02120-8 (pbk). **Hanson, Howard**(1960).*Harmonic Materials of Modern Music: Resources of the Tempered Scale*. New York: Appleton-Century-Crofts.- Rahn, John (1980).
*Basic Atonal Theory*. New York: Schirmer Books; London and Toronto: Prentice Hall International. ISBN 0-02-873160-3. - Schuijer, Michael (2008).
*Analyzing Atonal Music: Pitch-Class Set Theory and Its Contexts*. ISBN 978-1-58046-270-9. - 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 0-300-05686-9. 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 0-300-03493-8. Reprinted, New York: Oxford University Press, 2007. ISBN 9780195317138 - Morris, Robert (1987).
*Composition With Pitch-Classes: A Theory of Compositional Design*. New Haven: Yale University Press. ISBN 0-300-03684-1. - 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) - Straus, Joseph N. (2005).
*Introduction to Post-Tonal Theory*, 3rd edition. Upper Saddle River, NJ: Prentice-Hall. ISBN 0-13-189890-6.
## External links- Tucker, Gary (2001) "A Brief Introduction to Pitch-Class 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 Post-Functional Music Analysis: Post-Functional Theory Terminology",
*RobertKelleyPhd.com*. - Kelley (2002). "Introduction to Post-Functional Music Analysis: Set Theory, The Matrix, and the Twelve-Tone Method".
- "SetClass View (SCv)",
*Flexatone.net*. An athenaCL netTool for on-line, web-based 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.Andreas-Helmberger.de*. (German) - "Pitch-Class Set Theory and Perception",
*Ohio-State.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, two-set relationship calculators, and theory tutorial.
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