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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. Mathematicians and computer scientists most often order combinations using either alphabetical ordering, binary (base two) ordering, or
1970:
561:). 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.
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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.
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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.
696:. 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". Every set has at least one symmetry, as it maps onto itself under the identity operation T
750:
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
601:
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
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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
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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
407:
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. This can be considered the central postulate of musical set theory. In practice, set-theoretic musical analysis often
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The fundamental concept of musical set theory is the (musical) set, which is an unordered collection of pitch classes. More exactly, a pitch-class set is a numerical representation consisting of distinct integers (i.e., without duplicates). The elements of a set may be manifested in music as
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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
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is the ordinal number. 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. The augmented trichord {0, 4, 8}, receives the label 3–12, which happens to be the last trichord in Forte's list.
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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,
419:
as well. The complement of set X is the set consisting of all the pitch classes not contained in X. The product of two pitch classes is the product of their pitch-class numbers modulo 12. Since complementation and multiplication are not
730:
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 = 0 through 11) divided by the degree of symmetry of
597:
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.
275:
to denote ordered sequences, while others distinguish ordered sets by separating the numbers with spaces. Thus one might notate the unordered set of pitch classes 0, 1, and 2 (corresponding in this case to C,
245:. 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
306:
Though set theorists usually consider sets of equal-tempered pitch classes, it is possible to consider sets of pitches, non-equal-tempered pitch classes, rhythmic onsets, or "beat classes".
38:
on two pitch sets analyzable as or derivable from Z17, with intervals between pitch classes labeled for ease of comparison between the two sets and their common interval vector, 212320
268:
chords, successive tones (as in a melody), or both. Notational conventions vary from author to author, but sets are typically enclosed in curly braces: {}, or square brackets: .
535:...". "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."
229:
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
424:
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
214:. Furthermore, where musical set theory refers to ordered sets, mathematics would normally refer to tuples or sequences (though mathematics does speak of
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549:). Two sets related by transposition or inversion are said to belong to the same transpositional/inversional set class (inversion being written T
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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
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432:—but are not transpositionally or inversionally equivalent. Another name for this relationship, used by Hanson, is "isomeric".
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to music, there are numerous differences between the methods and terminology of the two. For example, musicians use the terms
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consists in the identification of non-obvious transpositional or inversional relationships between sets found in a piece.
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439:. Retrograding an ordered sequence reverses the order of its elements. Rotation of an ordered sequence is equivalent to
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572:(1973), is one of the first works in musical set theory. Forte provided each set class with a number of the form
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73:. The concepts of musical set theory are very general and can be applied to tonal and atonal styles in any
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218:, and although these can be seen to include the musical kind in some sense, they are far more involved).
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Since transpositionally related sets share the same normal form, normal forms can be used to label the T
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Operations on ordered sequences of pitch classes also include transposition and inversion, as well as
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1449:"Introduction to Post-Functional Music Analysis: Set Theory, The Matrix, and the Twelve-Tone Method"
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1141:. 1992. "Transpositional Combination of Beat-Class Sets in Steve Reich's Phase-Shifting Music".
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Two transpositionally related sets are said to belong to the same transpositional set class (T
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Alegant, Brian. 2001. "Cross-Partitions as
Harmony and Voice Leading in Twelve-Tone Music".
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1125:. Contributions to the Study of Music and Dance 66. Westport, Conn. and London: Praeger.
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1283:. Reprinted, with a foreword by Edward Gollin, New York: Oxford University Press, 2007.
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There are two main conventions for naming equal-tempered set classes. One, known as the
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Although musical set theory is often thought to involve the application of mathematical
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Transposition and inversion can be represented as elementary arithmetic operations. If
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1512:. Javascript PC Set calculator, two-set relationship calculators, and theory tutorial.
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The number of distinct operations in a system that map a set into itself is the set's
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1438:"Introduction to Post-Functional Music Analysis: Post-Functional Theory Terminology"
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does not equal 0 (mod 12). Inversionally symmetric sets map onto themselves under T
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1342:, second edition, revised and expanded. Berkeley: University of California Press.
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1461:. An athenaCL netTool for on-line, web-based pitch class analysis and reference.
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1201:. New York: Schirmer Books; London and Toronto: Prentice Hall International.
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116:. Some theorists apply the methods of musical set theory to the analysis of
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Harmonic
Materials of Modern Music: Resources of the Tempered Scale
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1416:"Twentieth Century Pitch Theory: Some Useful Terms and Techniques"
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Compare these two normal forms to see which is most "left packed."
1231:
Warburton, Dan. 1988. "A Working
Terminology for Minimal Music".
790:
1406:"Uniqueness of pitch class spaces, minimal bases and Z partners"
1374:, third edition. Upper Saddle River, New Jersey: Prentice-Hall.
1322:
1969:
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1216:
Analyzing Atonal Music: Pitch-Class Set Theory and Its
Contexts
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Pitch class inversion: 234te reflected around 0 to become t9821
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is a number representing a pitch class, its transposition by
46:
702:. Transpositionally symmetric sets map onto themselves for T
622:, each of which lead to differing but logical normal forms.
391:. Sets related by transposition or inversion are said to be
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tuning system, and to some extent more generally than that.
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The second notational system labels sets in terms of their
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One branch of musical set theory deals with collections (
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Starr, Daniel. 1978. "Sets, Invariance and
Partitions".
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The basic operations that may be performed on a set are
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Moreover, musical set theory is more closely related to
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and G. (For the use of numbers to represent notes, see
1306:. Reprinted, New York: Oxford University Press, 2007.
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1534:. Pitch class set library and prime form calculator.
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first elaborated many of the concepts for analyzing
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523:, it has to satisfy three conditions: it has to be
104:, and can be related by musical operations such as
1395:"A Brief Introduction to Pitch-Class Set Analysis"
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1296:Generalized Musical Intervals and Transformations
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1159:. New Haven and London: Yale University Press.
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282:, and D) as {0,1,2}. The ordered sequence C-C
61:, further developed the theory for analyzing
2165:List of dodecaphonic and serial compositions
1399:Mount Allison University Department of Music
672:The resulting set labels the initial set's T
468: mod 12. Inversion corresponds to
49:objects and describing their relationships.
539:Transpositional and inversional set classes
321:). Sets of higher cardinalities are called
162:. Unsourced material may be challenged and
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659:Invert the set and find the inversion's T
584:indicates the cardinality of the set and
182:Learn how and when to remove this message
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411:Some authors consider the operations of
403:. Since transposition and inversion are
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1496:"Pitch-Class Set Theory and Perception"
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593:sets (as opposed to pitch-class sets),
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124:Comparison with mathematical set theory
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480:is a pitch class, the inversion with
1324:. New Haven: Yale University Press.
1298:. New Haven: Yale University Press.
1275:. New Haven: Yale University Press.
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1189:. New York: Appleton-Century-Crofts.
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1123:Howard Hanson in Theory and Practice
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160:adding citations to reliable sources
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45:provides concepts for categorizing
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609:, which depends on the concept of
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271:Some theorists use angle brackets
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1372:Introduction to Post-Tonal Theory
759:I, and there are 12 sets in the T
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1968:
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57:music. Other theorists, such as
206:where mathematicians would use
1846:Structure implies multiplicity
1830:Generic and specific intervals
1506:"Software Tools for Composers"
13:
1:
1427:"Set Theory Primer for Music"
1157:The Structure of Atonal Music
1121:Cohen, Allen Laurence. 2004.
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570:The Structure of Atonal Music
2089:All-interval twelve-tone row
1483:"Pitch Class Set Calculator"
1147:30, no. 2 (Summer): 146–177.
309:Two-element sets are called
247:the vocabulary of set theory
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472:around some fixed point in
249:to talk about finite sets.
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1809:Cardinality equals variety
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399:and to belong to the same
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1476:"Java Set Theory Machine"
1447:Kelley, Robert T (2002).
1436:Kelley, Robert T (2001).
1367:22, no. 1 (Spring): 1–42.
1214:Schuijer, Michiel. 2008.
1144:Perspectives of New Music
1116:23, no. 1 (Spring): 1–40.
511:Equivalence class (music)
393:transpositionally related
1526:Taylor, Stephen Andrew.
1370:Straus, Joseph N. 2005.
1693:All-interval tetrachord
1642:List of music theorists
1532:stephenandrewtaylor.net
1425:Solomon, Larry (2005).
1364:Journal of Music Theory
1350:. (First edition 1977,
796:Transformational theory
515:"For a relation in set
437:retrograde and rotation
2094:All-trichord hexachord
2042:Second Viennese School
1698:All-trichord hexachord
1466:"All About Set Theory"
454:semitones is written T
397:inversionally related,
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371:Transformation (music)
98:pitch-class set theory
65:music, drawing on the
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27:Branch of music theory
1888:Twelve-tone technique
1816:(Deep scale property)
1455:"SetClass View (SCv)"
1114:Music Theory Spectrum
771:I equivalence class.
634:To identify a set's T
378:
313:, three-element sets
290:⟨0,1,2⟩
33:
2037:Josef Matthias Hauer
2002:Retrograde inversion
1841:Rothenberg propriety
1825:Generated collection
1748:Pitch-interval class
1393:Tucker, Gary (2001)
1340:Twelve-Tone Tonality
1294:Lewin, David. 1987.
650:Identify the set's T
521:equivalence relation
505:Equivalence relation
357:, and, finally, the
288:-D would be notated
156:improve this section
102:ordered or unordered
2119:Formula composition
1832:(Myhill's property)
1765:Similarity relation
1442:RobertKelleyPhd.com
1199:Basic Atonal Theory
501: mod 12.
2184:Musical set theory
1682:Musical set theory
718:I. For any given T
694:degree of symmetry
613:. To put a set in
441:cyclic permutation
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43:Musical set theory
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2124:Modernism (music)
2052:Arnold Schoenberg
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1517:PC Set Calculator
1510:ComposerTools.com
1312:978-0-19-531713-8
1289:978-0-19-531712-1
1224:978-1-58046-270-9
474:pitch class space
273:⟨ ⟩
233:objects, such as
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110:melodic inversion
75:equal temperament
18:Operation (music)
16:(Redirected from
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417:multiplication
369:Main article:
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329:(or pentads),
325:(or tetrads),
257:Main article:
254:
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253:Types of sets
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172:November 2023
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141:This section
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51:Howard Hanson
48:
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37:
32:
19:
2138:
2069:
2057:Anton Webern
1979:Permutations
1899:Fundamentals
1708:Forte number
1681:
1625:
1569:Music theory
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1255:Harmony Book
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569:
566:Forte number
563:
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487:is written I
482:index number
445:
434:
410:
396:
392:
382:
355:undecachords
308:
305:
270:
266:simultaneous
262:
242:
223:group theory
220:
216:ordered sets
193:
178:
169:
154:Please help
142:
97:
86:permutations
79:
42:
41:
2129:Punctualism
2114:Equivalence
1814:Common tone
1738:Pitch class
1733:Permutation
1606:Mathematics
1595:Composition
1528:"SetFinder"
1491:(in German)
973:Hanson 1960
826:Hanson 1960
684:set class.
646:set class:
620:Gray coding
607:normal form
529:symmetrical
359:dodecachord
335:heptachords
327:pentachords
323:tetrachords
301:pitch class
259:Set (music)
243:permutation
239:combination
208:translation
67:twelve-tone
59:Allen Forte
34:Example of
2144:Time point
2139:Set theory
2047:Alban Berg
1992:Retrograde
1931:Invariance
1916:Derivation
1796:set theory
1775:Z-relation
1703:Complement
1626:Set theory
1621:Psychology
1616:Philosophy
1611:Musicology
1600:Definition
1580:Aesthetics
1410:Sonic Arts
1237:2:135–159.
1195:Rahn, John
1035:, 179–181.
1033:Forte 1973
1021:Forte 1973
985:Cohen 2004
961:Forte 1973
949:Forte 1973
901:Forte 1973
891:, 21, 134.
865:Forte 1973
838:Forte 1973
802:References
688:Symmetries
665:set class.
656:set class.
533:transitive
509:See also:
470:reflection
426:Z-relation
422:isometries
405:isometries
353:(decads),
351:decachords
349:(nonads),
347:nonachords
345:(octads),
343:octachords
331:hexachords
212:reflection
196:set theory
69:theory of
36:Z-relation
2099:Atonality
2019:composers
1997:Inversion
1987:Prime row
1953:Aggregate
1936:Partition
1921:Hexachord
1892:serialism
1096:Rahn 1980
1081:Rahn 1980
1057:Rahn 1980
1045:Rahn 1980
937:Rahn 1980
925:Cohn 1992
889:Rahn 1980
877:Rahn 1980
850:Rahn 1980
595:multisets
531:..., and
525:reflexive
519:to be an
401:set class
389:inversion
315:trichords
204:inversion
143:does not
120:as well.
2178:Category
2134:Semitone
1948:Tone row
1804:Bisector
1794:Diatonic
1713:Identity
1585:Analysis
1338:. 1996.
1320:. 1987.
1271:. 1993.
1253:. 2002.
1234:Intégral
1197:. 1980.
1183:. 1960.
1167:(cloth)
1155:. 1973.
1047:, 33–38.
999:, 29–30.
951:, 73–74.
903:, 60–61.
775:See also
743:I type.
580:, where
296:♯
285:♯
279:♯
2017:Notable
1590:Aspects
1105:Sources
791:Tonnetz
164:removed
149:sources
90:pitches
47:musical
1631:Tuning
1521:MtA.Ca
1378:
1354:
1346:
1328:
1310:
1302:
1287:
1279:
1261:
1222:
1205:
1175:(pbk).
1171:
1163:
1129:
1098:, 148.
939:, 140.
927:, 149.
915:, 148.
708:where
555:I or I
118:rhythm
112:, and
63:atonal
2149:Trope
1083:, 91.
1059:, 90.
1023:, 12.
1011:, 85.
987:, 33.
975:, 22.
963:, 21.
879:, 28.
852:, 27.
816:, 99.
755:and T
527:...,
476:. If
319:triad
311:dyads
88:) of
55:tonal
2071:more
1958:List
1890:and
1758:List
1376:ISBN
1352:ISBN
1344:ISBN
1326:ISBN
1308:ISBN
1300:ISBN
1285:ISBN
1277:ISBN
1259:ISBN
1220:ISBN
1203:ISBN
1169:ISBN
1161:ISBN
1127:ISBN
1071:, 5.
867:, 3.
415:and
387:and
339:Rahn
225:and
210:and
202:and
147:any
145:cite
92:and
84:and
82:sets
2074:...
2068:...
1753:Set
1519:",
395:or
341:),
303:.)
158:by
2180::
1530:,
1508:,
1498:,
1489:.
1485:,
1468:.
1457:,
1440:,
1429:,
1418:,
1408:,
1397:,
1218:.
1088:^
857:^
765:/T
737:/T
724:/T
678:/I
640:/I
443:.
361:.
108:,
1880:e
1873:t
1866:v
1674:e
1667:t
1660:v
1561:e
1554:t
1547:v
1523:.
1515:"
1502:.
1472:.
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1444:.
1433:.
1422:.
1412:.
1401:.
1382:.
1358:)
1332:.
1314:.
1291:.
1265:.
1226:.
1209:.
1133:.
840:.
828:.
768:n
762:n
757:2
753:0
740:n
734:n
731:T
727:n
721:n
715:n
710:n
705:n
699:0
681:n
675:n
662:n
653:n
643:n
637:n
628:n
586:d
582:c
578:d
576:–
574:c
558:n
552:n
546:n
517:S
499:x
495:n
490:n
485:n
478:x
466:n
462:x
457:n
452:n
448:x
293:F
276:C
231:n
185:)
179:(
174:)
170:(
166:.
152:.
96:(
20:)
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