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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
1959:
550:). 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.
685:. 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
739:
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
590:
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
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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
252:
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
735:
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.
281:
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,
408:
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
719:
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
586:
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.
264:
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,
234:. 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
295:
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".
27:
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
257:
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: .
524:...". "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."
218:
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
413:
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
203:. Furthermore, where musical set theory refers to ordered sets, mathematics would normally refer to tuples or sequences (though mathematics does speak of
1222:
1867:
538:). 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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421:—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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428:. Retrograding an ordered sequence reverses the order of its elements. Rotation of an ordered sequence is equivalent to
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561:(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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62:. The concepts of musical set theory are very general and can be applied to tonal and atonal styles in any
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207:, 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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1438:"Introduction to Post-Functional Music Analysis: Set Theory, The Matrix, and the Twelve-Tone Method"
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1130:. 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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1114:. Contributions to the Study of Music and Dance 66. Westport, Conn. and London: Praeger.
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1272:. 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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1501:. 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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1427:"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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1331:, second edition, revised and expanded. Berkeley: University of California Press.
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1450:. An athenaCL netTool for on-line, web-based pitch class analysis and reference.
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1190:. New York: Schirmer Books; London and Toronto: Prentice Hall International.
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105:. Some theorists apply the methods of musical set theory to the analysis of
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1246:, edited by Nicholas Hopkins and John F. Link. New York: Carl Fischer.
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Harmonic
Materials of Modern Music: Resources of the Tempered Scale
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1405:"Twentieth Century Pitch Theory: Some Useful Terms and Techniques"
657:
Compare these two normal forms to see which is most "left packed."
1220:
Warburton, Dan. 1988. "A Working
Terminology for Minimal Music".
779:
1395:"Uniqueness of pitch class spaces, minimal bases and Z partners"
1363:, third edition. Upper Saddle River, New Jersey: Prentice-Hall.
1311:
1958:
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1205:
Analyzing Atonal Music: Pitch-Class Set Theory and Its
Contexts
368:
Pitch class inversion: 234te reflected around 0 to become t9821
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is a number representing a pitch class, its transposition by
35:
691:. Transpositionally symmetric sets map onto themselves for T
611:, each of which lead to differing but logical normal forms.
380:. 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 (
1350:
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
1295:. Reprinted, New York: Oxford University Press, 2007.
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1523:. Pitch class set library and prime form calculator.
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first elaborated many of the concepts for analyzing
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512:, it has to satisfy three conditions: it has to be
93:, and can be related by musical operations such as
1384:"A Brief Introduction to Pitch-Class Set Analysis"
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1285:Generalized Musical Intervals and Transformations
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1148:. New Haven and London: Yale University Press.
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271:, and D) as {0,1,2}. The ordered sequence C-C
50:, further developed the theory for analyzing
2154:List of dodecaphonic and serial compositions
1388:Mount Allison University Department of Music
661:The resulting set labels the initial set's T
457: mod 12. Inversion corresponds to
38:objects and describing their relationships.
528:Transpositional and inversional set classes
310:). Sets of higher cardinalities are called
151:. Unsourced material may be challenged and
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648:Invert the set and find the inversion's T
573:indicates the cardinality of the set and
171:Learn how and when to remove this message
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400:Some authors consider the operations of
392:. Since transposition and inversion are
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1485:"Pitch-Class Set Theory and Perception"
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582:sets (as opposed to pitch-class sets),
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113:Comparison with mathematical set theory
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1409:Form and Analysis: A Virtual Textbook
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469:is a pitch class, the inversion with
1313:. New Haven: Yale University Press.
1287:. New Haven: Yale University Press.
1264:. New Haven: Yale University Press.
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1178:. New York: Appleton-Century-Crofts.
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1112:Howard Hanson in Theory and Practice
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149:adding citations to reliable sources
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34:provides concepts for categorizing
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598:, which depends on the concept of
557:, derives from Allen Forte, whose
260:Some theorists use angle brackets
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1361:Introduction to Post-Tonal Theory
748:I, and there are 12 sets in the T
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1957:
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46:music. Other theorists, such as
195:where mathematicians would use
1835:Structure implies multiplicity
1819:Generic and specific intervals
1495:"Software Tools for Composers"
1:
1416:"Set Theory Primer for Music"
1146:The Structure of Atonal Music
1110:Cohen, Allen Laurence. 2004.
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559:The Structure of Atonal Music
2078:All-interval twelve-tone row
1472:"Pitch Class Set Calculator"
1136:30, no. 2 (Summer): 146–177.
298:Two-element sets are called
236:the vocabulary of set theory
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461:around some fixed point in
238:to talk about finite sets.
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1798:Cardinality equals variety
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388:and to belong to the same
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2004:
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1465:"Java Set Theory Machine"
1436:Kelley, Robert T (2002).
1425:Kelley, Robert T (2001).
1356:22, no. 1 (Spring): 1–42.
1203:Schuijer, Michiel. 2008.
1133:Perspectives of New Music
1105:23, no. 1 (Spring): 1–40.
500:Equivalence class (music)
382:transpositionally related
1515:Taylor, Stephen Andrew.
1359:Straus, Joseph N. 2005.
1682:All-interval tetrachord
1631:List of music theorists
1521:stephenandrewtaylor.net
1414:Solomon, Larry (2005).
1353:Journal of Music Theory
1339:. (First edition 1977,
785:Transformational theory
504:"For a relation in set
426:retrograde and rotation
2083:All-trichord hexachord
2031:Second Viennese School
1687:All-trichord hexachord
1455:"All About Set Theory"
443:semitones is written T
386:inversionally related,
369:
360:Transformation (music)
87:pitch-class set theory
54:music, drawing on the
28:
16:Branch of music theory
1877:Twelve-tone technique
1805:(Deep scale property)
1444:"SetClass View (SCv)"
1103:Music Theory Spectrum
760:I equivalence class.
623:To identify a set's T
367:
302:, three-element sets
279:⟨0,1,2⟩
22:
2026:Josef Matthias Hauer
1991:Retrograde inversion
1830:Rothenberg propriety
1814:Generated collection
1737:Pitch-interval class
1382:Tucker, Gary (2001)
1329:Twelve-Tone Tonality
1283:Lewin, David. 1987.
639:Identify the set's T
510:equivalence relation
494:Equivalence relation
346:, and, finally, the
277:-D would be notated
145:improve this section
91:ordered or unordered
2108:Formula composition
1821:(Myhill's property)
1754:Similarity relation
1431:RobertKelleyPhd.com
1188:Basic Atonal Theory
490: mod 12.
2173:Musical set theory
1671:Musical set theory
707:I. For any given T
683:degree of symmetry
602:. To put a set in
430:cyclic permutation
370:
32:Musical set theory
29:
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2113:Modernism (music)
2041:Arnold Schoenberg
1843:
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1506:PC Set Calculator
1499:ComposerTools.com
1301:978-0-19-531713-8
1278:978-0-19-531712-1
1213:978-1-58046-270-9
463:pitch class space
262:⟨ ⟩
222:objects, such as
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99:melodic inversion
64:equal temperament
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242:Types of sets
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216:combinatorics
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189:transposition
186:
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161:November 2023
154:
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130:This section
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92:
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83:pitch classes
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53:
49:
45:
41:
40:Howard Hanson
37:
33:
26:
21:
2127:
2058:
2046:Anton Webern
1968:Permutations
1888:Fundamentals
1697:Forte number
1670:
1614:
1558:Music theory
1520:
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1430:
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1261:
1258:Lewin, David
1244:Harmony Book
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1142:Forte, Allen
1131:
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600:normal order
599:
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555:Forte number
552:
531:
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476:is written I
471:index number
434:
423:
399:
385:
381:
371:
344:undecachords
297:
294:
259:
255:simultaneous
251:
231:
212:group theory
209:
205:ordered sets
182:
167:
158:
143:Please help
131:
86:
75:permutations
68:
31:
30:
2118:Punctualism
2103:Equivalence
1803:Common tone
1727:Pitch class
1722:Permutation
1595:Mathematics
1584:Composition
1517:"SetFinder"
1480:(in German)
962:Hanson 1960
815:Hanson 1960
673:set class.
635:set class:
609:Gray coding
596:normal form
518:symmetrical
348:dodecachord
324:heptachords
316:pentachords
312:tetrachords
290:pitch class
248:Set (music)
232:permutation
228:combination
197:translation
56:twelve-tone
48:Allen Forte
23:Example of
2133:Time point
2128:Set theory
2036:Alban Berg
1981:Retrograde
1920:Invariance
1905:Derivation
1785:set theory
1764:Z-relation
1692:Complement
1615:Set theory
1610:Psychology
1605:Philosophy
1600:Musicology
1589:Definition
1569:Aesthetics
1399:Sonic Arts
1226:2:135–159.
1184:Rahn, John
1024:, 179–181.
1022:Forte 1973
1010:Forte 1973
974:Cohen 2004
950:Forte 1973
938:Forte 1973
890:Forte 1973
880:, 21, 134.
854:Forte 1973
827:Forte 1973
791:References
677:Symmetries
654:set class.
645:set class.
522:transitive
498:See also:
459:reflection
415:Z-relation
411:isometries
394:isometries
342:(decads),
340:decachords
338:(nonads),
336:nonachords
334:(octads),
332:octachords
320:hexachords
201:reflection
185:set theory
58:theory of
25:Z-relation
2088:Atonality
2008:composers
1986:Inversion
1976:Prime row
1942:Aggregate
1925:Partition
1910:Hexachord
1881:serialism
1085:Rahn 1980
1070:Rahn 1980
1046:Rahn 1980
1034:Rahn 1980
926:Rahn 1980
914:Cohn 1992
878:Rahn 1980
866:Rahn 1980
839:Rahn 1980
584:multisets
520:..., and
514:reflexive
508:to be an
390:set class
378:inversion
304:trichords
193:inversion
132:does not
109:as well.
2167:Category
2123:Semitone
1937:Tone row
1793:Bisector
1783:Diatonic
1702:Identity
1574:Analysis
1327:. 1996.
1309:. 1987.
1260:. 1993.
1242:. 2002.
1223:Intégral
1186:. 1980.
1172:. 1960.
1156:(cloth)
1144:. 1973.
1036:, 33–38.
988:, 29–30.
940:, 73–74.
892:, 60–61.
764:See also
732:I type.
569:, where
285:♯
274:♯
268:♯
2006:Notable
1579:Aspects
1094:Sources
780:Tonnetz
153:removed
138:sources
79:pitches
36:musical
1620:Tuning
1510:MtA.Ca
1367:
1343:
1335:
1317:
1299:
1291:
1276:
1268:
1250:
1211:
1194:
1164:(pbk).
1160:
1152:
1118:
1087:, 148.
928:, 140.
916:, 149.
904:, 148.
697:where
544:I or I
107:rhythm
101:, and
52:atonal
2138:Trope
1072:, 91.
1048:, 90.
1012:, 12.
1000:, 85.
976:, 33.
964:, 22.
952:, 21.
868:, 28.
841:, 27.
805:, 99.
744:and T
516:...,
465:. If
308:triad
300:dyads
77:) of
44:tonal
2060:more
1947:List
1879:and
1747:List
1365:ISBN
1341:ISBN
1333:ISBN
1315:ISBN
1297:ISBN
1289:ISBN
1274:ISBN
1266:ISBN
1248:ISBN
1209:ISBN
1192:ISBN
1158:ISBN
1150:ISBN
1116:ISBN
1060:, 5.
856:, 3.
404:and
376:and
328:Rahn
214:and
199:and
191:and
136:any
134:cite
81:and
73:and
71:sets
2063:...
2057:...
1742:Set
1508:",
384:or
330:),
292:.)
147:by
2169::
1519:,
1497:,
1487:,
1478:.
1474:,
1457:.
1446:,
1429:,
1418:,
1407:,
1397:,
1386:,
1207:.
1077:^
846:^
754:/T
726:/T
713:/T
667:/I
629:/I
432:.
350:.
97:,
1869:e
1862:t
1855:v
1663:e
1656:t
1649:v
1550:e
1543:t
1536:v
1512:.
1504:"
1491:.
1461:.
1440:.
1433:.
1422:.
1411:.
1401:.
1390:.
1371:.
1347:)
1321:.
1303:.
1280:.
1254:.
1215:.
1198:.
1122:.
829:.
817:.
757:n
751:n
746:2
742:0
729:n
723:n
720:T
716:n
710:n
704:n
699:n
694:n
688:0
670:n
664:n
651:n
642:n
632:n
626:n
617:n
575:d
571:c
567:d
565:–
563:c
547:n
541:n
535:n
506:S
488:x
484:n
479:n
474:n
467:x
455:n
451:x
446:n
441:n
437:x
282:F
265:C
220:n
174:)
168:(
163:)
159:(
155:.
141:.
85:(
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