Example of Z-relation 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
Musical set theory provides concepts for categorizing musical objects and describing their relationships. Howard Hanson first elaborated many of the concepts for analyzing tonal music.[2] Other theorists, such as Allen Forte, further developed the theory for analyzing atonal music,[3] drawing on the 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.
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 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 n objects, such as pitch classes, 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 the vocabulary of set theory to talk about finite sets.
The fundamental concept of musical set theory is the (musical) set, which is an unordered collection of pitch classes.[4] More exactly, a pitch-class set is a numerical representation consisting of distinct integers (i.e., without duplicates).[5] 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: {},[6] or square brackets: [].[5]
Some theorists use angle brackets ⟨⟩ to denote ordered sequences,[7] while others distinguish ordered sets by separating the numbers with spaces.[8] 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 C-C♯-D would be notated ⟨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♯ 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] rhythmic onsets, or "beat classes".[9][10]
Pitch class inversion: 234te reflected around 0 to become t9821
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 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] 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 complementation and multiplication as well. The complement of set X is the set consisting of all the pitch classes not contained in X.[12] 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.[13] Another name for this relationship, used by Hanson,[14] is "isomeric".[15]
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 Tn=x+nmod12. Inversion corresponds to reflection around some fixed point in pitch class space. If x is a pitch class, the inversion with index numbern is written In=n-xmod12.
"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 ...".[16] "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."[17]
Transpositional and inversional set classes
Two transpositionally related sets are said to belong to the same transpositional set class (Tn). Two sets related by transposition or inversion are said to belong to the same transpositional/inversional set class (inversion being written TnI or In). 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 c–d, where c indicates the cardinality of the set and d is the ordinal number.[18] 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.[19] 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 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.[20] 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 Tn set classes.
To identify a set's Tn/In set class:
Identify the set's Tn set class.
Invert the set and find the inversion's Tn set class.
Compare these two normal forms to see which is most "left packed."
The resulting set labels the initial set's Tn/In set class.
Symmetries
The number of distinct operations in a system that map a set into itself is the set's degree of symmetry.[21] 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".[22] Every set has at least one symmetry, as it maps onto itself under the identity operation T0.[23] Transpositionally symmetric sets map onto themselves for Tn where n does not equal 0 (mod 12). Inversionally symmetric sets map onto themselves under TnI. For any given Tn/TnI 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 Tn/TnI 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.[24]
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 T0 and T2I, and there are 12 sets in the Tn/TnI equivalence class.[24]
Atonality in its broadest sense is music that lacks a tonal center, or key. Atonality, in this sense, usually describes compositions written from about the early 20th-century to the present day, where a hierarchy of harmonies focusing on a single, central triad is not used, and the notes of the chromatic scale function independently of one another. More narrowly, the term atonality describes music that does not conform to the system of tonal hierarchies that characterized European classical music between the seventeenth and nineteenth centuries. "The repertory of atonal music is characterized by the occurrence of pitches in novel combinations, as well as by the occurrence of familiar pitch combinations in unfamiliar environments".
The twelve-tone technique—also known as dodecaphony, twelve-tone serialism, and twelve-note composition—is a method of musical composition first devised by Austrian composer Josef Matthias Hauer, who published his "law of the twelve tones" in 1919. In 1923, Arnold Schoenberg (1874–1951) developed his own, better-known version of 12-tone technique, which became associated with the "Second Viennese School" composers, who were the primary users of the technique in the first decades of its existence. The technique is a means of ensuring that all 12 notes of the chromatic scale are sounded as often as one another in a piece of music while preventing the emphasis of any one note through the use of tone rows, orderings of the 12 pitch classes. All 12 notes are thus given more or less equal importance, and the music avoids being in a key. Over time, the technique increased greatly in popularity and eventually became widely influential on 20th-century composers. Many important composers who had originally not subscribed to or actively opposed the technique, such as Aaron Copland and Igor Stravinsky, eventually adopted it in their music.
In musical set theory, an interval class, also known as unordered pitch-class interval, interval distance, undirected interval, or "(even completely incorrectly) as 'interval mod 6'", is the shortest distance in pitch class space between two unordered pitch classes. For example, the interval class between pitch classes 4 and 9 is 5 because 9 − 4 = 5 is less than 4 − 9 = −5 ≡ 7 (mod 12). See modular arithmetic for more on modulo 12. The largest interval class is 6 since any greater interval n may be reduced to 12 − n.
In music theory, a trichord is a group of three different pitch classes found within a larger group. A trichord is a contiguous three-note set from a musical scale or a twelve-tone row.
In music using the twelve-tone technique, derivation is the construction of a row through segments. A derived row is a tone row whose entirety of twelve tones is constructed from a segment or portion of the whole, the generator. Anton Webern often used derived rows in his pieces. A partition is a segment created from a set through partitioning.
In music using the twelve tone technique, combinatoriality is a quality shared by twelve-tone tone rows whereby each section of a row and a proportionate number of its transformations combine to form aggregates. Much as the pitches of an aggregate created by a tone row do not need to occur simultaneously, the pitches of a combinatorially created aggregate need not occur simultaneously. Arnold Schoenberg, creator of the twelve-tone technique, often combined P-0/I-5 to create "two aggregates, between the first hexachords of each, and the second hexachords of each, respectively."
In music theory, complement refers to either traditional interval complementation, or the aggregate complementation of twelve-tone and serialism.
In music, a permutation (order) of a set is any ordering of the elements of that set. A specific arrangement of a set of discrete entities, or parameters, such as pitch, dynamics, or timbre. Different permutations may be related by transformation, through the application of zero or more operations, such as transposition, inversion, retrogradation, circular permutation, or multiplicative operations. These may produce reorderings of the members of the set, or may simply map the set onto itself.
In music, transposition refers to the process or operation of moving a collection of notes up or down in pitch by a constant interval.
The shifting of a melody, a harmonic progression or an entire musical piece to another key, while maintaining the same tone structure, i.e. the same succession of whole tones and semitones and remaining melodic intervals.
The mathematical operations of multiplication have several applications to music. Other than its application to the frequency ratios of intervals, it has been used in other ways for twelve-tone technique, and musical set theory. Additionally ring modulation is an electrical audio process involving multiplication that has been used for musical effect.
A set in music theory, as in mathematics and general parlance, is a collection of objects. In musical contexts the term is traditionally applied most often to collections of pitches or pitch-classes, but theorists have extended its use to other types of musical entities, so that one may speak of sets of durations or timbres, for example.
In musical set theory, an interval vector is an array of natural numbers which summarize the intervals present in a set of pitch classes. Other names include: ic vector, PIC vector and APIC vector
An all-interval tetrachord is a tetrachord, a collection of four pitch classes, containing all six interval classes. There are only two possible all-interval tetrachords, when expressed in prime form. In set theory notation, these are [0,1,4,6] (4-Z15) and [0,1,3,7] (4-Z29). Their inversions are [0,2,5,6] (4-Z15b) and [0,4,6,7] (4-Z29b). The interval vector for all all-interval tetrachords is [1,1,1,1,1,1].
A Klumpenhouwer Network, named after its inventor, Canadian music theorist and former doctoral student of David Lewin's at Harvard, Henry Klumpenhouwer, is "any network that uses T and/or I operations to interpret interrelations among pcs". According to George Perle, "a Klumpenhouwer network is a chord analyzed in terms of its dyadic sums and differences," and "this kind of analysis of triadic combinations was implicit in," his "concept of the cyclic set from the beginning", cyclic sets being those "sets whose alternate elements unfold complementary cycles of a single interval."
In musical set theory, a Forte number is the pair of numbers Allen Forte assigned to the prime form of each pitch class set of three or more members in The Structure of Atonal Music. The first number indicates the number of pitch classes in the pitch class set and the second number indicates the set's sequence in Forte's ordering of all pitch class sets containing that number of pitches.
In music theory, equivalence class is an equality (=) or equivalence between properties of sets (unordered) or twelve-tone rows. A relation rather than an operation, it may be contrasted with derivation. "It is not surprising that music theorists have different concepts of equivalence [from each other]..." "Indeed, an informal notion of equivalence has always been part of music theory and analysis. Pitch class set theory, however, has adhered to formal definitions of equivalence." Traditionally, octave equivalency is assumed, while inversional, permutational, and transpositional equivalency may or may not be considered.
In music, the "Ode-to-Napoleon" hexachord is the hexachord named after its use in the twelve-tone piece Ode to Napoleon Buonaparte Op. 41 (1942) by Arnold Schoenberg. Containing the pitch-classes 014589 it is given Forte number 6–20 in Allen Forte's taxonomic system. The primary form of the tone row used in the Ode allows the triads of G minor, E♭ minor, and B minor to easily appear.
In musical set theory, there are four kinds of interval:
The tone clock, and its related compositional theory tone-clock theory, is a post-tonal music composition technique, developed by composers Peter Schat and Jenny McLeod. Music written using tone-clock theory features a high economy of musical intervals within a generally chromatic musical language. This is because tone-clock theory encourages the composer to generate all their harmonic and melodic material from a limited number of intervallic configurations. Tone-clock theory is also concerned with the way that the three-note pitch-class sets can be shown to underlie larger sets, and considers these triads as a fundamental unit in the harmonic world of any piece. Because there are twelve possible triadic prime forms, Schat called them the "hours", and imagined them arrayed in a clock face, with the smallest hour in the one o'clock position, and the largest hour in the 12 o'clock position. A notable feature of tone-clock theory is tone-clock steering: transposing or inverting hours so that each note of the chromatic aggregate is generated once and once only.
Alegant, Brian. 2001. "Cross-Partitions as Harmony and Voice Leading in Twelve-Tone Music". Music Theory Spectrum 23, no. 1 (Spring): 1–40.
Cohen, Allen Laurence. 2004. Howard Hanson in Theory and Practice. Contributions to the Study of Music and Dance 66. Westport, Conn. and London: Praeger. ISBN0-313-32135-3.
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.
Lewin, David. 1993. Musical Form and Transformation: Four Analytic Essays. New Haven: Yale University Press. ISBN0-300-05686-9. Reprinted, with a foreword by Edward Gollin, New York: Oxford University Press, 2007. ISBN978-0-19-531712-1.
Lewin, David. 1987. Generalized Musical Intervals and Transformations. New Haven: Yale University Press. ISBN0-300-03493-8. Reprinted, New York: Oxford University Press, 2007. ISBN978-0-19-531713-8.
Morris, Robert. 1987. Composition With Pitch-Classes: A Theory of Compositional Design. New Haven: Yale University Press. ISBN0-300-03684-1.
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