Musical set theory is an atonal or post-tonal method of musical analysis and composition which is based on explaining and proving musical phenomena, taken as "sets" and subsets, using mathematical rules and notation and using that information to gain insight to compositions or their creation. Atonality in a general sense describes music that departs from the system of tonal hierarchies that are said to characterized the sound of classical European music from the sixteenth through the nineteenth centuries. ... Tonality is a system of writing music according to certain hierarchical pitch relationships around a center or tonic. ... Musical analysis can be defined as a process attempting to answer the question how does this music work?. The method employed to answer this question, and indeed exactly what is meant by the question, differs from analyst to analyst. ... A musical composition is a piece of original music designed for repeated performance (as opposed to strictly improvisational music, in which each performance is unique). ...

Mathematical set theory and musical set theory

Although musical set theory is often assumed to be the application of mathematical set theory to music, there is little coincidence between the terminology and even less between the methods of the two. In fact musical set theory is better characterized as the application of group theory and combinatorics to certain aspects of music theory. In musical set theory what is called a set is often in fact a tuple, an ordered collection of things (such as the term set form for tone row). Musical set theory also uses the terms linear and nonlinear for ordered and unordered sets, which has nothing to do with what these terms mean in mathematics. Allen Forte's book, The Structure of Atonal Music (ISBN 0300021208), one of the primary developments in musical set theory, is sometimes criticised for its supposedly faulty calculations and terminology. Musical set theory is best regarded however as an unrelated field from mathematical set theory, with its own vocabulary, whose only connection to mathematical set theory is in sometimes using the language of naive set theory to talk about finite sets. Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... Group theory is that branch of mathematics concerned with the study of groups. ... Combinatorics is a odd branch of mathematics that studies collections (usually finite) then constructing and analyzing objects meeting the criteria (as in combinatorial designs and matroid theory), with finding largest, smallest, or optimal objects (extremal combinatorics and combinatorial optimization), and with finding algebraic structures these objects may have (algebraic combinatorics). ... In mathematics, a tuple is a finite sequence of objects, that is, a list of a limited number of objects (an infinite sequence is a family). ... Allen Forte (born December 23, 1926) is a music theorist and musicologist. ... Naive set theory1 is distinguished from axiomatic set theory by the fact that the former regards sets as collections of objects, called the elements or members of the set, whereas the latter regards sets only as that which satisfies certain axioms. ...

Assumptions of atonal theory

In addition to octave and enharmonic equivalency assumed in twelve tone theory and equal tempered tonal theory, set theory also makes use of inversional and transpositional equivalency, though the degree of equivalency varies among theorists. Set theory does not, however, use diatonic functionality that is assumed in tonal theory, and this is the reason for the use of integer notation and modulo 12. Since the structures of tonal theory may then be constructed rather than assumed, tonal theory can be regarded as a specific area of atonal theory. In music, an octave (sometimes abbreviated 8ve or 8va) is the interval between one musical note and another with half or double the frequency. ... In music, an enharmonic is a note which is the equivalent of some other note, but spelled differently. ... In mathematics, an equivalence relation on a set X is a binary relation on X that is reflexive, symmetric and transitive, i. ... Twelve-tone technique is a system of musical composition devised by Arnold Schoenberg. ... Equal temperament is a scheme of musical tuning in which the octave is divided into a series of equal steps (equal frequency ratios). ... The adjective tonal can refer to: tonality in music a tonal language the opposite of Nagual, in the specific context of Carlos Castaneda, the tonal is what makes the world. ... In music theory, the word inversion has several meanings. ... In music transposition is moving a note or collection of notes (or pitches) up or down in pitch by a constant interval. ... See also: function and functional. ... Music notation is a system of writing for music. ... Modular arithmetic is a system of arithmetic for integers, where numbers wrap around after they reach a certain value — the modulus. ...

The set and set types

The fundamental concept of musical set theory is the set. A set is a collection of any musical materials or qualities, ordered or unordered, although most often sets of pitch classes are considered. Sets may be simultaneities or successions. A set is indicated by being enclosed in brackets: {}, an ordered set is indicated by <>, and an unordered set by (). Thus the set of pitch classes 0, 1, and 2 is {0,1,2}, the ordered set <0,1,2>, and the unordered set (0,1,2). In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... Order is the opposite of anarchy and chaos. ... In music and music theory a pitch class contains all notes that have the same name; for example, all Es, no matter which octave they are in, are in the same pitch class. ... Simultaneity is the property of two events happening at the same time in at least ONE Reference frame. ... Succession is the act or process of following in order or sequence. ...

The domain of all pitch class sets may be partitioned into types or equivalence classes based on cardinality or number of pitch classes, or other criteria. There are thirteen cardinalities from 0-12: the null set, monad, dyad, trichord, tetrachord, pentachord, hexachord, septachord, octachord, nonachord, decachord, undecachord, and aggregate or dodecachord. Domain has several meanings: // General some kind of territory, such as (for example) a demesne or a realm synonymous with field, e. ... In general, a partition is a splitting into parts. ... Type has historically had the following uses: In biology, a type is the specimen or specimens upon which an original species description is based. ... In mathematics, given a set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a: [a] = { x ∈ X | x ~ a } The notion of equivalence classes is useful for constructing sets out... In mathematics, the cardinality of a set is a measure of the number of elements of the set. There are two approaches to cardinality – one which compares sets directly using bijections, injections, and surjections, and another which uses cardinal numbers. ... KK Null, a Japanese musician Null, a special value in computer programming. ... The word monad comes from the Greek word μονάς (from the word μόνος, which means one, single, unique) and has had many meanings in different contexts in philosophy, mathematics, computing and music: Among the Pythagoreans (followers of Pythagoras) the monad was the first thing that came... In music, a dyad is any two notes or pitches, more commonly known as an interval. ... In music, especially in musical set theory, a trichord is a collection of three pitch classes, often one of the four ordered trichords in a tone row or set form. ... The tetrachord is a concept of music theory borrowed from ancient Greece. ... In music, a hexachord is a collection of six tones. ... Look up Aggregate in Wiktionary, the free dictionary The term aggregate may refer to— in communication, to collect messages from multiple sources for presentation together, as in an RSS (file format) aggregator or News aggregator. ...

Basic operations

The basic operations that may be performed on a set are transposition and inversion and multiplication. Order operations include retrograde and rotation. Compound operations, the result of two basic operations, may be performed and the product of operations X and Y on z is written "Y(X(z))" with X performed on z, and then Y performed on that result. These operations may also be called transformations, mappings, morphisms, or permutations; and in music theory, but not in mathematics, derivations. Taking all combinations of a certain number of basic operations (for example taking all combinations of transposition, inversion, and multiplication by 7) produces permutation groups. The word operation can mean any of several things: The method, act, process, or effect of using a device or system. ... In music transposition is moving a note or collection of notes (or pitches) up or down in pitch by a constant interval. ... In music theory, the word inversion has several meanings. ... In music and musical set theory, multiplication modulo 12 is a basic operation which may be performed on pitch or pitch class sets. ... This article is about retrograde motion. ... The term rotation can be used in several way and includes various topics. ... Transformation may refer to: In molecular biology: In genetics transformation involves the genetic alteration of a cell resulting from the introduction, uptake and expression of foreign DNA. In cell division, the transformation process converts normal cells into cells that will continue to divide without limit. ... The word mapping has several senses: In mathematics and related technical fields, it is some kind of function: see map (mathematics). ... In mathematics, a morphism is an abstraction of a structure-preserving process between two mathematical structures. ... In mathematics, especially in abstract algebra and related areas, a permutation is a bijection from a finite set X onto itself. ... There are several meanings of derivation: A derivation in abstract algebra is a linear map that satisfies Leibniz law. ... In mathematics, a permutation group is a group G whose elements are permutations of a given set M, and whose operation is the composition of permutations in G (which are thought of as bijective functions from the set M to itself); the relationship is often written as (G,M). ...

Transposition is moving a set up or down in pitch by a constant interval. If x is the original pitch transposed n semitones, T_n=x+n (mod12). Inversion is turning a set upside-down reversing the order of the intervals between pitch classes. More specifically the compound operation transpositional inversion is T_nI(x)=-x+n (mod12). Multiplication is multiplying the pitch class numbers of a set, the most useful multipliers are 1, 5, 7, 11, as multiplication by 1 is the same, multiplication by 11 is inversion, multiplication of the chromatic scale by 5 produces the circle of fourths and multiplication by 7 produces the circle of fifths. Retrograde is reversing the order of the set so the first member is last and the last is first. Rotation is placing the last member of the set first. In music transposition is moving a note or collection of notes (or pitches) up or down in pitch by a constant interval. ... In music theory, an interval is the relationship between two notes or pitches, the lower and higher members of the interval. ... In music, see: Perfect fourth Augmented fourth or tritone The subdominant, and the chord built on the subdominant, is often simply called the fourth as it is the fourth scale degree. ... In music theory, the circle of fifths is a model of pitch space. ...

Normal form

Another useful concept used in musical set theory is that of normal form. Since sets may be listed in any order without changing their identity, normal form is used as a way to compare sets (sometimes called normal order). Normal order is that which is stacked to the left, rises from left to right, within one octave and fits within the smallest interval. In the event of any ties for what produces the smallest outside interval, one compares the next most outside interval until the tie is broken, or the ordering that starts on the smallest pitch class integer is chosen. Normal order can be used to quickly compare if two sets may be transposed onto each other. For example, it is harder to compare {4,8,1} and {7,0,3} as quickly as {0,3,7} and {1,4,8}. The term normal form is used in a variety of contexts. ... // Computer programming In object-oriented programming, object identity is a mechanism for distinguishing different objects from each other. ...

Transpositional and inversional types

Each of the cardinality types listed above may be further partitioned into transpositional type (T_n type) and/or inversional type (T_n/T_nI type). A list of all sets which are in the same transpositional type as a given set may be found by transposing the original set by all intervals. Thus the trichord {8,4,1}, {1,4,8} in normal form, is in the same transpositional type as {1,4,8}+1={2,5,9}, {1,4,8}+2={3,6,10}, {4,7,11}, {5,8,0}, {6,9,1}, {7,10,2}, {8,11,3}, {9,0,4}, {10,1,5}, {11,2,6}, and {0,3,7}. All of the above are in the transpositional type {0,3,7}_Tn, as the representative set is that which is in the most normal form. {0,3,7} is equivalent under transposition and/or inversion with twenty four rather than twelve sets, the twelve above and their inversions. It happens to be the representative set for its class: {0,3,7}_Tn/T_nI, as it is the most normal ordered form between the most normal ordered form uninverted, {0,3,7}, and the most normal ordered transposition of its inversion, {0,4,7} (T₇I{0,3,7}={0,4,7}).

Thus, to find the type of a set:

• List the set in normal form.
• Transpose the set so that the first pitch class is zero.

This is the representative form of the T_n type.

• Perform T_nI and repeat the steps above.

This is the representative form of the inversions T_n type.

• Compare the T_n type representative forms.

The most normal form of the two representative types above is the representative form of the set's T_nI type.

Given any set of numbers from zero to eleven, there is a corresponding indexing integer ranging from 0 to 4095, defined as the sum of the numbers 2ⁱ for each number i in the set. Transpositions, or transpositions with inversion, are examples of permutation groups. Given any such group on the numbers from 0 to 11, we can find a corresponding representative form by finding the smallest index in the orbit of the set under the transformations of the group. In mathematics, a permutation group is a group G whose elements are permutations of a given set M, and whose operation is the composition of permutations in G (which are thought of as bijective functions from the set M to itself); the relationship is often written as (G,M). ... In mathematics, groups are often used to describe symmetries of objects. ...

Symmetry

The number of times which a set may be mapped onto itself through different operations is its degree of symmetry. Every set has at least one degree of symmetry, as it maps onto itself under the identity operation T₀. Transpositional symmetry is the property of set which maps onto itself for T_n where n does not equal 0. Inversional symmetry is the property of a set which maps into itself under T_nI. For any given T_n/T_nI type all sets will 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 T_n/T_nI type. This article is about the term degree as used in mathematics. ... Square with symmetry group D4 Symmetry is a characteristic of geometrical shapes, equations, and other objects; we say that such an object is symmetric with respect to a given operation if this operation, when applied to the object, does not appear to change it. ...

Inversionally symmetrical sets have a canonical ordering. The canonical ordering is the ordering such that the interval series of the set is its own retrograde or rather is retrograde symmetrical. For each of these canonical orderings a set will map onto itself under T_xI, x being the inversional index which is the sum of the first and last members of each canonical ordering. The first and last members, and each pair of members farther in, of sets of even cardinality will all equal the inverional index. For odd cardinality sets the middle number is a 1/2 index and the center of inversional symmetry.

Transpositionally symmetrical sets in normal form may be partitioned into segments which under transposition map onto each other cyclically, so that the last segment maps onto the first.

Sums

Sums are also used in musical set theory. George Perle provides the following example: Addition is one of the basic operations of arithmetic. ... George Perle (born May 6, 1915 in Bayonne, New Jersey) is a composer and musicologist who has studied with Ernst Krenek. ...

"C-E, D-F♯, E♭-G, are different instances of the same interval… the other kind of identity… has to do with axes of symmetry. C-E belongs to a family of symmetrically related dyads as follows:"

Axis pitches italicized, the axis is pitch class determined.

Thus in addition to being part of the interval-4 family, C-E is also a part of the sum-2 family (with G♯ equal to 0). In music theory, an interval is the relationship between two notes or pitches, the lower and higher members of the interval. ... Square with symmetry group D4 Symmetry is a characteristic of geometrical shapes, equations, and other objects; we say that such an object is symmetric with respect to a given operation if this operation, when applied to the object, does not appear to change it. ... In music, a dyad is any two notes or pitches, more commonly known as an interval. ... In music and music theory a pitch class contains all notes that have the same name; for example, all Es, no matter which octave they are in, are in the same pitch class. ...

The tone row to Alban Berg's Lyric Suite, {0,11,7,4,2,9,3,8,10,1,5,6}, is a series of six dyads, all sum 11. If the row is rotated and retrograded, so it runs , the dyads are all sum 6. In music, a tone row or note row is a permutation, an arrangement or ordering, of the twelve notes of the chromatic scale. ... Alban Maria Johannes Berg (February 9, 1885 – December 24, 1935) was an Austrian composer. ... Lyric Suite is a string quartet written by Alban Berg from 1925 to 1926 and (publically) dedicated to Alexander von Zemlinsky . ...

Axis pitches italicized, the axis is dyad (interval 1) determined

Theorists and books

• John Rahn: Basic Atonal Theory (ISBN 0028731603)
• Allen Forte: Structure of Atonal Music (ISBN 0300021208)
• David Lewin: Musical Form and Transformation: 4 Analytic Essays (ISBN 0300056869), Generalized Musical Intervals and Transformations (ISBN 0300034938)
• Joseph N. Straus: Introduction to Post-Tonal Theory (ISBN 0130143316)
• George Perle: Twelve Tone Tonality (ISBN 0520033876)

• Important terms: pitch (music), Z-relation, interval vector, permutation, identity (music), identity function

Allen Forte (born December 23, 1926) is a music theorist and musicologist. ... In music, pitch is the perception of the frequency of a note. ... In musical set theory, a Z-relation is a relation between two sets in which the two sets have the same interval content (i. ... Categories: Wikipedia cleanup | Stub ... In music, identity is similar to identity in universal algebra. ... An identity function f is a function which doesnt have any effect: it always returns the same value that was used as its argument. ...

External links

• A Brief Introduction to Pitch-Class Set Analysis
• Nick Collins : Uniqueness of pitch class spaces, minimal bases and Z partners
• Twentieth Century Pitch Theory: Some Useful Terms and Techniques
• Introduction to Set Theory by Larry Solomon
• Introduction to Post-Functional Music Analysis: Post-Functional Theory Terminology by Robert T. Kelley
• Introduction to Post-Functional Music Analysis: Set Theory, The Matrix, and the Twelve-Tone Method by Robert T. Kelley
• SetClass View (SCv) An athenaCL netTool for on-line, web-based pitch class analysis and reference.