Musical set theory provides concepts for categorizing

music Music is generally defined as the art of arranging sound to create some combination of form, harmony, melody, rhythm or otherwise expressive content. Exact definitions of music vary considerably around the world, though it is an aspect ...

al objects and describing their relationships.

Howard Hanson Howard Harold Hanson (October 28, 1896 – February 26, 1981)''The New York Times'' – Obituaries. Harold C. Schonberg. February 28, 1981 p. 1011/ref> was an American composer, conductor, educator, music theorist, and champion of American class ...

first elaborated many of the concepts for analyzing tonal music. Other theorists, such as

Allen Forte Allen, Allen's or Allens may refer to: Buildings * Allen Arena, an indoor arena at Lipscomb University in Nashville, Tennessee * Allen Center, a skyscraper complex in downtown Houston, Texas * Allen Fieldhouse, an indoor sports arena on the Univ ...

, further developed the theory for analyzing

atonal 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 ...

music, drawing on the

twelve-tone The twelve-tone technique—also known as dodecaphony, twelve-tone serialism, and (in British usage) twelve-note composition—is a method of musical composition first devised by Austrian composer Josef Matthias Hauer, who published his "law o ...

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 An equal temperament is a musical temperament or tuning system, which approximates just intervals by dividing an octave (or other interval) into equal steps. This means the ratio of the frequencies of any adjacent pair of notes is the same, wh ...

tuning system, and to some extent more generally than that. One branch of musical set theory deals with collections ( sets and

permutations In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or pr ...

) of pitches and

pitch class In music, a pitch class (p.c. or pc) is a set of all pitches that are a whole number of octaves apart; for example, the pitch class C consists of the Cs in all octaves. "The pitch class C stands for all possible Cs, in whatever octave positio ...

es (pitch-class set theory), which may be ordered or unordered, and can be related by musical operations such as transposition,

melodic inversion In music theory, an inversion is a type of change to intervals, chords, voices (in counterpoint), and melodies. In each of these cases, "inversion" has a distinct but related meaning. The concept of inversion also plays an important role in mu ...

, and complementation. Some theorists apply the methods of musical set theory to the analysis of

rhythm Rhythm (from Greek , ''rhythmos'', "any regular recurring motion, symmetry") generally means a " movement marked by the regulated succession of strong and weak elements, or of opposite or different conditions". This general meaning of regular recu ...

as well.

Mathematical set theory versus musical set theory

Although musical set theory is often thought to involve the application of mathematical

set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...

to music, there are numerous differences between the methods and terminology of the two. For example, musicians use the terms transposition and

inversion Inversion or inversions may refer to: Arts * , a French gay magazine (1924/1925) * ''Inversion'' (artwork), a 2005 temporary sculpture in Houston, Texas * Inversion (music), a term with various meanings in music theory and musical set theory * ...

where mathematicians would use

translation Translation is the communication of the Meaning (linguistic), meaning of a #Source and target languages, source-language text by means of an Dynamic and formal equivalence, equivalent #Source and target languages, target-language text. The ...

and

reflection Reflection or reflexion may refer to: Science and technology * Reflection (physics), a common wave phenomenon ** Specular reflection, reflection from a smooth surface *** Mirror image, a reflection in a mirror or in water ** Signal reflection, in ...

. Furthermore, where musical set theory refers to ordered sets, mathematics would normally refer to tuples or sequences (though mathematics does speak of

ordered set In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary r ...

s, 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 In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...

and

combinatorics Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many appl ...

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 objects, such as

es, is called a

combination In mathematics, a combination is a selection of items from a set that has distinct members, such that the order of selection does not matter (unlike permutations). For example, given three fruits, say an apple, an orange and a pear, there are th ...

, 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.

Set and set types

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 simultaneous 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: []. Some theorists use angle brackets 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, C, and D) as . The ordered sequence C-C-D would be notated 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 would represent F, F and G. (For the use of numbers to represent notes, see

.) 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". Two-element sets are called

dyad Dyad or dyade may refer to: Arts and entertainment * Dyad (music), a set of two notes or pitches * ''Dyad'' (novel), by Michael Brodsky, 1989 * ''Dyad'' (video game), 2012 * ''Dyad 1909'' and ''Dyad 1929'', ballets by Wayne McGregor Other uses ...

s, three-element sets

trichord 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 musical set theory there are twelve trichords give ...

s (occasionally "triads", though this is easily confused with the traditional meaning of the word triad). Sets of higher cardinalities are called

tetrachord In music theory, a tetrachord ( el, τετράχορδoν; lat, tetrachordum) is a series of four notes separated by three intervals. In traditional music theory, a tetrachord always spanned the interval of a perfect fourth, a 4:3 frequency pr ...

s (or tetrads),

pentachord A pentachord in music theory may be either of two things. In pitch-class set theory, a pentachord is defined as any five pitch classes, regarded as an unordered collection . In other contexts, a pentachord may be any consecutive five-note section ...

s (or pentads), hexachords (or hexads),

heptachord A set (pitch set, pitch-class set, set class, set form, set genus, pitch collection) in music theory, as in Set (mathematics), mathematics and general parlance, is a collection of objects. In Set theory (music), musical contexts the term is tradi ...

s (heptads or, sometimes, mixing Latin and Greek roots, "septachords"—e.g. Rahn), octachords (octads),

nonachord A set (pitch set, pitch-class set, set class, set form, set genus, pitch collection) 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 collect ...

s (nonads), decachords (decads),

undecachord A set (pitch set, pitch-class set, set class, set form, set genus, pitch collection) in music theory, as in Set (mathematics), mathematics and general parlance, is a collection of objects. In Set theory (music), musical contexts the term is tradi ...

s, and, finally, the

dodecachord A set (pitch set, pitch-class set, set class, set form, set genus, pitch collection) in music theory, as in Set (mathematics), mathematics and general parlance, is a collection of objects. In Set theory (music), musical contexts the term is tradi ...

Basic operations

The basic operations that may be performed on a set are transposition and

. Sets related by transposition or inversion are said to be ''transpositionally related'' or ''inversionally related,'' and to belong to the same

set class A set (pitch set, pitch-class set, set class, set form, set genus, pitch collection) 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 collect ...

. Since transposition and inversion are

isometries In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' mea ...

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 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 Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being additi ...

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

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 In musical set theory, an interval vector is an array of natural numbers which summarize the intervals present in a set of pitch classes. (That is, a set of pitches where octaves are disregarded.) Other names include: ic vector (or interva ...

, which obtains between two sets that share the same total interval content, or

interval vector In musical set theory, an interval vector is an array of natural numbers which summarize the intervals present in a set of pitch classes. (That is, a set of pitches where octaves are disregarded.) Other names include: ic vector (or inter ...

—but are not transpositionally or inversionally equivalent. Another name for this relationship, used by Hanson, is "isomeric". 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 In mathematics, and in particular in group theory, a cyclic permutation (or cycle) is a permutation of the elements of some set ''X'' which maps the elements of some subset ''S'' of ''X'' to each other in a cyclic fashion, while fixing (that is, ma ...

. Transposition and inversion can be represented as elementary arithmetic operations. If is a number representing a pitch class, its transposition by semitones is written T = + mod 12. Inversion corresponds to

around some fixed point in

pitch class space In music theory, pitch-class space is the circular space representing all the notes (pitch classes) in a musical octave. In this space, there is no distinction between tones that are separated by an integral number of octaves. For example, C4, ...

. If is a pitch class, the inversion with

index number In Statistics, Economics and Finance, an index is a statistical measure of change in a representative group of individual data points. These data may be derived from any number of sources, including company performance, prices, productivity, an ...

is written I = - mod 12.

Equivalence relation

"For a relation in set ''S'' to be an

equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relation ...

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Mathematical set theory versus musical set theory

Set and set types

Basic operations

Equivalence relation

Transpositional and inversional set classes

Symmetry

See also

References

Further reading

External links