Set theory (music)
Encyclopedia
Musical set theory provides concepts for categorizing music
al objects and describing their relationships. Many of the notions were first elaborated by Howard Hanson
(1960) in connection with tonal
music, and then mostly developed in connection with atonal music by theorists such as Allen Forte
(1973), drawing on the work in twelve-tone
theory of Milton Babbitt
. The concepts of set theory are very general and can be applied to tonal and atonal styles in any equally-tempered
tuning system, and to some extent more generally than that. One branch of musical set theory deals with collections (sets
and permutations
) of pitches
and pitch class
es (pitch-class set theory), which may be ordered or unordered
, and which can be related by musical operations such as transposition, inversion, and complementation. The methods of musical set theory are sometimes applied to the analysis of rhythm
as well.
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 set
s, and 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 best regarded as a field that is not so much related to mathematical set theory, as an application of combinatorics to music theory with its own vocabulary. The main connection to mathematical set theory is the use of the vocabulary of set theory
to talk about finite sets.
chords, successive tones (as in a melody), or both. Notational conventions vary from author to author, but sets are typically enclosed in curly braces: {} (Rahn 1980, 28), or square brackets:[] (Forte 1973, 3). Some theorists use angle brackets
to denote ordered sequences (Rahn 1980, 21 & 134), while others distinguish ordered sets by separating the numbers with spaces (Forte 1973, 60–61). 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 
or (0,1,2). Although C is considered to be 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, rhythmic onsets, or "beat classes" (Warburton 1988, 148; Cohn 1992, 149).
Two-element sets are called dyad
s, three-element sets trichord
s (occasionally "triads", though this is easily confused with the traditional meaning of the word triad
). Sets of higher cardinalities are called tetrachord
s (or tetrads), pentachord
s (or pentads), hexachord
s (or hexads), heptachords (heptads or, sometimes, mixing Latin and Greek roots, "septachords"—e.g., Rahn 1980, 140), octachords (octads), nonachords (nonads), decachords (decads), undecachords, and, finally, the dodecachord.
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, and hence its musical character. 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 (Forte 1973, 73–74).) However, 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 sharing the same total interval content, or interval vector
, but which are not transpositionally or inversionally equivalent (Forte 1973, 21). Another name for this relationship, used by Howard Hanson (1960), is "isomeric" (Cohen 2004, 33).
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 + n (mod12). Inversion corresponds to reflection
around some fixed point in pitch class space
. If "x" is a pitch class, the inversion with index number n is written In = n - x (mod12).
[in algebra
], it has to satisfy three conditions: it has to be reflexive
[...], symmetrical
[...], and transitive
[...]."(Schuijer 2008, p.29-30) "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" (Schuijer 2008, 85).
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 (Forte 1973, 12). 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 (Forte 1973, 179–81). 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 significantly different. Therefore there is a limitation in Forte's theory. However, the 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, 8, 5} (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 (Rahn 1980, 33–38). 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.
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:
The resulting set labels the initial set's Tn/In set class.
of symmetry
(Rahn 1980, 90). Every set has at least one symmetry, as it maps onto itself under the identity operation T0 (Rahn 1980, 91). Transpositionally symmetric sets map onto themselves for Tn where n does not equal 0. Inversionally symmetric sets map onto themselves under TnI. For any given Tn/TnI 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 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. 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 (Rahn 1980, 148).
Music
Music is an art form whose medium is sound and silence. Its common elements are pitch , rhythm , dynamics, and the sonic qualities of timbre and texture...
al objects and describing their relationships. Many of the notions were first elaborated by Howard Hanson
Howard Hanson
Howard Harold Hanson was an American composer, conductor, educator, music theorist, and champion of American classical music. As director for 40 years of the Eastman School of Music, he built a high-quality school and provided opportunities for commissioning and performing American music...
(1960) in connection with tonal
Tonality
Tonality is a system of music in which specific hierarchical pitch relationships are based on a key "center", or tonic. The term tonalité originated with Alexandre-Étienne Choron and was borrowed by François-Joseph Fétis in 1840...
music, and then mostly developed in connection with atonal music by theorists such as Allen Forte
Allen Forte
Allen Forte is a music theorist and musicologist. He was born in Portland, Oregon and fought in the Navy at the close of World War II before moving to the East Coast. He is now Battell Professor of Music, Emeritus at Yale University...
(1973), drawing on the work in twelve-tone
Twelve-tone technique
Twelve-tone technique is a method of musical composition devised by Arnold Schoenberg...
theory of Milton Babbitt
Milton Babbitt
Milton Byron Babbitt was an American composer, music theorist, and teacher. He is particularly noted for his serial and electronic music.-Biography:...
. The concepts of set theory are very general and can be applied to tonal and atonal styles in any equally-tempered
Equal temperament
An equal temperament is a musical temperament, or a system of tuning, in which every pair of adjacent notes has an identical frequency ratio. As pitch is perceived roughly as the logarithm of frequency, this means that the perceived "distance" from every note to its nearest neighbor is the same for...
tuning system, and to some extent more generally than that. One branch of musical set theory deals with collections (sets
Set (music)
A set in music theory, as in mathematics and general parlance, is a collection of objects...
and permutations
Permutation (music)
In music, a permutation of a set is any ordering of the elements of that set. Different permutations may be related by transformation, through the application of zero or more of certain operations, such as transposition, inversion, retrogradation, circular permutation , or multiplicative operations...
) of pitches
Pitch (music)
Pitch is an auditory perceptual property that allows the ordering of sounds on a frequency-related scale.Pitches are compared as "higher" and "lower" in the sense associated with musical melodies,...
and pitch class
Pitch class
In music, a pitch class is a set of all pitches that are a whole number of octaves apart, e.g., the pitch class C consists of the Cs in all octaves...
es (pitch-class set theory), which may be ordered or unordered
Order (mathematics)
Order in mathematics may refer to:-In algebra:*Order , the cardinality of a group or period of an element*Order, or degree of a polynomial*Order, or dimension of a matrix*Order , an algebraic structure*Ordered group...
, and which can be related by musical operations such as transposition, inversion, and complementation. The methods of musical set theory are sometimes applied to the analysis of rhythm
Rhythm
Rhythm may be generally defined as a "movement marked by the regulated succession of strong and weak elements, or of opposite or different conditions." This general meaning of regular recurrence or pattern in time may be applied to a wide variety of cyclical natural phenomena having a periodicity or...
as well.
Mathematical set theory versus musical set theory
Although musical set theory is often thought to involve the application of mathematical set theorySet theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
to music, there are numerous differences between the methods and terminology of the two. For example, musicians use the terms transposition
Transposition (music)
In music transposition refers to the process, or operation, of moving a collection of notes up or down in pitch by a constant interval.For example, one might transpose an entire piece of music into another key...
and inversion
Inversion (music)
In music theory, the word inversion has several meanings. There are inverted chords, inverted melodies, inverted intervals, and inverted voices...
where mathematicians would use translation
Translation (geometry)
In Euclidean geometry, a translation moves every point a constant distance in a specified direction. A translation can be described as a rigid motion, other rigid motions include rotations and reflections. A translation can also be interpreted as the addition of a constant vector to every point, or...
and reflection
Reflection (mathematics)
In mathematics, a reflection is a mapping from a Euclidean space to itself that is an isometry with a hyperplane as set of fixed points; this set is called the axis or plane of reflection. The image of a figure by a reflection is its mirror image in the axis or plane of reflection...
.
Furthermore, where musical set theory refers to ordered sets, mathematics would normally refer to tuples or sequences (though mathematics does speak of ordered set
Ordered set
In order theory in mathematics, a set with a binary relation R on its elements that is reflexive , antisymmetric and transitive is described as a partially ordered set or poset...
s, and 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
Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
and combinatorics
Combinatorics
Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...
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 best regarded as a field that is not so much related to mathematical set theory, as an application of combinatorics to music theory with its own vocabulary. The main connection to mathematical set theory is the use of the vocabulary of set theory
Naive set theory
Naive set theory is one of several theories of sets used in the discussion of the foundations of mathematics. The informal content of this naive set theory supports both the aspects of mathematical sets familiar in discrete mathematics , and the everyday usage of set theory concepts in most...
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 (Rahn 1980, 27). More exactly, a pitch-class set is a numerical representation consisting of distinct integers (i.e., without duplicates) (Forte 1973, 3). The elements of a set may be manifested in music as simultaneousSimultaneity (music)
In music, a simultaneity is more than one complete musical texture occurring at the same time, rather than in succession. This first appeared in the music of Charles Ives, and is common in the music of Conlon Nancarrow and others....
chords, successive tones (as in a melody), or both. Notational conventions vary from author to author, but sets are typically enclosed in curly braces: {} (Rahn 1980, 28), or square brackets:[] (Forte 1973, 3). Some theorists use angle brackets
to denote ordered sequences (Rahn 1980, 21 & 134), while others distinguish ordered sets by separating the numbers with spaces (Forte 1973, 60–61). 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 or (0,1,2). Although C is considered to be 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 classPitch class
In music, a pitch class is a set of all pitches that are a whole number of octaves apart, e.g., the pitch class C consists of the Cs in all octaves...
.)
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" (Warburton 1988, 148; Cohn 1992, 149).
Two-element sets are called dyad
Dyad (music)
In music, a dyad is a set of two notes or pitches. Although most chords have three or more notes, in certain contexts a dyad may be considered to be a chord. The most common two-note chord is made from the interval of a perfect fifth, which may be suggestive of music of the Medieval or Renaissance...
s, three-element sets trichord
Trichord
In music theory, a trichord is a group of three different pitch classes found within a larger group . For example a continguous three note set from a musical scale or twelve-tone row. The term is derived by analogy from the 20th-century use of the word "tetrachord"...
s (occasionally "triads", though this is easily confused with the traditional meaning of the word triad
Triad (music)
In music and music theory, a triad is a three-note chord that can be stacked in thirds. Its members, when actually stacked in thirds, from lowest pitched tone to highest, are called:* the Root...
). Sets of higher cardinalities are called tetrachord
Tetrachord
Traditionally, a tetrachord is a series of three intervals filling in the interval of a perfect fourth, a 4:3 frequency proportion. In modern usage a tetrachord is any four-note segment of a scale or tone row. The term tetrachord derives from ancient Greek music theory...
s (or tetrads), pentachord
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 of a diatonic scale...
s (or pentads), hexachord
Hexachord
In music, a hexachord is a collection of six pitch classes including six-note segments of a scale or tone row. The term was adopted in the Middle Ages and adapted in the twentieth-century in Milton Babbitt's serial theory.-Middle Ages:...
s (or hexads), heptachords (heptads or, sometimes, mixing Latin and Greek roots, "septachords"—e.g., Rahn 1980, 140), octachords (octads), nonachords (nonads), decachords (decads), undecachords, and, finally, the dodecachord.
Basic operations
The basic operations that may be performed on a set are transpositionTransposition (music)
In music transposition refers to the process, or operation, of moving a collection of notes up or down in pitch by a constant interval.For example, one might transpose an entire piece of music into another key...
and inversion
Inversion (music)
In music theory, the word inversion has several meanings. There are inverted chords, inverted melodies, inverted intervals, and inverted voices...
. 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
Isometry
In mathematics, an isometry is a distance-preserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent.Isometries are often used in constructions where one space is embedded in another space...
of pitch-class space, they preserve the intervallic structure of a set, and hence its musical character. 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
Complement (music)
In music the term complement refers to two distinct concepts.In traditional music theory a complement is the interval which, when added to the original interval, spans an octave in total. For example, a major 3rd is the complement of a minor 6th. The complement of any interval is also known as its...
and multiplication
Multiplication (music)
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...
as well. (The complement of set X is the set consisting of all the pitch classes not contained in X (Forte 1973, 73–74).) However, 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 sharing the same total interval content, or interval vector
Interval vector
In musical set theory, an interval vector is an array that expresses the intervallic content of a pitch-class set...
, but which are not transpositionally or inversionally equivalent (Forte 1973, 21). Another name for this relationship, used by Howard Hanson (1960), is "isomeric" (Cohen 2004, 33).
Operations on ordered sequences of pitch classes also include transposition and inversion, as well as retrograde and rotation
Permutation (music)
In music, a permutation of a set is any ordering of the elements of that set. Different permutations may be related by transformation, through the application of zero or more of certain operations, such as transposition, inversion, retrogradation, circular permutation , or multiplicative operations...
. Retrograding an ordered sequence reverses the order of its elements. Rotation of an ordered sequence is equivalent to cyclic permutation
Cyclic permutation
A cyclic permutation or circular permutation is a permutation built from one or more sets of elements in cyclic order.The notion "cyclic permutation" is used in different, but related ways:- Definition 1 :right|mapping of 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 + n (mod12). Inversion corresponds to reflection
Reflection (mathematics)
In mathematics, a reflection is a mapping from a Euclidean space to itself that is an isometry with a hyperplane as set of fixed points; this set is called the axis or plane of reflection. The image of a figure by a reflection is its mirror image in the axis or plane of reflection...
around some fixed point in pitch class space
Pitch class space
In music theory, pitch class space is the circular space representing all the notes in a musical octave.In this space, there is no distinction between tones that are separated by an integral number of octaves...
. If "x" is a pitch class, the inversion with index number n is written In = n - x (mod12).
Equivalence relation
"For a relation in set S to be an equivalence relationEquivalence relation
In mathematics, an equivalence relation is a relation that, loosely speaking, partitions a set so that every element of the set is a member of one and only one cell of the partition. Two elements of the set are considered equivalent if and only if they are elements of the same cell...
[in algebra
Algebra
Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...
], it has to satisfy three conditions: it has to be reflexive
Reflexive relation
In mathematics, a reflexive relation is a binary relation on a set for which every element is related to itself, i.e., a relation ~ on S where x~x holds true for every x in S. For example, ~ could be "is equal to".-Related terms:...
[...], symmetrical
Symmetry in mathematics
Symmetry occurs not only in geometry, but also in other branches of mathematics. It is actually the same as invariance: the property that something does not change under a set of transformations....
[...], and transitive
Transitive relation
In mathematics, a binary relation R over a set X is transitive if whenever an element a is related to an element b, and b is in turn related to an element c, then a is also related to c....
[...]."(Schuijer 2008, p.29-30) "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" (Schuijer 2008, 85).
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 are fairly similar sounding. Because of this, music theorists often consider set classes to be basic objects of musical interest.There are two main conventions for naming equal-tempered set classes. One, known as the Forte number
Forte number
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...
, 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 (Forte 1973, 12). 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 (Forte 1973, 179–81). 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 significantly different. Therefore there is a limitation in Forte's theory. However, the 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, 8, 5} (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 (Rahn 1980, 33–38). 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.
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.
Symmetry
The number of distinct operations in a system that map a set into itself is the set's degreeDegree (mathematics)
In mathematics, there are several meanings of degree depending on the subject.- Unit of angle :A degree , usually denoted by ° , is a measurement of a plane angle, representing 1⁄360 of a turn...
of symmetry
Symmetry
Symmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically pleasing proportionality and balance; such that it reflects beauty or perfection...
(Rahn 1980, 90). Every set has at least one symmetry, as it maps onto itself under the identity operation T0 (Rahn 1980, 91). Transpositionally symmetric sets map onto themselves for Tn where n does not equal 0. Inversionally symmetric sets map onto themselves under TnI. For any given Tn/TnI 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 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. 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 (Rahn 1980, 148).
See also
- Identity (music)Identity (music)In music, identity may refer to two different concepts, one in post-tonal theory and one in tuning theory.-Post-tonal theory:In post-tonal music theory, identity is similar to identity in universal algebra. An identity function is a permutation or transformation which transforms a pitch or pitch...
- Pitch intervalPitch intervalIn musical set theory, a pitch interval is the number of semitones that separates one pitch from another, upward or downward.They are notated as follows:For example C4 to D4 is 3 semitones:While C4 to D5 is 15 semitones:...
- TonnetzTonnetzIn musical tuning and harmony, the Tonnetz is a conceptual lattice diagram representing tonal space first described by Leonhard Euler in 1739....
- Transformational music theory
Further reading
- Carter, ElliottElliott CarterElliott Cook Carter, Jr. is a two-time Pulitzer Prize-winning American composer born and living in New York City. He studied with Nadia Boulanger in Paris in the 1930s, and then returned to the United States. After a neoclassical phase, he went on to write atonal, rhythmically complex music...
(2002). Harmony Book, edited by Nicholas Hopkins and John F. Link. New York: Carl Fischer. ISBN 0825845947. - Lewin, DavidDavid LewinDavid Lewin was an American music theorist, music critic and composer. Called "the most original and far-ranging theorist of his generation" , he did his most influential theoretical work on the development of transformational theory, which involves the application of mathematical group theory to...
(1993). Musical Form and Transformation: Four Analytic Essays. New Haven: Yale University Press. ISBN 0-300-05686-9. Reprinted, with a foreword by Edward Gollin, New York: Oxford University Press, 2007. ISBN 9780195317121 - Lewin, David (1987). Generalized Musical Intervals and Transformations. New Haven: Yale University Press. ISBN 0-300-03493-8. Reprinted, New York: Oxford University Press, 2007. ISBN 9780195317138
- Morris, Robert (1987). Composition With Pitch-Classes: A Theory of Compositional Design. New Haven: Yale University Press. ISBN 0-300-03684-1.
- Perle, GeorgeGeorge PerleGeorge Perle was a composer and music theorist. He was born in Bayonne, New Jersey. Perle was an alumnus of DePaul University...
(1996). Twelve-Tone Tonality, second edition, revised and expanded. Berkeley: University of California Press. ISBN 0-520-20142-6. (First edition 1977, ISBN 0-520-03387-6) - Starr, Daniel (1978). "Sets, Invariance and Partitions". Journal of Music Theory 22, no. 1 (Spring): 1–42.
- Straus, Joseph N. (2005). Introduction to Post-Tonal Theory, 3rd edition. Upper Saddle River, NJ: Prentice-Hall. ISBN 0-13-189890-6.
External links
- Tucker, Gary (2001) "A Brief Introduction to Pitch-Class Set Analysis", Mount Allison University Department of Music.
- Nick Collins "Uniqueness of pitch class spaces, minimal bases and Z partners", Sonic Arts.
- "Twentieth Century Pitch Theory: Some Useful Terms and Techniques", Form and Analysis: A Virtual Textbook.
- Solomon, Larry (2005). "Set Theory Primer for Music", SolomonMusic.net.
- Kelley, Robert T (2001). "Introduction to Post-Functional Music Analysis: Post-Functional Theory Terminology", RobertKelleyPhd.com.
- Kelley (2002). "Introduction to Post-Functional Music Analysis: Set Theory, The Matrix, and the Twelve-Tone Method".
- "SetClass View (SCv)", Flexatone.net. An athenaCL netTool for on-line, web-based pitch class analysis and reference.
- Tomlin, Jay. "All About {Musical} Set Theory", JayTomlin.com.
- "Java Set Theory Machine" or Calculator
- Helmberger, Andreas (2006). "Projekte: Pitch Class Set Calculator", www.Andreas-Helmberger.de.
- "Pitch-Class Set Theory and Perception", Ohio-State.edu.
- "Colors are sounds: How to See the Music", Creativelab. The method for transformation of music into an image.
- "Software Tools for Composers", ComposerTools.com. Javascript PC Set calculator, two-set relationship calculators, and theory tutorial.