Definitions

# Set theory (music)

Musical set theory provides concepts for categorizing musical 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; 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. Musical set theory deals with collections of pitches and pitch classes, 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.

## Mathematical set theory versus musical set theory

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 and between typical mathematical terminology and the terms used in musical set theory more generally. For example, musicians use the terms transposition and inversion where mathematicians would use translation and reflection. Furthermore, musicians talk about "ordered sets" where mathematicians would talk about permutations, sequences or tuples.

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, e.g., 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.

## Set and set types

The fundamental concept of musical set theory is the (musical) set, which typically refers to an unordered collection of equal tempered pitch classes. 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: {}. Some theorists use angle brackets <> to denote ordered sequences, while others use parentheses (). Thus one might notate the unordered set of pitch classes 0, 1, and 2 (e.g., C, C#, and D, or E, F, and F#) as {0,1,2}. The ordered sequence C-C#-D would be notated <0,1,2> or (0,1,2). C is not always considered to be zero; 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."

Two-element sets are called dyads, three-element sets trichords. Sets of higher cardinalities are called tetrachords, pentachords, hexachords, heptachords, octachords, nonachords, decachords, undecachords, and, finally, the dodecachord.

## Basic operations

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, 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.) 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 Forte, have emphasized the Z-relation which obtains between two sets sharing the same total interval content, or interval vector. However, Z-related sets can have very different musical characters, and not all music theorists feel that the relationship is musically significant.

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).

## 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 (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 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. Thus the chromatic trichord {0, 1, 2} belongs to set-class 3-1, indicating that it is the first three-note set class in Forte's list. The augmented trichord {0, 4, 8}, receives the label 3-12, which happens to be the last trichord in Forte's list.

Forte's nomenclature is a divisive issue in the music-theory community. The primary criticisms of the system are the following: (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, though there are some musical situations in which this is not desirable.

The second, and perhaps most popular notational system labels sets in terms of their normal form, which depends on the concept of normal order. (There are, in fact, competing definitions of normal order in the music-theoretical literature; we will adopt the simplest one here.) 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. 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 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 In set class.

## Symmetry

The number of transpositions and inversions mapping a set to itself is the set's degree of symmetry. Every set has at least one symmetry, as it maps onto itself under the identity operation T0. 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.

## Bibliography

• Forte, Allen (1973). The Structure of Atonal Music. New Haven and London: Yale University Press. ISBN 0-300-01610-7 (cloth) ISBN 0-300-02120-8 (pbk).
• Hanson, Howard (1960). Harmonic Materials of Modern Music: Resources of the Tempered Scale. New York: Appleton-Century-Crofts.
• Lewin, David (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, George (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)
• Rahn, John (1980). Basic Atonal Theory. New York: Schirmer Books; London and Toronto: Prentice Hall International. ISBN 0-02-873160-3.
• Straus, Joseph N. (2005). Introduction to Post-Tonal Theory, 3rd edition. Upper Saddle River, NJ: Prentice-Hall. ISBN 0-13-189890-6.