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Set theory (music)

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.

One branch of musical set theory deals with collections (sets and permutations) of pitches and pitch classes (pitch-class set theory), which may be ordered or unordered, and can be related by musical operations such as transposition, melodic inversion, and complementation. Some theorists apply the methods of musical set theory to the analysis of rhythm as well.

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."

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.


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.

Symmetries[edit]

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]

Identity (music)

Pitch interval

Tonnetz

Transformational theory

Tucker, Gary (2001) , Mount Allison University Department of Music.

"A Brief Introduction to Pitch-Class Set Analysis"

Nick Collins , Sonic Arts.

"Uniqueness of pitch class spaces, minimal bases and Z partners"

Form and Analysis: A Virtual Textbook.

"Twentieth Century Pitch Theory: Some Useful Terms and Techniques"

Solomon, Larry (2005). , SolomonMusic.net.

"Set Theory Primer for Music"

Kelley, Robert T (2001). , RobertKelleyPhd.com.

"Introduction to Post-Functional Music Analysis: Post-Functional Theory Terminology"

Kelley, Robert T (2002). .

"Introduction to Post-Functional Music Analysis: Set Theory, The Matrix, and the Twelve-Tone Method"

Flexatone.net. An athenaCL netTool for on-line, web-based pitch class analysis and reference.

"SetClass View (SCv)"

Tomlin, Jay. . JayTomlin.com.

"All About Set Theory"

or Calculator

"Java Set Theory Machine"

Kaiser, Ulrich. , musikanalyse.net. (in German)

"Pitch Class Set Calculator"

Ohio-State.edu.

"Pitch-Class Set Theory and Perception"

ComposerTools.com. Javascript PC Set calculator, two-set relationship calculators, and theory tutorial.

"Software Tools for Composers"

"", MtA.Ca.

PC Set Calculator

Taylor, Stephen Andrew. , stephenandrewtaylor.net. Pitch class set library and prime form calculator.

"SetFinder"