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.…

To analyze the structure of a piece of (typically atonal) music using musical set theory, one usually starts with a set of tones, which could form motives or chords.By applying simple operations such as transposition and inversion, one can discover deep structures in the music.Operations such as transposition and inversion are called isometries because they preserve the intervals between tones in a set.…

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.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 sets, and these can be seen to include the musical kind in some sense, they are far more involved).…

By applying simple operations such as transposition and inversion, one can discover deep structures in the music.Operations such as transposition and inversion are called isometries because they preserve the intervals between tones in a set.Expanding on the methods of musical set theory, some theorists have used abstract algebra to analyze music.…

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.…

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 T n = x + n (mod12).…

In the mid-1980s, McLeod encountered the work of Dutch composer Peter Schat, who had developed a post-tonal compositional technique called the Tone Clock.This technique emphasised the role on forming the chromatic aggregate through the transposition and inversion of three-note chords (trichords).McLeod expanded this technique to encompass all 223 possible set-classes (to use the terminology of Allan Forte's pitch-class set theory), and also developed an exceptionally coherent labelling, categorisation and analytical approach to the chromatic universe.…

Inversionally symmetric sets map onto themselves under T n I. For any given T n /T n I 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 n I 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.…