Set Theory (music) - Transpositional and Inversional Set Classes

Transpositional and Inversional Set Classes

Two transpositionally related sets are said to belong to the same transpositional set class (T_n). Two sets related by transposition or inversion are said to belong to the same transpositional/inversional set class (inversion being written T_nI or I_n). 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, 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 T_n set classes.

To identify a set's T_n/I_n set class:

• Identify the set's T_n set class.
• Invert the set and find the inversion's T_n set class.
• Compare these two normal forms to see which is most "left packed."

The resulting set labels the initial set's T_n/I_n set class.