Set (music)

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In musical set theory, a set is a collection of discrete entities, for example pitch sets, duration sets, and timbre sets (DeLone et al., 1975, p.475). A set form is the arrangement of an ordered set: the prime form (original order), inverse (upside down), retrograde (backwards), and retrograde inverse (backwards and upside down) (ibid).

A derived set is one which is generated or derived from consistent operations on a subset, for example Webern's Concerto, Op.24, in which the last three sets are derived from the first (ibid, p.474):

B Bb D Eb G F# G# E F C C# A

Represented numerically as the integers 0 to 11:

0 11 3 4  8 7  9  5 6 1 2  10

The first set being:

0 11 3 prime-form, interval-string = <-1 +4>

The second set being the retrograde-inverse of the first, transposed up one semitone:

  3 11 0 retrograde, interval-string = <-4 +1> mod 12
  
  3  7 6 inverse, interval-string = <+4 -1> mod 12
+ 1  1 1
  ------
= 4  8 7

The third set being the retrograde of the first, transposed up (or down) six semitones:

  3 11 0 retrograde
+ 6  6 6
  ------
  9  5 6

And the fourth set being the inverse of the first, transposed up one semitone:

  0 11  3 prime form, interval-vector = <-1 +4> mod 12

  0  1  9 inverse, interval-string = <+1 -4> mod 12
+ 1  1  1
  -------
  1  2 10

Each of the four trichords (3-note sets) thus displays a relationship which can be made obvious by any of the four serial row operations, and thus creates certain invariances. These invariances in serial music are analogous to the use of common-tones and common-chords in tonal music.

A time-point set is a duration set where the distance in time units between attack points, or time-points, is the distance in semitones between pitch classes (ibid, p.476).