In the theory of
serial music, however, some authors (notably
Milton Babbitt) use the term
set where others would use
row or
series, namely to denote an ordered collection (such as a
twelve-tone row) used to structure a work. These authors speak of
twelve-tone sets,
time-point sets,
derived sets, etc. (see below.) This is a different usage of the term
set from that described above (and referred to in the term "
set theory"). For these authors, a
set form (or
row form) is a particular arrangement of such an ordered set: the
prime form (original order),
inverse (upside down),
retrograde (backwards), and
retrograde inverse (backwards and upside down). : { \override Score.TimeSignature • 'stencil = ##f \override Score.SpacingSpanner.strict-note-spacing = ##t \set Score.proportionalNotationDuration = #(ly:make-moment 1/1) \relative c'' { \time 3/1 \set Score.tempoHideNote = ##t \tempo 1 = 60 b1 bes d es, g fis aes e f c' cis a } } This can be represented numerically as the integers 0 to 11: 0 11 3 4 8 7 9 5 6 1 2 10 The first subset (B B D) being: 0 11 3 prime-form, interval-string = The second subset (E G F) being the retrograde-inverse of the first, transposed up one semitone: 3 11 0 retrograde, interval-string = mod 12 3 7 6 inverse, interval-string = mod 12 + 1 1 1 ------ = 4 8 7 The third subset (G E F) 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 subset (C C A) being the inverse of the first, transposed up one semitone: 0 11 3 prime form, interval-vector = mod 12 0 1 9 inverse, interval-string = 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. ==Non-serial== {{Image frame|content= { \override Score.TimeSignature • 'stencil = ##f \relative c' { \time 4/4 \set Score.tempoHideNote = ##t \tempo 1 = 60 1 } } |width=240|caption=Sets (0,2), (0,10), and (10,0)}} The fundamental concept of a non-serial set is that it is an unordered collection of
pitch classes. The
normal form of a set is the most compact ordering of the pitches in a set. Tomlin defines the "most compact" ordering as the one where, "the largest of the intervals between any two consecutive pitches is between the first and last pitch listed". Forte (1973) and Rahn (1980) both list the prime forms of a set as the most left-packed possible version of the set. Forte packs from the left and Rahn packs from the right ("making the small numbers smaller," versus making, "the larger numbers ... smaller"). For many years, it was accepted that there were only five instances in which the two algorithms differ. However, in 2017, music theorist Ian Ring discovered that there is a sixth set class where Forte and Rahn's algorithms arrive at different prime forms. Ian Ring also established a much simpler algorithm for computing the prime form of a set, which produces the same results as the more complicated algorithm previously published by John Rahn. ==Vectors==