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. These invariances in serial music are analogous to the use of common-tones and common-chords in tonal music. Non-serial 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". For example, the set (0,2) (a major second) is in normal form while the set (0,10) (a minor seventh, the inversion of a major second) is not, its normal form being (10,0). Rather than the "original" (untransposed, uninverted) form of the set, the prime form may be considered either the normal form of the set or the normal form of its inversion, whichever is more tightly packed. 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 See also Forte number Pitch interval Similarity relation