A set in music theory, as in mathematics and general parlance, is a collection of objects. In musical contexts the term is traditionally applied most often to collections of pitches or pitch-classes, but theorists have extended its use to other types of musical entities, so that one may speak of sets of durations or timbres, for example. 's In memoriamDylan Thomas A set by itself does not necessarily possess any additional structure, such as an or permutation. Nevertheless, it is often musically important to consider sets that are equipped with an order relation ; in such contexts, bare sets are often referred to as "unordered", for the sake of emphasis. Two-element sets are called dyads, three-element sets trichords. Sets of higher cardinalities are called tetrachords, pentachords, hexachords, heptachords, octachords, nonachords, decachords, undecachords, and, finally, the dodecachord. 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.
Serial
In the theory ofserial music, however, some authors use the term "set" where others would use "row" or "series", namely to denote an ordered collection used to structure a work. These authors speak of "twelve tone sets", "time-point sets", "derived sets", etc. This is a different usage of the term "set" from that described above.
For these authors, a set form is a particular arrangement of such an ordered set: the prime form, inverse, retrograde, and retrograde inverse. 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 subsets are derived from the first:
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 being: 0 11 3 prime-form, interval-string = The second subset 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 being the retrograde of the first, transposed up six semitones: 3 11 0 retrograde + 6 6 6 ------ 9 5 6 And the fourth subset 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 thus displays a relationship which can be made obvious by any of the four serialrow 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 is in normal form while the set is not, its normal form being. Rather than the "original" 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 and Rahn 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. 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.
