A crash course of pc set analysis
Pitch class set analysis aims, not unexpectedly, to establish relations between pitch class sets . A pitch class set ('pc set') is simply an unordered set of classes of pitches. Note that we are talking about classes of pitches, rather than individual notes -- we discuss types rather than tokens. Surface structures like repetitions, octave, enharmonic spelling etc of individual notes are not properties of pitch classes and will not influence the analysis. Any individual token of, say, 'E' is considered equivalent as all other 'E' tokens (that is, a representative of the 'E' class) regardless of its duration, octave, or spelling.
The central concept to Forte's brand of pc set analysis is that of inclusion. Pitch set A is included in a set B if each and every member of A is also a member of B. Usually, the inclusion relation itself is of more interest than which of the sets that actually is included in the other. A more generally stated definition, then, is that there is a relation of inclusion between set A and set B if the smaller of them is included in the larger.
The inclusion may be literal -- each and every pitch class of A is also a member of B, or vice versa -- but more often, the smaller set may be included in the larger only if one of them first is transposed. This is permitted and indeed very common (just like any tonal tune may appear in twelve keys). Such a relation is called inclusion under transposition. (Coming to think about it, literal inclusion may also be described as inclusion under transposition -- only that the transposition is made by zero semitones).
The relation of inclusion between two sets is relevant also when one of them is inverted, that is, when ascending intervals are replaced with descending of the same size and vice versa. In that case the relation is called inclusion under inversion. In Forte's system, inversion always implies a subsequent transposition.
According to Forte, the concepts of "inclusion under transposition" and "inclusion under inversion [and transposition]" are relevant not only between a set A and a set B, but also between set A and the complement of set B. As in set logic, the complement is "the rest of the universe in question"; in this case, the universe comprises the twelve semitones of an octave, and the complement of a set X is thus all pitch classes not included in X. It is easy to show that set A is included in set Bc (the complement of set B) if and only if set B also is included in Ac (the complement of set A).
Forte now defines two important relations. If there is a relation of inclusion (under transposition and/or inversion) between sets A and B or between sets A and Bc (but not both), the sets are said to be members of the same set complex , and the relation between them is designated 'K'. If such relation holds between sets A and B and between sets A and Bc, they are said to pertain to the same subcomplex , and the relation between them is designated 'Kh'. The K- and Kh-relations play important structural roles in Forte's system.
According to Forte, no inclusion or transposition relations are relevant between sets that are of equal or complementary size (that is, the size of the complement of the set -- the sum of the sizes of a given set and its complement equals the size of the universe, in this case 12). Two equally sized sets where one is included in the other must be identical, and the relations between them are of course trivial. However, for sets of complementary size, this is not as obvious. Cook (1987:137) suggests a more generous interpretation, which does not exclude relations between complementary sized sets. Cook's definition of the 'K' and 'Kh' relations is adjusted accordingly.
A pitch class may also be characterized by the interval content it offers: the numbers of minor seconds, major thirds, perfect fifths etc. Since octaves are not a property of a pitch class, any interval larger than six semitones (i. e., a tritone) may be transposed to one of the following: prime, minor second, major second, minor third, major third, perfect fourth, tritone. The prime is analytically uninteresting; the number of others occurring in a given pc set are conveniently rendered as an interval vector . Any pc set has an interval vector; The ordinary major scale, for instance, has the interval vector 254361, meaning that it contains two different minor seconds, five different major seconds, four different minor thirds, three different major thirds, six different perfect fourths, and one tritone. Forte employs the interval vectors to define an additional set relation. Two sets that share the same interval vector are said to be z-related, and the relation between them is known as a z-relation .
It is useful to have a standardized way of ordering the pitches of a pc set, so they may conveniently be referred to. Since reordering and transposing never affects the relations described, such reorderings may be performed whenever they are convenient. First and foremost, sets are most practically represented in integer notation , counting C as 0. The set {D, G, Bb}, for instance, is known as [2, 7, 10]. The normal order of a pc set is its most 'compact' representation -- the set rearranged (and octavized as needed -- that is, multiples of 12 may always be added or subtracted arbitrarily to any element of the set, to keep it in the interval 0 - 11) so that the outer interval is as small as possible; if there are more than one possibility, rearranged again so that the lowest (second lowest, third lowest... ) interval is as small as possible. The best normal order is either the normal order of a given set or the normal order of its inversion, whatever gives the most compact representation. For [2, 7, 10], the normal order is [7, 10, 2] and the normal order of the inversion is [10, 2, 5] -- the best normal order is [7, 10, 2]. The prime form is the best normal order transposed to start with zero -- something like a dictionary look-up form. For [2, 7, 10] it would be [0, 3, 7].