In music, a cyclic set is a set, "whose alternate elements unfold complementary cycles of a single interval." [1] Those cycles are ascending and descending, being related by inversion since complementary:
In the above example, as explained, one interval (7) and its complement (-7 = +5), creates two series of pitches starting from the same note (8):
P7: 8 +7= 3 +7= 10 +7= 5...1 +7= 8 I5: 8 +5= 1 +5= 6 +5= 11...3 +5= 8
According to George Perle, "a Klumpenhouwer network is a chord analyzed in terms of its dyadic sums and differences," and, "this kind of analysis of triadic combinations was implicit in," his, "concept of the cyclic set from the beginning". [2]
A cognate set is a set created from joining two sets related through inversion such that they share a single series of dyads. [3]
0 7 2 9 4 11 6 1 8 3 10 5 (0 + 0 5 10 3 8 1 6 11 4 9 2 7 (0 ________________________________________ = 0 0 0 0 0 0 0 0 0 0 0 0 (0
The two cycles may also be aligned as pairs of sum 7 or sum 5 dyads. [3] All together these pairs of cycles form a set complex, "any cyclic set of the set complex may be uniquely identified by its two adjacency sums," and as such the example above shows p0p7 and i5i0. [4]
References
In music, a cyclic set is a set, "whose alternate elements unfold complementary cycles of a single interval." [1] Those cycles are ascending and descending, being related by inversion since complementary:
In the above example, as explained, one interval (7) and its complement (-7 = +5), creates two series of pitches starting from the same note (8):
P7: 8 +7= 3 +7= 10 +7= 5...1 +7= 8 I5: 8 +5= 1 +5= 6 +5= 11...3 +5= 8
According to George Perle, "a Klumpenhouwer network is a chord analyzed in terms of its dyadic sums and differences," and, "this kind of analysis of triadic combinations was implicit in," his, "concept of the cyclic set from the beginning". [2]
A cognate set is a set created from joining two sets related through inversion such that they share a single series of dyads. [3]
0 7 2 9 4 11 6 1 8 3 10 5 (0 + 0 5 10 3 8 1 6 11 4 9 2 7 (0 ________________________________________ = 0 0 0 0 0 0 0 0 0 0 0 0 (0
The two cycles may also be aligned as pairs of sum 7 or sum 5 dyads. [3] All together these pairs of cycles form a set complex, "any cyclic set of the set complex may be uniquely identified by its two adjacency sums," and as such the example above shows p0p7 and i5i0. [4]