Cyclic set

For the mathematical cyclic sets, see cyclic category.

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:

Cyclic set (sum 9) from Berg's Lyric Suite, and complementary interval cycles (P7 and I5) producing the cyclic set^[1]

In the above example, as explained, one interval (7) and its compliment (-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"^[1].

"Overlapping three-note segments,"^[1] of the sum 9 cyclic set

A cognate set is a set created from joining two sets related through inversion such that they share a single series of dyads^[2].

Cognate set created from paired interval-7 cycles of sum 0^[2]

  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^[2]. 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^[3].

Sources

1. ^ ^{a b c d} Perle, George (1996). Twelve-Tone Tonality, p.21. ISBN 0-520-20142-6.
2. ^ ^{a b c} Perle (1996), p.22.
3. ^ Perle (1996), p.23.