page_title: Cyclic set
hierarchical_section_title: Cyclic set
caption_attribution_description: English: Cyclic set (sum 9) from Berg's Lyric Suite, and complementary interval cycles (P7 and I5) producing the cyclic set.
context_page_description: In music, a cyclic set is a set, "whose alternate elements unfold complementary cycles of a single interval." Those cycles are ascending and descending, being related by inversion since complementary:
  In the above example, as explained, one interval and its complement, creates two series of pitches starting from the same note:
  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".
  A cognate set is a set created from joining two sets related through inversion such that they share a single series of dyads.
  0 7 2 9 4 11 6 1 8 3 10 5
context_section_description: In music, a cyclic set is a set, "whose alternate elements unfold complementary cycles of a single interval." 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".
  A cognate set is a set created from joining two sets related through inversion such that they share a single series of dyads.
    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. 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.
caption: Cyclic set (sum 9) from Berg's Lyric Suite, and complementary interval cycles (P7 and I5) producing the cyclic set[1]
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