music Music is generally defined as the art of arranging sound to create some combination of form, harmony, melody, rhythm or otherwise expressive content. Exact definitions of music vary considerably around the world, though it is an aspect ...

, a cyclic set is a

set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...

, "whose alternate elements unfold

complementary A complement is something that completes something else. Complement may refer specifically to: The arts * Complement (music), an interval that, when added to another, spans an octave ** Aggregate complementation, the separation of pitch-class ...

cycles of a single interval."Perle, George (1996). ''Twelve-Tone Tonality'', p.21. . 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 A Klumpenhouwer Network, named after its inventor, Canadian music theorist and former doctoral student of David Lewin's at Harvard, Henry Klumpenhouwer, is "any network that uses T and/or I operations ( transposition or inversion) to interpret i ...

is a

chord Chord may refer to: * Chord (music), an aggregate of musical pitches sounded simultaneously ** Guitar chord a chord played on a guitar, which has a particular tuning * Chord (geometry), a line segment joining two points on a curve * Chord ( ...

analyzed in terms of its dyadic

sums In mathematics, summation is the addition of a sequence of any kind of numbers, called ''addends'' or ''summands''; the result is their ''sum'' or ''total''. Beside numbers, other types of values can be summed as well: function (mathematics), fu ...

and differences," and, "this kind of analysis of triadic combinations was implicit in," his, "concept of the cyclic set from the beginning".Perle, George (1993). "Letter from George Perle", ''Music Theory Spectrum'', Vol. 15, No. 2 (Autumn), pp. 300-303. A cognate set is a set created from joining two sets related through

inversion Inversion or inversions may refer to: Arts * , a French gay magazine (1924/1925) * ''Inversion'' (artwork), a 2005 temporary sculpture in Houston, Texas * Inversion (music), a term with various meanings in music theory and musical set theory * ...

such that they share a single series of dyads.Perle (1996), p.22. 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.Perle (1996), p.23.

References

{{Atonality Intervals (music) Post-tonal music theory