In music

Music

Music is an art form whose medium is sound and silence. Its common elements are pitch , rhythm , dynamics, and the sonic qualities of timbre and texture...

, a cyclic set is a set

Set (music)

A set in music theory, as in mathematics and general parlance, is a collection of objects...

, "whose alternate elements unfold complementary cycles

Interval cycle

In music, an interval cycle is a collection of pitch classes created from a sequence of the same interval class. In other words a collection of pitches by starting with a certain note and going up by a certain interval until the original note is reached In music, an interval cycle is a collection...

 of a single interval

Interval (music)

In music theory, an interval is a combination of two notes, or the ratio between their frequencies. Two-note combinations are also called dyads...

." Those cycles are ascending and descending, being related by inversion since complementary:
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

George Perle

George Perle was a composer and music theorist. He was born in Bayonne, New Jersey. Perle was an alumnus of DePaul University...

, "a Klumpenhouwer network

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 interrelations among pcs" [ pitch class sets]...

 is a chord

Chord (music)

A chord in music is any harmonic set of two–three or more notes that is heard as if sounding simultaneously. These need not actually be played together: arpeggios and broken chords may for many practical and theoretical purposes be understood as chords...

 analyzed

Musical analysis

Musical analysis is the attempt to answer the question how does this music work?. The method employed to answer this question, and indeed exactly what is meant by the question, differs from analyst to analyst, and according to the purpose of the analysis. According to Ian Bent , analysis is "an...

 in terms of its dyadic

Dyad (music)

In music, a dyad is a set of two notes or pitches. Although most chords have three or more notes, in certain contexts a dyad may be considered to be a chord. The most common two-note chord is made from the interval of a perfect fifth, which may be suggestive of music of the Medieval or Renaissance...

 sums and differences

Interval (music)

In music theory, an interval is a combination of two notes, or the ratio between their frequencies. Two-note combinations are also called dyads...

," and, "this kind of analysis of triadic

Triad (music)

In music and music theory, a triad is a three-note chord that can be stacked in thirds. Its members, when actually stacked in thirds, from lowest pitched tone to highest, are called:* the Root...

 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

Inversion (music)

In music theory, the word inversion has several meanings. There are inverted chords, inverted melodies, inverted intervals, and inverted voices...

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.