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 (e.g. starting from C, going up by 3 semitones repeatedly until eventually C is again reached - the cycle is the collection of all the notes met on the way). In other words, interval cycles "unfold a single recurrent interval in a series that closes with a return to the initial pitch class". See: wikt:cycle.

Interval cycles are notated by George Perle using the letter "C" (for cycle), with an interval class integer to distinguish the interval. Thus the diminished seventh chord would be C3 and the augmented triad would be C4. A superscript may be added to distinguish between transpositions, using 0–11 to indicate the lowest pitch class in the cycle. "These interval cycles play a fundamental role in the harmonic organization of post-diatonic music and can easily be identified by naming the cycle."

Here are interval cycles C2, C3, C4 and C6:

: <score sound="1"> {

\override Score.TimeSignature #'stencil = ##f

\relative c' {

\clef treble \time 4/4

<c d e fis gis ais c>1^\markup { "C2" }

<c es ges a c>1^\markup { "C3" }

<c e gis c>1^\markup { "C4" }

<c fis c'>1^\markup { "C6" }

} }

</score>

[[Image:Twelve-tone interval cycles.png|thumb|Twelve-tone interval cycles Thus an interval cycle or pair of cycles may be reducible to a representation of the chromatic scale.

As such, interval cycles may be differentiated as ascending or descending, with, "the ascending form of the semitonal scale [called] a 'P cycle' and the descending form [called] an 'I cycle'," while, "inversionally related dyads [are called] 'P/I' dyads." P/I dyads will always share a sum of complementation. Cyclic sets are those "sets whose alternate elements unfold complementary cycles of a single interval," that is an ascending and descending cycle:

thumb|center|660x660px|Cyclic set (sum 9) from [[Alban Berg|Berg's Lyric Suite]]

In 1920 Berg discovered/created a "master array" of all twelve interval cycles:

Berg's Master Array of Interval Cycles

Cycles P 0 11 10 9 8 7 6 5 4 3 2 1 0

P I I 0 1 2 3 4 5 6 7 8 9 10 11 0

_______________________________________

0 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0

11 1 | 0 11 10 9 8 7 6 5 4 3 2 1 0

10 2 | 0 10 8 6 4 2 0 10 8 6 4 2 0

9 3 | 0 9 6 3 0 9 6 3 0 9 6 3 0

8 4 | 0 8 4 0 8 4 0 8 4 0 8 4 0

7 5 | 0 7 2 9 4 11 6 1 8 3 10 5 0

6 6 | 0 6 0 6 0 6 0 6 0 6 0 6 0

5 7 | 0 5 10 3 8 1 6 11 4 9 2 7 0

4 8 | 0 4 8 0 4 8 0 4 8 0 4 8 0

3 9 | 0 3 6 9 0 3 6 9 0 3 6 9 0

2 10 | 0 2 4 6 8 10 0 2 4 6 8 10 0

1 11 | 0 1 2 3 4 5 6 7 8 9 10 11 0

0 0 | 0 0 0 0 0 0 0 0 0 0 0 0 0

Source:

See also

  • Equal-interval chord
  • Identity (music)
  • Interval vector
  • Octatonic scale

References

  • The "Giant Steps" Progression and Cycle Diagrams by Dan Adler