Interval cycle

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In music, an interval cycle is a collection of pitch classes created from a sequence of the same interval class.[1] 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

post-diatonic music and can easily be identified by naming the cycle."[2]

Here are interval cycles C1, C2, C3, C4 and C6:

Interval cycles C1–C4 and C6

Twelve-tone interval cycles[1] complete the aggregate: C1 once (top) or C6 six times (bottom).

Interval cycles assume the use of equal temperament and may not work in other systems such as just intonation. For example, if the C4 interval cycle used justly-tuned major thirds it would fall flat of an octave return by an interval known as the diesis. Put another way, a major third above G♯ is B♯, which is only enharmonically the same as C in systems such as equal temperament, in which the diesis has been tempered out.

Interval cycles are

diatonic major scale:[2]

7-note segment of C7

This is known also known as a generated collection. A minimum of three pitches are needed to represent an interval cycle.[2]

Cyclic tonal

progressions signal the end of tonality.[2]

Interval cycles are also important in jazz, such as in Coltrane changes.

"Similarly," to any pair of transpositionally related sets being reducible to two transpositionally related representations of the chromatic scale, "the pitch-class relations between any pair of inversionally related sets is reducible to the pitch-class relations between two inversionally related representations of the semitonal scale."[3] 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."

sum of complementation. Cyclic sets are those "sets whose alternate elements unfold complementary cycles of a single interval,"[5]
that is an ascending and descending cycle:

Cyclic set (sum 9) from 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:[6]

See also

References

  1. ^
    ISBN 978-0-521-68200-8
    (pbk).
  2. ^
    ISBN 0-520-06991-9
    .
  3. ISBN 0-520-20142-6
    .
  4. ^ Perle (1996), p. 8-9.
  5. ^ Perle (1996), p. 21.
  6. ^ Perle (1996), p. 80.

External links

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