Editing Interval cycle (section)

In [[music]], an '''interval cycle''' is a [[set (music)|collection]] of [[pitch class]]es created from a sequence of the same [[interval class]].<ref name="Whittall">Whittall, Arnold. 2008. ''The Cambridge Introduction to Serialism'', p. 273-74. New York: Cambridge University Press. {{ISBN|978-0-521-68200-8}} (pbk).</ref> In other words, a collection of [[pitch (music)|pitches]] by starting with a certain [[Musical note|note]] and going up by a certain [[interval (music)|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 [[transposition (music)|transpositions]], using 0&ndash;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."<ref name="Listening">[[George Perle|Perle, George]] (1990). ''The Listening Composer'', p. 21. California: University of California Press. {{ISBN|0-520-06991-9}}.</ref>

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

[[Image:Interval cycles C1-C4 and C6.PNG|400px|Interval cycles C1&ndash;C4 and C6]]

[[Image:Twelve-tone interval cycles.png|thumb|right|Twelve-tone interval cycles<ref name="Whittall"/> complete the [[tone row#total chromatic|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 third]]s it would fall flat of an octave return by an interval known as the [[diesis]]. Put another way, a major third above G{{music|sharp}} is B{{music|sharp}}, 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 [[symmetrical]] and thus non-[[diatonic]]. However, a seven-pitch segment of C7 will produce the [[diatonic major scale]]:<ref name="Listening"/>

[[Image:7-note segment of C5.svg|330px|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.<ref name="Listening"/>

Cyclic tonal [[chord progression|progressions]] in the works of Romantic and late Romantic composers (e.g., [[Richard Wagner]], [[Johannes Brahms]], [[Gustav Mahler]]) form a link with the cyclic pitch successions in the atonal music of Modernists such as [[Béla Bartók]], [[Alexander Scriabin]], [[Edgard Varèse]], and the [[Second Viennese School]] ([[Arnold Schoenberg]], [[Alban Berg]], and [[Anton Webern]]). At the same time, these [[Simultaneity succession|progressions]] signal the end of [[tonality]].<ref name="Listening"/>

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."<ref>Perle, George (1996). ''Twelve-Tone Tonality'', p. 7. {{ISBN|0-520-20142-6}}.</ref> 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'''."<ref>Perle (1996), p. 8-9.</ref> P/I dyads will always share a [[sum of complementation]]. [[Cyclic set]]s are those "[[set (music)|sets]] whose alternate elements unfold [[Complement (music)#Rule of twelve|complementary]] cycles of a single [[interval (music)|interval]],"<ref>Perle (1996), p. 21.</ref> that is an ascending and descending cycle:
[[Image:Berg's Lyric Suite cyclic set.png|thumb|center|400px|Cyclic set (sum 9) from [[Alban Berg|Berg's]] ''[[Lyric Suite (Berg)|Lyric Suite]]'']]

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

      Berg's Master [[array (music)|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:<ref>Perle (1996), p. 80.</ref>