# Interval cycle

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In [music](/source/music), an '''interval cycle''' is a [collection](/source/set_(music)) of [pitch class](/source/pitch_class)es created from a sequence of the same [interval class](/source/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 [pitches](/source/pitch_(music)) by starting with a certain [note](/source/Musical_note) and going up by a certain [interval](/source/interval_(music)) 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](/source/George_Perle) using the letter "C" (for ''cycle''), with an [interval class](/source/interval_class) integer to distinguish the interval. Thus the [diminished seventh chord](/source/diminished_seventh_chord) would be C3 and the [augmented triad](/source/augmented_triad) would be C4. A superscript may be added to distinguish between [transpositions](/source/transposition_(music)), using 0&ndash;11 to indicate the lowest pitch class in the cycle. "These interval cycles play a fundamental role in the [harmonic](/source/harmonic) organization of [post-diatonic music](/source/post-diatonic_music) and can easily be identified by naming the cycle."<ref name="Listening">[Perle, George](/source/George_Perle) (1990). ''The Listening Composer'', p. 21. California: University of California Press. {{ISBN|0-520-06991-9}}.</ref>

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<ref name="Whittall"/> complete the [aggregate](/source/tone_row): C1 once (top) or C6 six times (bottom).|447x447px]]

Interval cycles assume the use of [equal temperament](/source/equal_temperament) and may not work in other systems such as [just intonation](/source/just_intonation). For example, if the C4 interval cycle used justly-tuned [major third](/source/major_third)s it would fall flat of an octave return by an interval known as the [diesis](/source/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](/source/symmetrical) and thus non-[diatonic](/source/diatonic). However, a seven-pitch segment of C7 will produce the [diatonic major scale](/source/diatonic_major_scale):<ref name="Listening"/>

330px|7-note segment of C7

This is known also known as a [generated collection](/source/generated_collection).
A minimum of three pitches are needed to represent an interval cycle.<ref name="Listening"/>

Cyclic tonal [progressions](/source/chord_progression) in the works of Romantic and late Romantic composers (e.g., [Richard Wagner](/source/Richard_Wagner), [Johannes Brahms](/source/Johannes_Brahms), [Gustav Mahler](/source/Gustav_Mahler)) form a link with the cyclic pitch successions in the atonal music of Modernists such as [Béla Bartók](/source/B%C3%A9la_Bart%C3%B3k), [Alexander Scriabin](/source/Alexander_Scriabin), [Edgard Varèse](/source/Edgard_Var%C3%A8se), and the [Second Viennese School](/source/Second_Viennese_School) ([Arnold Schoenberg](/source/Arnold_Schoenberg), [Alban Berg](/source/Alban_Berg), and [Anton Webern](/source/Anton_Webern)). At the same time, these [progressions](/source/Simultaneity_succession) signal the end of [tonality](/source/tonality).<ref name="Listening"/>

Interval cycles are also important in [jazz](/source/jazz), such as in [Coltrane changes](/source/Coltrane_changes).

"Similarly," to any pair of transpositionally related sets being reducible to two transpositionally related representations of the [chromatic scale](/source/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](/source/sum_of_complementation). [Cyclic set](/source/Cyclic_set)s are those "[sets](/source/set_(music)) whose alternate elements unfold [complementary](/source/Complement_(music)) cycles of a single [interval](/source/interval_(music)),"<ref>Perle (1996), p. 21.</ref> that is an ascending and descending cycle:
[[Image:Berg's Lyric Suite cyclic set.png|thumb|center|660x660px|Cyclic set (sum 9) from [Berg's](/source/Alban_Berg) ''[Lyric Suite](/source/Lyric_Suite_(Berg))'']]

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

      Berg's Master [Array](/source/array_(music)) 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>

==See also==
*[Equal-interval chord](/source/Equal-interval_chord)
*[Identity (music)](/source/Identity_(music))
*[Interval vector](/source/Interval_vector)
*[Octatonic scale](/source/Octatonic_scale)

== References ==
{{reflist}}

== External links ==
* [http://danadler.com/misc/Cycles.pdf The "Giant Steps" Progression and Cycle Diagrams] by Dan Adler

{{DEFAULTSORT:Interval Cycle}}
Category:Intervals (music)
Category:Musical symmetry

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Adapted from the Wikipedia article [Interval cycle](https://en.wikipedia.org/wiki/Interval_cycle) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Interval_cycle?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.