music Music is generally defined as the art of arranging sound to create some combination of form, harmony, melody, rhythm or otherwise expressive content. Exact definitions of music vary considerably around the world, though it is an aspe ...

, an interval cycle is a collection of pitch classes created from a sequence of the same interval class.Whittall, Arnold. 2008. ''The Cambridge Introduction to Serialism'', p. 273-74. New York: Cambridge University Press. (pbk). 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 George Perle (6 May 1915 – 23 January 2009) was an American composer and music theorist. As a composer, his music was largely atonal, using methods similar to the twelve-tone technique of the Second Viennese School. This serialist style, and ...

using the letter "C" (for ''cycle''), with an interval class integer to distinguish the interval. Thus the

diminished seventh chord The diminished seventh chord is a four-note chord (a seventh chord) composed of a root note, together with a minor third, a diminished fifth, and a diminished seventh above the root: (1, 3, 5, 7). For example, the diminished seve ...

would be C3 and the

augmented triad Augment or augmentation may refer to: Language * Augment (Indo-European), a syllable added to the beginning of the word in certain Indo-European languages *Augment (Bantu languages), a morpheme that is prefixed to the noun class prefix of nouns ...

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 A harmonic is a wave with a frequency that is a positive integer multiple of the ''fundamental frequency'', the frequency of the original periodic signal, such as a sinusoidal wave. The original signal is also called the ''1st harmonic'', t ...

organization of post-diatonic music and can easily be identified by naming the cycle." Perle, George (1990). ''The Listening Composer'', p. 21. California: University of California Press. . Here are interval cycles C1, C2, C3, C4 and C6: Interval cycles assume the use of

equal temperament An equal temperament is a musical temperament or tuning system, which approximates just intervals by dividing an octave (or other interval) into equal steps. This means the ratio of the frequencies of any adjacent pair of notes is the same, ...

and may not work in other systems such as

just intonation In music, just intonation or pure intonation is the tuning of musical intervals as whole number ratios (such as 3:2 or 4:3) of frequencies. An interval tuned in this way is said to be pure, and is called a just interval. Just intervals (and ...

. For example, if the C4 interval cycle used justly-tuned

major third In classical music, a third is a Interval (music), musical interval encompassing three staff positions (see Interval (music)#Number, Interval number for more details), and the major third () is a third spanning four semitones.Allen Forte, ...

s it would fall flat of an octave return by an interval known as the

diesis In classical music from Western culture, a diesis ( , plural dieses ( , "difference"; Greek: δίεσις "leak" or "escape"Benson, Dave (2006). ''Music: A Mathematical Offering'', p.171. . Based on the technique of playing the aulos, where ...

. 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 symmetrical and thus non-

diatonic Diatonic and chromatic are terms in music theory that are most often used to characterize scales, and are also applied to musical instruments, intervals, chords, notes, musical styles, and kinds of harmony. They are very often used as a ...

. However, a seven-pitch segment of C7 will produce the

diatonic major scale In music theory, a diatonic scale is any heptatonic scale that includes five whole steps (whole tones) and two half steps (semitones) in each octave, in which the two half steps are separated from each other by either two or three whole step ...

: This is known also known as a

generated collection In diatonic set theory, a generated collection is a collection or scale formed by repeatedly adding a constant interval in integer notation, the generator, also known as an interval cycle, around the chromatic circle until a complete collect ...

. A minimum of three pitches are needed to represent an interval cycle. Cyclic tonal progressions in the works of Romantic composers such as

Gustav Mahler Gustav Mahler (; 7 July 1860 – 18 May 1911) was an Austro-Bohemian Romantic composer, and one of the leading conductors of his generation. As a composer he acted as a bridge between the 19th-century Austro-German tradition and the modernism ...

and

Richard Wagner Wilhelm Richard Wagner ( ; ; 22 May 181313 February 1883) was a German composer, theatre director, polemicist, and conductor who is chiefly known for his operas (or, as some of his mature works were later known, "music dramas"). Unlike most op ...

form a link with the cyclic pitch successions in the atonal music of Modernists such as

Béla Bartók Béla Viktor János Bartók (; ; 25 March 1881 – 26 September 1945) was a Hungarian composer, pianist, and ethnomusicologist. He is considered one of the most important composers of the 20th century; he and Franz Liszt are regarded as Hu ...

Alexander Scriabin Alexander Nikolayevich Scriabin (; russian: Александр Николаевич Скрябин ; – ) was a Russian composer and virtuoso pianist. Before 1903, Scriabin was greatly influenced by the music of Frédéric Chopin and composed ...

Edgard Varèse Edgard Victor Achille Charles Varèse (; also spelled Edgar; December 22, 1883 – November 6, 1965) was a French-born composer who spent the greater part of his career in the United States. Varèse's music emphasizes timbre and rhythm; he coine ...

, and the

Second Viennese School The Second Viennese School (german: Zweite Wiener Schule, Neue Wiener Schule) was the group of composers that comprised Arnold Schoenberg and his pupils, particularly Alban Berg and Anton Webern, and close associates in early 20th-century Vienn ...

(

Arnold Schoenberg Arnold Schoenberg or Schönberg (, ; ; 13 September 187413 July 1951) was an Austrian-American composer, music theorist, teacher, writer, and painter. He is widely considered one of the most influential composers of the 20th century. He was as ...

Alban Berg Alban Maria Johannes Berg ( , ; 9 February 1885 – 24 December 1935) was an Austrian composer of the Second Viennese School. His compositional style combined Romantic lyricism with the twelve-tone technique. Although he left a relatively sm ...

, and

Anton Webern Anton Friedrich Wilhelm von Webern (3 December 188315 September 1945), better known as Anton Webern (), was an Austrian composer and conductor whose music was among the most radical of its milieu in its sheer concision, even aphorism, and stead ...

). At the same time, these progressions signal the end of

tonality Tonality is the arrangement of pitches and/or chords of a musical work in a hierarchy of perceived relations, stabilities, attractions and directionality. In this hierarchy, the single pitch or triadic chord with the greatest stability is ca ...

. Interval cycles are also important in

jazz Jazz is a music genre that originated in the African-American communities of New Orleans, Louisiana in the late 19th and early 20th centuries, with its roots in blues and ragtime. Since the 1920s Jazz Age, it has been recognized as a m ...

, 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 chromatic scale (or twelve-tone scale) is a set of twelve pitches (more completely, pitch classes) used in tonal music, with notes separated by the interval of a semitone. Chromatic instruments, such as the piano, are made to produce th ...

, "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." 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 alleda 'P cycle' and the descending form alledan 'I cycle'," while, "inversionally related dyads re 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: In 1920 Berg discovered/created a "master array" of all twelve interval cycles: Berg's Master

Array An array is a systematic arrangement of similar objects, usually in rows and columns. Things called an array include: {{TOC right Music * In twelve-tone and serial composition, the presentation of simultaneous twelve-tone sets such that the ...

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:Perle (1996), p. 80.

References

External links

The "Giant Steps" Progression and Cycle Diagrams
by Dan Adler {{DEFAULTSORT:Interval Cycle Intervals (music) Musical symmetry

See also

References

External links