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music Music is generally defined as the The arts, art of arranging sound to create some combination of Musical form, form, harmony, melody, rhythm or otherwise Musical expression, expressive content. Exact definition of music, definitions of mu ...
, 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 Note, notes, or NOTE may refer to: Music and entertainment * Musical note, a pitched sound (or a symbol for a sound) in music * Notes (album), ''Notes'' (album), a 1987 album by Paul Bley and Paul Motian * ''Notes'', a common (yet unofficial) sho ...
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 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 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'', ...
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 musical interval encompassing three staff positions (see Interval number for more details), and the major third () is a third spanning four semitones. Forte, Allen (1979). ''Tonal Harmony in Concept and P ...
s 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 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 p ...
. However, a seven-pitch segment of C7 will produce the diatonic major scale: 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 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 o ...
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 Hun ...
,
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 coined ...
, and the Second Viennese School (
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 ste ...
). 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 cal ...
. 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 majo ...
, 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 the ...
, "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 Re or RE may refer to: Geography * Re, Norway, a former municipality in Vestfold county, Norway * Re, Vestland, a village in Gloppen municipality, Vestland county, Norway * Re, Piedmont, an Italian municipality * Île de Ré, an island off the w ...
'P/I' dyads." P/I dyads will always share a sum of complementation.
Cyclic set In music, a cyclic set is a set, "whose alternate elements unfold complementary cycles of a single interval."Perle, George (1996). ''Twelve-Tone Tonality'', p.21. . Those cycles are ascending and descending, being related by inversion since compl ...
s 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.


See also

* Equal-interval chord * Identity (music) * Interval vector *
Octatonic scale An octatonic scale is any eight- note musical scale. However, the term most often refers to the symmetric scale composed of alternating whole and half steps, as shown at right. In classical theory (in contrast to jazz theory), this symmetrica ...


References


External links


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