Interval cycle

In <a href="/facts/Music/hDVP0xsQ">music</a>, an interval cycle is a <a href="/facts/Set_(music)/NMoE2nSG">collection</a> of <a href="/facts/Pitch_class/40mqdJtN">pitch classes</a> created from a sequence of the same <a href="/facts/Interval_class/SU3AKfND">interval class</a>. In other words, a collection of <a href="/facts/Pitch_(music)/fQAK0JsC">pitches</a> by starting with a certain <a href="/facts/Musical_note/spIQyV4T">note</a> and going up by a certain <a href="/facts/Interval_(music)/aFwkkgpY">interval</a> 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 <a href="/facts/George_Perle/0zsEtdMY">George Perle</a> using the letter "C" (for cycle), with an <a href="/facts/Interval_class/SU3AKfND">interval class</a> integer to distinguish the interval. Thus the <a href="/facts/Diminished_seventh_chord/BsyETryS">diminished seventh chord</a> would be C3 and the <a href="/facts/Augmented_triad/2R7MpG0C">augmented triad</a> would be C4. A superscript may be added to distinguish between <a href="/facts/Transposition_(music)/qrGNnycc">transpositions</a>, using 0–11 to indicate the lowest pitch class in the cycle. "These interval cycles play a fundamental role in the <a href="/facts/Harmonic/9U9REzeR">harmonic</a> organization of <a href="/facts/Post-diatonic_music/LToI29Xu">post-diatonic music</a> and can easily be identified by naming the cycle."
Here are interval cycles C1, C2, C3, C4 and C6:

Interval cycles assume the use of <a href="/facts/Equal_temperament/8AwSblW3">equal temperament</a> and may not work in other systems such as <a href="/facts/Just_intonation/TM91PueU">just intonation</a>. For example, if the C4 interval cycle used justly-tuned <a href="/facts/Major_third/7CPRURg4">major thirds</a> it would fall flat of an octave return by an interval known as the <a href="/facts/Diesis/r9bKNAk9">diesis</a>. 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 <a href="/facts/Symmetrical/hpu7DK8p">symmetrical</a> and thus non-<a href="/facts/Diatonic/gyD81SHE">diatonic</a>. However, a seven-pitch segment of C7 will produce the <a href="/facts/Diatonic_major_scale/ejXSu379">diatonic major scale</a>:
This is known also known as a <a href="/facts/Generated_collection/9d9ySvH3">generated collection</a>.
A minimum of three pitches are needed to represent an interval cycle.
Cyclic tonal <a href="/facts/Chord_progression/zrB3N8Fc">progressions</a> in the works of Romantic and late Romantic composers (e.g., <a href="/facts/Richard_Wagner/XoFH4v69">Richard Wagner</a>, <a href="/facts/Johannes_Brahms/wr1Qo5Jh">Johannes Brahms</a>, <a href="/facts/Gustav_Mahler/yYTMSJb6">Gustav Mahler</a>) form a link with the cyclic pitch successions in the atonal music of Modernists such as <a href="/facts/B%25C3%25A9la_Bart%25C3%25B3k/C1DQu47U">Béla Bartók</a>, <a href="/facts/Alexander_Scriabin/dLD7t9nq">Alexander Scriabin</a>, <a href="/facts/Edgard_Var%25C3%25A8se/8RbYRSIL">Edgard Varèse</a>, and the <a href="/facts/Second_Viennese_School/mvSBXD92">Second Viennese School</a> (<a href="/facts/Arnold_Schoenberg/o9xuyDwC">Arnold Schoenberg</a>, <a href="/facts/Alban_Berg/7UWWkE06">Alban Berg</a>, and <a href="/facts/Anton_Webern/p4IvxKRt">Anton Webern</a>). At the same time, these <a href="/facts/Simultaneity_succession/2jRLBG7W">progressions</a> signal the end of <a href="/facts/Tonality/vBJp7QcT">tonality</a>.
Interval cycles are also important in <a href="/facts/Jazz/lmUzXGq8">jazz</a>, such as in <a href="/facts/Coltrane_changes/5EPuHaMt">Coltrane changes</a>.
"Similarly," to any pair of transpositionally related sets being reducible to two transpositionally related representations of the <a href="/facts/Chromatic_scale/ntfCjHK4">chromatic scale</a>, "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 [called] a 'P cycle' and the descending form [called] an 'I cycle'," while, "inversionally related dyads [are called] 'P/I' dyads." P/I dyads will always share a <a href="/facts/Sum_of_complementation/R5NpcqlD">sum of complementation</a>. <a href="/facts/Cyclic_set/Glwh1Y7w">Cyclic sets</a> are those "<a href="/facts/Set_(music)/NMoE2nSG">sets</a> whose alternate elements unfold <a href="/facts/Complement_(music)/R5NpcqlD">complementary</a> cycles of a single <a href="/facts/Interval_(music)/aFwkkgpY">interval</a>," that is an ascending and descending cycle:

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

 Berg's Master <a href="/facts/Array_(music)/qNVeLqzk">Array</a> 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:

Interval cycle open-in-new

Interval cycle