Interval cycle
Encyclopedia
In music
, an interval cycle is a collection
of pitch class
es created from a sequence of the same interval class
. 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".
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 transpositions
, using 0–11 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.".
Here are interval cycles C1, C2, C3, C4 and C6:
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 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. However, a seven-pitch segment of C7 will produce the diatonic major scale:
This is known also known as a generated collection
.
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
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 progressions
signal the end of tonality
.
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." 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 sum of complementation. Cyclic set
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 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
Music
Music is an art form whose medium is sound and silence. Its common elements are pitch , rhythm , dynamics, and the sonic qualities of timbre and texture...
, an interval cycle is a collection
Set (music)
A set in music theory, as in mathematics and general parlance, is a collection of objects...
of pitch class
Pitch class
In music, a pitch class is a set of all pitches that are a whole number of octaves apart, e.g., the pitch class C consists of the Cs in all octaves...
es created from a sequence of the same interval class
Interval class
In musical set theory, an interval class , also known as unordered pitch-class interval, interval distance, undirected interval, or interval mod...
. In other words a collection of pitches
Pitch (music)
Pitch is an auditory perceptual property that allows the ordering of sounds on a frequency-related scale.Pitches are compared as "higher" and "lower" in the sense associated with musical melodies,...
by starting with a certain note
Note
In music, the term note has two primary meanings:#A sign used in musical notation to represent the relative duration and pitch of a sound;#A pitched sound itself....
and going up by a certain interval
Interval (music)
In music theory, an interval is a combination of two notes, or the ratio between their frequencies. Two-note combinations are also called dyads...
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".
Interval cycles are notated by George Perle
George Perle
George Perle was a composer and music theorist. He was born in Bayonne, New Jersey. Perle was an alumnus of DePaul University...
using the letter "C" (for cycle), with an interval class
Interval class
In musical set theory, an interval class , also known as unordered pitch-class interval, interval distance, undirected interval, or interval mod...
integer to distinguish the interval. Thus the diminished seventh chord
Diminished seventh chord
A diminished seventh chord is a four note chord that comprises a diminished triad plus the interval of a diminished seventh above the root. Thus it is , or enharmonically , of any major scale; for example, C diminished-seventh would be , or enharmonically...
would be C3 and the augmented triad
Augmented triad
In music, an augmented triad is a triad, or chord, consisting of two major thirds . The term augmented triad arises from an augmented triad being a three note chord, or triad, whose top note is raised, or augmented...
would be C4. A superscript may be added to distinguish between transpositions
Transposition (music)
In music transposition refers to the process, or operation, of moving a collection of notes up or down in pitch by a constant interval.For example, one might transpose an entire piece of music into another key...
, using 0–11 to indicate the lowest pitch class in the cycle. "These interval cycles play a fundamental role in the harmonic
Harmonic
A harmonic of a wave is a component frequency of the signal that is an integer multiple of the fundamental frequency, i.e. if the fundamental frequency is f, the harmonics have frequencies 2f, 3f, 4f, . . . etc. The harmonics have the property that they are all periodic at the fundamental...
organization of post-diatonic music 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 equal temperament
Equal temperament
An equal temperament is a musical temperament, or a system of tuning, in which every pair of adjacent notes has an identical frequency ratio. As pitch is perceived roughly as the logarithm of frequency, this means that the perceived "distance" from every note to its nearest neighbor is the same for...
and may not work in other systems such as just intonation
Just intonation
In music, just intonation is any musical tuning in which the frequencies of notes are related by ratios of small whole numbers. Any interval tuned in this way is called a just interval. The two notes in any just interval are members of the same harmonic series...
. For example, if the C4 interval cycle used justly-tuned major third
Major third
In classical music from Western culture, a third is a musical interval encompassing three staff positions , and the major third is one of two commonly occurring thirds. It is qualified as major because it is the largest of the two: the major third spans four semitones, the minor third three...
s it would fall flat of an octave return by an interval known as the diesis
Diesis
In classical music from Western culture, a diesis is either an accidental , or a comma type of musical interval, usually defined as the difference between an octave and three justly tuned major thirds , equal to 128:125 or about 41.06 cents...
. 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. However, a seven-pitch segment of C7 will produce the diatonic major scale:
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 collection or scale is formed...
.
A minimum of three pitches are needed to represent an interval cycle.
Cyclic tonal progressions
Chord progression
A chord progression is a series of musical chords, or chord changes that "aims for a definite goal" of establishing a tonality founded on a key, root or tonic chord. In other words, the succession of root relationships...
in the works of Romantic composers such as Gustav Mahler
Gustav Mahler
Gustav Mahler was a late-Romantic Austrian composer and one of the leading conductors of his generation. He was born in the village of Kalischt, Bohemia, in what was then Austria-Hungary, now Kaliště in the Czech Republic...
and Richard Wagner
Richard Wagner
Wilhelm Richard Wagner was a German composer, conductor, theatre director, philosopher, music theorist, poet, essayist and writer primarily known for his operas...
form a link with the cyclic pitch successions in the atonal music of Modernists such as Béla Bartók
Béla Bartók
Béla Viktor János Bartók was a Hungarian composer and pianist. He is considered one of the most important composers of the 20th century and is regarded, along with Liszt, as Hungary's greatest composer...
, Alexander Scriabin
Alexander Scriabin
Alexander Nikolayevich Scriabin was a Russian composer and pianist who initially developed a lyrical and idiosyncratic tonal language inspired by the music of Frédéric Chopin. Quite independent of the innovations of Arnold Schoenberg, Scriabin developed an increasingly atonal musical system,...
, Edgard Varèse
Edgard Varèse
Edgard Victor Achille Charles Varèse, , whose name was also spelled Edgar Varèse , was an innovative French-born composer who spent the greater part of his career in the United States....
, and the Second Viennese School
Second Viennese School
The Second Viennese School is the group of composers that comprised Arnold Schoenberg and his pupils and close associates in early 20th century Vienna, where he lived and taught, sporadically, between 1903 and 1925...
(Arnold Schoenberg
Arnold Schoenberg
Arnold Schoenberg was an Austrian composer, associated with the expressionist movement in German poetry and art, and leader of the Second Viennese School...
, Alban Berg
Alban Berg
Alban Maria Johannes Berg was an Austrian composer. He was a member of the Second Viennese School with Arnold Schoenberg and Anton Webern, and produced compositions that combined Mahlerian Romanticism with a personal adaptation of Schoenberg's twelve-tone technique.-Early life:Berg was born in...
, and Anton Webern
Anton Webern
Anton Webern was an Austrian composer and conductor. He was a member of the Second Viennese School. As a student and significant follower of Arnold Schoenberg, he became one of the best-known exponents of the twelve-tone technique; in addition, his innovations regarding schematic organization of...
). At the same time, these progressions
Simultaneity succession
In music and music theory a simultaneity succession is a series of different groups of pitches or pitch classes, each of which is played at the same time as the other pitches of its group...
signal the end of tonality
Tonality
Tonality is a system of music in which specific hierarchical pitch relationships are based on a key "center", or tonic. The term tonalité originated with Alexandre-Étienne Choron and was borrowed by François-Joseph Fétis in 1840...
.
Interval cycles are also important in jazz
Jazz
Jazz is a musical style that originated at the beginning of the 20th century in African American communities in the Southern United States. It was born out of a mix of African and European music traditions. From its early development until the present, jazz has incorporated music from 19th and 20th...
, such as in Coltrane changes
Coltrane changes
In jazz harmony, the Coltrane changes are a harmonic progression variation using substitute chords over common jazz chord progressions...
.
"Similarly," to any pair of transpositionally related sets being reducible to two transpositionally related representations of the chromatic scale
Chromatic scale
The chromatic scale is a musical scale with twelve pitches, each a semitone apart. On a modern piano or other equal-tempered instrument, all the half steps are the same size...
, "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 sum of complementation. Cyclic set
Cyclic set
In music, a cyclic set is a set, "whose alternate elements unfold complementary cycles of a single interval." Those cycles are ascending and descending, being related by inversion since complementary:...
s are those, "sets
Set (music)
A set in music theory, as in mathematics and general parlance, is a collection of objects...
whose alternate elements unfold complementary cycles of a single interval
Interval (music)
In music theory, an interval is a combination of two notes, or the ratio between their frequencies. Two-note combinations are also called dyads...
," that is an ascending and descending cycle:
In 1920 Berg discovered/created a "master array" of all twelve interval cycles:
Berg's Master 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
External links
- The "Giant Steps" Progression and Cycle Diagrams by Dan Adler