Play (help·info)).An all-interval tetrachord is a tetrachord, a collection of four pitch classes, containing all six interval classes. [1] There are only two possible all-interval tetrachords (to within inversion), when expressed in prime form. In set theory notation, these are [0,1,4,6] (4-Z15) [2] and [0,1,3,7] (4-Z29). [3] Their inversions are [0,2,5,6] (4-Z15b) and [0,4,6,7] (4-Z29b). [4] The interval vector for all all-interval tetrachords is [1,1,1,1,1,1].
Play (help·info)).
In the examples below, the tetrachords [0,1,4,6] and [0,1,3,7] are built on E.
| ic | notes of [0,1,4,6] built on E | diatonic counterparts |
|---|---|---|
| 1 | E to F | minor 2nd and major 7th |
| 2 | A♭ to B♭ | major 2nd and minor 7th |
| 3 | F to A♭ | minor 3rd and major 6th |
| 4 | E to G♯ | major 3rd and minor 6th |
| 5 | F to B♭ | perfect 4th and perfect 5th |
| 6 | E to B♭ | augmented 4th and diminished 5th |
| ic | notes of [0,1,3,7] built on E | diatonic counterparts |
|---|---|---|
| 1 | E to F | minor 2nd and major 7th |
| 2 | F to G | major 2nd and minor 7th |
| 3 | E to G | minor 3rd and major 6th |
| 4 | G to B | major 3rd and minor 6th |
| 5 | E to B | perfect 4th and perfect 5th |
| 6 | F to B | augmented 4th and diminished 5th |
The unique qualities of the all-interval tetrachord have made it very popular in 20th-century music. Composers including Elliott Carter (First String Quartet) and George Perle used it extensively.[ citation needed ]
In music, a tone row or note row, also series or set, is a non-repetitive ordering of a set of pitch-classes, typically of the twelve notes in musical set theory of the chromatic scale, though both larger and smaller sets are sometimes found.
In music theory, a tetrachord is a series of four notes separated by three intervals. In traditional music theory, a tetrachord always spanned the interval of a perfect fourth, a 4:3 frequency proportion —but in modern use it means any four-note segment of a scale or tone row, not necessarily related to a particular tuning system.
The twelve-tone technique—also known as dodecaphony, twelve-tone serialism, and twelve-note composition—is a method of musical composition first devised by Austrian composer Josef Matthias Hauer, who published his "law of the twelve tones" in 1919. In 1923, Arnold Schoenberg (1874–1951) developed his own, better-known version of 12-tone technique, which became associated with the "Second Viennese School" composers, who were the primary users of the technique in the first decades of its existence. The technique is a means of ensuring that all 12 notes of the chromatic scale are sounded as often as one another in a piece of music while preventing the emphasis of any one note through the use of tone rows, orderings of the 12 pitch classes. All 12 notes are thus given more or less equal importance, and the music avoids being in a key. Over time, the technique increased greatly in popularity and eventually became widely influential on 20th-century composers. Many important composers who had originally not subscribed to or actively opposed the technique, such as Aaron Copland and Igor Stravinsky, eventually adopted it in their music.
In music, a pitch class (p.c. or pc) is a set of all pitches that are a whole number of octaves apart; for example, the pitch class C consists of the Cs in all octaves. "The pitch class C stands for all possible Cs, in whatever octave position." Important to musical set theory, a pitch class is "all pitches related to each other by octave, enharmonic equivalence, or both." Thus, using scientific pitch notation, the pitch class "C" is the set
Musical set theory provides concepts for categorizing musical objects and describing their relationships. Howard Hanson first elaborated many of the concepts for analyzing tonal music. Other theorists, such as Allen Forte, further developed the theory for analyzing atonal music, drawing on the twelve-tone theory of Milton Babbitt. The concepts of musical set theory are very general and can be applied to tonal and atonal styles in any equal temperament tuning system, and to some extent more generally than that.
In musical set theory, an interval class, also known as unordered pitch-class interval, interval distance, undirected interval, or "(even completely incorrectly) as 'interval mod 6'", is the shortest distance in pitch class space between two unordered pitch classes. For example, the interval class between pitch classes 4 and 9 is 5 because 9 − 4 = 5 is less than 4 − 9 = −5 ≡ 7 (mod 12). See modular arithmetic for more on modulo 12. The largest interval class is 6 since any greater interval n may be reduced to 12 − n.
In music theory, a trichord is a group of three different pitch classes found within a larger group. A trichord is a contiguous three-note set from a musical scale or a twelve-tone row.
In music using the twelve-tone technique, derivation is the construction of a row through segments. A derived row is a tone row whose entirety of twelve tones is constructed from a segment or portion of the whole, the generator. Anton Webern often used derived rows in his pieces. A partition is a segment created from a set through partitioning.
In music using the twelve tone technique, combinatoriality is a quality shared by twelve-tone tone rows whereby each section of a row and a proportionate number of its transformations combine to form aggregates. Much as the pitches of an aggregate created by a tone row do not need to occur simultaneously, the pitches of a combinatorially created aggregate need not occur simultaneously. Arnold Schoenberg, creator of the twelve-tone technique, often combined P-0/I-5 to create "two aggregates, between the first hexachords of each, and the second hexachords of each, respectively."
In music theory, complement refers to either traditional interval complementation, or the aggregate complementation of twelve-tone and serialism.
In music, a permutation (order) of a set is any ordering of the elements of that set. A specific arrangement of a set of discrete entities, or parameters, such as pitch, dynamics, or timbre. Different permutations may be related by transformation, through the application of zero or more operations, such as transposition, inversion, retrogradation, circular permutation, or multiplicative operations. These may produce reorderings of the members of the set, or may simply map the set onto itself.
A set in music theory, as in mathematics and general parlance, is a collection of objects. In musical contexts the term is traditionally applied most often to collections of pitches or pitch-classes, but theorists have extended its use to other types of musical entities, so that one may speak of sets of durations or timbres, for example.
In musical set theory, an interval vector is an array of natural numbers which summarize the intervals present in a set of pitch classes. Other names include: ic vector, PIC vector and APIC vector
Composition for Four Instruments (1948) is an early serial music composition written by American composer Milton Babbitt. It is Babbitt's first published ensemble work, following shortly after his Three Compositions for Piano (1947). In both these pieces, Babbitt expands upon the methods of twelve-tone composition developed by Arnold Schoenberg. He is notably innovative for his application of serial techniques to rhythm. Composition for Four Instruments is considered one of the early examples of “totally serialized” music. It is remarkable for a strong sense of integration and concentration on its particular premises—qualities that caused Elliott Carter, upon first hearing it in 1951, to persuade New Music Edition to publish it.
Fritz Heinrich Klein was an Austrian composer.
In musical set theory, a Forte number is the pair of numbers Allen Forte assigned to the prime form of each pitch class set of three or more members in The Structure of Atonal Music. The first number indicates the number of pitch classes in the pitch class set and the second number indicates the set's sequence in Forte's ordering of all pitch class sets containing that number of pitches.
In music, the all-trichord hexachord is a unique hexachord that contains all twelve trichords, or from which all twelve possible trichords may be derived. The prime form of this set class is {012478} and its Forte number is 6-Z17. Its complement is 6-Z43 and they share the interval vector of <3,2,2,3,3,2>.
In music, an all-interval twelve-tone row, series, or chord, is a twelve-tone tone row arranged so that it contains one instance of each interval within the octave, 1 through 11. A "twelve-note spatial set made up of the eleven intervals [between consecutive pitches]." There are 1,928 distinct all-interval twelve-tone rows. These sets may be ordered in time or in register. "Distinct" in this context means in transpositionally and rotationally normal form, and disregarding inversionally related forms. These 1,928 tone rows have been independently rediscovered several times, their first computation probably was by Andre Riotte in 1961, see.
The Tone Clock, and its related compositional theory Tone-Clock Theory, is a post-tonal music composition technique, developed by composers Peter Schat and Jenny McLeod. Music written using tone-clock theory features a high economy of musical intervals within a generally chromatic musical language. This is because tone-clock theory encourages the composer to generate all their harmonic and melodic material from a limited number of intervallic configurations. Tone-clock theory is also concerned with the way that the three-note pitch-class sets can be shown to underlie larger sets, and considers these triads as a fundamental unit in the harmonic world of any piece. Because there are twelve possible triadic prime forms, Schat called them the 'hours', and imagined them arrayed in a clock face, with the smallest hour in the 1 o'clock position, and the largest hour in the 12 o'clock position. A notable feature of Tone-Clock Theory is 'tone-clock steering': transposing and/or inverting hours so that each note of the chromatic aggregate is generated once and once only.
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