Atonality in its broadest sense is music that lacks a tonal center, or key. Atonality, in this sense, usually describes compositions written from about the early 20th-century to the present day, where a hierarchy of harmonies focusing on a single, central triad is not used, and the notes of the chromatic scale function independently of one another. More narrowly, the term atonality describes music that does not conform to the system of tonal hierarchies that characterized European classical music between the seventeenth and nineteenth centuries. "The repertory of atonal music is characterized by the occurrence of pitches in novel combinations, as well as by the occurrence of familiar pitch combinations in unfamiliar environments".
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, a hexachord is a six-note series, as exhibited in a scale or tone row. The term was adopted in this sense during the Middle Ages and adapted in the 20th century in Milton Babbitt's serial theory. The word is taken from the Greek: ἑξάχορδος, compounded from ἕξ and χορδή, and was also the term used in music theory up to the 18th century for the interval of a sixth.
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.
In music, transposition refers to the process or operation of moving a collection of notes up or down in pitch by a constant interval.
The shifting of a melody, a harmonic progression or an entire musical piece to another key, while maintaining the same tone structure, i.e. the same succession of whole tones and semitones and remaining melodic intervals.
The mathematical operations of multiplication have several applications to music. Other than its application to the frequency ratios of intervals, it has been used in other ways for twelve-tone technique, and musical set theory. Additionally ring modulation is an electrical audio process involving multiplication that has been used for musical effect.
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
An all-interval tetrachord is a tetrachord, a collection of four pitch classes, containing all six interval classes. There are only two possible all-interval tetrachords, when expressed in prime form. In set theory notation, these are [0,1,4,6] (4-Z15) and [0,1,3,7] (4-Z29). Their inversions are [0,2,5,6] (4-Z15b) and [0,4,6,7] (4-Z29b). The interval vector for all all-interval tetrachords is [1,1,1,1,1,1].
In music, a Klumpenhouwer Network is "any network that uses T and/or I operations to interpret interrelations among pcs". According to George Perle, "a Klumpenhouwer network is a chord analyzed in terms of its dyadic sums and differences," and "this kind of analysis of triadic combinations was implicit in," his "concept of the cyclic set from the beginning", cyclic sets being those "sets whose alternate elements unfold complementary cycles of a single interval." It is named for the Canadian music theorist Henry Klumpenhouwer, a former doctoral student of David Lewin's.
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 "Ode-to-Napoleon" hexachord is the hexachord named after its use in the twelve-tone piece Ode to Napoleon Buonaparte Op. 41 (1942) by Arnold Schoenberg. Containing the pitch-classes 014589 it is given Forte number 6–20 in Allen Forte's taxonomic system. The primary form of the tone row used in the Ode allows the triads of G minor, E♭ minor, and B minor to easily appear.
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.
6-Z44 (012569), known as the Schoenberg hexachord, is Arnold Schoenberg's signature hexachord, as one transposition contains the pitches [A], Es, C, H, B, E, G, E♭, B, and B♭ being Es, H, and B in German.
John Rahn, born on February 26, 1944, in New York City, is a music theorist, composer, bassoonist, and Professor of Music at the University of Washington School of Music, Seattle. A former student of Milton Babbitt and Benjamin Boretz, he was editor of Perspectives of New Music from 1983 to 1993 and since 2001 has been co-editor with Benjamin Boretz and Robert Morris.
![Set 3-1 has three possible versions: [01112T], [011TE1], and [0TT1E1], where the subscripts indicate adjacency intervals. The normal form is the smallest "slice of pie" (shaded) or most compact form, in this case: [01112T]. Set theory 3-1 in the chromatic circle fix.svg](http://upload.wikimedia.org/wikipedia/commons/thumb/e/ee/Set_theory_3-1_in_the_chromatic_circle_fix.svg/330px-Set_theory_3-1_in_the_chromatic_circle_fix.svg.png)
