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Permutation (music)
In music, a permutation (order) of a set (music), set is any ordering of the elements of that set. A specific arrangement of a set of discrete entities, or parameter (music), parameters, such as pitch (music), pitch, Dynamics (music), dynamics, or timbre. Different permutations may be related by Transformation (music), transformation, through the application of zero or more ''operations'', such as transposition (music), transposition, Melodic inversion, inversion, Retrograde (music), retrogradation, circular permutation (also called ''rotation''), or multiplication (music), multiplicative operations (such as the cycle of fourths and Circle of fifths, cycle of fifths transforms). These may produce reorderings of the members of the set, or may simply map the set onto itself. Order is particularly important in the theories of composition techniques originating in the 20th century such as the twelve-tone technique and serialism. Analytical techniques such as set theory (music), set th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Set Theory (music)
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. One branch of musical set theory deals with collections ( sets and permutations) of pitches and pitch classes (pitch-class set theory), which may be ordered or unordered, and can be related by musical operations such as transposition, melodic inversion, and complementation. Some theorists apply the methods of musical set theory to the analysis of rhythm as well. Comparison with mathematical set theory Although musical set theory is often thought to involve the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Secondary Set
In music, a tone row or note row ( or '), also series or set, is a non-repetitive ordering of a set of pitch-classes, typically of the twelve notes in set theory (music), musical set theory of the chromatic scale, though both larger and smaller sets are sometimes found. History and usage Tone rows are the basis of Arnold Schoenberg's twelve-tone technique and most types of serialism, serial music. Tone rows were widely used in 20th-century contemporary music, like Dmitri Shostakovich's use of twelve-tone rows, "without dodecaphonic transformations." A tone row has been identified in the A minor prelude, BWV 889, from book II of Johann Sebastian Bach, J.S. Bach's ''The Well-Tempered Clavier'' (1742) and by the late eighteenth century it is found in works such as Wolfgang Amadeus Mozart, Mozart's Milanese Quartets (Mozart)#Quartet No. 4 in C major, K. 157, C major String Quartet, K. 157 (1772), String Quartet No. 16 (Mozart), String Quartet in E-flat major, K. 428, String Quint ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sergei Rachmaninoff
Sergei Vasilyevich Rachmaninoff; in Russian pre-revolutionary script. (28 March 1943) was a Russian composer, virtuoso pianist, and Conducting, conductor. Rachmaninoff is widely considered one of the finest pianists of his day and, as a composer, one of the last great representatives of Romantic music, Romanticism in Russian classical music. Early influences of Pyotr Ilyich Tchaikovsky, Tchaikovsky, Nikolai Rimsky-Korsakov, Rimsky-Korsakov, and other Russian composers gave way to a thoroughly personal idiom notable for its song-like melody, melodicism, Music#Expression, expressiveness, dense Counterpoint, contrapuntal textures, and rich Orchestration, orchestral colours. The piano is featured prominently in Rachmaninoff's compositional output and he used his skills as a performer to fully explore the expressive and technical possibilities of the instrument. Born into a musical family, Rachmaninoff began learning the piano at the age of four. He studied piano and composition at ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Permutation
In mathematics, a permutation of a set can mean one of two different things: * an arrangement of its members in a sequence or linear order, or * the act or process of changing the linear order of an ordered set. An example of the first meaning is the six permutations (orderings) of the set : written as tuples, they are (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), and (3, 2, 1). Anagrams of a word whose letters are all different are also permutations: the letters are already ordered in the original word, and the anagram reorders them. The study of permutations of finite sets is an important topic in combinatorics and group theory. Permutations are used in almost every branch of mathematics and in many other fields of science. In computer science, they are used for analyzing sorting algorithms; in quantum physics, for describing states of particles; and in biology, for describing RNA sequences. The number of permutations of distinct objects is factorial, us ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and as an end to obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics and from evolutionary biology to computer science. Combinatorics is well known for the breadth of the problems it tackles. Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry, as well as in its many application areas. Many combinatorial questions have historically been considered in isolation, giving an ''ad hoc'' solution to a problem arising in some mathematical context. In the later twentieth century, however, powerful and general theoretical methods were developed, making combinatorics into an independent branch of mathematics in its own right. One of the oldest and most accessible parts of combinatorics ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pitch Class
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 : = . Although there is no formal upper or lower limit to this sequence, only a few of these pitches are audible to humans. Pitch class is important because human pitch-perception is periodic: pitches belonging to the same pitch class are perceived as having a similar quality or color, a property called " octave equivalence". Psychologists refer to the quality of a pitch as its "chroma". A ''chroma'' is an attribute of pitches (as opposed to ''tone height''), just like hue is an attribute of color. A ''pitch class'' is a set of all p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ordered Set
In mathematics, especially order theory, a partial order on a set is an arrangement such that, for certain pairs of elements, one precedes the other. The word ''partial'' is used to indicate that not every pair of elements needs to be comparable; that is, there may be pairs for which neither element precedes the other. Partial orders thus generalize total orders, in which every pair is comparable. Formally, a partial order is a homogeneous binary relation that is reflexive, antisymmetric, and transitive. A partially ordered set (poset for short) is an ordered pair P=(X,\leq) consisting of a set X (called the ''ground set'' of P) and a partial order \leq on X. When the meaning is clear from context and there is no ambiguity about the partial order, the set X itself is sometimes called a poset. Partial order relations The term ''partial order'' usually refers to the reflexive partial order relations, referred to in this article as ''non-strict'' partial orders. However some a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Concerto (Webern)
Anton Webern's Concerto for Nine Instruments, Op. 24 (German: Konzert für neun Instrumente) is a twelve-tone chamber piece composed in 1934. Its tone row is one of the most notable in history. The piece is admired for its extreme concision and is considered a hallmark in the development of total serialism. Composition By the late 1920s, Webern had developed an extraordinary application of Arnold Schoenberg's twelve-tone technique in works like ''String Trio'' (1927), '' Symphony'' (1928), and ''Quartet'' (1932).Puffett, Kathryn Bailey. "Webern, Anton". ''Grove Music Online''. 2001. Webern began sketching an orchestral work on January 16, 1931. In early February, Webern began attempting to create a melodic equivalent of a Sator Square. Webern had long been enamored of the square. In addition to writing "tenet" in his first sketch for the ''Concerto'', he ended his lectures about new music by quoting it to his audience.Webern, Anton. The Path to the New Music'. Edited by Willi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Anton Webern
Anton Webern (; 3 December 1883 – 15 September 1945) was an Austrian composer, conductor, and musicologist. His music was among the most radical of its milieu in its lyric poetry, lyrical, poetic concision and use of then novel atonality, atonal and twelve-tone technique, twelve-tone techniques. His approach was typically rigorous, inspired by his studies of the Franco-Flemish School under Guido Adler and by Arnold Schoenberg's emphasis on structure in teaching composition from the music of Johann Sebastian Bach, the First Viennese School, and Johannes Brahms. Webern, Schoenberg, and their colleague Alban Berg were at the core of what became known as the Second Viennese School. Webern was arguably the first and certainly the last of the three to write music in an Aphorism, aphoristic and Expressionist music, expressionist style, reflecting his instincts and the idiosyncrasy of his compositional process. He treated themes of love, loss, nature, and spirituality, working from his ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tone Row
In music, a tone row or note row ( or '), 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. History and usage Tone rows are the basis of Arnold Schoenberg's twelve-tone technique and most types of serial music. Tone rows were widely used in 20th-century contemporary music, like Dmitri Shostakovich's use of twelve-tone rows, "without dodecaphonic transformations." A tone row has been identified in the A minor prelude, BWV 889, from book II of J.S. Bach's '' The Well-Tempered Clavier'' (1742) and by the late eighteenth century it is found in works such as Mozart's C major String Quartet, K. 157 (1772), String Quartet in E-flat major, K. 428, String Quintet in G minor, K. 516 (1790), and the Symphony in G minor, K. 550 (1788). Beethoven also used the technique but, on the whole, "Mozart seems to have employed seria ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
