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Pitch Multiplication
The mathematical operations of multiplication have several applications to music. Other than its application to the frequency ratios of intervals (for example, Just intonation, and the twelfth root of two in equal temperament), 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 multiplicative operation is a mapping in which the argument is multiplied. Multiplication originated intuitively in interval expansion, including tone row order number rotation, for example in the music of Béla Bartók and Alban Berg. Pitch number rotation, ''Fünferreihe'' or "five-series" and ''Siebenerreihe'' or "seven-series", was first described by Ernst Krenek in ''Über neue Musik''. Princeton-based theorists, including James K. Randall, Godfrey Winham, and Hubert S. Howe "were the first to discuss and adopt them, not only with regards to twel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
James K
James is a common English language surname and given name: *James (name), the typically masculine first name James * James (surname), various people with the last name James James or James City may also refer to: People * King James (other), various kings named James * Saint James (other) * James (musician) * James, brother of Jesus Places Canada * James Bay, a large body of water * James, Ontario United Kingdom * James College, a college of the University of York United States * James, Georgia, an unincorporated community * James, Iowa, an unincorporated community * James City, North Carolina * James City County, Virginia ** James City (Virginia Company) ** James City Shire * James City, Pennsylvania * St. James City, Florida Arts, entertainment, and media * ''James'' (2005 film), a Bollywood film * ''James'' (2008 film), an Irish short film * ''James'' (2022 film), an Indian Kannada-language film * James the Red Engine, a character in ''Thomas the Tank En ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Charles Wuorinen
Charles Peter Wuorinen (; June 9, 1938 – March 11, 2020) was an American composer of contemporary classical music based in New York City. He performed his works and other 20th-century music as pianist and conductor. He composed more than 270 works, including orchestral music, chamber music, solo instrumental and vocal works, and operas such as ''Brokeback Mountain''. Salman Rushdie and Annie Proulx have collaborated with him. Wuorinen's work has been called serialist, but he came to disparage that term as meaningless. His ''Time's Encomium'', his only purely electronic piece, received the Pulitzer Prize for Music. Wuorinen also taught at several institutions, including Columbia University and Manhattan School of Music. Life and career Background Wuorinen was born on the Upper West Side of Manhattan in New York City. His father, John H. Wuorinen, the chair of the history department at Columbia University, was a noted scholar of Scandinavian affairs, who also worke ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robert Morris (composer)
Robert Daniel Morris (born October 19, 1943) is an American composer and music theorist. Work in music theory As a music theorist, Morris's work has bridged an important gap between the rigorously academic and the highly experimental. Born in Cheltenham, England, in 1943, Morris received his musical education at the Eastman School of Music (B.M. in composition with distinction) and the University of Michigan (M.M. and D.M.A. in composition and ethnomusicology), where he studied composition with John La Montaine, Leslie Bassett, Ross Lee Finney, and Eugene Kurtz. At Tanglewood, as a Margret Lee Crofts Fellow, he worked with Gunther Schuller. Morris has taught composition, electronic music, and music theory at the University of Hawaii and at Yale University, where he was Chairman of the Composition Department and Director of the Yale Electronic Music Studio. He was also Director of the Computer and Electronic Studio, Director of Graduate (music) Studies, and Associate Professor of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Milton Babbitt
Milton Byron Babbitt (May 10, 1916 – January 29, 2011) was an American composer, music theorist, mathematician, and teacher. He is particularly noted for his Serialism, serial and electronic music. Biography Babbitt was born in Philadelphia to Albert E. Babbitt and Sarah Potamkin, who were Jewish. He was raised in Jackson, Mississippi, and began studying the violin when he was four but soon switched to clarinet and saxophone. Early in his life he was attracted to jazz and theater music, and "played in every pit-orchestra that came to town". Babbitt was making his own arrangements of popular songs by age 7, "wrote a lot of pop tunes for school productions", and won a local songwriting contest when he was 13. A Jackson newspaper called Babbitt a "whiz kid" and noted "that he had perfect pitch and could add up his family’s grocery bills in his head. In his teens he became a great fan of jazz cornet player Bix Beiderbecke." Babbitt's father was a mathematician, and Babbitt inten ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Herbert Eimert
Herbert Eimert (8 April 1897 – 15 December 1972) was a German music theorist, musicologist, journalist, music critic, editor, radio producer, and composer. Education Herbert Eimert was born in Bad Kreuznach. He studied music theory and composition from 1919 to 1924 at the Cologne Musikhochschule with Hermann Abendroth, , and August von Othegraven. In 1924, while still a student, he published an ''Atonale Musiklehre'' (Atonal Music Theory Text) which, together with a twelve-tone string quartet composed for the end-of-term examination concert, led to an altercation with Bölsche, who withdrew the quartet from the program and expelled Eimert from his composition class. In 1924, he began studies in musicology at the University of Cologne with Ernst Bücken, Willi Kahl, and Georg Kinsky, and read philosophy with Max Scheler (a pupil of Husserl) and Nicolai Hartmann. He attained his doctorate in 1931 with a dissertation titled ''Musikalische Formstrukturen im 17. und 18. Jahrhu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transposition (music)
In music, transposition refers to the process or operation of moving a collection of notes ( pitches or pitch classes) up or down in pitch by a constant interval. For example, one might transpose an entire piece of music into another key. Similarly, one might transpose a tone row or an unordered collection of pitches such as a chord so that it begins on another pitch. The transposition of a set ''A'' by ''n'' semitones is designated by ''T''''n''(''A''), representing the addition ( mod 12) of an integer ''n'' to each of the pitch class integers of the set ''A''. Thus the set (''A'') consisting of 0–1–2 transposed by 5 semitones is 5–6–7 (''T''5(''A'')) since , , and . Scalar transpositions In scalar transposition, every pitch in a collection is shifted up or down a fixed number of scale steps within some scale. The pitches remain in the same scale before and after the shift. This term covers both chromatic and diatonic transpositions as follows. Chromatic transpo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tritone
In music theory, the tritone is defined as a musical interval composed of three adjacent whole tones (six semitones). For instance, the interval from F up to the B above it (in short, F–B) is a tritone as it can be decomposed into the three adjacent whole tones F–G, G–A, and A–B. Narrowly defined, each of these whole tones must be a step in the scale, so by this definition, within a diatonic scale there is only one tritone for each octave. For instance, the above-mentioned interval F–B is the only tritone formed from the notes of the C major scale. More broadly, a tritone is also commonly defined as any interval with a width of three whole tones (spanning six semitones in the chromatic scale), regardless of scale degrees. According to this definition, a diatonic scale contains two tritones for each octave. For instance, the above-mentioned C major scale contains the tritones F–B (from F to the B above it, also called augmented fourth) and B–F (from B to the F abo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Circle Of Fourths
In music theory, the circle of fifths is a way of organizing the 12 chromatic pitches as a sequence of perfect fifths. (This is strictly true in the standard 12-tone equal temperament system — using a different system requires one interval of diminished sixth to be treated as a fifth). If C is chosen as a starting point, the sequence is: C, G, D, A, E, B (=C), F (=G), C (=D), A, E, B, F. Continuing the pattern from F returns the sequence to its starting point of C. This order places the most closely related key signatures adjacent to one another. It is usually illustrated in the form of a circle. Definition The circle of fifths organizes pitches in a sequence of perfect fifths, generally shown as a circle with the pitches (and their corresponding keys) in a clockwise progression. Musicians and composers often use the circle of fifths to describe the musical relationships between pitches. Its design is helpful in composing and harmonizing melodies, building chords, and modul ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chromatic
Diatonic and chromatic are terms in music theory that are most often used to characterize scales, and are also applied to musical instruments, intervals, chords, notes, musical styles, and kinds of harmony. They are very often used as a pair, especially when applied to contrasting features of the common practice music of the period 1600–1900. These terms may mean different things in different contexts. Very often, ''diatonic'' refers to musical elements derived from the modes and transpositions of the "white note scale" C–D–E–F–G–A–B. In some usages it includes all forms of heptatonic scale that are in common use in Western music (the major, and all forms of the minor). ''Chromatic'' most often refers to structures derived from the twelve-note chromatic scale, which consists of all semitones. Historically, however, it had other senses, referring in Ancient Greek music theory to a particular tuning of the tetrachord, and to a rhythmic notational convention in me ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Relatively Prime
In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivalent to their greatest common divisor (GCD) being 1. One says also '' is prime to '' or '' is coprime with ''. The numbers 8 and 9 are coprime, despite the fact that neither considered individually is a prime number, since 1 is their only common divisor. On the other hand, 6 and 9 are not coprime, because they are both divisible by 3. The numerator and denominator of a reduced fraction are coprime, by definition. Notation and testing Standard notations for relatively prime integers and are: and . In their 1989 textbook ''Concrete Mathematics'', Ronald Graham, Donald Knuth, and Oren Patashnik proposed that the notation a\perp b be used to indicate that and are relatively prime and that the term "prime" be used instead of coprime (as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bijection
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set. There are no unpaired elements. In mathematical terms, a bijective function is a one-to-one (injective) and onto (surjective) mapping of a set ''X'' to a set ''Y''. The term ''one-to-one correspondence'' must not be confused with ''one-to-one function'' (an injective function; see figures). A bijection from the set ''X'' to the set ''Y'' has an inverse function from ''Y'' to ''X''. If ''X'' and ''Y'' are finite sets, then the existence of a bijection means they have the same number of elements. For infinite sets, the picture is more complicated, leading to the concept of cardinal number—a way to distinguish the various sizes of infinite sets. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
