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Forte Number
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'' (1973, ). 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 the 12-TET tuning system (or in any other system of tuning that splits the octave into twelve semitones), each pitch class may be denoted by an integer in the range from 0 to 11 (inclusive), and a pitch class set may be denoted by a set of these integers. The prime form of a pitch class set is the most compact (i.e., leftwards packed or smallest in lexicographic order) of either the normal form of a set or of its inversion. The normal form of a set is that which is transposed so as to be most compact. For example, a second inversion major chord contains the pitch classes 7, 0, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Set Theory 3-1 In The Chromatic Circle Fix
Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electronics and computing *Set (abstract data type), a data type in computer science that is a collection of unique values ** Set (C++), a set implementation in the C++ Standard Library * Set (command), a command for setting values of environment variables in Unix and Microsoft operating-systems * Secure Electronic Transaction, a standard protocol for securing credit card transactions over insecure networks * Single-electron transistor, a device to amplify currents in nanoelectronics * Single-ended triode, a type of electronic amplifier * Set!, a programming syntax in the scheme programming language Biology and psychology * Set (psychology), a set of expectations which shapes perception or thought *Set or sett, a badger's den *Set, a small tuber ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Major Chord
In music theory, a major chord is a chord (music), chord that has a root (chord), root, a major third, and a perfect fifth. When a chord comprises only these three notes, it is called a major Triad (music), triad. For example, the major triad built on C, called a C major triad, has pitches C–E–G: In harmonic analysis and on lead sheet, lead sheets, a C major chord can be notated as C, CM, CΔ, or Cmaj. A major triad is represented by the Pitch class#Integer notation, integer notation . A major triad can also be described by its Interval (music), intervals: the interval between the bottom and middle notes is a major third, and the interval between the middle and top notes is a minor third. By contrast, a minor triad has a minor third interval on the bottom and major third interval on top. They both contain fifths, because a major third (four semitones) plus a minor third (three semitones) equals a perfect fifth (seven semitones). Chords that are constructed of consecutive ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
David Schiff
David Schiff (born August 30, 1945 in New York City) is an American composer, writer and conductor whose music draws on elements of jazz, rock, and klezmer styles, showing the influence of composers as diverse as Stravinsky, Mahler, Charles Mingus, Eric Dolphy and Terry Riley. His music has been performed by major orchestras and festivals around the United States and by soloists David Shifrin, Regina Carter, David Taylor, Marty Ehrlich, David Krakauer, Nadine Asin and Peter Kogan. He is the author of books on the music of Elliott Carter, George Gershwin and Duke Ellington. His work has been honored by the League-ISCM National Composers Competition award and the ASCAP-Deems Taylor award for his book on Elliott Carter.Carol Oja, "David Schiff," The New Grove Dictionary of Opera Biography Schiff grew up in the Bronx and New Rochelle, New York, started playing piano when he was four and composing when he was nine. He received a B.A in English literature from Columbia Universi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elliott Carter
Elliott Cook Carter Jr. (December 11, 1908 – November 5, 2012) was an American modernist composer who was one of the most respected composers of the second half of the 20th century. He combined elements of European modernism and American "ultra-modernism" into a distinctive style with a personal harmonic and rhythmic language, after an early neoclassical phase. His compositions are performed throughout the world, and include orchestral, chamber music, solo instrumental, and vocal works. Carter was the recipient of many awards – he was twice awarded the Pulitzer Prize for his string quartets. He also wrote the large-scale orchestral triptych '' Symphonia: sum fluxae pretium spei''. Carter was born in New York City. He developed an interest in modern music in the 1920s. He was later introduced to Charles Ives, and he soon came to appreciate the American ultra-modernists. After studying at Harvard University with Edward Burlingame Hill, Gustav Holst and Walter Piston, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cyclic Permutation
In mathematics, and in particular in group theory, a cyclic permutation is a permutation consisting of a single cycle. In some cases, cyclic permutations are referred to as cycles; if a cyclic permutation has ''k'' elements, it may be called a ''k''-cycle. Some authors widen this definition to include permutations with fixed points in addition to at most one non-trivial cycle. In cycle notation, cyclic permutations are denoted by the list of their elements enclosed with parentheses, in the order to which they are permuted. For example, the permutation (1 3 2 4) that sends 1 to 3, 3 to 2, 2 to 4 and 4 to 1 is a 4-cycle, and the permutation (1 3 2)(4) that sends 1 to 3, 3 to 2, 2 to 1 and 4 to 4 is considered a 3-cycle by some authors. On the other hand, the permutation (1 3)(2 4) that sends 1 to 3, 3 to 1, 2 to 4 and 4 to 2 is not a cyclic permutation because it separately permutes the pairs and . For the wider definition of a cyclic permutation, allowing fixed points, these fixe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binary Sequence
A bitstream (or bit stream), also known as binary sequence, is a sequence of bits. A bytestream is a sequence of bytes. Typically, each byte is an 8-bit quantity, and so the term octet stream is sometimes used interchangeably. An octet may be encoded as a sequence of 8 bits in multiple different ways (see bit numbering) so there is no unique and direct translation between bytestreams and bitstreams. Bitstreams and bytestreams are used extensively in telecommunications and computing. For example, synchronous bitstreams are carried by SONET, and Transmission Control Protocol transports an asynchronous bytestream. Relationship to bytestreams In practice, bitstreams are not used directly to encode bytestreams; a communication channel may use a signalling method that does not directly translate to bits (for instance, by transmitting signals of multiple frequencies) and typically also encodes other information such as framing and error correction together with its data. Exam ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equivalence Class
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a and b belong to the same equivalence class if, and only if, they are equivalent. Formally, given a set S and an equivalence relation \sim on S, the of an element a in S is denoted /math> or, equivalently, to emphasize its equivalence relation \sim, and is defined as the set of all elements in S with which a is \sim-related. The definition of equivalence relations implies that the equivalence classes form a partition of S, meaning, that every element of the set belongs to exactly one equivalence class. The set of the equivalence classes is sometimes called the quotient set or the quotient space of S by \sim, and is denoted by S /. When the set S has some structure (such as a group operation or a topology) and the equivalence re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Necklace (combinatorics)
In combinatorics, a ''k''-ary necklace of length ''n'' is an equivalence class of ''n''-character strings over an alphabet of size ''k'', taking all rotations as equivalent. It represents a structure with ''n'' circularly connected beads which have ''k'' available colors. A ''k''-ary bracelet, also referred to as a turnover (or free) necklace, is a necklace such that strings may also be equivalent under reflection. That is, given two strings, if each is the reverse of the other, they belong to the same equivalence class. For this reason, a necklace might also be called a fixed necklace to distinguish it from a turnover necklace. Formally, one may represent a necklace as an orbit of the cyclic group acting on ''n''-character strings over an alphabet of size ''k'', and a bracelet as an orbit of the dihedral group. One can count these orbits, and thus necklaces and bracelets, using Pólya's enumeration theorem. Equivalence classes Number of necklaces There are :N_k(n) ... [...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] |
Interval Vector
In musical set theory, an interval vector is an array of natural numbers which summarize the intervals present in a set of pitch classes. (That is, a set of pitches where octaves are disregarded.) Other names include: ic vector (or interval-class vector), PIC vector (or pitch-class interval vector) and APIC vector (or absolute pitch-class interval vector, which Michiel Schuijer states is more proper.) While primarily an analytic tool, interval vectors can also be useful for composers, as they quickly show the sound qualities that are created by different collections of pitch class. That is, sets with high concentrations of conventionally dissonant intervals (i.e., seconds and sevenths) sound more dissonant, while sets with higher numbers of conventionally consonant intervals (i.e., thirds and sixths) sound more consonant. While the actual perception of consonance and dissonance involves many contextual factors, such as register, an interval vector can nevertheless be a he ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diatonic Scale
In music theory a diatonic scale is a heptatonic scale, heptatonic (seven-note) scale that includes five whole steps (whole tones) and two half steps (semitones) in each octave, in which the two half steps are separated from each other by either two or three whole steps. In other words, the half steps are maximally separated from each other. The seven pitch (music), pitches of any diatonic scale can also be obtained by using a Interval cycle, chain of six perfect fifths. For instance, the seven natural (music), natural pitch classes that form the C-major scale can be obtained from a stack of perfect fifths starting from F: :F–C–G–D–A–E–B. Any sequence of seven successive natural notes, such as C–D–E–F–G–A–B, and any Transposition (music), transposition thereof, is a diatonic scale. Modern musical keyboards are designed so that the white-key notes form a diatonic scale, though transpositions of this diatonic scale require one or more black keys. A diaton ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Viennese Trichord
In music theory, a Viennese trichord (also known as Viennese fourth chord and tritone-fourth chordDeLone, Richard, et al (1975). ''Aspects of 20th Century Music'', p. 348. Englewood Cliffs, New Jersey: Prentice-Hall .), named for the Second Viennese School, is a pitch set with prime form (0,1,6). Its Forte number is 3-5. The sets C–D–G and C–F–G are both examples of Viennese trichords, though they may be voiced in many ways. According to Henry Martin, " mposers such as Webern ... are partial to 016 trichords, given their 'more dissonant' inclusion of ics 1 and 6."Martin, Henry (Winter, 2000). "Seven Steps to Heaven: A Species Approach to Twentieth-Century Analysis and Composition", p. 149, ''Perspectives of New Music'', vol. 38, no. 1, pp. 129–168. In jazz and popular music, the chord formed by the inversion of the set usually has a dominant function, being the third, seventh, and added fourth/eleventh of a dominant chord with elided root Forte, Allen (2000). "Har ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
