1.
Tetrachord
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In music theory, traditionally, a tetrachord is a series of four notes separated by three smaller intervals that span the interval of a perfect fourth, a 4,3 frequency proportion. In modern usage a tetrachord is any four-note segment of a scale or tone row, the term tetrachord derives from ancient Greek music theory, where it signified a segment of the Greater and Lesser Perfect Systems bounded by unmovable notes, the notes between these were movable. It literally means four strings, originally in reference to instruments such as the lyre or the kithara. Modern music theory makes use of the octave as the unit for determining tuning. Ancient Greek theorists recognized that the octave is a fundamental interval, ancient Greek music theory distinguishes three genera of tetrachords. This characteristic interval is usually smaller, becoming a whole tone. Classically, the diatonic tetrachord consists of two intervals of a tone and one of a semitone, e. g. A–G–F–E. Chromatic A chromatic tetrachord has an interval that is greater than about half the total interval of the tetrachord. Classically, the interval is a minor third, and the two smaller intervals are equal semitones, e. g. A–G♭–F–E. Enharmonic An enharmonic tetrachord has an interval that is greater than about four-fifths the total tetrachord interval. Classically, the interval is a ditone or a major third. The hypate and mese, and the paramese and nete are unmovable, fixed a perfect fourth apart, while the position of the parhypate and lichanos, as the three genera simply represent ranges of possible intervals within the tetrachord, various shades with specific tunings were specified. Once the genus and shade of tetrachord are specified, their arrangement can produce three types of scales, depending on which note of the tetrachord is taken as the first note of the scale. The tetrachords themselves remain independent of the scales that they produce, larger scales are constructed from conjunct or disjunct tetrachords. Conjunct tetrachords share a note, while disjunct tetrachords are separated by a tone of 9/8. Alternating conjunct and disjunct tetrachords form a scale that repeats in octaves, the Greeks analyzed genera using various terms, including diatonic, enharmonic, and chromatic. Scales are constructed from conjunct or disjunct tetrachords and this is a partial table of the superparticular divisions by Chalmers after Hofmann. Tetrachords based upon equal temperament tuning were used to explain common heptatonic scales, the 19th-century theorists of ancient Greek music believed that this had also been the case in Antiquity, and imagined that there had existed Dorian, Phrygian or Lydian tetrachords
2.
Pitch class
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In music, a pitch class is a set of all pitches that are a whole number of octaves apart, e. g. 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. Psychologists refer to the quality of a pitch as its chroma, a chroma is an attribute of pitches, just like hue is an attribute of color. A pitch class is a set of all pitches that share the same chroma, because octave-related pitches belong to the same class, when an octave is reached, the numbers begin again at zero. Indeed, the mapping from pitch to real numbers defined in this forms the basis of the MIDI Tuning Standard. To represent pitch classes, we need to identify or glue together all pitches belonging to the same pitch class—i. e, all numbers p and p +12. The result is a quotient group that musicians call pitch class space. Points in this space can be labelled using real numbers in the range 0 ≤ x <12. These numbers provide numerical alternatives to the names of elementary music theory,0 = C,1 = C♯/D♭,2 = D,2.5 = D,3 = D♯/E♭. In this system, pitch classes represented by integers are classes of equal temperament. In music, integer notation is the translation of pitch classes and/or interval classes into whole numbers, thus if C =0, then C♯ =1. A♯ =10, B =11, with 10 and 11 substituted by t and e in some sources and this allows the most economical presentation of information regarding post-tonal materials. In the integer model of pitch, all classes and intervals between pitch classes are designated using the numbers 0 through 11. It is not used to music for performance, but is a common analytical and compositional tool when working with chromatic music, including twelve tone, serial. The C above this is not 12, but 0 again, thus arithmetic modulo 12 is used to represent octave equivalence. One advantage of this system is that it ignores the spelling of notes according to their diatonic functionality, there are a few disadvantages with integer notation. First, theorists have used the same integers to indicate elements of different tuning systems. Thus, the numbers 0,1,2,5, are used to notate pitch classes in 6-tone equal temperament
3.
Interval class
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For example, the interval class between pitch classes 4 and 9 is 5 because 9 −4 =5 is less than 4 −9 = −5 ≡7. 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. The concept of interval class accounts for octave, enharmonic, consider, for instance, the following passage, (To hear a MIDI realization, click the following,106 KB In the example above, all four labeled pitch-pairs, or dyads, share a common intervallic color. In atonal theory, this similarity is denoted by interval class—ic 5, tonal theory, however, classifies the four intervals differently, interval 1 as perfect fifth,2, perfect twelfth,3, diminished sixth, and 4, perfect fourth. The unordered pitch class interval i may be defined as i = the smaller of i ⟨ a, b ⟩ and i ⟨ b, a ⟩, where i <a, b> is an ordered pitch-class interval. While notating unordered intervals with parentheses, as in the example directly above, is perhaps the standard, some theorists, including Robert Morris, prefer to use braces, pitch interval Similarity relation Morris, Robert. Class Notes for Atonal Music Theory, ISBN 0-300-04536-0 ISBN 0-300-04537-9 Solomons Set Theory Primer
4.
Set theory (music)
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Musical set theory provides concepts for categorizing musical objects and describing their relationships. The concepts of set theory are general and can be applied to tonal and atonal styles in any equally tempered tuning system. The methods of set theory are sometimes applied to the analysis of rhythm as well. Although musical set theory is thought to involve the application of mathematical set theory to music. For example, musicians use the terms transposition and inversion where mathematicians would use translation and reflection, furthermore, where musical set theory refers to ordered sets, mathematics would normally refer to tuples or sequences. In combinatorics, a subset of n objects, such as pitch classes, is called a combination. Musical set theory is best regarded as a field that is not so much related to mathematical set theory, the main connection to mathematical set theory is the use of the vocabulary of set theory to talk about finite sets. The fundamental concept of set theory is the set, which is an unordered collection of pitch classes. More exactly, a set is a numerical representation consisting of distinct integers. The elements of a set may be manifested in music as simultaneous chords, successive tones, notational conventions vary from author to author, but sets are typically enclosed in curly braces, or square brackets. Some theorists use angle brackets ⟨ ⟩ to denote ordered sequences, thus one might notate the unordered set of pitch classes 0,1, and 2 as. The ordered sequence C-C♯-D would be notated ⟨0,1,2 ⟩ or, although C is considered zero in this example, this is not always the case. For example, a piece with a pitch center of F might be most usefully analyzed with F set to zero (in which case would represent F, F♯. Though set theorists usually consider sets of equal-tempered pitch classes, it is possible to consider sets of pitches, non-equal-tempered pitch classes, rhythmic onsets, two-element sets are called dyads, three-element sets trichords. Sets of higher cardinalities are called tetrachords, pentachords, hexachords, heptachords, octachords, nonachords, decachords, undecachords, and, finally, the basic operations that may be performed on a set are transposition and inversion. Sets related by transposition or inversion are said to be related or inversionally related. Since transposition and inversion are isometries of space, they preserve the intervallic structure of a set. This can be considered the central postulate of musical set theory, in practice, set-theoretic musical analysis often consists in the identification of non-obvious transpositional or inversional relationships between sets found in a piece
5.
Interval vector
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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, sets with high concentrations of conventionally dissonant intervals sound more dissonant, while the actual perception of consonance and dissonance involves many contextual factors, such as register, an interval vector can nevertheless be a helpful tool. In twelve-tone equal temperament, a vector has six digits. Because interval classes are used, the vector for a given set remains the same. The interval classes designated by each digit ascend from left to right, the modern notation, introduced by Allen Forte, has considerable advantages and is extendable to any equal division of the octave. A scale whose interval vector has six digits is said to have the deep scale property. Major, natural minor and modal scales have this property, for a practical example, the interval vector for a C major triad in the root position, is <001110>. This means that the set has one third or minor sixth, one minor third or major sixth. As the interval vector does not change with transposition or inversion, it belongs to the entire set class, some interval vectors correspond to more than one sets that cannot be transposed or inverted to produce the other. For a set of n pitch classes, the sum of all the numbers in the interval vector equals the triangular number Tn−1 = n×/2. An expanded form of the vector is also used in transformation theory, as set out in David Lewins Generalized Musical Intervals. For example, the two sets 4-z15A and 4-z29A have the interval vector but one can not transpose and/or invert the one set onto the other. In the case of each may be referred to as a Z-hexachord. Any hexachord not of the Z type is its own complement while the complement of a Z-hexachord is its Z-correspondent, for example 6-Z3, see, 6-Z44, 6-Z17, 6-Z11, and Forte number. The term, for zygotic, originated with Allen Forte in 1964, Hanson called this the isomeric relationship, and defined two such sets as isomeric. Mathematical proofs of the hexachord theorem where published by Kassler, Regener, though it is commonly observed that Z-related sets always occur in pairs, David Lewin noted that this is a result of twelve-tone equal temperament. In 16-ET, Z-related sets are found as triplets, Lewins student Jonathan Wild continued this work for other tuning systems, finding Z-related tuplets with up to 16 members in higher ET systems. Straus argues, in the Z-relation will sound similar because they have the interval content
6.
Elliott Carter
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Elliott Cook Carter Jr. was an American composer who was twice awarded the Pulitzer Prize. He studied with Nadia Boulanger in Paris in the 1930s, then returned to the United States, after an early neoclassical phase, his style shifted to an emphasis on atonality and rhythmic complexity. His compositions are known and performed throughout the world, they include orchestral, chamber music, solo instrumental and he was productive in his later years, publishing more than 40 works between the ages of 90 and 100, and over 20 more after he turned 100 in 2008. He completed his last work, Epigrams for piano trio, on August 13,2012, Elliott Cook Carter Jr. was born in Manhattan on December 11,1908, the son of a wealthy lace importer. Carters father was Elliott Carter Sr. and his mother was the former Florence Chambers, as a teenager, he developed an interest in music, and received encouragement in this regard from Charles Ives. While he was a student at the Horace Mann School, he wrote an letter to Ives. In 1924, a 15-year-old Carter was in the audience when Pierre Monteux conducted the Boston Symphony Orchestra in the New York première of The Rite of Spring, according to a 2008 report. When Carter attended Harvard, starting in 1926, Ives took him under his wing and made sure he went to the BSO concerts conducted by Serge Koussevitzky, although Carter majored in English at Harvard College, he also studied music there and at the nearby Longy School of Music. His professors at Harvard included Walter Piston and Gustav Holst and he sang with the Harvard Glee Club and did graduate work in music at Harvard, from which he received a masters degree in music in 1932. He then went to Paris to study with Nadia Boulanger at the École Normale de Musique de Paris, Carter worked with Boulanger from 1932 to 1935, and in that year received a doctorate in music. Later that same year, he returned to the US and wrote music for the Ballet Caravan, on July 6,1939, Carter married Helen Frost-Jones. They had one child, a son, David Chambers Carter, from 1940 to 1944, he taught in the program, including music, at St. Johns College in Annapolis, Maryland. During World War II, he worked for the Office of War Information and he later held teaching posts at the Peabody Conservatory, Columbia University, Queens College, New York, Yale University, Cornell University and the Juilliard School. In 1967, he was appointed a member of the American Academy of Arts, in 1981, he was awarded the Ernst von Siemens Music Prize, in 1985 the National Medal of Arts. In 1983 Carter was awarded the Edward MacDowell Medal for outstanding contribution to the arts by the MacDowell Colony and he lived with his wife in the same apartment in Greenwich Village from the time they bought it in 1945 to her death in 2003. Carter composed his only opera What Next. in 1997-1998 at the behest of conductor Daniel Barenboim for the Berlin State Opera. The work premiered in Berlin in 1999 and had its first staging in the United States at the Tanglewood Music Festival in 2006 under the baton of James Levine. He later considered writing operas on the themes of suicide and a story by Henry James
7.
George Perle
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George Perle was a composer and music theorist. Perle was born in Bayonne, New Jersey and he graduated from DePaul University, where he studied with Wesley LaViolette and received private lessons from Ernst Krenek. Later, he served as a fifth grade in the United States Army during World War II. He earned his doctorate at New York University in 1956, Perle composed with a technique of his own devising called twelve-tone tonality. The system similarly creates a hierarchy among intervals and finally, among larger collections of notes, the main debt of this system to the 12-tone system lies in its use of an ordered linear succession in the same way that a 12-tone set does. In 1968, Perle cofounded the Alban Berg Society with Igor Stravinsky and Hans F. Redlich, after retiring from Queens College in 1985, he became a professor emeritus at the Aaron Copland School of Music. In 1986, Perle was awarded a Pulitzer Prize for Music for his Fourth Wind Quintet, in about 1989 Perle became composer-in-residence for the San Francisco Symphony, a three-year appointment. It was also around this time that he had published his book entitled The Listening Composer. He died aged 93 in his home in New York City in January 2009 and he was subsequently buried in Calverton National Cemetery. On his headstone are inscribed the words An die Musik, a growing number of younger artists have come to appreciate Perle as a composer ahead of his time. In the run-up to his 100th birthday celebrations the composer-pianist Michael Brown released a well received CD of a sampling of Perles work for piano, Perle was married to the sculptor Laura Slobe from 1940 to 1952, the couple were members of the Socialist Workers Party. His second wife, Barbara Philips, died in 1978, Perle was survived at his death by his third wife, the former Shirley Gabis Rhoads, two daughters, and a stepdaughter. Swift differentiates between Perles free or intuitive, tone-centered, and twelve-tone modal music, serial Composition and Atonality, An Introduction to the Music of Schoenberg, Berg, and Webern. Symmetry, the Twelve-Tone Scale, and Tonality, Contemporary Music Review 6, pp. 81–96
8.
All-interval twelve-tone row
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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, 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. Since the sum of numbers 1 through 11 equals 66, a row must contain a tritone between its first and last notes, as well as in their middle. The Pyramid chord consists of every interval stacked, low to high, from 12 to 1 and while it contains all intervals, it does not contain all pitch classes and is thus not a tone row. Klein chose the name Mutterakkord in order to avoid a term such as all-interval twelve-tone row. The Mother chord row was used by Alban Berg in his Lyric Suite. The Grandmother chord is an eleven-interval, twelve-note, invertible chord with all of the properties of the Mother chord and it was invented by Nicolas Slonimsky on February 13,1938. Found by John F. Link, they have used by Elliott Carter in pieces such as Symphonia. On Eleven-Interval Twelve-Tone Rows, Perspectives of New Music 3/2, 93–103, a Re-examination of All-Interval Rows, Proceedings of the American Society of University Composers 7/8, 73–74. Archived from the original on March 8,2012
9.
All-trichord hexachord
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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 set class is and its Forte number is 6-Z17. Its complement is 6-Z43 and they share the interval vector of <3,2,2,3,3 and it appears in pieces by Robert Morris and Elliott Carter. Carter uses all-interval twelve-tone sets consisting of all-trichord hexachords in his Symphonia, all-combinatorial hexachord All-interval tetrachord All-interval twelve-tone row Boland, Marguerite. The All-trichord Hexachord, Compositional Strategies in Elliott Carters Con leggerezza pensosa and Gra, linking and Morphing, Harmonic Flow in Elliott Carters Con Leggerezza Pensosa. Some Properties of the All-Trichord Hexachord, In Theory Only 11/6, the Complement Union Property in the Music of Elliott Carter. Journal of Music Theory 48, no, registral Constraints on All-Interval Rows in Elliott Carters Changes. Morris, Robert D. Pitch-Class Complementation and Its Generalizations, journal of Music Theory 34, no. Perspectives of New Music 33, nos.1 &2, setting the Pace, The Role of Speeds in Elliott Carters A Mirror on Which to Dwell. A Transformational Space for Elliott Carters Recent Complement-union Music, in Mathematics and Computation in Music, edited by Timour Klouche and Thomas Noll, 303–310. Communications in Computer and Information Science 37, listening to the Music Itself, Breaking Through the Shell of Elliott Carters In Genesis, Music Theory Online 13/3
10.
Serialism
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In music, serialism is a method or technique of composition that uses a series of values to manipulate different musical elements. Serialism began primarily with Arnold Schoenbergs twelve-tone technique, though some of his contemporaries were working to establish serialism as a form of post-tonal thinking. The idea of serialism is also applied in ways in the visual arts, design, and architecture. Integral serialism or total serialism is the use of series for such as duration, dynamics. Other terms, used especially in Europe to distinguish post–World War II serial music from music and its American extensions, are general serialism. Serialism is a method, highly specialized technique, or way of composition and it may also be considered, a philosophy of life, a way of relating the human mind to the world and creating a completeness when dealing with a subject. However, serialism is not by itself a system of composition, neither is pitch serialism necessarily incompatible with tonality, though it is most often used as a means of composing atonal music. It is sometimes used specifically to apply only to music where at least one element other than pitch is subjected to being treated as a row or series. In such usages post-Webernian serialism will be used to denote works that extend serial techniques to other elements of music, other terms used to make the distinction are twelve-note serialism for the former, and integral serialism for the latter. A row may be assembled pre-compositionally, or it may be derived from a spontaneously invented thematic or motivic idea, the structure of the row, however, does not in itself define the structure of a composition, which requires development of a comprehensive strategy. The basic set may have restrictions, such as the requirement that it use each interval only once. The series is not an order of succession, but indeed a hierarchy—which may be independent of order of succession. Stockhausen, for example, in early serial compositions such as Kreuzspiel and Formel and this provides an exemplary demonstration of that logical principle of seriality, every situation must occur once and only once. Prohibited intervals, like the octave, and prohibited successional relations, such as premature note repetitions, frequently occur, the number twelve no longer plays any governing, defining rôle, the pitch constellations no longer hold to the limitation determined by their formation. The dodecaphonic series loses its significance as a model of shape is played out. And the chromatic total remains active only, and provisionally, as a general reference, in the 1960s Pousseur took this a step further, applying a consistent set of predefined transformations to pre-existent music. In his opera Votre Faust Pousseur used a number of different quotations, themselves arranged into a scale for serial treatment, so as to bring coherence. He extended this serial polyphony of styles in a series of works in the late 1960s, as well as later in portions of Licht
11.
Trichord
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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 scale or a twelve-tone row. In late-19th to early 20th-century Russian musicology, the term trichord meant something more specific, several of these pitch sets interlocking could form a larger set such as a pentatonic scale. The term is derived by analogy from the 20th-century use of the word tetrachord, triad All-trichord hexachord Viennese trichord Babbitt, Milton. The Score and I. M. A. Magazine, no, set Structure as a Compositional Determinant. Journal of Music Theory 5, no, in The Collected Essays of Milton Babbitt, edited by Stephen Peles, Stephen Dembski, Andrew Mead, and Joseph Straus, 55–69. New Haven and London, Yale University Press, friedmann, Michael L. Ear Training for Twentieth-Century Music. Some Combinational Resources of Equal-Tempered Systems, journal of Music Theory 11, no. Harmonic Materials of Modern Music, Resources of the Tempered Scale, kodály Today, A Cognitive Approach to Elementary Music Education. Oxford and New York, Oxford University Press, Особенности народно-русской музыкальной системы, edited by T. V. Popova. The New Grove Dictionary of Music and Musicians, second edition, edited by Stanley Sadie, the Trichord, An Analytic Outlook for Twentieth-Century Music
12.
International Standard Book Number
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The International Standard Book Number is a unique numeric commercial book identifier. An ISBN is assigned to each edition and variation of a book, for example, an e-book, a paperback and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, the method of assigning an ISBN is nation-based and varies from country to country, often depending on how large the publishing industry is within a country. The initial ISBN configuration of recognition was generated in 1967 based upon the 9-digit Standard Book Numbering created in 1966, the 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108. Occasionally, a book may appear without a printed ISBN if it is printed privately or the author does not follow the usual ISBN procedure, however, this can be rectified later. Another identifier, the International Standard Serial Number, identifies periodical publications such as magazines, the ISBN configuration of recognition was generated in 1967 in the United Kingdom by David Whitaker and in 1968 in the US by Emery Koltay. The 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108, the United Kingdom continued to use the 9-digit SBN code until 1974. The ISO on-line facility only refers back to 1978, an SBN may be converted to an ISBN by prefixing the digit 0. For example, the edition of Mr. J. G. Reeder Returns, published by Hodder in 1965, has SBN340013818 -340 indicating the publisher,01381 their serial number. This can be converted to ISBN 0-340-01381-8, the check digit does not need to be re-calculated, since 1 January 2007, ISBNs have contained 13 digits, a format that is compatible with Bookland European Article Number EAN-13s. An ISBN is assigned to each edition and variation of a book, for example, an ebook, a paperback, and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, a 13-digit ISBN can be separated into its parts, and when this is done it is customary to separate the parts with hyphens or spaces. Separating the parts of a 10-digit ISBN is also done with either hyphens or spaces, figuring out how to correctly separate a given ISBN number is complicated, because most of the parts do not use a fixed number of digits. ISBN issuance is country-specific, in that ISBNs are issued by the ISBN registration agency that is responsible for country or territory regardless of the publication language. Some ISBN registration agencies are based in national libraries or within ministries of culture, in other cases, the ISBN registration service is provided by organisations such as bibliographic data providers that are not government funded. In Canada, ISBNs are issued at no cost with the purpose of encouraging Canadian culture. In the United Kingdom, United States, and some countries, where the service is provided by non-government-funded organisations. Australia, ISBNs are issued by the library services agency Thorpe-Bowker
13.
Set (music)
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A set in music theory, as in mathematics and general parlance, is a collection of objects. A set by itself not necessarily possess any additional structure. Nevertheless, it is often important to consider sets that are equipped with an order relation, in such contexts, bare sets are often referred to as unordered. Two-element sets are called dyads, three-element sets trichords, sets of higher cardinalities are called tetrachords, pentachords, hexachords, heptachords, octachords, nonachords, decachords, undecachords, and, finally, the dodecachord. A time-point set is a set where the distance in time units between attack points, or time-points, is the distance in semitones between pitch classes. In the theory of music, however, some authors use the term set where others would use row or series. These authors speak of twelve sets, time-point sets, derived sets. This is a different usage of the set from that described above. For these authors, a set form is an arrangement of such an ordered set, the prime form, inverse, retrograde. A derived set is one which is generated or derived from consistent operations on a subset, for example Weberns Concerto and these invariances in serial music are analogous to the use of common-tones and common-chords in tonal music. The fundamental concept of a set is that it is an unordered collection of pitch classes. The normal form of a set is the most compact ordering of the pitches in a set, tomlin defines the most compact ordering as the one where, the largest of the intervals between any two consecutive pitches is between the first and last pitch listed. For example, the set is in normal form while the set is not, rather than the original form of the set the prime form may be considered either the normal form of the set or the normal form of its inversion, whichever is more tightly packed. Forte and Rahn both list the prime forms of a set as the most left-packed possible version of the set, Forte packs from the left and Rahn packs from the right. However, these differ in five instances and are the result of different algorithms. Forte number Permutation Pitch interval Similarity relation Schuijer, Michiel, analyzing Atonal Music, Pitch-Class Set Theory and Its Contexts. Calculates normal form, prime form, Forte number, and interval class vector for a given set and vice versa
14.
Cardinality (music)
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A set in music theory, as in mathematics and general parlance, is a collection of objects. A set by itself not necessarily possess any additional structure. Nevertheless, it is often important to consider sets that are equipped with an order relation, in such contexts, bare sets are often referred to as unordered. Two-element sets are called dyads, three-element sets trichords, sets of higher cardinalities are called tetrachords, pentachords, hexachords, heptachords, octachords, nonachords, decachords, undecachords, and, finally, the dodecachord. A time-point set is a set where the distance in time units between attack points, or time-points, is the distance in semitones between pitch classes. In the theory of music, however, some authors use the term set where others would use row or series. These authors speak of twelve sets, time-point sets, derived sets. This is a different usage of the set from that described above. For these authors, a set form is an arrangement of such an ordered set, the prime form, inverse, retrograde. A derived set is one which is generated or derived from consistent operations on a subset, for example Weberns Concerto and these invariances in serial music are analogous to the use of common-tones and common-chords in tonal music. The fundamental concept of a set is that it is an unordered collection of pitch classes. The normal form of a set is the most compact ordering of the pitches in a set, tomlin defines the most compact ordering as the one where, the largest of the intervals between any two consecutive pitches is between the first and last pitch listed. For example, the set is in normal form while the set is not, rather than the original form of the set the prime form may be considered either the normal form of the set or the normal form of its inversion, whichever is more tightly packed. Forte and Rahn both list the prime forms of a set as the most left-packed possible version of the set, Forte packs from the left and Rahn packs from the right. However, these differ in five instances and are the result of different algorithms. Forte number Permutation Pitch interval Similarity relation Schuijer, Michiel, analyzing Atonal Music, Pitch-Class Set Theory and Its Contexts. Calculates normal form, prime form, Forte number, and interval class vector for a given set and vice versa
15.
Dyad (music)
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In music, a dyad is a set of two notes or pitches that, in particular contexts, may imply a chord. To understand dyads it is necessary first to understand the intervals between the notes, take the notes C and E for example, we know that the interval between these two pitches is a major third, which can imply a C major chord. The most common chord is the interval of a perfect fifth. The reason fifths are so common is because of the harmonic series, the harmonic series is built over a fundamental pitch, and the rest of the partials in the series are called overtones. The second partial is an octave above the fundamental and the pitch is a fifth, so if C is the fundamental pitch the second note is C an octave higher. Since an interval is the distance between two pitches, a dyad can be classified by the interval it represents, when the pitches of a dyad occur in succession, they form a melodic interval. When they occur simultaneously, they form a harmonic interval, double stop Power chord Harmonic series Counterpoint Polyphony
16.
Viennese trichord
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In music theory, a Viennese trichord, named for the Second Viennese School, is prime form <0,1, 6>. As opposed to Hindemith and 037, Composers such as Webern. are partial to 016 trichords, in jazz and popular music, the chord usually has a dominant function, being the third, seventh, and added sixth/thirteenth of a dominant chord with elided root. All About Set Theory, Java Set Theory Machine
17.
Pentachord
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A pentachord in music theory may be either of two things. In pitch-class set theory, a pentachord is defined as any five pitch classes, in other contexts, a pentachord may be any consecutive five-note section of a diatonic scale. A pentad is a five-note chord, the name pentachord was also given to a musical instrument, now in disuse, built to the specifications of Sir Edward Walpole. It was demonstrated by Karl Friedrich Abel at his first public concert in London, on 5 April 1759, in the dedication to Walpole of his cello sonatas op. 3, the cellist/composer James Cervetto praised the pentachord, declaring, performances on the instrument are documented as late as 1783, after which it seems to have fallen out of use. It appears to have similar to a five-string violoncello. The Twelve-note Music of Anton Webern, Old Forms in a New Language, Music in the Twentieth Century 2. Cambridge and New York, Cambridge University Press, reprinted Cambridge, Cambridge University Press,2004. Digital paperback reprint, Cambridge and New York, Cambridge University Press,2006, knape, Walter, Murray R. Charters, and Simon McVeigh. The New Grove Dictionary of Music and Musicians, second edition, edited by Stanley Sadie, Oxford and New York, Oxford University Press. By the Will and Order of Providence, The Wesley Family Concerts, royal Musical Association Research Chronicle, no. The New Grove Dictionary of Music and Musicians, second edition, edited by Stanley Sadie, New York and London, Longman Inc
18.
Hexachord
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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, 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. The hexachord as a device was first described by Guido of Arezzo. In each hexachord, all adjacent pitches are a tone apart. These six pitches are named ut, re, mi, fa, sol and these six names are derived from the first syllable of each half-verse of the first stanza of the 8th-century Vesper hymn Ut queant laxis resonare fibris / Mira gestorum famuli tuorum, etc. Melodies with a wider than a major sixth required the device of mutation to a new hexachord. For example, the beginning on C and rising to A. A melody moving a semitone higher than la required changing the la to mi, because B♭ was named by the soft or rounded letter B, the hexachord with this note in it was called the hexachordum molle. Similarly, the hexachord with mi and fa expressed by the notes B♮ and C was called the hexachordum durum, because the B♮ was represented by a squared-off, or hard B. Starting in the 14th century, these three hexachords were extended in order to accommodate the use of signed accidentals on other notes. The new notes, being outside the gamut of those available, had to be imagined, or feigned. David Lewin used the term in this sense as early as 1959, carlton Gamer uses both terms interchangeably. Hexatonic scale Musica ficta Guidonian hand Combinatoriality Hexachordal complementation 6-20, 6-34, 6-Z43, New York and London, Longman Inc. The New Grove Dictionary of Music and Musicians, second edition, edited by Stanley Sadie and John Tyrrell
19.
Chromatic hexachord
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In music theory, the chromatic hexachord is the hexachord consisting of a consecutive six-note segment of the chromatic scale. It is the first hexachord as ordered by Forte number, for example, zero through five and six through eleven. On C, C, C♯, D, D♯, E, F and F♯, G, G♯, A, A♯, in the larger context of thirty-five source hexachords catalogued by Donald Martino, it is designated Type A. Applying the circle of fifths transformation to the chromatic hexachord produces the diatonic hexachord, as with the diatonic scale, the chromatic hexachord is, hierarchical in interval makeup, and may also be produced by, or contains, 3-1, 3-2, 3-3, 3-6, and 3-7. Babbitts Second Quartet and Reflections for piano and tape feature the hexachord,2, Diario polacco ’58 in 1959, is built from two chromatic hexachords. Chromatic fourth Chromatic tetrachord Lament bass Tone cluster Babbitt, Milton, words About Music, The Madison Lectures, edited by Stephen Dembski and Joseph Straus. New Haven and London, Yale University Press, new Haven and London, Yale University Press. The Source Set and Its Aggregate Formations, Journal of Music Theory 5, no. Addendum in Journal of Music Theory 6, no, the Song Unsung, Luigi Nonos Il canto sospeso, Journal of the Royal Musical Association 131, no. The Music of Elliott Carter, second edition
20.
Diatonic hexachord
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It is the thirty-second hexachord as ordered by Forte number, and its complement is the diatonic hexachord at the tritone. If the circle of fifths transformation is applied to the diatonic hexachord the chromatic hexachord results, hugo Riemann points out that the hexachord consists of three overlapping tetrachords, Lydian, Phrygian, and Dorian, as well as two overlapping pentatonic scales. More generally diatonic hexachord may refer to any subset of the diatonic septad, 6-Z25, 6-Z26, 6-32
21.
Mystic chord
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Scriabin, however, did not use the chord directly but rather derived material from its transpositions. It consists of the classes, C, F♯, B♭. This is often interpreted as a quartal hexachord consisting of a fourth, diminished fourth, augmented fourth. However, the chord may be spelled in a variety of ways, the term mystic chord, appears to derive from Scriabins intense interest in Theosophy, and the chord is imagined to reflect this mysticism. It was coined by Arthur Eaglefield Hull in 1916 and it is also known as the Prometheus chord, after its extensive use in his work Prometheus, The Poem of Fire, Op.60. The term was invented by Leonid Sabaneyev, Scriabin himself called it the chord of the pleroma, which was designed to afford instant apprehension of -that is, to reveal- what was in essence beyond the mind of man to conceptualize. Its preternatural stillness was a gnostic intimation of a hidden otherness, jim Samson points out that it fits in well with Scriabins predominantly dominant quality sonorities and harmony as it may take on a dominant quality on C or F♯. The pitch collection is related to the scale, the whole tone scale. For example, the chord is a tone scale with one note raised a semitone. The notes of the chord also conform to a Lydian dominant quality, more often than not, the notes are reordered in order to supply a variety of harmonic or melodic material. Certain of Scriabins late pieces are based on other synthetic chords or scales that do not rely on the mystic chord, there seems today to be a general consensus that the mystic chord is neither the key nor the generating element in Scriabins method. Other sources suggest that Scriabins method of organization is based on ordered scales that feature scale degrees. For example, a group of miniatures are governed by the acoustic and/or the octatonic scales. A rare example of purely quartal spacing can be found in the Fifth Piano Sonata, incomplete versions of the chord spaced entirely in fourths are considerably more common, for example, in Deux Morceaux, Op.57. According to George Perle, Scriabin used this chord in what he calls a pre-serial manner, producing harmonies, chords, with the increasing use of more dissonant sonorities, some composers of the 20th and 21st centuries have used this chord in various ways. Elektra chord Petrushka chord Psalms chord Tristan chord Hewitt, Michael, some occurrences of the Mystic chord in the scores of the Petrucci Music Library
22.
"Ode-to-Napoleon" hexachord
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In music, the Ode-to-Napoleon hexachord is the hexachord named after its use in the twelve-tone piece Ode to Napoleon Buonaparte by Arnold Schoenberg. Containing the pitch-classes 014589 it is given Forte number 6-20 in Allen Fortes 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. The Ode-to-Napoleon hexachord is the six-member set-class with the highest number of interval classes 3 and 4 yet lacks 2s and its only five-note subset is 5-21, the complement of which is 7-21, the only superset of 6-20. The only more redundant hexachord is 6-35 and it is also Ernő Lendvais 1,3 Model scale and one of Milton Babbitts six all-combinatorial hexachord source sets. The hexachord has been used by composers including Bruno Maderna and Luigi Nono,41 di Arnold Schönberg, Weberns Concerto Op.24, Schoenbergs Suite Op.29, Babbitts Composition for Twelve Instruments and Composition for Four Instruments third and fourth movements. The hexachord has also used by Alexander Scriabin and Béla Bartók but is not featured in the music of Igor Stravinsky. The Music of Alexander Scriabin, p.214, cited in Van den Toorn, p. 128-29. Re, Intervallic Relations Between Two Collections of Notes, journal of Music Theory 3, no. Wason, Robert W. Tonality and Atonality in Frederic Rzewskis Variations on The People United Will Never Be Defeated, perspectives of New Music 26, no.1. Cited in Van den Toorn, p. 128-29
23.
Petrushka chord
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The Petrushka chord is a recurring polytonic device used in Igor Stravinskys ballet Petrushka and in later music. These two major triads, C major and F♯ major – a tritone apart – clash, horribly with each other, the Petrushka chord is defined as two simultaneous major triads separated by a tritone. In Petrushka Stravinsky used C major on top of F♯ major, Listen to this segment The device uses tones that, together, when enharmonically spelled C D♭ E G♭ G B♭, it is called the tritone scale. Alternatively, when spelled C D♭ E F♯ G B♭ it can be read as the diminished scale. It may also be categorized as a lydian dominant♭9 omit 13 scale. possess for Stravinsky an a priori conceptual status, although attributed to Stravinsky, the chord was present much earlier in Franz Liszts Malédiction Concerto. Maurice Ravel uses this chord in his piano work Jeux deau to create flourishing, Stravinsky heard Jeux deau and several other works by Ravel no later than 1907 at the Evenings for Contemporary Music program. Stravinsky used the chord repeatedly throughout the ballet Petrushka to represent the puppet, jazz musicians utilize this chord as an upper structure to colorize a dominant chord. The Petrushka chord is used in the track Above the Clouds. Polychord Elektra chord Mystic chord Psalms chord Tristan chord
24.
Sacher hexachord
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The Sacher hexachord is a hexachord notable for its use in twelve compositions created at the invitation of Mstislav Rostropovich for Sachers seventieth birthday in 1976. The twelve compositions include Pierre Boulezs Messagesquisse, Hans Werner Henzes Capriccio, Witold Lutosławskis Sacher Variation, Messagesquisse is dedicated to Sacher, but Boulezs Répons, Dérive 1, Incises, and Sur Incises all use rows with the same pitches. The hexachords complement is its Z-relation, 6-Z40
25.
Schoenberg hexachord
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6-Z44, known as the Schoenberg hexachord, is Arnold Schoenbergs signature hexachord, as one transposition contains the pitches, Es, C, H, B, E, G, E♭, B, and B♭ being Es, H, and B in German. Its Z-related hexachord and complement is 6-Z19 and they have the interval vector of <3,1,3,4,3, 1> in common. 6-Z44 lacks prime and inversional combinatoriality, 6-Z44 contains set 3-3 twice and set 3-4 twice. Set 7-22 contains 6-Z44 twice and 6-Z19 twice, Schoenberg used the hexachord in the song Seraphita and the monodrama Die glückliche Hand. 6-Z44 is associated with the character Hauptmann in Alban Bergs Wozzeck, each movement of Bergs 1913 Four Pieces for Clarinet and Piano begins with a statement of 6-Z44 or 6-Z19. 6-Z44 is one of the harmonies in the last movement, of Igor Stravinskys The Rite of Spring
26.
Tone row
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Tone rows are the basis of Arnold Schoenbergs twelve-tone technique and most types of serial music. Beethoven also used the technique but, on the whole, Mozart seems to have employed serial technique far more often than Beethoven, there are also examples in the works of Liszt. Tone rows are designated by letters and subscript numbers, the numbers indicate the initial or final pitch-class number of the given row form, most often with c=0. P indicates prime, a forward-directed right-side up form, I indicates inversion, a forward-directed upside-down form. R indicates retrograde, a backwards right-side up form, RI indicates retrograde-inversion, a backwards upside-down form. Transposition is indicated by a T number, for example P8 is a T transposition of P4, a twelve-tone composition will take one or more tone rows, called the prime form, as its basis plus their transformations. It is then varied again through inversion, untransposed, taking form I-0 and this whole tone scale reappears in the second movement when the chorale It is enough from Bachs cantata no. 60, which opens with consecutive whole tones, is quoted literally in the woodwinds, some tone rows have a high degree of internal organisation. A row created in this manner, through variants of a trichord or tetrachord called the generator, is called a derived row, the tone rows of many of Weberns other late works are similarly intricate. The set-complex is the forms of the set generated by stating each aspect or transformation on each pitch class. The all-interval twelve-tone row is a tone row arranged so that it contains one instance of each interval within the octave,0 through 11, the total chromatic is the set of all twelve pitch classes. An array is a succession of aggregates, an aggregate may be achieved through complementation or combinatoriality, such as with hexachords. A secondary set is a row which is derived from or, results from the reversed coupling of hexachords. First used in Babbitts String Quartet No.4, an aggregate may be vertically or horizontally weighted. An all-partition array is created by combining a collection of hexachordally combinatorial arrays, schoenberg specified many strict rules and desirable guidelines for the construction of tone rows such as number of notes and intervals to avoid. Each permutation contains a just chromatic scale, however, transformations produce pitches outside of the primary row form, the pitches of each hexachord are drawn from different otonality or utonality on A+ utonality, C otonality and utonality, and E♭- otonality, outlining a diminished triad. A literary parallel of the row is found in Georges Perecs poems which use each of a particular set of letters only once. Tone row may also be used to other musical collections or scales such as in Arabic music
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Complement (music)
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In music theory, complement refers to either traditional interval complementation, or the aggregate complementation of twelve-tone and serialism. In interval complementation a complement is the interval which, when added to the original interval, for example, a major 3rd is the complement of a minor 6th. The complement of any interval is known as its inverse or inversion. Note that the octave and the unison are each others complements, in the aggregate complementation of twelve-tone music and serialism the complement of one set of notes from the chromatic scale contains all the other notes of the scale. For example, A-B-C-D-E-F-G is complemented by B♭-C♯-E♭-F♯-A♭, note that musical set theory broadens the definition of both senses somewhat. The rule of nine is a way to work out which intervals complement each other. Taking the names of the intervals as cardinal numbers, we have for example 4+5=9, hence the fourth and the fifth complement each other. Where we are using more generic names this rule cannot be applied, however, octave and unison are not generic but specifically refer to notes with the same name, hence 8+1=9. Using integer notation and modulo 12, any two intervals which add up to 0 are complements, in this case the unison,0, is its own complement, while for other intervals the complements are the same as above. Thus the #Sum of complementation is 12, the complement of a pitch-class set consists, in the literal sense, of all the notes remaining in the twelve-note chromatic that are not in that set. In the twelve-tone technique this is often the separation of the total chromatic of twelve pitch classes into two hexachords of six classes each. Hexachordal complementation is the use of the potential for pairs of hexachords to each contain six different pitch classes, the first set may be called P0, in which case the second set would be P1. In contrast, where transpositionally related sets show the difference for every pair of corresponding pitch classes. Thus for P0 and I11 the sum of complementation is 11. This is because since P is equivalent to M, and M is the complement of M, P is also the complement of M, from a logical and musical point of view, even though not its literal pc complement. Originator Allen Forte describes this as, significant extension of the complement relation, though George Perle describes this as, as a further example take the chromatic sets 7-1 and 5-1. If the pitch-classes of 7-1 span C-F♯ and those of 5-1 span G-B then they are literal complements, however, if 5-1 spans C-E, C♯-F, or D-F♯, then it is an abstract complement of 7-1. As these examples make clear, once sets or pitch-class sets are labeled, the complement relation is easily recognized by the identical ordinal number in pairs of sets of complementary cardinalities
28.
Forte number
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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. In the 12-TET tuning system, each class may be denoted by an integer in the range from 0 to 11. The prime form of a class set is the most compact 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 major chord contains the pitch classes 7,0, and 4. The normal form would then be 0,4 and 7 and its inversion, which happens to be the minor chord, contains the pitch classes 0,3, and 7, and is the prime form. The major and minor chords are both given Forte # 3-11, indicating that it is the eleventh in Fortes ordering of pitch class sets with three pitches. In contrast, the Viennese trichord, with pitch classes 0,1, Sets of pitches which share the same Forte number have identical interval vectors. Those that have different Forte numbers have different interval vectors with the exception of z-related sets. In this correspondence, a one in a sequence corresponds to a pitch that is present in a pitch class set. The rotation of binary sequences corresponds to transposition of chords, the most compact form of a pitch class set is the lexicographically maximal sequence within the corresponding equivalence class of sequences. There are two methods of computing Forte number and prime form, the second introduced in John Rahns Basic Atonal Theory and this affects sets 5-20, 6-Z29, 6-31, 7-20, and 8-26. The article, List of pitch-class sets, appears to use the Rahn algorithm, for example, the Forte prime for 6-31 is. Elliott Carter had earlier produced a numbered listing of pitch class sets, or chords, as Carter referred to them, List of pitch-class sets All About Set Theory, What is a Forte Number. The Table of Pitch Class Sets, SolomonsMusic. net
29.
Identity (music)
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In post-tonal music theory, identity is similar to identity in universal algebra. An identity function is a permutation or transformation which transforms a pitch or pitch class set into itself, for instance, inverting an augmented triad or C4 interval cycle,048, produces itself,084. Performing a retrograde operation upon the class set 01210 produces 01210. In addition to being a property of a set, identity is, by extension. George Perle provides the example, C-E, D-F♯, E♭-G, are different instances of the same interval. Other kind of identity. has to do with axes of symmetry. C-E belongs to a family of symmetrically related dyads as follows, C=0, so in mod12, Thus, in addition to being part of the interval-4 family, C-E is also a part of the sum-4 family
30.
Multiplication (music)
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The mathematical operations of multiplication have several applications to music. Other than its application to the ratios of intervals, it has been used in other ways for twelve-tone technique. Additionally ring modulation is an electrical audio process involving multiplication that has 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, when dealing with pitch-class sets, multiplication modulo 12 is a common operation. Dealing with all tones, or a tone row, there are only a few numbers which one may multiply a row by. This is because each of these numbers is prime to 12. This kind of multiplication is frequently combined with a transposition operation and it was first described in print by Herbert Eimert, under the terms Quartverwandlung and Quintverwandlung, and has been used by the composers Milton Babbitt, Robert Morris, and Charles Wuorinen. This operation also accounts for certain harmonic transformations in jazz, thus multiplication by the two meaningful operations may be designated with M5 and M7 or M and IM. M1 = Identity M5 = Cycle of fourths transform M7 = Cycle of fifths transform M11 = Inversion M11M5 = M7 M7M5 = M11 M5M5 = M1 M7M11M5 = M1. Pierre Boulez described an operation he called pitch multiplication, which is akin to the Cartesian product of pitch-class sets. Given two sets, the result of multiplication will be the set of sums of all possible pairings of elements between the original two sets. Given the limited space of modulo 12 arithmetic, when using this very often duplicate tones are produced. Howard Hanson called this operation of commutative mathematical convolution superposition or @-projection, thus p@m or p/m means perfect fifth at major third, e. g. He specifically noted that two triad forms could be so multiplied, or a triad multiplied by itself, to produce a resultant scale, the latter squaring of a triad produces a particular scale highly saturated in instances of the source triad. Thus pmn, Hansons name for common the major triad, when squared, is PMN and this second sonority, multiplied by the first, gives his formula for generating scales and their harmonizations. Joseph Schillinger used the idea, undeveloped, to categorize common 19th- and early 20th-century harmonic styles as product of horizontal harmonic root-motion, some of the composers styles which he cites appear in the following multiplication table
31.
Permutation (music)
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In music, a permutation of a set is any ordering of the elements of that set. A specific arrangement of a set of entities, or parameters, such as pitch, dynamics. 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 compositional techniques originating in the 20th century such as the twelve-tone technique and serialism. Analytical techniques such as set theory take care to distinguish between ordered and unordered collections, in traditional theory concepts such voicing and form include ordering. For example, many forms, such as rondo, are defined by the order of their sections. The permutations resulting from applying the inversion or retrograde operations are categorized as the prime forms inversions and retrogrades, likewise, applying both inversion and retrograde to a prime form produces its retrograde-inversions, which are considered a distinct type of permutation. Permutation may be applied to sets as well. However, the use of operations to such smaller sets do not necessarily result in permutation of the original set. One technique facilitating twelve-tone permutation is the use of values corresponding with musical letter names. The first note of the first of the primes, actually prime zero, is represented by 0, prime zero is retrieved entirely by choice of the composer. To receive the retrograde of any prime, the numbers are simply rewritten backwards. To receive the inversion of any prime, each value is subtracted from 12. The retrograde inversion is the values of the numbers read backwards. In that regard, a permutation is a combinatorial permutation from mathematics as it applies to music. Cyclical permutation is the maintenance of the order of the tone row with the only change being that of the initial pitch-class. A secondary set may be considered a cyclical permutation beginning on the member of a hexachordally combinatorial row
32.
Similarity relation (music)
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In music, a similarity relation or pitch-class similarity is a comparison between sets of the same cardinality, based upon shared pitch class and/or interval class content. Allen Forte originally designated four types, Rp, R0, R1, in Rp one pitch class is different, in R0 all are different, and in R1 and R2 four interval classes are the same. Rp is defined for sets S1 and S2 of cardinal number n and S3 of cardinal number n-1 as, Rp iff Meaning that S1 and S2 each have all the pitch-classes of S3, plus one. He pc similarity relation Rp is not especially significant when taken alone, when Rp is combined with the similarity measures, however, a considerable reduction is effected. Similarity of Interval-class Content Between Pitch-class Sets, The IcVSIM Relation, Journal of Music Theory 34, 1-28
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Cardinality equals variety
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The musical operation of scalar transposition shifts every note in a melody by the same number of scale steps. The musical operation of chromatic transposition shifts every note in a melody by the distance in pitch class space. In general, for a given scale S, the scalar transpositions of a line L can be grouped into categories, or transpositional set classes, whose members are related by chromatic transposition. In diatonic set theory cardinality equals variety when, for any melodic line L in a particular scale S, for example, the melodic line C-D-E has three distinct pitch classes. When transposed diatonically to all degrees in the C major scale. Melodic lines in the C major scale with n distinct pitch classes always generate n distinct patterns, the property was first described by John Clough and Gerald Myerson in Variety and Multiplicity in Diatonic Systems. Cardinality equals variety in the collection and the pentatonic scale. Nondegenerate well-formed scales are those that possess Myhills property, Variety and Multiplicity in Diatonic Systems, Journal of Music Theory 29, 249-70. Aspects of Well-Formed Scales, Music Theory Spectrum 29, 249-70, a Mathematical Model of the Diatonic System, Journal of Music Theory 33, 1-25. Coherent Tone-Systems, A Study in the Theory of Diatonicism, Journal of Music Theory 40, foundations of Diatonic Theory, A Mathematically Based Approach to Music Fundamentals
34.
Diatonic scale
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This pattern ensures that, in a diatonic scale spanning more than one octave, all the half steps are maximally separated from each other. The word diatonic comes from the Greek διατονικός, meaning progressing through tones, the seven pitches of any diatonic scale can be obtained using a chain of six perfect fifths. Musical keyboards are designed to play notes, and hence diatonic scales. A diatonic scale can be described as two tetrachords separated by a whole tone. The term diatonic originally referred to the genus, one of the three genera of the ancient Greeks. In musical set theory, Allen Forte classifies diatonic scales as set form 7–35 and this article does not concern alternative seven-note diatonic scales such as the harmonic minor or the melodic minor. Western music from the Middle Ages until the late 19th century is based on the diatonic scale, there is one claim that the 45, 000-year-old Divje Babe Flute uses a diatonic scale, but there is no proof or consensus it is even a musical instrument. There is evidence that the Sumerians and Babylonians used some version of the diatonic scale and this derives from surviving inscriptions that contain a tuning system and musical composition. Despite the conjectural nature of reconstructions of the known as the Hurrian songs from the surviving score. The scales corresponding to the medieval Church modes were diatonic, one of these, the one starting on B, has no pure fifth above its reference note, it is probably for this reason that it was not used. Of the six remaining scales, two were described as corresponding to two others with a B♭ instead of a B♮, A–B–C–D–E–F–G–A was described as D–E–F–G–A–B♭C–D and C–D–E–F–G–A–B–C as F–G–A–B♭–C–D–E–F, as a result, medieval theory described the Church modes as corresponding to four diatonic scales only. Heinrich Glarean considered that the modal scales including a B♭ had to be the result of a transposition, by the beginning of the Baroque period, the notion of musical key was established, describing additional possible transpositions of the diatonic scale. Major and minor scales came to dominate until at least the start of the 20th century, some Church modes survived into the early 18th century, as well as appearing occasionally in classical and 20th-century music, and later in modal jazz. Glareans six natural scales could be transposed not only to include one flat in the signature, of these six scales, three are major scales, three are minor. To these may be added the seventh diatonic scale, with a fifth above the reference note. Transposing each of seven scales on the twelve degrees of the chromatic scale results in a total of eighty-four diatonic scales. The modern musical keyboard originated as a keyboard with only lower keys. The major scale, for instance, can be represented as T–T–S–T–T–T–S or 2–2–1–2–2–2–1 The major scale or Ionian scale is one of the diatonic scales and it is made up of seven distinct notes, plus an eighth that duplicates the first an octave higher
35.
Generated collection
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All scales with the deep scale property can be generated by any interval coprime with twelve. The C major diatonic collection can be generated by adding a cycle of fifths starting at F. Using integer notation and modulo 12,5 +7 =0,0 +7 =7,7 +7 =2,2 +7 =9,9 +7 =4,4 +7 =11. The C major scale could also be generated using cycle of perfect fourths, a generated collection for which a single generic interval corresponds to the single generator or interval cycle used is a MOS or well formed generated collection. For example, the collection is well formed, for the perfect fifth corresponds to the generator 7. The pentatonic scale is well formed. While unpublished, this became widely known and used in the microtonal music community. A bisector is a weaker substitute used to create collections that cannot be generated, distance model circle of fifths Pythagorean tuning Handwritten letter of Erv Wilson Carey, Norman and Clampitt, David. Aspects of Well-Formed Scales, Music Theory Spectrum 11, 187–206, Scales, Sets, and Interval Cycles,79. Foundations of Diatonic Theory, A Mathematically Based Approach to Music Fundamentals
36.
Maximal evenness
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This property was first described by music theorist John Clough and mathematician Jack Douthett in Maximally Even Sets. The whole-tone scale is also even, for instance adjacent notes are separated only by whole-tones. Second-order maximal evenness is maximal evenness of a subcollection of a collection that is maximally even. In a dynamical approach, spinning concentric circles and iterated maximally even sets have been constructed as an approach to Riemann theory. This approach leads to some interesting connections between diatonic and chromatic theory, distributional evenness is a property of musical scales. A scale is distributionally even if for each generic interval there are one or two specific intervals, Maximally Even Sets, Journal of Music Theory 35, 93-173. Maximally Even Sets and Configurations, Common Threads in Mathematics, Physics, dinner Tables and Concentric Circles, A Harmony in Mathematics, Music, and Physics, College Mathematics Journal 39, 203-211. Foundations of Diatonic Theory, A Mathematically Based Approach to Music Fundamentals, cognitive Constraints on Compositional Systems, Contemporary Music Review 6, pp. 97-121. Filter Point-Symmetry and Dynamical Voice-Leading, Music Theory and Mathematics, Chords, Collections, ed. J. Douthett, M. Hyde, and C. Smith
37.
Structure implies multiplicity
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In diatonic set theory structure implies multiplicity is a quality of a collection or scale. Structure being the intervals in relation to the circle of fifths, the property was first described by John Clough and Gerald Myerson in Variety and Multiplicity in Diatonic Systems. Structure implies multiplicity is true of the collection and the pentatonic scale. For example, cardinality equals variety dictates that a three member diatonic subset of the C major scale, C-D-E transposed to all scale degrees gives three interval patterns, M2-M2, M2-m2, m2-M2. On the circle of fifths, C G D A E B F1212123 E and C are three notes apart, C and D are two notes apart, D and E two notes apart. Just as the distance around the circle of fifths between forms the interval pattern 3-2-2, M2-M2 occurs three times, M2-m2 occurs twice, and m2-M2 occurs twice, cardinality equals variety and structure implies multiplicity are true of all collections with Myhills property or maximal evenness. Variety and Multiplicity in Diatonic Systems, Journal of Music Theory 29, a Mathematical Model of the Diatonic System, Journal of Music Theory 33, 1-25. Coherent Tone-Systems, A Study in the Theory of Diatonicism, Journal of Music Theory 40, foundations of Diatonic Theory, A Mathematically Based Approach to Music Fundamentals
38.
Music theory
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Music theory is the study of the practices and possibilities of music. The term is used in three ways in music, though all three are interrelated. The first is what is otherwise called rudiments, currently taught as the elements of notation, of key signatures, of time signatures, of rhythmic notation, Theory in this sense is treated as the necessary preliminary to the study of harmony, counterpoint, and form. The second is the study of writings about music from ancient times onwards, Music theory is frequently concerned with describing how musicians and composers make music, including tuning systems and composition methods among other topics. However, this medieval discipline became the basis for tuning systems in later centuries, Music theory as a practical discipline encompasses the methods and concepts composers and other musicians use in creating music. The development, preservation, and transmission of music theory in this sense may be found in oral and written music-making traditions, musical instruments, and other artifacts. In ancient and living cultures around the world, the deep and long roots of music theory are clearly visible in instruments, oral traditions, and current music making. Many cultures, at least as far back as ancient Mesopotamia and ancient China, have also considered music theory in more formal ways such as written treatises, in modern academia, music theory is a subfield of musicology, the wider study of musical cultures and history. Etymologically, music theory is an act of contemplation of music, from the Greek θεωρία, a looking at, viewing, contemplation, speculation, theory, also a sight, a person who researches, teaches, or writes articles about music theory is a music theorist. University study, typically to the M. A. or Ph. D level, is required to teach as a music theorist in a US or Canadian university. Methods of analysis include mathematics, graphic analysis, and especially analysis enabled by Western music notation, comparative, descriptive, statistical, and other methods are also used. See for instance Paleolithic flutes, Gǔdí, and Anasazi flute, several surviving Sumerian and Akkadian clay tablets include musical information of a theoretical nature, mainly lists of intervals and tunings. The scholar Sam Mirelman reports that the earliest of these dates from before 1500 BCE. Further, All the Mesopotamian texts are united by the use of a terminology for music that, much of Chinese music history and theory remains unclear. The earliest texts about Chinese music theory are inscribed on the stone and they include more than 2800 words describing theories and practices of music pitches of the time. The bells produce two intertwined pentatonic scales three tones apart with additional pitches completing the chromatic scale, Chinese theory starts from numbers, the main musical numbers being twelve, five and eight. Twelve refers to the number of pitches on which the scales can be constructed, the Lüshi chunqiu from about 239 BCE recalls the legend of Ling Lun. On order of the Yellow Emperor, Ling Lun collected twelve bamboo lengths with thick, blowing on one of these like a pipe, he found its sound agreeable and named it huangzhong, the Yellow Bell