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Structure Implies Multiplicity
In diatonic set theory structure implies multiplicity is a quality of a collection or scale (music), scale. For collections or scales which have this property, the interval series formed by the shortest distance around a diatonic circle of fifths between members of a series indicates the number of unique interval (music), interval patterns (adjacently, rather than around the circle of fifths) formed by diatonic transpositions of that series. Structure refers to the intervals in relation to the circle of fifths; multiplicity refers to the number of times each different (adjacent) interval pattern occurs. The property was first described by John Clough and Gerald Myerson in "Variety and Multiplicity in Diatonic Systems" (1985). () Structure implies multiplicity is true of the diatonic collection and the pentatonic scale, and any subset. 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diatonic Set Theory
Diatonic set theory is a subdivision or application of set theory (music), musical set theory which applies the techniques and musical analysis, analysis of discrete mathematics to properties of the diatonic collection such as maximal evenness, Myhill's property, well formed generated collection, well formedness, the deep scale property, cardinality equals variety, and structure implies multiplicity. The name is something of a misnomer as the concepts involved usually apply much more generally, to any periodically repeating scale. Music theorists working in diatonic set theory include Eytan Agmon, Gerald J. Balzano, Norman Carey, David Clampitt, John Clough, Jay Rahn, and mathematician Jack Douthett. A number of key concepts were first formulated by David Rothenberg (the Rothenberg propriety), who published in the journal ''Mathematical Systems Theory'', and Erv Wilson, working entirely outside of the academic world. See also *Bisector (music), Bisector *Diatonic and chromatic *Gen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scale (music)
In music theory, a scale is "any consecutive series of notes that form a progression between one note and its octave", typically by order of pitch or fundamental frequency. The word "scale" originates from the Latin ''scala'', which literally means "ladder". Therefore, any scale is distinguishable by its "step-pattern", or how its intervals interact with each other. Often, especially in the context of the common practice period, most or all of the melody and harmony of a musical work is built using the notes of a single scale, which can be conveniently represented on a staff with a standard key signature. Due to the principle of octave equivalence, scales are generally considered to span a single octave, with higher or lower octaves simply repeating the pattern. A musical scale represents a division of the octave space into a certain number of scale steps, a scale step being the recognizable distance (or interval) between two successive notes of the scale. However, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Circle Of Fifths
In music theory, the circle of fifths (sometimes also cycle of fifths) is a way of organizing pitches as a sequence of perfect fifths. Starting on a C, and using the standard system of tuning for Western music (12-tone equal temperament), the sequence is: C, G, D, A, E, B, F/G, C/D, G/A, D/E, A/B, F, and C. This order places the most closely related key signatures adjacent to one another. Twelve-tone equal temperament tuning divides each octave into twelve equivalent semitones, and the circle of fifths leads to a C seven octaves above the starting point. If the fifths are tuned with an exact frequency ratio of 3:2 (the system of tuning known as just intonation), this is not the case (the circle does not "close"). 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 clockwise order. It can be viewed in a counterclockwise direction as a circle of fourths. Harmonic progres ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Interval (music)
In music theory, an interval is a difference in pitch between two sounds. An interval may be described as horizontal, linear, or melodic if it refers to successively sounding tones, such as two adjacent pitches in a melody, and vertical or harmonic if it pertains to simultaneously sounding tones, such as in a chord. In Western music, intervals are most commonly differences between notes of a diatonic scale. Intervals between successive notes of a scale are also known as scale steps. The smallest of these intervals is a semitone. Intervals smaller than a semitone are called microtones. They can be formed using the notes of various kinds of non-diatonic scales. Some of the very smallest ones are called commas, and describe small discrepancies, observed in some tuning systems, between enharmonically equivalent notes such as C and D. Intervals can be arbitrarily small, and even imperceptible to the human ear. In physical terms, an interval is the ratio between two sonic fr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diatonic Transposition
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, a music transposer 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. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John Clough
John Clough (born 13 September 1984 in St. Helens) is a former rugby league footballer playing over 250 games for Salford City Reds (2001–06), London Broncos, Halifax (2006), Leigh Centurions (2007), Blackpool Panthers (2007–10), Oldham Oldham is a town in Greater Manchester, England. It lies amongst the Pennines on elevated ground between the rivers River Irk, Irk and River Medlock, Medlock, southeast of Rochdale, and northeast of Manchester. It is the administrative cent ... (2011–14) and Oxford (2015) as a . John Clough is a former Lancashire and Great Britain Academy representative. Genealogical information John Clough is brother of the rugby league footballer, Paul Clough. References External linksStatistics at rugbyleagueproject.org 1984 births Living people Blackpool Panthers players English rugby league players Halifax Panthers players Leigh Leopards players London Broncos players Oldham R.L.F.C. players Oxford Rugby League players Rugby ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gerald Myerson
Gerald is a masculine given name derived from the Germanic languages prefix ''ger-'' ("spear") and suffix ''-wald'' ("rule"). Gerald is a Norman French variant of the Germanic name. An Old English equivalent name was Garweald, the likely original name of Gerald of Mayo, a British Roman Catholic monk who established a monastery in Mayo, Ireland in 670. Nearly two centuries later, Gerald of Aurillac Gerald of Aurillac (or Saint Gerald) ( 855 – c. 909) is a French saint of the Roman Catholic Church, also recognized by other religious denominations of Christianity. Life Gerald was born into the Gallo-Roman nobility, counting Cesarius of Ar ..., a French count, took a vow of celibacy and later became known as the Roman Catholic patron saint of bachelors. The name was in regular use during the Middle Ages but declined after 1300 in England. It remained a common name in Ireland, where it was a common name among the powerful FitzGerald dynasty. The name was revived in the Anglosp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diatonic Collection
In music theory a diatonic scale is a 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 pitches of any diatonic scale can also be obtained by using a chain of six perfect fifths. For instance, the seven 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 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 diatonic scale can be also described as two tetrachords separated by a whole tone. In musical ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pentatonic Scale
A pentatonic scale is a musical scale with five notes per octave, in contrast to heptatonic scales, which have seven notes per octave (such as the major scale and minor scale). Pentatonic scales were developed independently by many ancient civilizations and are still used in various musical styles to this day. As Leonard Bernstein put it: "The universality of this scale is so well known that I'm sure you could give me examples of it, from all corners of the earth, as from Scotland, or from China, or from Africa, and from American Indian cultures, from East Indian cultures, from Central and South America, Australia, Finland ...now, that is a true musico-linguistic universal." There are two types of pentatonic scales: Those with semitones (hemitonic) and those without (anhemitonic). Types Hemitonic and anhemitonic Musicology commonly classifies pentatonic scales as either ''hemitonic'' or ''anhemitonic''. Hemitonic scales contain one or more semitones and anhemitonic s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cardinality Equals Variety
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 same 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, the number of these classes is equal to the number of distinct pitch classes in the line L. For example, the melodic line C-D-E has three distinct pitch classes. When transposed diatonically to all scale degrees in the C major scale, we obtain three interval patterns: M2-M2, M2-m2, m2-M2. 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 Geral ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scale Degree
In music theory, the scale degree is the position of a particular note on a scale relative to the tonic—the first and main note of the scale from which each octave is assumed to begin. Degrees are useful for indicating the size of intervals and chords and whether an interval is major or minor. In the most general sense, the scale degree is the number given to each step of the scale, usually starting with 1 for tonic. Defining it like this implies that a tonic is specified. For instance, the 7-tone diatonic scale may become the major scale once the proper degree has been chosen as tonic (e.g. the C-major scale C–D–E–F–G–A–B, in which C is the tonic). If the scale has no tonic, the starting degree must be chosen arbitrarily. In set theory, for instance, the 12 degrees of the chromatic scale are usually numbered starting from C=0, the twelve pitch classes being numbered from 0 to 11. In a more specific sense, scale degrees are given names that indicate their ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cardinality Equals Variety CDE
In mathematics, the cardinality of a finite set is the number of its elements, and is therefore a measure of size of the set. Since the discovery by Georg Cantor, in the late 19th century, of different sizes of infinite sets, the term ''cardinality'' was coined for generalizing to infinite sets the concept of the number of elements. More precisely, two sets have the same cardinality if there exists a one-to-one correspondence between them. In the case of finite sets, the common operation of ''counting'' consists of establishing a one-to-one correspondence between a given set and the set of the first natural numbers, for some natural number . In this case, is the cardinality of the set. A set is ''infinite'' if the counting operation never finishes, that is, if its cardinality is not a natural number. The basic example of an infinite set is the set of all natural numbers. The great discovery of Cantor is that infinite sets of apparently different size may have the same cardina ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
