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Euler–Fokker Genus
In music theory and tuning, an Euler–Fokker genus (plural: genera), named after Leonhard Euler and Adriaan Fokker,Rasch, Rudolph (2000). ''Harry Partch'', p.31-2. Dunn, David, ed. . is a musical scale in just intonation whose pitches can be expressed as products of some of the members of some multiset of generating prime factors. Powers of two are usually ignored, because of the way the human ear perceives octaves as equivalent. An x-dimensional tone-dimension contains x factors. "An Euler-Fokker genus with two dimensions may be represented in a two-dimensional (rectangular) tone-grid, one with three dimensions in a three-dimensional (block-shaped) tone-lattice. Euler-Fokker genera are characterized by a listing of the number of steps in each dimension. The number of steps is represented by a repeated mention of the dimension, so that there arise descriptions such as 3 5 5 5 7 3 5 5 7 7 11 11 etc." For example, the multiset yields the Euler–Fokker genus , 3,&n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler Genus 337
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in many other branches of mathematics such as analytic number theory, complex analysis, and infinitesimal calculus. He introduced much of modern mathematical terminology and Mathematical notation, notation, including the notion of a function (mathematics), mathematical function. He is also known for his work in mechanics, fluid dynamics, optics, astronomy and music theory. Euler is held to be one of the greatest mathematicians in history and the greatest of the 18th century. A statement attributed to Pierre-Simon Laplace expresses Euler's influence on mathematics: "Read Euler, read Euler, he is the master of us all." Carl Friedrich Gauss remarked: "The study of Euler's works will remain the best school for the different fields of mathematics, a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perfect Fifth
In music theory, a perfect fifth is the musical interval corresponding to a pair of pitches with a frequency ratio of 3:2, or very nearly so. In classical music from Western culture, a fifth is the interval from the first to the last of five consecutive notes in a diatonic scale. The perfect fifth (often abbreviated P5) spans seven semitones, while the diminished fifth spans six and the augmented fifth spans eight semitones. For example, the interval from C to G is a perfect fifth, as the note G lies seven semitones above C. The perfect fifth may be derived from the harmonic series as the interval between the second and third harmonics. In a diatonic scale, the dominant note is a perfect fifth above the tonic note. The perfect fifth is more consonant, or stable, than any other interval except the unison and the octave. It occurs above the root of all major and minor chords (triads) and their extensions. Until the late 19th century, it was often referred to by one of i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hexany
In musical tuning systems, the hexany, invented by Erv Wilson, represents one of the simplest structures found in his combination product sets. It is referred to as an uncentered structure, meaning that it implies no tonic. It achieves this by using consonant relations as opposed to the dissonance methods normally employed by atonality. While it is often and confusingly overlapped with the Euler–Fokker genus, the subsequent stellation of Wilson's combination product sets (CPS) are outside of that Genus. The Euler Fokker Genus fails to see 1 as a possible member of a set except as a starting point. The numbers of vertices of his combination sets follow the numbers in Pascal's triangle. In this construction, the hexany is the third cross-section of the four-factor set and the first uncentered one. hexany is the name that Erv Wilson gave to the six notes in the 2-out-of-4 combination product set, abbreviated as 2*4 CPS. Simply, the hexany is the 2 out of 4 set. It is construc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Just Augmented Fifth
In classical music from Western culture, an augmented fifth () is an interval produced by widening a perfect fifth by a chromatic semitone.Benward & Saker (2003). ''Music: In Theory and Practice, Vol. I'', p.54. . For instance, the interval from C to G is a perfect fifth, seven semitones wide, and both the intervals from C to G, and from C to G are augmented fifths, spanning eight semitones. Being augmented, it is considered a dissonant interval. Its inversion is the diminished fourth, and its enharmonic equivalent is the minor sixth. The augmented fifth only began to make an appearance at the beginning of the common practice period of music as a consequence of composers seeking to strengthen the normally weak seventh degree when composing music in minor modes. This was achieved by chromatically raising the seventh degree (or subtonic) to match that of the unstable seventh degree (or leading tone) of the major mode (an increasingly widespread practice that led to the cr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Just Major Seventh
Just or JUST may refer to: __NOTOC__ People * Just (surname) * Just (given name) Arts and entertainment * ''Just'', a 1998 album by Dave Lindholm * "Just" (song), a song by Radiohead * "Just", a song from the album ''Lost and Found'' by Mudvayne * ''Just!'' (series), a series of short-story collections for children by Andy Griffiths JUST * Jordan University of Science and Technology, Jordan * Jessore University of Science & Technology, Bangladesh * Jinwen University of Science and Technology, New Taipei, Taiwan Businesses * Just Group plc, a British company specialising in retirement products and services * Just Group, an Australian owner and operator of seven retail brands * JUST, Inc., an American food manufacturing company See also * * List of people known as the Just * Saint-Just (other) * Justice Justice, in its broadest sense, is the principle that people receive that which they deserve, with the interpretation of what then constitutes "deserving" bein ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Otonality And Utonality
''Otonality'' and ''utonality'' are terms introduced by Harry Partch to describe chords whose pitch classes are the harmonics or subharmonics of a given fixed tone ( identity), respectively. For example: , , ,... or , , ,.... Definition An otonality is a collection of pitches which can be expressed in ratios, expressing their relationship to the fixed tone, that have equal denominators and consecutive numerators. For example, , , and ( just major chord) form an otonality because they can be written as , , . This in turn can be written as an extended ratio 4:5:6. Every otonality is therefore composed of members of a harmonic series. Similarly, the ratios of a utonality share the same numerator and have consecutive denominators. , , , and () form a utonality, sometimes written as , or as . Every utonality is therefore composed of members of a subharmonic series. This term is used extensively by Harry Partch in ''Genesis of a Music''. An otonality correspond ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lowest Common Multiple
In arithmetic and number theory, the least common multiple, lowest common multiple, or smallest common multiple of two integers ''a'' and ''b'', usually denoted by lcm(''a'', ''b''), is the smallest positive integer that is divisible by both ''a'' and ''b''. Since division of integers by zero is undefined, this definition has meaning only if ''a'' and ''b'' are both different from zero. However, some authors define lcm(''a'',0) as 0 for all ''a'', since 0 is the only common multiple of ''a'' and 0. The lcm is the "lowest common denominator" (lcd) that can be used before fractions can be added, subtracted or compared. The least common multiple of more than two integers ''a'', ''b'', ''c'', . . . , usually denoted by lcm(''a'', ''b'', ''c'', . . .), is also well defined: It is the smallest positive integer that is divisible by each of ''a'', ''b'', ''c'', . . . Overview A multiple of a number is the product of that number and an integer. For example, 10 is a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Greatest Common Divisor
In mathematics, the greatest common divisor (GCD) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers ''x'', ''y'', the greatest common divisor of ''x'' and ''y'' is denoted \gcd (x,y). For example, the GCD of 8 and 12 is 4, that is, \gcd (8, 12) = 4. In the name "greatest common divisor", the adjective "greatest" may be replaced by "highest", and the word "divisor" may be replaced by "factor", so that other names include highest common factor (hcf), etc. Historically, other names for the same concept have included greatest common measure. This notion can be extended to polynomials (see Polynomial greatest common divisor) and other commutative rings (see below). Overview Definition The ''greatest common divisor'' (GCD) of two nonzero integers and is the greatest positive integer such that is a divisor of both and ; that is, there are integers and such that and , and is the larges ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alan Ridout
Alan Ridout (9 December 1934 – 19 March 1996) was a British composer and teacher. Life Born in West Wickham, Kent, England, Alan Ridout studied briefly at the Guildhall School of Music before commencing four years of study at the Royal College of Music, London with Herbert Howells and Gordon Jacob. He was later taught by Michael Tippett, Peter Fricker and (under a Dutch government scholarship) Henk Badings.Miall, Peter.Obituary: Alan Ridout in ''The Independent'', 23 October, 2011 He went on to teach at the Royal College of Music, the University of Birmingham, the University of Cambridge, the University of London, and at The King's School, Canterbury. He also broadcast musical talks on the radio. Alan Ridout lived for much of his life in Canterbury, but after a serious heart attack in 1990 he moved to France, settling in the town of Vitré, Brittany, before moving on to Caen at the very end of his life. Music Ridout's style is mostly tonal, though in younger life ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
31 Equal Temperament
In music, 31 equal temperament, 31-ET, which can also be abbreviated 31-TET (31 tone ET) or 31-EDO (equal division of the octave), also known as tricesimoprimal, is the tempered scale derived by dividing the octave into 31 equal-sized steps (equal frequency ratios). Each step represents a frequency ratio of , or 38.71 cents (). 31-ET is a very good approximation of quarter-comma meantone temperament. More generally, it is a regular diatonic tuning in which the tempered perfect fifth is equal to 696.77 cents, as shown in Figure 1. On an isomorphic keyboard, the fingering of music composed in 31-ET is precisely the same as it is in any other syntonic tuning (such as 12-ET), so long as the notes are spelled properly — that is, with no assumption of enharmonicity. History and use Division of the octave into 31 steps arose naturally out of Renaissance music theory; the lesser diesis — the ratio of an octave to three major thirds, 128:125 or 41.06 cents — was approximat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Septimal Comma
A septimal comma is a small musical interval in just intonation that contains the number seven in its prime factorization. There is more than one such interval, so the term ''septimal comma'' is ambiguous, but it most commonly refers to the interval 64/63 (27.26 cents). Use of septimal commas introduces new intervals that extend tuning beyond common-practice, extending music to the 7-limit, including the 7/6 septimal minor third, the 7/5 septimal tritone and the 8/7 septimal major second. Composers who made extensive use of these intervals include Harry Partch and Ben Johnston. Johnston uses a "7" as an accidental to indicate a note is lowered 49 cents, or an upside down seven ("ㄥ" or "") to indicate a note is raised 49 cents (36/35).John Fonville. "Ben Johnston's Extended Just Intonation – A Guide for Interpreters", p. 113, '' Perspectives of New Music'', vol. 29, no. 2 (Summer 1991), pp. 106–137. Specific commas The 64/63 septimal comma, also known as '' Archytas' Co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
