1.
Tuplet
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In music a tuplet is any rhythm that involves dividing the beat into a different number of equal subdivisions from that usually permitted by the time-signature. This is indicated by a number, indicating the fraction involved, the notes involved are also often grouped with a bracket or a slur. The most common type is the triplet, an alternative modern term, irrational rhythm, was originally borrowed from Greek prosody where it referred to a syllable having a metrical value not corresponding to its actual time-value, or. A metrical foot containing such a syllable, the term would be incorrect if used in the mathematical sense or in the more general sense of unreasonable, utterly illogical, absurd. Alternative terms found occasionally are artificial division, abnormal divisions, irregular rhythm, the term polyrhythm, sometimes incorrectly used of tuplets, actually refers to the simultaneous use of opposing time signatures. Besides triplet, the terms duplet, quadruplet, quintuplet, sextuplet, septuplet, the terms nonuplet, decuplet, undecuplet, dodecuplet, and tredecuplet had been suggested but up until 1925 had not caught on. By 1964 the terms nonuplet and decuplet were usual, while subdivisions by greater numbers were commonly described as group of eleven notes, group of twelve notes. The most common tuplet is the triplet, shown at right, similarly, three triplet eighth notes are equal in duration to one quarter note. If several note values appear under the bracket, they are all affected the same way. If the notes of the tuplet are beamed together, the bracket may be omitted, for other tuplets, the number indicates a ratio to the next lower normal value in the prevailing meter. Some numbers are used inconsistently, for example septuplets usually indicate 7 notes in the duration of 4—or in compound meter 7 for 6—but may sometimes be used to mean 7 notes in the duration of 8. Thus, a septuplet lasting a whole note can be written with either quarter notes or eighth notes, a French alternative is to write pour or de in place of the colon, or above the bracketed irregular number. This reflects the French usage of, for example, six-pour-quatre as a name for the sextolet. There are disagreements about the sextuplet —which is also called sestole, sestolet, sextole, some go so far as to call the latter, when written with a numeral 6, a false sextuplet. In compound meter, even-numbered tuplets can indicate that a value is changed in relation to the dotted version of the next higher note value. Thus, two eighth notes take the time normally totaled by three eighth notes, equal to a dotted quarter note. Four quadruplet eighth notes would also equal a dotted quarter note, the duplet eighth note is thus exactly the same duration as a dotted eighth note, but the duplet notation is far more common in compound meters. A duplet in compound time is often written as 2,3 than 2, 1 1⁄2
2.
Octuple scull
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An octuple scull is a racing shell or a rowing boat used in the sport of competitive rowing. The octuple is directed by a coxswain and propelled by eight rowers who move the boat by sculling with two oars, one in each hand, like a coxed eight, an octuple is typically 65.2 feet long and weighs 211.2 pounds. Racing boats are long, narrow, and broadly semi-circular in cross-section in order to drag to a minimum. They usually have a fin towards the rear, to prevent roll. Originally made from wood, shells are now almost always made from a material for strength. The riggers in sculling apply the forces symmetrically to each side of the boat, when there are eight rowers in a boat, each with only one sweep oar and rowing on opposite sides, the combination is referred to as a coxed eight. In sweep oared racing the rigging means the forces are staggered alternately along the boat, the symmetrical forces in sculling make the boat more efficient and so the octuple scull is faster than the coxed eight. Thames Ditton Regatta,11, Final, Junior Octuple Sculls Plate – Hampton Court
3.
Twelve-tone technique
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All 12 notes are thus given more or less equal importance, and the music avoids being in a key. Over time, the technique increased greatly in popularity and eventually became widely influential on 20th century composers, many important composers who had originally not subscribed to or even actively opposed the technique, such as Aaron Copland and Igor Stravinsky, eventually adopted it in their music. Schoenberg himself described the system as a Method of composing with twelve tones which are related only with one another and it is commonly considered a form of serialism. Schoenbergs countryman and contemporary Josef Matthias Hauer also developed a system using unordered hexachords or tropes—but with no connection to Schoenbergs twelve-tone technique. Other composers have created systematic use of the scale. The twelve tone technique was preceded by freely atonal pieces of 1908–23 which, though free, the twelve-tone technique was also preceded by nondodecaphonic serial composition used independently in the works of Alexander Scriabin, Igor Stravinsky, Béla Bartók, Carl Ruggles, and others. Oliver Neighbour argues that Bartók was the first composer to use a group of twelve notes consciously for a structural purpose, essentially, Schoenberg and Hauer systematized and defined for their own dodecaphonic purposes a pervasive technical feature of modern musical practice, the ostinato. In Hauers breakthrough piece Nomos, Op. Schoenbergs idea in developing the technique was for it to replace those structural differentiations provided formerly by tonal harmonies, Some of these composers extended the technique to control aspects other than the pitches of notes, thus producing serial music. Some even subjected all elements of music to the serial process, the basis of the twelve-tone technique is the tone row, an ordered arrangement of the twelve notes of the chromatic scale. There are four postulates or preconditions to the technique which apply to the row, on which a work or section is based, no note is repeated within the row. The row may be subjected to interval-preserving transformations -—that is, it may appear in inversion, retrograde, or retrograde-inversion, the row in any of its four transformations may begin on any degree of the chromatic scale, in other words it may be freely transposed. A particular transformation together with a choice of transpositional level is referred to as a set form or row form, every row thus has up to 48 different row forms. However, not all prime series will yield so many variations because transposed transformations may be identical to each other. A simple case is the chromatic scale, the retrograde inversion of which is identical to the prime form. In the above example, as is typical, the retrograde inversion contains three points where the sequence of two pitches are identical to the prime row, thus the generative power of even the most basic transformations is both unpredictable and inevitable. Motivic development can be driven by such internal consistency, note that rules 1–4 above apply to the construction of the row itself, and not to the interpretation of the row in the composition. While a row may be expressed literally on the surface as thematic material, it need not be, however, individual composers have constructed more detailed systems in which matters such as these are also governed by systematic rules. The tone row chosen as the basis of the piece is called the prime series, untransposed, it is notated as P0
4.
Mathematics
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Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times
5.
Element (mathematics)
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In mathematics, an element, or member, of a set is any one of the distinct objects that make up that set. Writing A = means that the elements of the set A are the numbers 1,2,3 and 4, sets of elements of A, for example, are subsets of A. For example, consider the set B =, the elements of B are not 1,2,3, and 4. Rather, there are three elements of B, namely the numbers 1 and 2, and the set. The elements of a set can be anything, for example, C =, is the set whose elements are the colors red, green and blue. The relation is an element of, also called set membership, is denoted by the symbol ∈, writing x ∈ A means that x is an element of A. Equivalent expressions are x is a member of A, x belongs to A, x is in A and x lies in A, another possible notation for the same relation is A ∋ x, meaning A contains x, though it is used less often. The negation of set membership is denoted by the symbol ∉, writing x ∉ A means that x is not an element of A. The symbol ϵ was first used by Giuseppe Peano 1889 in his work Arithmetices principia nova methodo exposita, here he wrote on page X, Signum ϵ significat est. Ita a ϵ b legitur a est quoddam b. which means The symbol ϵ means is, so a ϵ b is read as a is a b. The symbol itself is a stylized lowercase Greek letter epsilon, the first letter of the word ἐστί, the Unicode characters for these symbols are U+2208, U+220B and U+2209. The equivalent LaTeX commands are \in, \ni and \notin, mathematica has commands \ and \. The number of elements in a set is a property known as cardinality, informally. In the above examples the cardinality of the set A is 4, an infinite set is a set with an infinite number of elements, while a finite set is a set with a finite number of elements. The above examples are examples of finite sets, an example of an infinite set is the set of positive integers =. Using the sets defined above, namely A =, B = and C =,2 ∈ A ∈ B3,4 ∉ B is a member of B Yellow ∉ C The cardinality of D = is finite, the cardinality of P = is infinite. Halmos, Paul R. Naive Set Theory, Undergraduate Texts in Mathematics, NY, Springer-Verlag, ISBN 0-387-90092-6 - Naive means that it is not fully axiomatized, not that it is silly or easy. Jech, Thomas, Set Theory, Stanford Encyclopedia of Philosophy Suppes, Patrick, Axiomatic Set Theory, NY, Dover Publications, Inc
6.
Sequence
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In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed. Like a set, it contains members, the number of elements is called the length of the sequence. Unlike a set, order matters, and exactly the elements can appear multiple times at different positions in the sequence. Formally, a sequence can be defined as a function whose domain is either the set of the numbers or the set of the first n natural numbers. The position of an element in a sequence is its rank or index and it depends on the context or of a specific convention, if the first element has index 0 or 1. For example, is a sequence of letters with the letter M first, also, the sequence, which contains the number 1 at two different positions, is a valid sequence. Sequences can be finite, as in these examples, or infinite, the empty sequence is included in most notions of sequence, but may be excluded depending on the context. A sequence can be thought of as a list of elements with a particular order, Sequences are useful in a number of mathematical disciplines for studying functions, spaces, and other mathematical structures using the convergence properties of sequences. In particular, sequences are the basis for series, which are important in differential equations, Sequences are also of interest in their own right and can be studied as patterns or puzzles, such as in the study of prime numbers. There are a number of ways to denote a sequence, some of which are useful for specific types of sequences. One way to specify a sequence is to list the elements, for example, the first four odd numbers form the sequence. This notation can be used for sequences as well. For instance, the sequence of positive odd integers can be written. Listing is most useful for sequences with a pattern that can be easily discerned from the first few elements. Other ways to denote a sequence are discussed after the examples, the prime numbers are the natural numbers bigger than 1, that have no divisors but 1 and themselves. Taking these in their natural order gives the sequence, the prime numbers are widely used in mathematics and specifically in number theory. The Fibonacci numbers are the integer sequence whose elements are the sum of the two elements. The first two elements are either 0 and 1 or 1 and 1 so that the sequence is, for a large list of examples of integer sequences, see On-Line Encyclopedia of Integer Sequences
7.
Integer
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An integer is a number that can be written without a fractional component. For example,21,4,0, and −2048 are integers, while 9.75, 5 1⁄2, the set of integers consists of zero, the positive natural numbers, also called whole numbers or counting numbers, and their additive inverses. This is often denoted by a boldface Z or blackboard bold Z standing for the German word Zahlen, ℤ is a subset of the sets of rational and real numbers and, like the natural numbers, is countably infinite. The integers form the smallest group and the smallest ring containing the natural numbers, in algebraic number theory, the integers are sometimes called rational integers to distinguish them from the more general algebraic integers. In fact, the integers are the integers that are also rational numbers. Like the natural numbers, Z is closed under the operations of addition and multiplication, that is, however, with the inclusion of the negative natural numbers, and, importantly,0, Z is also closed under subtraction. The integers form a ring which is the most basic one, in the following sense, for any unital ring. This universal property, namely to be an object in the category of rings. Z is not closed under division, since the quotient of two integers, need not be an integer, although the natural numbers are closed under exponentiation, the integers are not. The following lists some of the properties of addition and multiplication for any integers a, b and c. In the language of algebra, the first five properties listed above for addition say that Z under addition is an abelian group. As a group under addition, Z is a cyclic group, in fact, Z under addition is the only infinite cyclic group, in the sense that any infinite cyclic group is isomorphic to Z. The first four properties listed above for multiplication say that Z under multiplication is a commutative monoid. However, not every integer has an inverse, e. g. there is no integer x such that 2x =1, because the left hand side is even. This means that Z under multiplication is not a group, all the rules from the above property table, except for the last, taken together say that Z together with addition and multiplication is a commutative ring with unity. It is the prototype of all objects of algebraic structure. Only those equalities of expressions are true in Z for all values of variables, note that certain non-zero integers map to zero in certain rings. The lack of zero-divisors in the means that the commutative ring Z is an integral domain
8.
Recursive definition
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A recursive definition in mathematical logic and computer science is used to define the elements in a set in terms of other elements in the set. A recursive definition of a function defines values of the functions for some inputs in terms of the values of the function for other inputs. For example, the function n. is defined by the rules 0. This definition is valid for all n, because the recursion eventually reaches the base case of 0. The definition may also be thought of as giving a procedure describing how to construct the function n. starting from n =0, the recursion theorem states that such a definition indeed defines a function. An inductive definition of a set describes the elements in a set in terms of elements in the set. For example, one definition of the set N of natural numbers is,1 is in N, if an element n is in N then n+1 is in N. N is the intersection of all sets satisfying and, there are many sets that satisfy and - for example, the set satisfies the definition. However, condition specifies the set of numbers by removing the sets with extraneous members. Properties of recursively defined functions and sets can often be proved by a principle that follows the recursive definition. Most recursive definitions have two foundations, a case and an inductive clause. In contrast, a circular definition may have no base case, such a situation would lead to an infinite regress. That recursive definitions are valid - meaning that a recursive definition identifies a unique function - is a theorem of set theory, more generally, recursive definitions of functions can be made whenever the domain is a well-ordered set, using the principle of transfinite recursion. The formal criteria for what constitutes a valid recursive definition are more complex for the general case, an outline of the general proof and the criteria can be found in Munkres. It is chiefly in logic or computer programming that recursive definitions are found, for example, a well formed formula can be defined as, a symbol which stands for a proposition - like p means Connor is a lawyer. The negation symbol, followed by a wff - like Np means It is not true that Connor is a lawyer, any of the four binary connectives followed by two wffs. The symbol K means both are true, so Kpq may mean Connor is a lawyer, and Mary likes music, the value of such a recursive definition is that it can be used to determine whether any particular string of symbols is well formed. Kpq is well formed, because its K followed by the atomic wffs p and q, nKpq is well formed, because its N followed by Kpq, which is in turn a wff