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	Musical set theory - Wikipedia, the free encyclopedia
		Musical set theory is an atonal or post-tonal method of musical analysis and composition which is based on explaining and proving musical phenomena, taken as "sets" and subsets, using mathematical rules and notation and using that information to gain insight to compositions or their creation.
		Although musical set theory is often assumed to be the application of mathematical set theory to music, there is little coincidence between the terminology and even less between the methods of the two.
		Set theory does not, however, use diatonic functionality that is assumed in tonal theory, and this is the reason for the use of integer notation and modulo 12.
	en.wikipedia.org /wiki/Musical_set_theory (1450 words)

	Set theory - Wikipedia, the free encyclopedia
		Naive set theory is the original set theory developed by mathematicians at the end of the 19th century.
		Axiomatic set theory is a rigorous axiomatic theory developed in response to the discovery of serious flaws (such as Russell's paradox) in naïve set theory.
		Musical set theory concerns the application of combinatorics and group theory to music; beyond the fact that it uses finite sets it has nothing to do with mathematical set theory of any kind.
	en.wikipedia.org /wiki/Set_theory (380 words)

	Musical set theory article - Musical set theory theory sequence linear Allen Forte ISBN 0300021208 Assumptions - ... (Site not responding. Last check: 2007-10-20)
		Although musical set theory may be considered the application of mathematical set theory to music, there is often little coincidence between the terminology and possibly the methods of the two.
		Both theories make use of sets, but in the mathematical theory a set is always an unordered collection of things, while in music theory what is called a set is often, in the mathematical theory, a sequence, an ordered collection of things (such as the term set form for tone row).
		The fundamental concept of musical set theory is, of course, the set.
	www.what-means.com /encyclopedia/Musical_set_theory (1390 words)

	Learn more about Set theory in the online encyclopedia. (Site not responding. Last check: 2007-10-20)
		Naive set theory is the original set theory developed by mathematicians.
		Axiomatic set theory is a rigorous axiomatic theory developed in response to the discovery of serious flaws (such as Russell's Paradox) in naive set theory.
		Musical set theory may be considered the application of mathematical set theory to music.
	www.onlineencyclopedia.org /s/se/set_theory_1.html (224 words)

	set (Site not responding. Last check: 2007-10-20)
		Cantor's theorem states that the cardinality of the set of all subsets of a set A is strictly greater than the cardinality of A itself.
		The "number of elements" in a certain set is called the cardinal number of the set and denoted $A$ for a set $A$ (for a finite set this is an ordinary number, for an infinite set it differentiates between different "degrees of infiniteness", named $\aleph_0$ (aleph zero), $\aleph_1, \aleph_2...$ ).
		The set of functions from a set A to a set B is sometimes denoted by B
	www.yourencyclopedia.net /Set.html (1086 words)

	Encyclopedia: Musical set theory
		Naive set theory1 is distinguished from axiomatic set theory by the fact that the former regards sets as collections of objects, called the elements or members of the set, whereas the latter regards sets only as that which satisfies certain axioms.
		In music, especially in musical set theory, a trichord is a collection of three pitch classes, often one of the four ordered trichords in a tone row or set form.
		In music theory, the circle of fifths is a model of pitch space and is the series encompassing all of the notes in the equally tempered chromatic scale.
	www.nationmaster.com /encyclopedia/Musical-set-theory (2906 words)

	Musical set theory Article, Musicalsettheory Information (Site not responding. Last check: 2007-10-20)
		Although musical set theory may be considered the application of mathematical set theory to music, there is often little coincidence between the terminology and possibly the methods ofthe two.
		Musical settheory may, however, be considered as an unrelated field from mathematical set theory that, at the most, adapted some techniquesfrom mathematical set theory for its own uses.
		Multiplication is multiplying the pitch class numbers of a set, the most usefulmultipliers are 1, 5, 7, 11, as multiplication by 1 is the same, multiplication by 11 is inversion, multiplication of thechromatic scale by 5 produces the circle of fourths and multiplication by 7 producesthe circle of fifths.
	www.anoca.org /form/normal/musical_set_theory.html (1251 words)

	Set - Encyclopedia.WorldSearch (Site not responding. Last check: 2007-10-20)
		Sets are one of the most important and fundamental concepts in modern mathematics.
		Basic set theory, having only been invented at the end of the 19th century, is now a ubiquitous part of mathematics education, being introduced as early as elementary school.
		A set can also have an infinite number of members; for example, the set of natural numbers is infinite.
	encyclopedia.worldsearch.com /set.htm (1519 words)

	Set Theory Calculator Help - WebCalc
		Keep in mind that sets and set classes determined pitch content only; the composers remained free to fashion all other aspects of the music according to their artistic desires (at least until super-serialism, a philosophy of subjecting every aspect of the music to serial techniques, came into fashion in the 1950s).
		The set (2,9,10), for example, is not in normal form because the interval between 2 and 9 (7) is larger than the intervals between 9 and 10 (1) or between 10 and 2 (4).
		Sets with the same prime form contain the same number of pitches and the same collection of intervals between its pitches, hence they are in some sense aurally "equivalent," in much the same way that all major chords are aurally equivalent in tonal music.
	www.webcalc.net /calc/0515_help.php (2159 words)

	Type article - Type biology type species computer science datatype printing typesetting - What-Means.com (Site not responding. Last check: 2007-10-20)
		In the decades of the 1900s and 1910s, in philosophy, Bertrand Russell used his Theory of Types to further discuss the mapping of mathematics to logic.
		In set theory and musical set theory, see equivalence class.
		Martin Löf's intuitionistic type theory is a constructive alternative to set theory.
	www.what-means.com /encyclopedia/Type (188 words)

	Set (disambiguation) - Wikipedia, the free encyclopedia - TESTVERSION (Site not responding. Last check: 2007-10-20)
		The word set, which is among the words with the most numerous definitions in the English language (at 464 definitions according to the Oxford English Dictionary), may have one of the following meanings.
		Sets are a formalized notion of a collection of objects and are part of mathematics.
		Musical set theory for tone row, see collection for diatonic set theory.
	www.wissen-im-web.net /wiki/Sets (187 words)

	All About Musical Set Theory
		Notice that the clock-face graph of the inverted set is a mirror image of the original.
		The complement of a set consists of all notes not in the set.
		Allen Forte is on the music faculty at Yale University.
	www.jaytomlin.com /music/settheory/help.html (2147 words)

	Musical set theory (Site not responding. Last check: 2007-10-20)
		In addition to octave and enharmonic equivalency assumed in twelve tone theory and equal tempered tonal theory, set theory also makes use of inversionalal and transpositionalal equivalency.
		Thus the set of pitch classes 0, 1, and 2 is {0,1,2}, the ordered set
		These operations may also be called transformations, mappings or permutations; and in music theory, but not in mathematics, derivations.
	www.sciencedaily.com /encyclopedia/musical_set_theory (1373 words)

	Flexistentialism * More on Set Theory (Site not responding. Last check: 2007-10-20)
		Set Theory is related to Serialism, but revolves around the idea of organizing pitches into unique sets, and then organizing the sets by their properties, and then using related sets to make your music.
		Set Theory is not a contrived theory, but was based originally on analysis of compositions by Schoenberg and other Serialist composers as an attempt to glean more order and understanding from their methods.
		The inverse and complement of a set aren't very useful by themselves, but they are necessary to determine the two important values for a set, that being the Normal Form, and the Prime Form.
	www.flexistentialist.org /archives/000267.shtml (2455 words)

	Forte Set Theory: Pitch Content Analysis (Site not responding. Last check: 2007-10-20)
		An interval vector is a 6-digit number which enumerates the appearances of each interval class (within a chord, melody, musical event).
		Sets of cardinality of less than 2 or more than 10 are not considered as all sets are subset-related to them.
		While these conditions make classification possible, they are also the primary weaknesses of the system in the eyes of its critics.
	music.acu.edu /www/reid/class/caim/forteset.html (186 words)

	Amazon.com: Books: Axiomatic Set Theory (Site not responding. Last check: 2007-10-20)
		Set theory, the theory of types, and mathematical logic are still very important though in computer science and in artificial intelligence, due to the needs in these fields for knowledge representation, computational models of intelligence, and automated reasoning.
		The notion of a set is defined formally, and then the axiom of extensionality, which gives a criterion for two sets being equal, and the axiom schema schema of separation.
		The theory of denumerable sets is then discussed, followed by one of the most fascinating concepts in all of mathematics: the theory of transfinite and infinite cardinals.
	www.amazon.com /exec/obidos/tg/detail/-/0486616304?v=glance (2708 words)

	Musical set theory (Site not responding. Last check: 2007-10-20)
		Both theories make use of sets, but in the mathematical theory a set is an unordered collection of things, while in music theory what is called a set is, in the mathematical theory, a sequence, an ordered collection of things.
		In addition to octave and enharmonic equivalency assumed in twelve tone theory, set theory also makes use of inversionalal and transpositional equivalency.
		Set theory, like the twelve tone technique, makes use of integer notation and modulo 12.
	portaljuice.com /musical_set_theory.html (321 words)

	Articles - Inversion (music) (Site not responding. Last check: 2007-10-20)
		(iv) A notation for chord inversion often used in popular music is to write the name of a chord, followed by a forward slash, and then the name of the note that is to sound in the bass.
		Inversional equivalency is used little in tonal theory, though it is assumed a set which may be inverted onto another are remotely in common.
		In musical set theory inversion may be usefully thought of as the compound operation transpositional inversion, which is the same sense of inversion as in the Inverted melodies section above, with transposition carried out after inversion.
	lastring.com /articles/Inversion_(music)?mySession=a9e439fce5ce65417... (1234 words)

	Amazon.com: Books: Set Theory (Perspectives in Mathematical Logic) (Site not responding. Last check: 2007-10-20)
		This introduction to modern set theory covers all aspects of its two main general areas: classical set theory including large cardinals, infinitary combinatorics, desriptive set theory, and independence proofs starting with Goedel's proof around 1938 followed by Cohen's proof in 1963, whereby Cohen's method of forcing probably had a greater influence on mathematics.
		Set Theory (Studies in Logic and the Foundations of Mathematics) by Kenneth Kunen
		Set Theory and Logic by Robert R Stoll
	www.amazon.com /exec/obidos/tg/detail/-/3540630481?v=glance (1354 words)

	MU 458 Links
		Brief, easy-to-understand introduction to the fundamental concepts of set theory.
		Part I covers "nonlinear" set theory (in which the order of the pitch-classes in the set is not predetermined); Part II covers "linear" set theory, also known as serialism (in which the order of the pitch-classes in the set is significant).
		How (and why) atonal music and serialism came to be; step by step descriptions of how a composer might use serial techniques to create some music.
	www.dpo.uab.edu /~clemmons/458links.html (320 words)

	Music Theory Corner
		While I learned a lot about such topics as musical structure and counterpoint, I discovered that much of the harmonic theory that was taught in the program was unnecessarily complicated, and in some cases inconsistent or just plain incorrect.
		I don't claim to be a genius at music theory, but I have worked with one and I think I picked up a few things from him.
		Standard music theory as taught today is based on what classical composers did in the 18th century.
	www.standingstones.com /theorcnr.html (649 words)

	Set theory information online from schoolnewstoday.com (Site not responding. Last check: 2007-10-20)
		Set Theory for the Working Mathematician Krzysztof Ciesielski, CUP (1997).
		All About Musical Set Theory Set theory is not the same as serialism, but the two share many of the same methods...
		Set theory encompasses the notion of defining sets of pitches and...
	www.schoolnewstoday.com /set_theory-help.html (1621 words)

	Musical set theory - Education - Information - Educational Resources - Encyclopedia - Music
		A set is indicated by being enclosed in brackets:, an ordered set is indicated by
		Thus the set of pitch classes 0, 1, and 2 is, the ordered set
		is equivalent under transposition and/or inversion with twenty four rather than twelve sets, the twelve above and their inversions.
	www.music.us /education/M/Musical-set-theory.htm (1595 words)

	George Perle - Wikiquote
		It occurs most obviously in the music of Scriabin and the Vienna circle, Schoenberg, Webern, and Berg, in 1909-1910, and very soon afterwards, though less obviously, in the music of Bartok and Stravinsky.
		In Schoenberg's compositional practice, however, the concept of a segmental pitch-class content is represented as well, as a basis for the association of paired inversionally related set forms.
		The eigth-note chords of the flute and clarinets form alternately, with the sustaining oboes and horns, the six-tone sonorities labeled A and B. The sonorities A and B are both representatives of the same set class (6-Z19) and are thus made up of precisely the same intervals.
	en.wikiquote.org /wiki/George_Perle (620 words)

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