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Topic: Inversional equivalency

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	equivalence relation - Article and Reference from OnPedia.com
		Equivalence relations are often used to group together objects that are similar in some sense.
		Green's relations are five equivalence relations on the elements of a semigroup.
		Consider for instance the square X = 0,1x0,1 and the equivalence relation on X generated by the requirements (a,0) ~ (a,1) for all a in 0,1 and (0,b) ~ (1,b) for all b in 0,1.
	www.onpedia.com /encyclopedia/equivalence-relation (784 words)

	Inversion (music) (Site not responding. Last check: 2007-11-05)
		Under inversion, perfect intervals remain perfect, major intervals become minor and the reverse, augmented intervals become diminished and the reverse, seconds become sevenths and the reverse, thirds become sixes and the reverse, and fourths become fifths and the reverse.
		Inversional equivalency is the concept that intervalss, chordss, and other sets of pitches are the same when inverted.
		Inversional equivalency is used little in tonal theory, though it is assumed a set which may be inverted onto another are remotely in common.
	uncover.us /en/wikipedia/i/in/inversion__music_.html (1095 words)

	equivalence class - Article and Reference from OnPedia.com
		The homotopy class of a continuous map f is the equivalence class of all maps homotopic to f.
		In natural language processing, an equivalence class is a set of all references to a single person, place, thing, or event, either real or conceptual.
		Because of the properties of an equivalence relation it holds that a is in a" title="a">a and that any two equivalence classes are either equal or disjoint.
	www.onpedia.com /encyclopedia/equivalence-class (729 words)

	Inversion (music) (Site not responding. Last check: 2007-11-05)
		Inversional equivalency is the concept that intervalss and chordss are the same when inverted.
		It is similar to enharmonic equivalency and octave equivalency and even transpositional equivalency but unlike these is used almost exclusively only in musical set theory.
		When applied to counterpoint, a contrapuntal inversion of two melodies simultaneously being played by two voices is the switching of the melodies between voices, so that the upper-voice melody is now played in the lower voice, and vice versa.
	encyclopedia.jigyasa.in /wikipedia/i/in/inversion__music_.html (668 words)

	CONK! Encyclopedia: Musical_set_theory (Site not responding. Last check: 2007-11-05)
		In addition to octave and enharmonic equivalency assumed in twelve tone theory and equal tempered tonal theory, set theory also makes use of inversional and transpositional equivalency, though the degree of equivalency varies among theorists.
		Inversion is turning a set upside-down reversing the order of the intervals between pitch classes.
		Multiplication is multiplying the pitch class numbers of a set, the most useful multipliers are 1, 5, 7, 11, as multiplication by 1 is the same, multiplication by 11 is inversion, multiplication of the chromatic scale by 5 produces the circle of fourths and multiplication by 7 produces the circle of fifths.
	www.conk.com /search/encyclopedia.cgi?q=Musical_set_theory (1416 words)

	Inversion (music) - Wikpedia (Site not responding. Last check: 2007-11-05)
		An interval is inverted by raising or lowering either of the notes the necessary number of octaves, so that both retain their names (pitch class) and the one which was higher is now lower and vice versa, changing the perspective or relation between the pitch classes.
		Under inversion, perfect intervals remain perfect, major intervals become minor and the reverse, augmented intervals become diminished and the reverse.
		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.
	www.bostoncoop.net /~tpryor/wiki/index.php?title=Inversional_equivalency (1205 words)

	Read about Equivalence relation at WorldVillage Encyclopedia. Research Equivalence relation and learn about Equivalence ... (Site not responding. Last check: 2007-11-05)
		Green's relations are five equivalence relations on the elements of a
		natural numbers is not an equivalence relation, because although it is reflexive and symmetric, it is not transitive (2 and 6 have a common factor greater than 1, and 6 and 3 have a common factor greater than 1, but 2 and 3 do not have a common factor greater than 1).
		equivalence classes: all elements equivalent to each other are put into one class.
	encyclopedia.worldvillage.com /s/b/Equivalence_relation (794 words)

	CONK! Encyclopedia: Octave (Site not responding. Last check: 2007-11-05)
		This is similar to enharmonic equivalency, and less so transpositional equivalency and, less still, inversional equivalency, the latter two of which are generally used only in musical set theory or atonal theory.
		Octave equivalency is a part of most musics, but is far from universal in "primitive" and early music (e.g., Nettl, 1956; Sachs and Kunst, 1962).
		As well as being used to describe the relationship between two notes, the word is also used when speaking of a range of notes that fall between a pair an octave apart.
	www.conk.com /search/encyclopedia.cgi?q=Octave (564 words)

	Read about Octave at WorldVillage Encyclopedia. Research Octave and learn about Octave here! (Site not responding. Last check: 2007-11-05)
		equivalency, and is closely related to the concept of harmonics.
		inversional equivalency, the latter two of which are generally used only in
		Octave equivalency is a part of most musics, but is far from universal in "primitive" and
	encyclopedia.worldvillage.com /s/b/Octave (455 words)

	Equivalence relation (Site not responding. Last check: 2007-11-05)
		The relation "has a common factor with" between natural numbers is not an equivalence relation, because although it is reflexive and symmetric, it is not transitive (2 and 6 have a common factor, and 6 and 3 have a common factor, but 2 and 3 do not have a common factor).
		The relation "is the mother of" on the set of all human beings is not and equivalence relation, because although it is reflexive and symmetric, it is not transitive (if A is the mother of B, and B is the mother of C, it does not necessarily mean A is the mother of C)
		In music see octave equivalency, transpositional equivalency, inversional equivalency, enharmonic equivalency, rotationalal equivalency.
	www.brainyencyclopedia.com /encyclopedia/e/eq/equivalence_relation.html (872 words)

	Tonality (Site not responding. Last check: 2007-11-05)
		The first inversion would be E-G-C, and the second inversion would be G-C-E. To summarize, traditional tonal music is described in terms of a scale of notes.
		This is generally held to imply that a note which has different places in a chord will be heard differently, and that therefore there is not enharmonic equivalency.
		In tonal music chords which are moved to different keys, or played with different root notes are not perceived as being the same, and thus transpositional equivalency and far less still inversional equivalency are not generally held to apply.
	uncover.us /en/wikipedia/t/to/tonality.html (3448 words)

	Musical set theory (Site not responding. Last check: 2007-11-05)
		transpositional equivalency, though the degree of equivalency varies among theorists.
		Set theory does not, however, use diatonic functionality that is assumed in tonal theory, and this is the reason for the use of
		equivalence classes based on cardinality or number of pitch classes, or other criteria.
	en.efactory.pl /Musical_set_theory (1347 words)

	Upto11.net - Wikipedia Article for Interval class
		In music, specifically, musical set theory an interval class, or unordered pitch-class interval, is an interval measured by the distance between its two pitch classes ordered so they are as close as possible.
		It was created to account for octave and inversional equivalency.
		Since there are 12 pitch classes in the equal tempered scale the largest interval class is 6 semitones, since any interval larger than that is not the closest ordering, see complement.
	upto11.net /generic_wiki.php?q=interval_class (118 words)

	The Aural Perception of Pitch-Class Set Relations: A Computer-Assisted Investigation (Site not responding. Last check: 2007-11-05)
		The perception test was designed to measure how recognition of set equivalency may be affected by the perceptual ability of the listener, the set types used and the particular set manipulation involved in the equivalency relationship.
		The test paradigm consisted of the presentation of an original set followed by three comparison sets, one of which was transpositionally or inversionally equivalent to the original, the other two being non-equivalent lures in an Rp, R1 or R2 similarity relation with the original.
		Statistical tests of subject-response data showed that: subjects with absolute-pitch recognition performed with more accuracy; ordered transformations were more recognizable than reordered transformations; transpositional equivalencies were more discernable than inversional equivalencies; octave displacement disguised set equivalency; and non-equivalent sets with similarity through contour or successive interval invariance were easily confused with equivalent sets.
	www.music-cog.ohio-state.edu /Gibson/millar.abstract.html (323 words)

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