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Topic: Enharmonic equivalency

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	Octave
		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.brainyencyclopedia.com /encyclopedia/o/oc/octave.html (698 words)

	CONK! Encyclopedia: Enharmonic_equivalency (Site not responding. Last check: 2007-10-23)
		For example, in twelve-tone equal temperament (the modern system of musical tuning in the west), the notes C sharp and D flat are enharmonically equivalent - that is, they are represented by the same key (on a musical keyboard, for example), and thus are identical in pitch, although they have different names and diatonic functionality.
		Thus, G sharp and A flat are not the same note; G sharp is, in fact 41 "cents" lower in pitch (41% of a semitone, not quite a quarter of a tone).
		An enharmonic is also one of the three Greek genera in music, in which the tetrachords are divided (descending) as a ditone plus two microtones.
	conk.com /search/encyclopedia.cgi?q=Enharmonic_equivalency (510 words)

	Equivalence relation - Wikipedia, the free encyclopedia
		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.
		Every equivalence relation on X defines a partition of X into subsets called equivalence classes: all elements equivalent to each other are put into one class.
	en.wikipedia.org /wiki/Graphing_equivalence (869 words)

	Equivalence class - Wikipedia, the free encyclopedia
		The equivalence classes are known as right cosets of H in G; one of them is H itself.
		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] and that any two equivalence classes are either equal or disjoint.
	www.wikipedia.org /wiki/Equivalence_class (764 words)

	Equivalence class -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-23)
		This equivalence relation is known as the (The choicest or most essential or most vital part of some idea or experience) kernel of f.
		Because of the properties of an equivalence relation it holds that a is in [a] and that any two equivalence classes are either equal or (Click link for more info and facts about disjoint) disjoint.
		It follows that the set of all equivalence classes of X forms a (A vertical structure that divides or separates (as a wall divides one room from another)) partition of X: every element of X belongs to one and only one equivalence class.
	www.absoluteastronomy.com /encyclopedia/E/Eq/Equivalence_class.htm (956 words)

	Equivalence relation -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-23)
		Every equivalence relation on X defines a (A vertical structure that divides or separates (as a wall divides one room from another)) partition of X into subsets called (Click link for more info and facts about equivalence class) equivalence classes: all elements equivalent to each other are put into one class.
		The equivalence classes of this relation are the right (Click link for more info and facts about coset) cosets of H in G.
		If an equivalence relation ~ on X is given, then the set of all its equivalence classes is the (Click link for more info and facts about quotient set) quotient set of X by ~ and is denoted by X/~.
	www.absoluteastronomy.com /encyclopedia/e/eq/equivalence_relation.htm (1103 words)

	Octave - Wikipedia, the free encyclopedia
		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.
	en.wikipedia.org /wiki/Octave (590 words)

	Equivalence class - Encyclopedia, History, Geography and Biography (Site not responding. Last check: 2007-10-23)
		The rational numbers can be constructed as the set of equivalence classes of ordered pairs of integers (a,b) with b not zero, where the equivalence relation is defined by
		The equivalence classes are known as right cosets of H in G.
		Every group can be partitioned into equivalence classes called conjugacy classes.
	www.arikah.net /encyclopedia/Equivalence_class (765 words)

	CONK! Encyclopedia: Musical_set_theory (Site not responding. Last check: 2007-10-23)
		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.
		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.
		The domain of all pitch class sets may be partitioned into types or equivalence classes based on cardinality or number of pitch classes, or other criteria.
	www.conk.com /search/encyclopedia.cgi?q=Musical_set_theory (1416 words)

	Enharmonic Equivalents (Site not responding. Last check: 2007-10-23)
		The word, enharmonic, originates in Greek music theory, but in its modern meaning, applies to music written for instruments tuned in so-called equal temperament.
		According to this system, at least three different names may be given to each pitch, so that, for example, F-sharp, E-double-sharp, and G-flat are all tuned exactly alike, and played by the same key on the piano keyboard.
		Thus, F-sharp is considered the enharmonic equivalent of G-flat and E-double-sharp.
	www.wholarts.com /music/ed/enharm.htm (94 words)

	Pitch (music) (Site not responding. Last check: 2007-10-23)
		In atonal, twelve tone, or set theory a "pitch" is a specific frequency while a pitch class is all the octaves of a frequency.
		Pitches are named with integers because of octave and enharmonic equivalency (for example, C# and Db are the same pitch while C4 and C5 are functionally the same).
		As well as various systems of musical tuning being used to determine the relative frequency of notes in a scale, various pitch standards have been used historically to fix the absolute position of the scale.
	www.bidprobe.com /en/wikipedia/p/pi/pitch__music_.html (1595 words)

	Modular arithmetic - Wikipedia, the free encyclopedia
		This is an equivalence relation, and the equivalence class of the integer a is denoted by [a]
		However, remainders (and so the modulo operation) are sometimes defined so that the remainder of a number has the same sign as that number, and using that definition this rule does not work.
		In music, modular arithmetic is used in the consideration of the twelve tone equally tempered scale, where octave and enharmonic equivalency occurs (that is, pitches in a 1∶2 or 2∶1 ratio are equivalent, and C-sharp is the same as D-flat).
	www.wikipedia.org /wiki/Modular_arithmetic (869 words)

	Station Information - Pitch class
		The term is often used in reference to serial music and the twelve-tone technique, which regulates the pitch class of notes but not their octave.
		In this case, because of enharmonic equivalency, it is more accurate to say that a pitch class is all of octaves of a pitch, which would be labeled by a number, not a traditional letter name.
		In both cases, when "pitch class" is used, the use of "pitch" indicates a specific frequency or note and does not include its octaves.
	www.stationinformation.com /encyclopedia/p/pi/pitch_class.html (128 words)

	Encyclopedia: Musical set theory (Site not responding. Last check: 2007-10-23)
		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 mathematics, an equivalence relation on a set X is a binary relation on X that is reflexive, symmetric and transitive, i.
		In mathematics, given a set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a: [a] = { x âˆˆ X
	www.nationmaster.com /encyclopedia/Musical-set-theory (2906 words)

	Tonality - Wikipedia, the free encyclopedia
		The majority of tonal music assumes that notes spaced over several octaves are perceived the same way as if they were played in one octave or octave equivalency.
		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.
	www.lighthousepoint.us /project/wikipedia/index.php/Tonality (3849 words)

	math lessons - Advanced modular arithmetic theory
		We call two integers a and b congruent modulo n if their difference is divisible by n, or equivalently, if they leave the same remainder when divided by n.
		This is an equivalence relation, and the equivalence class of an integer a is denoted by [a]
		In music, because of octave and enharmonic equivalency (that is, pitches in a 1/2 or 2/1 ratio are equivalent, and C# is the same as Db), modular arithmetic is used in the consideration of the twelve tone equally tempered scale, especially in twelve tone music.
	www.mathdaily.com /lessons/Advanced_modular_arithmetic_theory (1206 words)

	Enharmonic - SmartyBrain Encyclopedia and Dictionary (Site not responding. Last check: 2007-10-23)
		An elementary treatise on musical intervals and temperament,: With an account of an enharmonic harmonium exhibited in the loan collection of scientific...
		An essay on perfect intonation;: With remarks, showing the practicability of attaining it in the organ; together with a brief description of the enharmonic organ, of Messrs.
		The modern enharmonic scale as the basis of the chromatic element in music: (a new phase in the science of harmony) a lecture delivered at the Toronto...
	www.smartybrain.com /index.php/Enharmonic_equivalency (614 words)

	Musical set theory (Site not responding. Last check: 2007-10-23)
		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.
		Settheory 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.
		The domain of all pitch class sets may be partitioned into types or equivalence classes based on cardinality or number ofpitch classes, or other criteria.
	www.therfcc.org /musical-set-theory-18932.html (1223 words)

	Station Information - Musical set theory
		Musical set theory may, however, be considered as an unrelated field from mathematical set theory that, at the most, adapted some techniques from mathematical set theory for its own uses.
		In addition to octave and enharmonic equivalency assumed in twelve tone theory, set theory also makes use of inversionalal and transpositional equivalency.
		However, many theorists question or do not use inversional equivalency while Solomon simply adds a distinction to the end of the Forte prime form number of any sets which would be considered inversionally equivalent in his own Solomon Prime form number.
	www.stationinformation.com /encyclopedia/m/mu/musical_set_theory.html (312 words)

	Enharmonic Equivalency Encyclopedia Article, Definition, History, Biography (Site not responding. Last check: 2007-10-23)
		Looking For enharmonic equivalency - Find enharmonic equivalency and more at Lycos Search.
		Find enharmonic equivalency - Your relevant result is a click away!
		Look for enharmonic equivalency - Find enharmonic equivalency at one of the best sites the Internet has to offer!
	www.karr.net /encyclopedia/Enharmonic_equivalency (704 words)

	Inversion (music) (Site not responding. Last check: 2007-10-23)
		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 melodies, the inversion of a given melody is the melody turned upside-down - so if the original melody has a rising major third (see interval), the inverted melody has a falling major third.
	portaljuice.com /inversion__music_.html (644 words)

	Tonalsoft Encyclopaedia of Tuning -- W. S. B. Woolhouse's 'Essay on musical intervals', (c)1999 by Joe Monzo
		For the practical tuning of a keyed instrument, such as the organ, in which the full enharmonic scale is to be introduced, perhaps the best method after all would be to divide the octave into 19 equal intervals by 20 keys.
		He then analyzes the resources of a 53-EDO 'enharmonic organ', built by J. Robson and Son, St. Martin's-lane, but says that the number of keys is too much to be practicable, and settles again on 19-EDO.
		He then analyzes the resources of a 53-tET 'enharmonic organ', built by J. Robson and Son, St. Martin's-lane, but says that the number of keys is too much to be practicable, and settles again on 19-tET.
	sonic-arts.org /monzo/woolhouse/essay.htm (8660 words)

	equivalence relation - Article and Reference from OnPedia.com
		equivalence relation - Article and Reference from OnPedia.com
		If an equivalence relation ~ on X is given, then the set of all its equivalence classes is the quotient set of X by ~ and is denoted by X/~.
		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)

	Modular arithmetic (Site not responding. Last check: 2007-10-23)
		Modular arithmetic is applied in number theory, abstract algebra, cryptography, and visual and musical art.
		In music, because of octave and enharmonic equivalency (that is, pitches in a 1/2 or 2/1 ratio are equivalent, and C# is the same as Db), modular arithmetic is used in the consideration of the twelve tone equal temperamentequally tempered scale, especially in twelve tone techniquetwelve tone/ music.
		In visual art modular arithmetic can be used to create artistic patterns based on the multiplication and addition tables modulo ''n'' (see link below).
	www.infothis.com /find/Modular_arithmetic (415 words)

	Inversion (music) - Encyclopedia, History, Geography and Biography (Site not responding. Last check: 2007-10-23)
		Similarly, in twelve-tone technique, the inversion of the tone row is the so-called prime series turned upside-down.
		This page was last modified 20:08, 10 Mar 2005.
		The article about Inversion (music) contains information related to Inversion (music), Inverted chords, Notations for inverted chords, Inverted intervals, Inversion in counterpoint, Inverted melodies, Inversional equivalency and Inversion in musical set theory.
	www.arikah.com /encyclopedia/Inversional_equivalency (1262 words)

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