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In set theory, the axiom of extensionality states that two sets are equal if and only if they contain the same elements. In mathematics formalized in set theory, it is common to identify relations—and, most importantly, functions —with their extension as stated above, so that it is impossible for two relations or functions with the same ...
Although musical set theory is often thought to involve the application of mathematical set theory to music, there are numerous differences between the methods and terminology of the two. For example, musicians use the terms transposition and inversion where mathematicians would use translation and reflection .
The axiom of extensionality, [1] [2] also called the axiom of extent, [3] [4] is an axiom used in many forms of axiomatic set theory, such as Zermelo–Fraenkel set theory. [5] [6] The axiom defines what a set is. [1] Informally, the axiom means that the two sets A and B are equal if and only if A and B have the same members.
A set (pitch set, pitch-class set, set class, set form, set genus, pitch collection) in music theory, as in mathematics and general parlance, is a collection of objects. In musical contexts the term is traditionally applied most often to collections of pitches or pitch-classes , but theorists have extended its use to other types of musical ...
Pitch class set theory, however, has adhered to formal definitions of equivalence." [ 1 ] Traditionally, octave equivalency is assumed, while inversional , permutational , and transpositional equivalency may or may not be considered ( sequences and modulations are techniques of the common practice period which are based on transpositional ...
The axioms of Zermelo set theory are stated for objects, some of which (but not necessarily all) are sets, and the remaining objects are urelements and not sets. Zermelo's language implicitly includes a membership relation ∈, an equality relation = (if it is not included in the underlying logic), and a unary predicate saying whether an object is a set.
In music theory, the spiral array model is an extended type of pitch space. A mathematical model involving concentric helices (an "array of spirals"), it represents human perceptions of pitches, chords, and keys in the same geometric space. It was proposed in 2000 by Elaine Chew in her MIT doctoral thesis Toward a Mathematical Model of Tonality ...
For example, the Oberschelp set theory developed the Zermelo–Fraenkel set theory within the framework of class logic. [5] Three principles guarantee that cumbersome ZF formulas are translatable into convenient classes formulas; guarantee a class logical increase in the ZF language they form without quantities axioms together with the axioms ...