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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.
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 .
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 ...
The first two, the "dependency locality theory" and the "expectancy theory" refer to syntactic processing in language, whereas the third one, the "tonal pitch space theory", relates to the syntactic processing in music. The language theories contribute to the concept that in order to conceive the structure of a sentence, resources are consumed.
(That set might be empty, currently.) For example, the extension of a function is a set of ordered pairs that pair up the arguments and values of the function; in other words, the function's graph. The extension of an object in abstract algebra, such as a group, is the underlying set of the object. The extension of a set is the set itself.
The temporal dynamics of music and language describes how the brain coordinates its different regions to process musical and vocal sounds. Both music and language feature rhythmic and melodic structure. Both employ a finite set of basic elements (such as tones or words) that are combined in ordered ways to create complete musical or lingual ideas.
Extensionalism, in the philosophy of language, in logic and semantics, is the view that all languages or at least all scientific languages should be extensional.It has been described as the default option for the scientism in the nineteenth century and the result of the application of empiricistic inductive methodology to the problem of semantics.
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 ...