Music Group Theory

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Music Group Theory is an abstract system of isomorphisms, built on Music Set Theory in conjunction with Modern Algebra, which seeks to form a cohesive analytical system whereby meaningful general observations can be made about music groups, either individually or collectively. Prior to the advent of Music Group Theory, music fans were limited to making individual observations about specific music groups or, even, specific musical works. Music Group Theory allows groups whose music is substantially the same to be lumped together in an equivalence class and appraised and reviewed collectively.

The set of all black musicians.

[edit] Music Sets

Music Set Theory forms the basis of Music Group Theory. A Music Set is a notional collection of musical items (such as musicians, music groups, notes, chords, songs, albums, keys, modes, or genres). A Music Set may be defined either by listing each item that belongs to the set, or by defining some rule which may be applied to any musical item to determine whether it belongs to the set.

Examples of music sets:

• The set containing Bach, Vivaldi, Pachelbel, and Weird Al Yankovic
• The set of all major and harmonic minor keys whose key signature contains no flats and no more than seven sharps
• The set of all songs whose titles begin with a vowel

[edit] Music Operations

In addition to sets, Music Group Theory allows one to define Music Operations, or things which may be done (or which one may imagine being done) to music items. Each operation is defined with respect to a particular music set and may be used to create a new set containing the results of performing the operation on each member of the original set.

Examples of music sets with operations:

• The set of all songs written in C Major
  • The operation whereby the song is transcribed into G Major
• The set containing Industrial Metal, Hard Rock, and Polka
  • The operation whereby the number of people who regularly listen to the genre is computed
• The set containing Michael Jackson, Barry Manilow, and Lance Bass
  • The operation whereby the gender of the singer is changed

[edit] Definition of a Music Group

The set of all blues musicians whose talent came from a pact with Satan.

For any given music set with one or more music operations defined on it, if certain things can be proven about the operations with respect to the set, then the set and its operations will form a music group. If it can be once proven that a given set and its operations form a music group, then various useful deductions can be made about the group's music.

In order for a music set and its operations to be a music group, the following must be proven:

• The operation and the set are well-defined, i.e., the set contains each member of the group exactly once, and the operation may be performed on every member of the group.
• The operation is associative, i.e., it associates each member of the group with a musical item (typically, an instrument).
• The operation is communative, i.e., it builds a community of fans who will follow the group's music.
• The operation is distributive, i.e., it provides a way to distribute the group's music to the public. Traditionally a major record label is used for the distributive operation, but more recently some groups have been successfully formed using an internet-based distributive operation.

It has been proposed that additional requirements be enacted, e.g., to require it to be proven that every member of the group has musical talent and/or musical training before the group may be formed. However, no consensus has been reached about what government body would be responsible for enforcement of these additional requirements, and so as of 2007 they have not been put in place.

[edit] Real and Imaginary Groups

While it is useful to analyze existing music groups via music group theory, it is also possible to use the theory to construct and analyze groups that currently do not exist. For instance, it is possible to define the set containing Garth Brooks and Doctor Dre and, by defining suitable operations, to determine what music group they could form together, what the music would be like, who would listen to it, and so forth.

[edit] Musical Isomorphisms

The set of all psychedelic drugs favoured by musicians.

An important part of music group theory is the establishment of isomorphisms, or formally defined similarities between musical items.

Some examples of isomorphisms:

• Michael Jackson and Barney both sing kinda funny; they both dance kinda funny; both are of an indeterminate colour; and they both like little kids. Thus, in these four ways, Michael Jackson is isomorphic to Barney.
• Amazing Grace and the theme song from Gilligan's Island are written in the same time signature, with the same pattern of measures per stanza (discounting the repeating last line). As a result, the two tunes are interchangeable, and the words from one may be sung to the other tune.

[edit] Equivalence Classes

Building on the concept of isomorphisms, it is also possible to define music equivalence classes, wherein two or more music groups are found to be isomorphic in such significant ways that their music may be considered equivalent.

Examples of music equivalence classes:

• Britney Spears and Christina Aguilera belong to an equivalence class. Because their styles are isomorphic, it is not possible (except by listing specific song titles) to create a rule that distinguishes their music. Thus, either of them is interchangeable with the other.

[edit] See Also

• Music Topology