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In
In music
Music is generally defined as the The arts, art of arranging sound to create some combination of Musical form, form, harmony, melody, rhythm or otherwise Musical expression, expressive content. Exact definition of music, definitions of mu ...
, a similarity relation or pitch-class similarity is a comparison between sets of the same cardinality (two sets containing the same number of pitch classes), based upon shared pitch class
In music, a pitch class (p.c. or pc) is a set of all pitches that are a whole number of octaves apart; for example, the pitch class C consists of the Cs in all octaves. "The pitch class C stands for all possible Cs, in whatever octave positi ...
and/or interval class
In musical set theory, an interval class (often abbreviated: ic), also known as unordered pitch-class interval, interval distance, undirected interval, or "(even completely incorrectly) as 'interval mod 6'" (; ), is the shortest distance in pitc ...
content.
Allen Forte Allen, Allen's or Allens may refer to:
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* Allen Center, a skyscraper complex in downtown Houston, Texas
* Allen Fieldhouse, an indoor sports arena on the Unive ...
originally designated four types: Rp (maximal similarity with respect to pitch class), R0 (minimal similarity), R1 (first order maximal similarity), and R2 (second order maximal similarity). In Rp one pitch class is different, in R0 all are different, and in R1 and R2 four interval classes are the same.
Rp is defined for sets S1 and S2 of cardinal number
In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. T ...
n and S3 of cardinal number n-1 as:Forte, Allen (1977). ''The Structure of Atonal Music'', p.47. .
:Rp(S1,S2) iff
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bicondi ...
(S3 ⊂
In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
S1, S3 ⊂ S2)
Meaning that S1 and S2 each have all the pitch-classes of S3 (transposed or inverted), plus one.
See also
*Equivalence class (music)
In music theory, equivalence class is an equality ( =) or equivalence between properties of sets (unordered) or twelve-tone rows (ordered sets). A relation rather than an operation, it may be contrasted with derivation.Schuijer (2008). ''An ...
References
Further reading
*Isaacson, Eric. (1990). "Similarity of Interval-class Content Between Pitch-class Sets: The IcVSIM ''V''ector ''SIM''ilarity"">Interval_vector.html" ;"title="''I''nterval ''c''lass Interval vector">''V''ector ''SIM''ilarity"Relation", ''Journal of Music Theory'' 34: 1-28.External links
{{Set theory (music) Musical set theory