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'" (Rahn 1980, 29; Whittall 2008, 273–74), is the shortest distance in pitch class space between two unordered pitch classes. For example, the interval class between pitch classes 4 and 9 is 5 because 9−4 = 5 is less than 4−9=−5≡7(mod12). See modular arithmetic for more on modulo 12. The largest interval class is 6 since any greater interval n may be reduced to 12−n.
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In the example above, all four labeled pitch-pairs, or dyads, share a common "intervallic color." In atonal theory, this similarity is denoted by interval class—ic 5, in this case. Tonal theory, however, classifies the four intervals differently: interval 1 as perfect fifth; 2, perfect twelfth; 3, diminished sixth; and 4, perfect fourth.
Notation of interval classes
The unordered pitch class intervali(a,b) may be defined as
where i⟨a,b⟩ is an ordered pitch-class interval (Rahn 1980, 28).
While notating unordered intervals with parentheses, as in the example directly above, is perhaps the standard, some theorists, including Robert Morris,[1] prefer to use braces, as in i{a,b}. Both notations are considered acceptable.
Whittall, Arnold (2008). The Cambridge Introduction to Serialism. New York: Cambridge University Press. ISBN978-0-521-68200-8 (pbk).
Further reading
Friedmann, Michael (1990). Ear Training for Twentieth-Century Music. New Haven: Yale University Press. ISBN0-300-04536-0 (cloth) ISBN0-300-04537-9 (pbk)
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