In
musical set theory
Musical set theory provides concepts for categorizing musical objects and describing their relationships. Howard Hanson first elaborated many of the concepts for analyzing tonality, tonal music. Other theorists, such as Allen Forte, further devel ...
, 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
pitch class space
In music theory, pitch-class space is the circular space representing all the Musical note, notes (pitch classes) in a musical octave. In this space, there is no distinction between tones separated by an integral number of octaves. For example, C4, ...
between two unordered
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 positio ...
es. For example, the interval class between pitch classes 4 and 9 is 5 because 9 − 4 = 5 is less than 4 − 9 = −5 ≡ 7 (mod 12). See
modular arithmetic
In mathematics, modular arithmetic is a system of arithmetic operations for integers, other than the usual ones from elementary arithmetic, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to mo ...
for more on
modulo
In computing and mathematics, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another, the latter being called the '' modulus'' of the operation.
Given two positive numbers and , mo ...
12. The largest interval class is 6 since any greater interval ''n'' may be reduced to 12 − ''n''.
Use of interval classes
The concept of interval class accounts for
octave
In music, an octave (: eighth) or perfect octave (sometimes called the diapason) is an interval between two notes, one having twice the frequency of vibration of the other. The octave relationship is a natural phenomenon that has been referr ...
,
enharmonic
In music, two written notes have enharmonic equivalence if they produce the same pitch but are notated differently. Similarly, written intervals, chords, or key signatures are considered enharmonic if they represent identical pitches that ar ...
, and
inversional equivalency
In music theory, an inversion is a rearrangement of the top-to-bottom elements in an interval, a chord, a melody, or a group of contrapuntal lines of music. In each of these cases, "inversion" has a distinct but related meaning. The concept of i ...
. Consider, for instance, the following passage:
(To hear a MIDI realization, click the following:
In the example above, all four labeled pitch-pairs, or
dyads, share a common "intervallic color." In
atonal
Atonality in its broadest sense is music that lacks a tonal center, or key. ''Atonality'', in this sense, usually describes compositions written from about the early 20th-century to the present day, where a hierarchy of harmonies focusing on ...
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 interval ''i''(''a'', ''b'') may be defined as
:
where ''i'' is an ordered pitch-class interval .
While notating unordered intervals with parentheses, as in the example directly above, is perhaps the standard, some theorists, including
Robert Morris,
prefer to use braces, as in ''i''. Both notations are considered acceptable.
Table of interval class equivalencies
See also
*
Pitch interval
*
Similarity relation
References
Sources
*
*
*
Further reading
*Friedmann, Michael (1990). ''Ear Training for Twentieth-Century Music''. New Haven: Yale University Press. (cloth) (pbk)
{{Twelve-tone technique
Musical set theory