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'" (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 (mod 12). 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.

Use of interval classes

The concept of interval class accounts for octave, enharmonic, and inversional equivalency. Consider, for instance, the following passage:

(To hear a MIDI realization, click the following: 106 KB^ⓘ

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 interval i(a, b) may be defined as

i(a,b)={\text{ the smaller of }}i\langle a,b\rangle {\text{ and }}i\langle b,a\rangle ,

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.

Table of interval class equivalencies

**Interval Class Table**
ic	included intervals	tonal counterparts	extended intervals
0	0	unison and octave	diminished 2nd and augmented 7th
1	1 and 11	minor 2nd and major 7th	augmented unison and diminished octave
2	2 and 10	major 2nd and minor 7th	diminished 3rd and augmented 6th
3	3 and 9	minor 3rd and major 6th	augmented 2nd and diminished 7th
4	4 and 8	major 3rd and minor 6th	diminished 4th and augmented 5th
5	5 and 7	perfect 4th and perfect 5th	augmented 3rd and diminished 6th
6	6	augmented 4th and diminished 5th

References

^ Morris (1991)

Sources

Morris, Robert (1991). Class Notes for Atonal Music Theory. Hanover, NH: Frog Peak Music.
Rahn, John (1980). Basic Atonal Theory. ISBN 0-02-873160-3.
Whittall, Arnold (2008). The Cambridge Introduction to Serialism. New York: Cambridge University Press. ISBN 978-0-521-68200-8 (pbk).

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