# 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 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:

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

:$i\; (a,b)\; =\; ext\{\; the\; smaller\; of\; \}i\; langle\; a,b\; angle\; ext\{\; and\; \}i\; langle\; b,a\; angle,$

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 , prefer to use braces, as in

*i*{

*a*,

*b*}. Both notations are considered acceptable.