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(ab) 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{ab}. Both notations are considered acceptable.

Table of interval class equivalencies