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 &minus; 4 = 5 is less than 4 &minus; 9 = &minus;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 &minus; 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


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

where i$⟨a, b⟩$ is an ordered pitch-class interval.

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{a, b}. Both notations are considered acceptable.