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*.

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.

The **unordered pitch class interval** *i*(*a*, *b*) may be defined as

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.

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 |