{{Short description|Distance between unordered pitch classes}} thumb|275px|right|Interval class {{audio|Interval class.mid|Play}}.

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'" ({{harvnb|Rahn|1980|loc=29}}; {{harvnb|Whittall|2008|loc=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:

[[Image:Octatonic ic7.JPG|400px|Octatonic motif]]

(To hear a MIDI realization, click the following: {{Audio|Octatonic_ic7.ogg|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

:<math>i (a,b) =\text{ the smaller of }i \langle a,b\rangle\text{ and }i \langle b,a\rangle,</math>

where ''i''{{angbr|''a'',&nbsp;''b''}} is an ordered pitch-class interval {{harv|Rahn|1980|loc=28}}.

While notating unordered intervals with parentheses, as in the example directly above, is perhaps the standard, some theorists, including Robert Morris,<ref>{{harvtxt|Morris|1991}}</ref> prefer to use braces, as in ''i''{''a'',&nbsp;''b''}. Both notations are considered acceptable.

== Table of interval class equivalencies ==

{| class="wikitable" |+'''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 || |}

==See also== *Pitch interval *Similarity relation

==References== {{reflist}}

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

==Further reading== *Friedmann, Michael (1990). ''Ear Training for Twentieth-Century Music''. New Haven: Yale University Press. {{ISBN|0-300-04536-0}} (cloth) {{ISBN|0-300-04537-9}} (pbk)

{{Set theory (music)}} {{Twelve-tone technique}}

Category:Musical set theory