View source for Interval class

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

In [[set theory (music)|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 class]]es. For example, the interval class between pitch classes 4 and 9 is 5 because 9&nbsp;&minus;&nbsp;4 = 5 is less than 4&nbsp;&minus;&nbsp;9&nbsp;=&nbsp;&minus;5&nbsp;≡&nbsp;7&nbsp;(mod&nbsp;12). See [[modular arithmetic]] for more on [[modulo operation|modulo]] 12. The largest interval class is 6 since any greater interval ''n'' may be reduced to 12&nbsp;&minus;&nbsp;''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 [[Dyad (music)|dyads]], share a common "intervallic color." In [[atonal]] theory, this similarity is denoted by interval class—ic 5, in this case. [[tonality|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'',&nbsp;''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 (composer)|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]]