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[[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 '''interval mod 6''' (Rahn 1980, 29; Whittall 2008, 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 − 4 = 5 is less than 4 − 9 = −5 ≡ 7 (mod 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 − ''n''.


== Use of interval classes ==
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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.
 
Incidentally, the example's pitch [[Set theory (music)|collection]] forms an [[octatonic]] set.
 
== 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'' <''a'',&nbsp;''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 (composer)|Robert Morris]] (1991), prefer to use braces, as in ''i'' {''a'',''b}''. Both notations are considered acceptable.
 
== Table of interval class equivalencies ==
 
{| class="wikitable"
|+'''Interval Class Table'''
|-
! ic !! included intervals !! tonal counterparts
|-
!  0
| 0 || unison and octave
|-
! 1
| 1 and 11 || minor 2nd and major 7th
|-
!  2
| 2 and 10 || major 2nd and minor 7th
|-
!  3
| 3 and 9 || minor 3rd and major 6th
|-
!  4
| 4 and 8 || major 3rd and minor 6th
|-
!  5
| 5 and 7 || perfect 4th and perfect 5th
|-
!  6
| 6 || augmented 4th and diminished 5th
|}
 
==See also==
*[[Pitch interval]]
*[[Similarity relation]]
 
==Sources==
{{reflist}}
*Morris, Robert (1991). ''Class Notes for Atonal Music Theory''. Hanover, NH: Frog Peak Music.
*Rahn, John (1980). ''Basic Atonal Theory''. ISBN 0-02-873160-3. For forumala definitions only.
*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)
 
==External links==
*[http://solomonsmusic.net/setheory.htm#Basic%20Definition Solomon's Set Theory Primer]
 
{{Set theory (music)}}
 
[[Category:Musical set theory]]

Latest revision as of 14:11, 26 October 2014


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