Interval class

Distance between unordered pitch classes
Interval class 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'" (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.

Use of interval classes

The concept of interval class accounts for octave, enharmonic, and inversional equivalency. Consider, for instance, the following passage:

Octatonic motif

(To hear a MIDI realization, click the following: 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

i ( a , b ) =  the smaller of  i ⟨ a , b ⟩  and  i ⟨ b , a ⟩ , {\displaystyle 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 (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.

Table of interval class equivalencies

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

Sources

  • 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.
  • 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)
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Musical set theory
  • All-interval tetrachord
  • All-trichord hexachord
  • Complement
  • Forte number
  • Identity
  • Interval class
  • Interval vector
  • Multiplication
  • Permutation
  • Pitch class
  • Pitch interval
  • Pitch-interval class
  • Set
  • Similarity relation
  • Transformation
  • Z-relation
Diatonic
set theory