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Interval Class

 
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In musical set theory, an interval class (usual abbreviation: ic) is the shortest distance in pitch class space
Pitch class space

In music theory, pitch class space is the circular space that results when we ignore the difference between octave-related pitches. Mathematically, it is a quotient space that results from identifying or "gluing together" pitches sharing the same pitch class....
 between two unordered pitch class
Pitch class

In music, a pitch class is a set of all Pitch that are a whole number of octaves apart, e.g. the pitch class C consists of the Cs in all octaves....
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
Modular arithmetic

In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value — the modulus....
 for more on modulo
Modulo operation

In computing, the modulo operation finds the remainder of division of one number by another.Given two numbers, and , a modulo n is the remainder, on division of a by n....
 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 was created to account for octave
Octave

In music, an octave The octave is occasionally referred to as a diapason.The octave above an indicated note is sometimes abbreviated 8va, and the octave below 8vb....
, enharmonic
Enharmonic

In modern music and musical notation, an enharmonic equivalent is a note , interval , or key signature which is equivalence to some other note, interval, or key signature, but "spelled", or named, differently....
, and inversional equivalency.






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In musical set theory, an interval class (usual abbreviation: ic) is the shortest distance in pitch class space
Pitch class space

In music theory, pitch class space is the circular space that results when we ignore the difference between octave-related pitches. Mathematically, it is a quotient space that results from identifying or "gluing together" pitches sharing the same pitch class....
 between two unordered pitch class
Pitch class

In music, a pitch class is a set of all Pitch that are a whole number of octaves apart, e.g. the pitch class C consists of the Cs in all octaves....
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
Modular arithmetic

In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value — the modulus....
 for more on modulo
Modulo operation

In computing, the modulo operation finds the remainder of division of one number by another.Given two numbers, and , a modulo n is the remainder, on division of a by n....
 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 was created to account for octave
Octave

In music, an octave The octave is occasionally referred to as a diapason.The octave above an indicated note is sometimes abbreviated 8va, and the octave below 8vb....
, enharmonic
Enharmonic

In modern music and musical notation, an enharmonic equivalent is a note , interval , or key signature which is equivalence to some other note, interval, or key signature, but "spelled", or named, differently....
, and inversional equivalency. Consider, for instance, the following passage:

Octatonic Ic7
(To hear a MIDI realization, click the following:

In the example above, all four labeled pitch-pairs, or dyad
Dyad

Etymology: Late Latin dyad-, dyas, from Greek d?a?, from dyodyad may refer to:*dyad , according to the Pythagoreans, is the principle of "twoness" or "otherness"...
s, share a common "intervallic color." In atonal
Atonal

Atonal may refer to:*AtonalityAtonal or Atonaltzin may refer to:*Atonal I*Atonal II...
 theory, this similarity is denoted by interval class -- ic 5, in this case. Tonal
Tonal

Tonal may refer to:* Tonal , a concept appearing in the belief systems and traditions of Mesoamerican cultures, involving a spiritual link between a person and an animal...
 theory, however, classifies the four intervals differently: interval 1 as perfect fifth; 2, perfect twelfth; 3, diminished sixth; and 4, perfect fourth. Thus we see that in a dodecaphonic (i.e., chromatic) context, terminology tailored for the analysis of heptatonic (i.e., diatonic) music is often no longer suitable.

Incidentally, the example's pitch collection
Set theory (music)

Musical set theory provides concepts for categorizing musical objects and describing their relationships. Many of the notions were first elaborated by Howard Hanson in connection with tonality music, and then mostly developed in connection with atonal music; the concepts of set theory are very general and can be applied to tonal and atonal...
 forms an octatonic set.

Notation of interval classes


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

where i <a, b> is an ordered pitch class interval.

While notating unordered intervals with parentheses, as in the example directly above, is perhaps the standard, some theorists, including Robert Morris
Robert Morris (composer)

Robert Morris is an United States composer and music theorist....
 (1991), prefer to use braces, as in i . Both notations are considered acceptable.

Table of interval class equivalencies


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


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. For forumala definitions only.


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