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interval class

{{Short description|Distance between unordered pitch classes}}

Image:Interval class.png

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 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

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

where i{{angbr|a, 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,{{harvtxt|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

icincluded intervalstonal counterpartsextended 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

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