Interval class
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In musical set theory, an interval class (usual abbreviation: ic) 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 minus n.
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[edit] Use of interval classes
The concept of interval class was created to account for octave, enharmonic, and inversional equivalency. Consider, for instance, the following passage:
(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. 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 forms an octatonic set.
[edit] 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 > , 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 (1991), prefer to use braces, as in i {a,b}. Both notations are considered acceptable.
[edit] Table of interval class equivalencies
| 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 |
[edit] Sources
- Morris, Robert (1991). Class Notes for Atonal Music Theory. ASIN B0006DHW9I.
- Rahn, John (1980). Basic Atonal Theory. ISBN 0-02-873160-3. For forumala definitions only.
[edit] Further reading
- Friedmann, Michael (1990). Ear Training for Twentieth Century Music. ISBN 0-300-04537-9.
