Interval class
Encyclopedia
In musical set theory, an interval class (often abbreviated: ic), also known as unordered pitch-class interval, interval distance, undirected interval, or (completely incorrectly, since this would mean, e.g., that a perfect fourth and a minor second are inversionally equivalent and belong to the same interval class, as would a unison
and a tritone
) 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
12. The largest interval class is 6 since any greater interval n may be reduced to 12 − n.
, enharmonic
, and inversional equivalency. Consider, for instance, the following passage:
(To hear a MIDI realization, click the following:
In the example above, all four labeled pitch-pairs, or dyad
s, 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.
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
(1991), prefer to use braces, as in i {a,b}. Both notations are considered acceptable.
Unison
In music, the word unison can be applied in more than one way. In general terms, it may refer to two notes sounding the same pitch, often but not always at the same time; or to the same musical voice being sounded by several voices or instruments together, either at the same pitch or at a distance...
and a tritone
Tritone
In classical music from Western culture, the tritone |tone]]) is traditionally defined as a musical interval composed of three whole tones. In a chromatic scale, each whole tone can be further divided into two semitones...
) interval mod 6 (Rahn 1980, 29; Whittall 2008, 273–74), is the shortest distance in pitch class space
Pitch class space
In music theory, pitch class space is the circular space representing all the notes in a musical octave.In this space, there is no distinction between tones that are separated by an integral number of octaves...
between two unordered pitch class
Pitch class
In music, a pitch class is a set of all pitches 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 positive numbers, and , a modulo n can be thought of as 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 octaveOctave
In music, an octave is the interval between one musical pitch and another with half or double its frequency. The octave relationship is a natural phenomenon that has been referred to as the "basic miracle of music", the use of which is "common in most musical systems"...
, enharmonic
Enharmonic
In modern musical notation and tuning, an enharmonic equivalent is a note , interval , or key signature which is equivalent to some other note, interval, or key signature, but "spelled", or named, differently...
, and inversional equivalency. Consider, for instance, the following passage:
In the example above, all four labeled pitch-pairs, or dyad
Dyad
Dyad may refer to:*Dyad , a pair of sister chromatids occurring in prophase I of meiosis; may also be used to describe protein morphology*Dyad , Greek philosophers' principle of "twoness" or "otherness"...
s, share a common "intervallic color." In atonal theory, this similarity is denoted by interval class—ic 5, in this case. Tonal
Tonality
Tonality is a system of music in which specific hierarchical pitch relationships are based on a key "center", or tonic. The term tonalité originated with Alexandre-Étienne Choron and was borrowed by François-Joseph Fétis in 1840...
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 tonal music, and then mostly developed in connection with atonal music by theorists such as Allen Forte , drawing...
forms an octatonic set.
Notation of interval classes
The unordered pitch class interval i (a, b) may be defined aswhere 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
Robert Morris (composer)
Robert Morris is an American composer and music theorist.-Work in music theory:As a music theorist, Morris' work has bridged an important gap between the rigorously academic and the highly experimental. Born in Cheltenham, England in 1943, Morris received his musical education at the Eastman...
(1991), prefer to use braces, as in i {a,b}. Both notations are considered acceptable.
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 |
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.
- 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)