Integer notation

Integer notation

In integer notation, or the integer model of pitch, all pitch classes and intervals between pitch classes are designated using the numbers 0 through 11. It is not used to notate music for performance, but is a common analytical and compositional tool when working with chromatic music, including twelve tone, serial, or otherwise atonal music.

Pitch classes can be notated in this way by assigning the number 0 to some note - C natural by convention - and assigning consecutive integers to consecutive semitones; so if 0 is C natural, 1 is C sharp, 2 is D natural and so on up to 11 which is B natural. (See pitch class.) The C above this is not 12, but 0 again (12-12=0). Thus arithmetic modulo 12 is used to represent octave equivalence. One advantage of this system is that it ignores the "spelling" of notes (B sharp, C natural and D double-flat are all 0) according to their diatonic functionality.

There are a few drawbacks with integer notation. First, theorists have traditionally used the same integers to indicate elements of different tuning systems. Thus, the numbers 0, 1, 2, ... 5, are used to notate pitch classes in 6-tone equal temperament. This means that the meaning of a given integer changes with the underlying tuning system: "1" can refer to C♯ in 12-tone equal temperament, but D in 6-tone equal temperament. Second, integer notation does not seem to allow for the notation of microtones, or notes not belonging to the underlying equal division of the octave. For these reasons, some theorists have recently advocated using rational numbers to represent pitches and pitch classes, in a way that is not dependent on any underlying division of the octave. See the articles on pitch and pitch class for more information.

Another drawback with integer notation is that the same numbers are used to represent both pitches and intervals. For example, the number "4" serves both as a label for the pitch class E (if C=0) and as a label for the "distance" between the pitch classes D and F♯. (In much the same way, the term "10 degrees" can function as a label both for a temperature, and for the distance between two temperatures.) Only one of these labelings is sensitive to the (arbitrary) choice of pitch class 0. For example, if one makes a different choice about which pitch class is labeled 0, then the pitch class E will no longer be labelled "4." However, the distance between D and F♯ will still be assigned the number 4. The late music theorist David Lewin was particularly sensitive to the confusions that this can cause.

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