Integer Notation
To avoid the problem of enharmonic spellings, theorists typically represent pitch classes using numbers beginning from zero, with each successively larger integer representing a pitch class a semitone higher than the preceding one. Because octave-related pitches belong to the same class, when an octave is reached, the numbers begin again at zero. This cyclical system is referred to as modular arithmetic and, in the usual case of chromatic twelve-tone scales, pitch-class numbering is regarded as "mod 12"—that is, the twelfth member is identical to the first. One can map a pitch's fundamental frequency (measured in hertz) to a real number using the equation
This creates a linear pitch space in which octaves have size 12, semitones (the distance between adjacent keys on the piano keyboard) have size 1, and middle C is assigned the number 60. Indeed, the mapping from pitch to real numbers defined in this manner forms the basis of the MIDI Tuning Standard, which uses the real numbers from 0 to 127 to represent the pitches C-1 to G9. To represent pitch classes, we need to identify or "glue together" all pitches belonging to the same pitch class—i.e. all numbers p and p + 12. The result is a circular quotient space that musicians call pitch class space and mathematicians call R/12Z. Points in this space can be labelled using real numbers in the range 0 ≤ x < 12. These numbers provide numerical alternatives to the letter names of elementary music theory:
- 0 = C, 1 = C♯/D♭, 2 = D, 2.5 = "D quarter tone sharp", 3 = D♯/E♭,
and so on. In this system, pitch classes represented by integers are classes of twelve-tone equal temperament (assuming standard concert A).
To avoid confusing 10 with 1 and 0, some theorists assign pitch classes 10 and 11 the letters "t" (after "ten") and "e" (after "eleven"), respectively (or A and B, as in the writings of Allen Forte and Robert Morris).
In music, integer notation is the translation of pitch classes and/or interval classes into whole numbers. Thus C=0, C#=1 ... A#=10, B=11, with "10" and "11" substituted by "t" and "e" in some sources. This allows the most economical presentation of information regarding post-tonal materials.
In 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. 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 disadvantages 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.
Also, 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, and both this and the above may be viewed as disadvantages.
Read more about this topic: Pitch Class