Definitions

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. Thus, using scientific pitch notation the pitch class "C" is the infinite set

{Cn} = {..., C-2, C-1, C0, C1, C2, C3 ...}

Pitch class is important because human pitch-perception is periodic: pitches belonging to the same pitch class are perceived as having a similar "quality" or "color." Psychologists refer to the quality of a pitch as its "chroma"; music theorists use the term "pitch class" instead.

There is a subtle difference between the concepts "chroma" and "pitch class." A "chroma" is an attribute of pitches, just like whiteness is an attribute of white things. A "pitch class" is a set of all pitches sharing the same chroma, just like "the set of all white things" is the collection of all white objects. Music theory's use of the term "pitch class" rather than "chroma" reflects the influence of logical positivism on its founders, particularly Milton Babbitt.

Note that in standard Western equal temperament, distinct spellings can refer to the same sounding object: B#3, C4, and Dbb4 all refer to the same pitch, hence share the same chroma, and therefore belong to the same pitch class; a phenomenon called enharmonic equivalence.

To avoid the problem of enharmonic spellings, theorists typically represent pitch classes using numbers. One can map a pitch's fundamental frequency $f$ (measured in hertz) to a real number $p$ using the equation


p = 69 + 12log_2 {(f/440)}

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#/Db, 2 = D, 2.5 = "D quarter-tone sharp"

and so on. In this system, pitch classes which are represented by integers are pitch 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).

Other ways to label pitch classes

Pitch class
Pitch
class
Tonal counterparts
0 C (also B sharp, D double-flat)
1 C sharp, D flat (also B double-sharp)
2 D (also C double-sharp, E double-flat)
3 D sharp, E flat (also F double-flat)
4 E (also D double-sharp, F flat)
5 F (also E sharp, G double-flat)
6 F sharp, G flat (also E double-sharp)
7 G (also F double-sharp, A double-flat)
8 G sharp, A flat
9 A (also G double-sharp, B double-flat)
t or A A sharp, B flat (also C double-flat)
e or B B (also A double-sharp, C flat)
The system described above is flexible enough to describe any pitch class in any tuning system: for example, one can use the numbers {0, 2.4, 4.8, 7.2, 9.6} to refer to the five-tone scale that divides the octave evenly. However, in some contexts, it is convenient to use alternative labeling systems. For example, in just intonation, we may express pitches in terms of positive rational numbers p/q, expressed by reference to a 1 (often written "1/1") which represents a fixed pitch. If a and b are two positive rational numbers, they belong to the same pitch class if and only if

$a/b = 2^n,$

for some integer n. Therefore, we can represent pitch classes in this system using ratios p/q where neither p or q is divisible by 2, that is, as ratios of odd integers. Alternatively, we can represent just intonation pitch classes by reducing to the octave, $1 le p/q < 2$.

It is also very common to label pitch classes with reference to some scale. For example, one can label the pitch classes of n-tone equal temperament using the integers 0 to n-1. In much the same way, one could label the pitch classes of the C major scale, C-D-E-F-G-A-B using the numbers from 0 to 6. This system has two advantages over the continuous labeling system described above. First, it eliminates any suggestion that there is something natural about a 12-fold division of the octave. Second, it avoids pitch-class universes with unwieldy decimal expansions when considered relative to 12; for example, in the continuous system, the pitch-classes of 19-tet are labeled 0.63158... , 1.26316... , etc. Labeling these pitch classes {0,1,2,3 ... , 18} simplifies the arithmetic used in pitch-class set manipulations.

The disadvantage of the scale-based system is that it assigns an infinite number of different names to chords that sound identical. For example, in twelve-tone equal-temperament the C major triad is notated {0, 4, 7}. In twenty-four-tone equal-temperament, this same triad is labeled {0, 8, 14}. Moreover, the scale-based system appears to suggest that different tuning systems use steps of the same size ("1") but have octaves of differing size ("12" in 12-tone equal-temperament, "19" in 19-tone equal temperament, and so on), whereas in fact the opposite is true: different tuning systems divide the same octave into different-sized steps.

In general, it is often more useful to use the traditional integer system when one is working within a single temperament; when one is comparing chords in different temperaments, the continuous system can be more useful.