Pitch class

From Free net encyclopedia

Human pitch-perception is periodic: pitches separated by an integral number of octaves are perceived as having a similar "quality" or "color." Psychologists refer to the quality of a pitch as its "chroma." Music theorists typically use the term "pitch class" rather than "chroma."

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.

The pitch class "C" is an infinite set containing all pitches with the chroma C, no matter what octave they are in. Thus, using scientific pitch notation it is the infinite set

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

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.

To avoid the problem of enharmonic spellings, theorists typically represent pitch classes using numbers. One can map a pitch's fundamental frequency <math> f </math> to a real number <math> p </math> using the equation

<math>

p = 69 + 12\log_2 {(f/440)} </math>

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. 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 12 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).

**PITCH CLASS TABLE**
	pc	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)

[edit]

Other ways to label pitch classes

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. Indeed, the real numbers defined in this manner form the basis of the MIDI Tuning Standard, which simply uses the result translated into base 128. 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

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, <math>1 \le p/q < 2</math>.

[edit]