Pitch class

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  {
\override Score.TimeSignature #'stencil = ##f
\relative c' {
  \clef treble \key c \major \time 4/4
  <c c'>1
} }
Perfect octave
  {
\override Score.TimeSignature #'stencil = ##f
\new PianoStaff <<
\new Staff \relative c' {
  \clef treble \key c \major \time 4/4
  <c' c' c'>1 \bar "|."
}

\new Staff \relative c' {
  \clef bass \key c \major \time 4/4
  <c c, c, c,>1
} >> }
All Cs from C1 to C7 inclusive

In

enharmonic equivalence, or both."[2] Thus, using scientific pitch notation
, the pitch class "C" is the set

{Cn : n is an integer} = {..., C−2, C−1, C0, C1, C2, C3 ...}.

Although there is no formal upper or lower limit to this sequence, only a few of these pitches are audible to humans. Pitch class is important because human

octave equivalence
".

Psychologists refer to the quality of a pitch as its "chroma".[3] A chroma is an attribute of pitches (as opposed to tone height), just like hue is an attribute of color. A pitch class is a set of all pitches that share the same chroma, just like "the set of all white things" is the collection of all white objects.[4]

In standard Western

enharmonic equivalence
.

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 that would be one semitone higher than the preceding one, if they were all realised as actual pitches in the same octave. 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 "modulo 12" (customarily abbreviated "mod 12" in the music-theory literature)—that is, every twelfth member is identical. One can map a pitch's fundamental frequency f (measured in hertz) to a real number p using the equation

This creates a linear

MIDI Tuning Standard, which uses the real numbers from 0 to 127 to represent the pitches C−1 to G9 (thus, middle C is 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 cyclical quotient group that music theorists 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 = Dhalf sharp (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).

  {
\override Score.TimeSignature #'stencil = ##f
\relative c' {
  \clef treble \key c \major
  c1 cis d dis e f |\break
  fis g gis a ais b \bar "||"
} }
\addlyrics { "0" "1" "2" "3" "4" "5" "6" "7" "8" "9" t e }
\layout { \context {\Score \omit BarNumber} line-width = #100 }
Integer notation.

In

post-tonal materials.[5]

In the integer model of pitch, all pitch classes and

twelve tone, serial, or otherwise atonal
music.

Pitch classes can be notated in this way by assigning the number 0 to some note and assigning consecutive integers to consecutive

diatonic functionality
.

Disadvantages

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 label both a temperature and 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 labeled "4". However, the distance between D and F will still be assigned the number 4. Both this and the issue in the paragraph directly above may be viewed as disadvantages (though mathematically, an element "4" should not be confused with the function "+4").

Other ways to label pitch classes

Pitch class
Pitch
class
Tonal counterparts Solfege
0 C (also B, Ddouble flat) do
1 C, D (also Bdouble sharp)
2 D (also Cdouble sharp, Edouble flat) re
3 D, E (also Fdouble flat)
4 E (also Ddouble sharp, F) mi
5 F (also E, Gdouble flat) fa
6 F, G (also Edouble sharp)
7 G (also Fdouble sharp, Adouble flat) sol
8 G, A
9 A (also Gdouble sharp, Bdouble flat) la
10, t or A A, B (also Cdouble flat)
11, e or B B (also Adouble sharp, C) si

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

for some integer n. Therefore, we can represent pitch classes in this system using ratios p/q where neither p nor 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 ≤ 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 twelvefold 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 equal temperament 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.

See also

References

  1. (pbk).
  2. .
  3. .
  4. . "A pitch class is defined to be the set of all pitches that share the same chroma."
  5. ^ a b c Whittall (2008), p.273.
  6. ^ Robert D. Morris, "Generalizing Rotational Arrays", Journal of Music Theory 32, no. 1 (Spring 1988): 75–132, citation on 83.

Further reading