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In music, a pitch class ( p.c. or pc) is a set of all pitches that are a whole number of apart; for example, the pitch class C consists of the Cs in all octaves. "The pitch class C stands for all possible Cs, in whatever octave position."Arnold Whittall, The Cambridge Introduction to Serialism (New York: Cambridge University Press, 2008): 276. (pbk). Important to musical set theory, a pitch class is "all pitches related to each other by octave, Enharmonic, or both."Don Michael Randel, ed. (2003). "Set theory", The Harvard Dictionary of Music, p.776. Harvard. . Thus, using scientific pitch notation, the pitch class "C" is the set
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 pitch-perception is periodic: pitches belonging to the same pitch class are perceived as having a similar quality or color, a property called "octave equivalence".
Psychologists refer to the quality of a pitch as its "chroma".Dmitri Tymoczko (2011). A Geometry of Music: Harmony and Counterpoint in the Extended Common Practice, p.30. Oxford Studies in Music Theory. . A chroma is an attribute of pitches (as opposed to tone height), just as 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.Müller, Meinard (2007). Information Retrieval for Music and Motion, p.60. . "A pitch class is defined to be the set of all pitches that share the same chroma."
In standard Western equal temperament, distinct spellings can refer to the same sounding object: B3, C4, and D4 all refer to the same pitch, hence share the same chroma, and therefore belong to the same pitch class. This phenomenon is called Enharmonic.
This creates a linear pitch space in which octaves have size 12, (the distance between adjacent keys on the piano keyboard) have size 1, and middle C (C4) is assigned the number 0 (thus, the pitches on piano are −39 to +48). 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 (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/12 Z. Points in this space can be labelled using in the range 0 ≤ x < 12. These numbers provide numerical alternatives to the letter names of elementary music theory:
and so on. In this system, pitch classes represented by integers are classes of twelve-tone equal temperament (assuming standard concert A).
In music, integer notation is the translation of pitch classes or into integer.Whittall (2008), p.273. Thus if C = 0, then C = 1 ... A = 10, B = 11, with "10" and "11" substituted by "t" and "e" in some sources, A and B in othersRobert D. Morris, "Generalizing Rotational Arrays", Journal of Music Theory 32, no. 1 (Spring 1988): 75–132, citation on 83. (like the duodecimal numeral system, which also uses "t" and "e", or A and B, for "10" and "11"). 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 musical analysis and compositional tool when working with chromatic music, including twelve tone, serialism, or otherwise atonality music.
Pitch classes can be notated in this way by assigning the number 0 to some note and assigning consecutive integers to consecutive ; so if 0 is C natural, 1 is C, 2 is D and so on up to 11, which is B. 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, C and D are all 0) according to their diatonic functionality.
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").
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for some integer n. Therefore, we can represent pitch classes in this system using ratios 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 ≤ < 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.
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