Dictionary## Pitch class |

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In music, a - {C
_{n}: n is an integer} = {..., C_{-2}, C_{-1}, C_{0}, C_{1}, C_{2}, C_{3}...}
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 phenomenon called octave equivalence. Psychologists refer to the quality of a pitch as its "chroma". A "chroma" is an attribute of pitches, just like hue is an attribute of color. 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. Note that in standard Western
## Integer notationTo avoid the problem of enharmonic spellings, theorists typically represent pitch classes using numbers. One can map a pitch's fundamental frequency *p*= 69 + 12log_{2}(*f*/ 440)
This creates a linear pitch space in which octaves have size 12, - 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, In the integer model of pitch, all pitch classes and Pitch classes can be notated in this way by assigning the number 0 to some note—C natural by convention 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 Additionally, integer notation does not seem to allow for the notation of microtones, or notes not belonging to the underlying equal division of the octave. For these reasons, some theorists have recently advocated using rational numbers to represent pitches and pitch classes, in a way that is not dependent on any underlying division of the octave. ## 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. 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, . It is also very common to label pitch classes with reference to some 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## Sources**^**Arnold Whitall,*The Cambridge Introduction to Serialism*(New York: Cambridge University Press, 2008): 276. ISBN 978-0-521-68200-8 (pbk).- ^
^{a}^{b}^{c}Whittall (2008), p.273.
## Further reading- Purwins, Hendrik (2005). "Profiles of Pitch Classes: Circularity of Relative Pitch and Key—Experiments, Models, Computational Music Analysis, and Perspectives". Ph.D. Thesis. Berlin: Technische Universität Berlin.
- Rahn, John (1980).
*Basic Atonal Theory*. New York: Longman; London and Toronto: Prentice Hall International. ISBN 0-02-873160-3. Reprinted 1987, New York: Schirmer Books; London: Collier Macmillan. - Schuijer, Michiel (2008).
*Analyzing Atonal Music: Pitch-Class Set Theory and Its Contexts*. Eastman Studies in Music 60. Rochester, NY: University of Rochester Press. ISBN 978-1-58046-270-9.
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