Pitch class

In music, a pitch class (p.c. or pc) 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. 'The pitch class C stands for all possible Cs, in whatever octave position.' Important to musical set theory, a pitch class is, 'all pitches related to each other by octave, enharmonic equivalence, or both.' Thus, using scientific pitch notation, the pitch class 'C' is the set In music, a pitch class (p.c. or pc) 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. 'The pitch class C stands for all possible Cs, in whatever octave position.' Important to musical set theory, a pitch class is, 'all pitches related to each other by octave, enharmonic equivalence, or both.' 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 the human ear. 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'. 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. Note that in standard Western equal temperament, distinct spellings can refer to the same sounding object: B♯3, C4, and D4 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 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 pitch space in which octaves have size 12, semitones (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 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: 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 and/or interval classes into whole numbers. 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 others (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 analytical and compositional tool when working with chromatic music, including twelve tone, serial, or otherwise atonal music.