Inmusic, apitch class (p.c. orpc) is aset of allpitches that are a whole number ofoctaves 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."[1] Important tomusical set theory, a pitch class is "all pitches related to each other by octave,enharmonic equivalence, or both."[2] Thus, usingscientific 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 humanpitch-perception isperiodic: 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".[3] Achroma is an attribute of pitches (as opposed totone height), just likehue is an attribute ofcolor. Apitch 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 Westernequal temperament, distinct spellings can refer to the same sounding object: B♯3, C4, and D
4 all refer to the same pitch, hence share the same chroma, and therefore belong to the same pitch class. This phenomenon is calledenharmonic 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 asmodular arithmetic and, in the usual case of chromatic twelve-tone scales, pitch-class numbering is "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 frequencyf (measured inhertz) to a real numberp using the equation
This creates a linearpitch space in which octaves have size 12,semitones (the distance between adjacent keys on the piano keyboard) have size 1, andmiddle C (C4) is assigned the number 0 (thus, the pitches onpiano are −39 to +48). Indeed, the mapping from pitch to real numbers defined in this manner forms the basis of theMIDI 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 pitchclasses, we need to identify or "glue together" all pitches belonging to the same pitch class—i.e. all numbersp andp + 12. The result is a cyclicalquotient group that music theorists callpitch class space and mathematicians callR/12Z. Points in this space can be labelled usingreal numbers in the range 0 ≤ x < 12. These numbers provide numerical alternatives to the letter names of elementary music theory:
(quarter tone sharp), 3 = D♯/E♭,and so on. In this system, pitch classes represented by integers are classes oftwelve-tone equal temperament (assuming standard concert A).
Inmusic,integer notation is the translation of pitch classes orinterval classes intowhole numbers.[5] Thus if C = 0, then C♯ = 1 ... A♯ = 10, B = 11, with "10" and "11" substituted by "t" and "e" in some sources,[5]A andB in others[6] (like theduodecimal numeral system, which also uses "t" and "e", orA andB, for "10" and "11"). This allows the most economical presentation of information regardingpost-tonal materials.[5]
In the integer model of pitch, all pitch classes andintervals between pitch classes are designated using the numbers 0 through 11. It is not used to notate music for performance, but is a commonanalytical andcompositional tool when working with chromatic music, includingtwelve tone,serial, or otherwiseatonal music.
Pitch classes can be notated in this way by assigning the number 0 to some note and assigning consecutive integers to consecutivesemitones; 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 arithmeticmodulo 12 is used to representoctaveequivalence. One advantage of this system is that it ignores the "spelling" of notes (B♯, C♮ and D are all 0) according to theirdiatonic functionality.
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 bothpitches andintervals. For example, the number 4 serves both as a label for the pitch class E (if C = 0) and as a label for thedistance 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").
| Pitch class | Tonal counterparts | Solfege |
|---|---|---|
| 0 | C (also B♯, D, A ) | do |
| 1 | C♯,D♭ (also B , E ) | |
| 2 | D (also C, E, B, F) | re |
| 3 | D♯,E♭ (also F, C) | |
| 4 | E (also D, F♭, G) | mi |
| 5 | F (also E♯, G, D) | fa |
| 6 | F♯,G♭ (also E, A) | |
| 7 | G (also F, A, E) | sol |
| 8 | G♯,A♭ (also F, B) | |
| 9 | A (also G, B, C) | la |
| 10, t or A | A♯,B♭ (also C, G) | |
| 11, e or B | B (also A, C♭, D) | 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, injust intonation, we may express pitches in terms of positive rational numbersp/q, expressed by reference to a 1 (often written "1/1"), which represents a fixed pitch. Ifa andb are two positive rational numbers, they belong to the same pitch class if and only if
for someintegern. Therefore, we can represent pitch classes in this system using ratiosp/q where neitherp norq 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 somescale. For example, one can label the pitch classes ofn-toneequal temperament using the integers 0 ton − 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 of19 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.
