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In
music Music is generally defined as the art of arranging sound to create some combination of form, harmony, melody, rhythm or otherwise expressive content. Exact definitions of music vary considerably around the world, though it is an aspe ...
, a pitch class (p.c. or pc) is a
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
of all pitches that are a whole number of
octave In music, an octave ( la, octavus: eighth) or perfect octave (sometimes called the diapason) is the interval between one musical pitch and another with double its frequency. The octave relationship is a natural phenomenon that has been refer ...
s 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." 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 Scientific pitch notation (SPN), also known as American standard pitch notation (ASPN) and international pitch notation (IPN), is a method of specifying musical pitch by combining a musical note name (with accidental if needed) and a number id ...
, 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". A ''chroma'' is an attribute of pitches (as opposed to ''tone height''), just like hue is an attribute of
color Color (American English) or colour (British English) is the visual perceptual property deriving from the spectrum of light interacting with the photoreceptor cells of the eyes. Color categories and physical specifications of color are associ ...
. 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 An equal temperament is a musical temperament or tuning system, which approximates just intervals by dividing an octave (or other interval) into equal steps. This means the ratio of the frequencies of any adjacent pair of notes is the same, ...
, 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 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 In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his boo ...
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 The hertz (symbol: Hz) is the unit of frequency in the International System of Units (SI), equivalent to one event (or cycle) per second. The hertz is an SI derived unit whose expression in terms of SI base units is s−1, meaning that o ...
) to a real number ''p'' using the equation :p = 9 + 12\log_2 \frac. This creates a linear pitch space in which octaves have size 12,
semitone A semitone, also called a half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically. It is defined as the interval between two adjacent no ...
s (the distance between adjacent keys on the piano keyboard) have size 1, and
middle C C or Do is the first note and semitone of the C major scale, the third note of the A minor scale (the relative minor of C major), and the fourth note (G, A, B, C) of the Guidonian hand, commonly pitched around 261.63  Hz. The actual fr ...
(C4) is assigned the number 0 (thus, the pitches on
piano The piano is a stringed keyboard instrument in which the strings are struck by wooden hammers that are coated with a softer material (modern hammers are covered with dense wool felt; some early pianos used leather). It is played using a keyboa ...
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 A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored" out). For exam ...
that musicians call pitch class space and mathematicians call R/12Z. Points in this space can be labelled using
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s 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 = D (
quarter tone A quarter tone is a pitch halfway between the usual notes of a chromatic scale or an interval about half as wide (aurally, or logarithmically) as a semitone, which itself is half a whole tone. Quarter tones divide the octave by 50 cents each ...
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). In
music Music is generally defined as the art of arranging sound to create some combination of form, harmony, melody, rhythm or otherwise expressive content. Exact definitions of music vary considerably around the world, though it is an aspe ...
, integer notation is the translation of pitch classes and/or interval classes into whole numbers.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 The duodecimal system (also known as base 12, dozenal, or, rarely, uncial) is a positional notation numeral system using twelve as its base. The number twelve (that is, the number written as "12" in the decimal numerical system) is instead wr ...
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 In semantics, mathematical logic and related disciplines, the principle of compositionality is the principle that the meaning of a complex expression is determined by the meanings of its constituent expressions and the rules used to combine them. ...
tool when working with chromatic music, including twelve tone, serial, or otherwise
atonal Atonality in its broadest sense is music that lacks a tonal center, or key. ''Atonality'', in this sense, usually describes compositions written from about the early 20th-century to the present day, where a hierarchy of harmonies focusing on a s ...
music. Pitch classes can be notated in this way by assigning the number 0 to some note and assigning consecutive integers to consecutive
semitone A semitone, also called a half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically. It is defined as the interval between two adjacent no ...
s; 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 In computing, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another (called the '' modulus'' of the operation). Given two positive numbers and , modulo (often abbreviated as ) is ...
12 is used to represent
octave In music, an octave ( la, octavus: eighth) or perfect octave (sometimes called the diapason) is the interval between one musical pitch and another with double its frequency. The octave relationship is a natural phenomenon that has been refer ...
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.


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

The system described above is flexible enough to describe any pitch class in any tuning system: for example, one can use the numbers 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 In music, just intonation or pure intonation is the tuning of musical intervals as whole number ratios (such as 3:2 or 4:3) of frequencies. An interval tuned in this way is said to be pure, and is called a just interval. Just intervals (and ...
, we may express pitches in terms of positive rational numbers , expressed by reference to a 1 (often written ""), 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 :\frac = 2^n for some
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
''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 An equal temperament is a musical temperament or tuning system, which approximates just intervals by dividing an octave (or other interval) into equal steps. This means the ratio of the frequencies of any adjacent pair of notes is the same, ...
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 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 . In twenty-four-tone equal-temperament, this same triad is labeled . 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

*
Flat (music) In music, flat (Italian bemolle for "soft B") means "lower in pitch". Flat is the opposite of sharp, which is a raising of pitch. In musical notation, flat means "lower in pitch by one semitone (half step)", notated using the symbol which is der ...
* Sharp (music) * Pitch circularity * Pitch interval *
Tone row In music, a tone row or note row (german: Reihe or '), also series or set, is a non-repetitive ordering of a set of pitch-classes, typically of the twelve notes in musical set theory of the chromatic scale, though both larger and smaller sets ...
(
List A ''list'' is any set of items in a row. List or lists may also refer to: People * List (surname) Organizations * List College, an undergraduate division of the Jewish Theological Seminary of America * SC Germania List, German rugby unio ...
)


References


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. . 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. . {{DEFAULTSORT:Pitch Class Musical notation Musical set theory Pitch (music)