Pitch class

# Pitch class

Discussion

Encyclopedia
In music
Music
Music is an art form whose medium is sound and silence. Its common elements are pitch , rhythm , dynamics, and the sonic qualities of timbre and texture...

, a pitch class is a set
Set (music)
A set in music theory, as in mathematics and general parlance, is a collection of objects...

of all pitches
Pitch (music)
Pitch is an auditory perceptual property that allows the ordering of sounds on a frequency-related scale.Pitches are compared as "higher" and "lower" in the sense associated with musical melodies,...

that are a whole number of octave
Octave
In music, an octave is the interval between one musical pitch and another with half or double its frequency. The octave relationship is a natural phenomenon that has been referred to as the "basic miracle of music", the use of which is "common in most musical systems"...

s 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." Thus, using scientific pitch notation
Scientific pitch notation
Scientific pitch notation is one of several methods that name the notes of the standard Western chromatic scale by combining a letter-name, accidentals, and a number identifying the pitch's octave...

, the pitch class "C" is the set
{Cn : n is an integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...

} = {..., C-2, C-1, C0, C1, C2, C3 ...};

although there is no formal limit to this sequence on either end, only a limited number of these pitches will actually be audible to the human ear.
Pitch class is important because human pitch-perception is periodic
Periodic function
In mathematics, a periodic function is a function that repeats its values in regular intervals or periods. The most important examples are the trigonometric functions, which repeat over intervals of length 2π radians. Periodic functions are used throughout science to describe oscillations,...

: 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, just like hue
Hue
Hue is one of the main properties of a color, defined technically , as "the degree to which a stimulus can be describedas similar to or different from stimuli that are described as red, green, blue, and yellow,"...

is an attribute of color
Color
Color or colour is the visual perceptual property corresponding in humans to the categories called red, green, blue and others. Color derives from the spectrum of light interacting in the eye with the spectral sensitivities of the light receptors...

. 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 equal temperament
Equal temperament
An equal temperament is a musical temperament, or a system of tuning, in which every pair of adjacent notes has an identical frequency ratio. As pitch is perceived roughly as the logarithm of frequency, this means that the perceived "distance" from every note to its nearest neighbor is the same for...

, 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; a phenomenon called enharmonic equivalence
Enharmonic
In modern musical notation and tuning, an enharmonic equivalent is a note , interval , or key signature which is equivalent to some other note, interval, or key signature, but "spelled", or named, differently...

.

## Integer notation

To avoid the problem of enharmonic spellings, theorists typically represent pitch classes using numbers. One can map a pitch's fundamental frequency (measured in hertz
Hertz
The hertz is the SI unit of frequency defined as the number of cycles per second of a periodic phenomenon. One of its most common uses is the description of the sine wave, particularly those used in radio and audio applications....

) to a real number using the equation

This creates a linear pitch space
Pitch space
In music theory, pitch spaces model relationships between pitches. These models typically use distance to model the degree of relatedness, with closely related pitches placed near one another, and less closely related pitches placed farther apart. Depending on the complexity of the relationships...

in which octaves have size 12, semitone
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....

s (the distance between adjacent keys on the piano keyboard) have size 1, and middle C
Middle C
C or Do is the first note of the fixed-Do solfège scale. Its enharmonic is B.-Middle C:Middle C is designated C4 in scientific pitch notation because of the note's position as the fourth C key on a standard 88-key piano keyboard...

is assigned the number 60. Indeed, the mapping from pitch to real numbers defined in this manner forms the basis of the MIDI Tuning Standard
MIDI Tuning Standard
MIDI Tuning Standard is a specification of precise musical pitch agreed to by the MIDI Manufacturers Association in the MIDI protocol. MTS allows for both a bulk tuning dump message, giving a tuning for each of 128 notes, and a tuning message for individual notes as they are played.-Frequency...

, which uses the real numbers from 0 to 127 to represent the pitches C-1 to G9. 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 circular quotient space
Quotient space
In topology and related areas of mathematics, a quotient space is, intuitively speaking, the result of identifying or "gluing together" certain points of a given space. The points to be identified are specified by an equivalence relation...

that musicians call pitch class space
Pitch class space
In music theory, pitch class space is the circular space representing all the notes in a musical octave.In this space, there is no distinction between tones that are separated by an integral number of octaves...

and mathematicians call R/12Z. Points in this space can be labelled using real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

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
Quarter tone
A quarter tone , is a pitch halfway between the usual notes of a chromatic scale, an interval about half as wide as a semitone, which is half a whole 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
Allen Forte
Allen Forte is a music theorist and musicologist. He was born in Portland, Oregon and fought in the Navy at the close of World War II before moving to the East Coast. He is now Battell Professor of Music, Emeritus at Yale University...

and Robert Morris
Robert Morris (composer)
Robert Morris is an American composer and music theorist.-Work in music theory:As a music theorist, Morris' work has bridged an important gap between the rigorously academic and the highly experimental. Born in Cheltenham, England in 1943, Morris received his musical education at the Eastman...

).
In music
Music
Music is an art form whose medium is sound and silence. Its common elements are pitch , rhythm , dynamics, and the sonic qualities of timbre and texture...

, integer notation is the translation of pitch classes and/or interval class
Interval class
In musical set theory, an interval class , also known as unordered pitch-class interval, interval distance, undirected interval, or interval mod...

es into whole numbers
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...

. Thus C=0, C#=1 ... A#=10, B=11, with "10" and "11" substituted by "t" and "e" in some sources This allows the most economical presentation of information regarding post-tonal materials.

In the integer model of pitch, all pitch classes and interval
Interval (music)
In music theory, an interval is a combination of two notes, or the ratio between their frequencies. Two-note combinations are also called dyads...

s 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
Musical analysis
Musical analysis is the attempt to answer the question how does this music work?. The method employed to answer this question, and indeed exactly what is meant by the question, differs from analyst to analyst, and according to the purpose of the analysis. According to Ian Bent , analysis is "an...

and compositional
Musical composition
Musical composition can refer to an original piece of music, the structure of a musical piece, or the process of creating a new piece of music. People who practice composition are called composers.- Musical compositions :...

tool when working with chromatic music, including twelve tone, serial
Serialism
In music, serialism is a method or technique of composition that uses a series of values to manipulate different musical elements. Serialism began primarily with Arnold Schoenberg's twelve-tone technique, though his contemporaries were also working to establish serialism as one example of...

, or otherwise atonal
Atonality
Atonality in its broadest sense describes music that lacks a tonal center, or key. Atonality in this sense usually describes compositions written from about 1908 to the present day where a hierarchy of pitches focusing on a single, central tone is not used, and the notes of the chromatic scale...

music.

Pitch classes can be notated in this way by assigning the number 0 to some note—C natural by convention—and assigning consecutive integers to consecutive semitone
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....

s; so if 0 is C natural, 1 is C sharp, 2 is D natural and so on up to 11, which is B natural. The C above this is not 12, but 0 again (12-12=0). Thus arithmetic modulo
Modular arithmetic
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus....

12 is used to represent octave
Octave
In music, an octave is the interval between one musical pitch and another with half or double its frequency. The octave relationship is a natural phenomenon that has been referred to as the "basic miracle of music", the use of which is "common in most musical systems"...

equivalence
Equivalence
Equivalence or equivalent may refer to:*In chemistry:**Equivalent **Equivalence point**Equivalent weight*In computing:**Turing equivalence *In ethics:**Moral equivalence*In history:...

. One advantage of this system is that it ignores the "spelling" of notes (B sharp, C natural and D double-flat are all 0) according to their diatonic 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 both pitches
Pitch (music)
Pitch is an auditory perceptual property that allows the ordering of sounds on a frequency-related scale.Pitches are compared as "higher" and "lower" in the sense associated with musical melodies,...

and intervals
Interval (music)
In music theory, an interval is a combination of two notes, or the ratio between their frequencies. Two-note combinations are also called dyads...

. 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 function as a label both for a temperature, and for 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 labelled "4." However, the distance between D and F will still be assigned the number 4. The late music theorist David Lewin
David Lewin
David Lewin was an American music theorist, music critic and composer. Called "the most original and far-ranging theorist of his generation" , he did his most influential theoretical work on the development of transformational theory, which involves the application of mathematical group theory to...

was particularly sensitive to the confusions that this can cause, and both this and the above may be viewed as disadvantages.

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

Pitch class
Pitch
class
Tonal counterparts
0 C
C (musical note)
C or Do is the first note of the fixed-Do solfège scale. Its enharmonic is B.-Middle C:Middle C is designated C4 in scientific pitch notation because of the note's position as the fourth C key on a standard 88-key piano keyboard...

(also B, D)
1 C
In music
Music
Music is an art form whose medium is sound and silence. Its common elements are pitch , rhythm , dynamics, and the sonic qualities of timbre and texture...

, a pitch class is a set
Set (music)
A set in music theory, as in mathematics and general parlance, is a collection of objects...

of all pitches
Pitch (music)
Pitch is an auditory perceptual property that allows the ordering of sounds on a frequency-related scale.Pitches are compared as "higher" and "lower" in the sense associated with musical melodies,...

that are a whole number of octave
Octave
In music, an octave is the interval between one musical pitch and another with half or double its frequency. The octave relationship is a natural phenomenon that has been referred to as the "basic miracle of music", the use of which is "common in most musical systems"...

s 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." Thus, using scientific pitch notation
Scientific pitch notation
Scientific pitch notation is one of several methods that name the notes of the standard Western chromatic scale by combining a letter-name, accidentals, and a number identifying the pitch's octave...

, the pitch class "C" is the set
{Cn : n is an integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...

} = {..., C-2, C-1, C0, C1, C2, C3 ...};

although there is no formal limit to this sequence on either end, only a limited number of these pitches will actually be audible to the human ear.
Pitch class is important because human pitch-perception is periodic
Periodic function
In mathematics, a periodic function is a function that repeats its values in regular intervals or periods. The most important examples are the trigonometric functions, which repeat over intervals of length 2π radians. Periodic functions are used throughout science to describe oscillations,...

: 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, just like hue
Hue
Hue is one of the main properties of a color, defined technically , as "the degree to which a stimulus can be describedas similar to or different from stimuli that are described as red, green, blue, and yellow,"...

is an attribute of color
Color
Color or colour is the visual perceptual property corresponding in humans to the categories called red, green, blue and others. Color derives from the spectrum of light interacting in the eye with the spectral sensitivities of the light receptors...

. 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.{{citation needed|date=March 2010}}

Note that in standard Western equal temperament
Equal temperament
An equal temperament is a musical temperament, or a system of tuning, in which every pair of adjacent notes has an identical frequency ratio. As pitch is perceived roughly as the logarithm of frequency, this means that the perceived "distance" from every note to its nearest neighbor is the same for...

, distinct spellings can refer to the same sounding object: B{{music|sharp}}3, C4, and D{{music|bb}}4 all refer to the same pitch, hence share the same chroma, and therefore belong to the same pitch class; a phenomenon called enharmonic equivalence
Enharmonic
In modern musical notation and tuning, an enharmonic equivalent is a note , interval , or key signature which is equivalent to some other note, interval, or key signature, but "spelled", or named, differently...

.

## Integer notation

To avoid the problem of enharmonic spellings, theorists typically represent pitch classes using numbers. One can map a pitch's fundamental frequency (measured in hertz
Hertz
The hertz is the SI unit of frequency defined as the number of cycles per second of a periodic phenomenon. One of its most common uses is the description of the sine wave, particularly those used in radio and audio applications....

) to a real number using the equation

This creates a linear pitch space
Pitch space
In music theory, pitch spaces model relationships between pitches. These models typically use distance to model the degree of relatedness, with closely related pitches placed near one another, and less closely related pitches placed farther apart. Depending on the complexity of the relationships...

in which octaves have size 12, semitone
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....

s (the distance between adjacent keys on the piano keyboard) have size 1, and middle C
Middle C
C or Do is the first note of the fixed-Do solfège scale. Its enharmonic is B.-Middle C:Middle C is designated C4 in scientific pitch notation because of the note's position as the fourth C key on a standard 88-key piano keyboard...

is assigned the number 60. Indeed, the mapping from pitch to real numbers defined in this manner forms the basis of the MIDI Tuning Standard
MIDI Tuning Standard
MIDI Tuning Standard is a specification of precise musical pitch agreed to by the MIDI Manufacturers Association in the MIDI protocol. MTS allows for both a bulk tuning dump message, giving a tuning for each of 128 notes, and a tuning message for individual notes as they are played.-Frequency...

, which uses the real numbers from 0 to 127 to represent the pitches C-1 to G9. 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 circular quotient space
Quotient space
In topology and related areas of mathematics, a quotient space is, intuitively speaking, the result of identifying or "gluing together" certain points of a given space. The points to be identified are specified by an equivalence relation...

that musicians call pitch class space
Pitch class space
In music theory, pitch class space is the circular space representing all the notes in a musical octave.In this space, there is no distinction between tones that are separated by an integral number of octaves...

and mathematicians call R/12Z. Points in this space can be labelled using real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

s in the range 0 ≤ x < 12. These numbers provide numerical alternatives to the letter names of elementary music theory:
0 = C, 1 = C{{music|#}}/D{{music|b}}, 2 = D, 2.5 = "D quarter tone
Quarter tone
A quarter tone , is a pitch halfway between the usual notes of a chromatic scale, an interval about half as wide as a semitone, which is half a whole tone....

sharp", 3 = D{{music|#}}/E{{music|b}},

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
Allen Forte
Allen Forte is a music theorist and musicologist. He was born in Portland, Oregon and fought in the Navy at the close of World War II before moving to the East Coast. He is now Battell Professor of Music, Emeritus at Yale University...

and Robert Morris
Robert Morris (composer)
Robert Morris is an American composer and music theorist.-Work in music theory:As a music theorist, Morris' work has bridged an important gap between the rigorously academic and the highly experimental. Born in Cheltenham, England in 1943, Morris received his musical education at the Eastman...

).
In music
Music
Music is an art form whose medium is sound and silence. Its common elements are pitch , rhythm , dynamics, and the sonic qualities of timbre and texture...

, integer notation is the translation of pitch classes and/or interval class
Interval class
In musical set theory, an interval class , also known as unordered pitch-class interval, interval distance, undirected interval, or interval mod...

es into whole numbers
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...

. Thus C=0, C#=1 ... A#=10, B=11, with "10" and "11" substituted by "t" and "e" in some sources This allows the most economical presentation of information regarding post-tonal materials.

In the integer model of pitch, all pitch classes and interval
Interval (music)
In music theory, an interval is a combination of two notes, or the ratio between their frequencies. Two-note combinations are also called dyads...

s 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
Musical analysis
Musical analysis is the attempt to answer the question how does this music work?. The method employed to answer this question, and indeed exactly what is meant by the question, differs from analyst to analyst, and according to the purpose of the analysis. According to Ian Bent , analysis is "an...

and compositional
Musical composition
Musical composition can refer to an original piece of music, the structure of a musical piece, or the process of creating a new piece of music. People who practice composition are called composers.- Musical compositions :...

tool when working with chromatic music, including twelve tone, serial
Serialism
In music, serialism is a method or technique of composition that uses a series of values to manipulate different musical elements. Serialism began primarily with Arnold Schoenberg's twelve-tone technique, though his contemporaries were also working to establish serialism as one example of...

, or otherwise atonal
Atonality
Atonality in its broadest sense describes music that lacks a tonal center, or key. Atonality in this sense usually describes compositions written from about 1908 to the present day where a hierarchy of pitches focusing on a single, central tone is not used, and the notes of the chromatic scale...

music.

Pitch classes can be notated in this way by assigning the number 0 to some note—C natural by convention{{Citation needed|date=July 2010}}—and assigning consecutive integers to consecutive semitone
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....

s; so if 0 is C natural, 1 is C sharp, 2 is D natural and so on up to 11, which is B natural. The C above this is not 12, but 0 again (12-12=0). Thus arithmetic modulo
Modular arithmetic
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus....

12 is used to represent octave
Octave
In music, an octave is the interval between one musical pitch and another with half or double its frequency. The octave relationship is a natural phenomenon that has been referred to as the "basic miracle of music", the use of which is "common in most musical systems"...

equivalence
Equivalence
Equivalence or equivalent may refer to:*In chemistry:**Equivalent **Equivalence point**Equivalent weight*In computing:**Turing equivalence *In ethics:**Moral equivalence*In history:...

. One advantage of this system is that it ignores the "spelling" of notes (B sharp, C natural and D double-flat are all 0) according to their diatonic 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 both pitches
Pitch (music)
Pitch is an auditory perceptual property that allows the ordering of sounds on a frequency-related scale.Pitches are compared as "higher" and "lower" in the sense associated with musical melodies,...

and intervals
Interval (music)
In music theory, an interval is a combination of two notes, or the ratio between their frequencies. Two-note combinations are also called dyads...

. 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{{music|#}}. (In much the same way, the term "10 degrees" can function as a label both for a temperature, and for 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 labelled "4." However, the distance between D and F{{music|#}} will still be assigned the number 4. The late music theorist David Lewin
David Lewin
David Lewin was an American music theorist, music critic and composer. Called "the most original and far-ranging theorist of his generation" , he did his most influential theoretical work on the development of transformational theory, which involves the application of mathematical group theory to...

was particularly sensitive to the confusions that this can cause{{example needed|date=March 2010}}, and both this and the above may be viewed as disadvantages.

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

Pitch class
Pitch
class
Tonal counterparts
0 C
C (musical note)
C or Do is the first note of the fixed-Do solfège scale. Its enharmonic is B.-Middle C:Middle C is designated C4 in scientific pitch notation because of the note's position as the fourth C key on a standard 88-key piano keyboard...

(also B{{music|sharp}}, D{{music|doubleflat}})
1 C
In music
Music
Music is an art form whose medium is sound and silence. Its common elements are pitch , rhythm , dynamics, and the sonic qualities of timbre and texture...

, a pitch class is a set
Set (music)
A set in music theory, as in mathematics and general parlance, is a collection of objects...

of all pitches
Pitch (music)
Pitch is an auditory perceptual property that allows the ordering of sounds on a frequency-related scale.Pitches are compared as "higher" and "lower" in the sense associated with musical melodies,...

that are a whole number of octave
Octave
In music, an octave is the interval between one musical pitch and another with half or double its frequency. The octave relationship is a natural phenomenon that has been referred to as the "basic miracle of music", the use of which is "common in most musical systems"...

s 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." Thus, using scientific pitch notation
Scientific pitch notation
Scientific pitch notation is one of several methods that name the notes of the standard Western chromatic scale by combining a letter-name, accidentals, and a number identifying the pitch's octave...

, the pitch class "C" is the set
{Cn : n is an integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...

} = {..., C-2, C-1, C0, C1, C2, C3 ...};

although there is no formal limit to this sequence on either end, only a limited number of these pitches will actually be audible to the human ear.
Pitch class is important because human pitch-perception is periodic
Periodic function
In mathematics, a periodic function is a function that repeats its values in regular intervals or periods. The most important examples are the trigonometric functions, which repeat over intervals of length 2π radians. Periodic functions are used throughout science to describe oscillations,...

: 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, just like hue
Hue
Hue is one of the main properties of a color, defined technically , as "the degree to which a stimulus can be describedas similar to or different from stimuli that are described as red, green, blue, and yellow,"...

is an attribute of color
Color
Color or colour is the visual perceptual property corresponding in humans to the categories called red, green, blue and others. Color derives from the spectrum of light interacting in the eye with the spectral sensitivities of the light receptors...

. 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.{{citation needed|date=March 2010}}

Note that in standard Western equal temperament
Equal temperament
An equal temperament is a musical temperament, or a system of tuning, in which every pair of adjacent notes has an identical frequency ratio. As pitch is perceived roughly as the logarithm of frequency, this means that the perceived "distance" from every note to its nearest neighbor is the same for...

, distinct spellings can refer to the same sounding object: B{{music|sharp}}3, C4, and D{{music|bb}}4 all refer to the same pitch, hence share the same chroma, and therefore belong to the same pitch class; a phenomenon called enharmonic equivalence
Enharmonic
In modern musical notation and tuning, an enharmonic equivalent is a note , interval , or key signature which is equivalent to some other note, interval, or key signature, but "spelled", or named, differently...

.

## Integer notation

To avoid the problem of enharmonic spellings, theorists typically represent pitch classes using numbers. One can map a pitch's fundamental frequency (measured in hertz
Hertz
The hertz is the SI unit of frequency defined as the number of cycles per second of a periodic phenomenon. One of its most common uses is the description of the sine wave, particularly those used in radio and audio applications....

) to a real number using the equation

This creates a linear pitch space
Pitch space
In music theory, pitch spaces model relationships between pitches. These models typically use distance to model the degree of relatedness, with closely related pitches placed near one another, and less closely related pitches placed farther apart. Depending on the complexity of the relationships...

in which octaves have size 12, semitone
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....

s (the distance between adjacent keys on the piano keyboard) have size 1, and middle C
Middle C
C or Do is the first note of the fixed-Do solfège scale. Its enharmonic is B.-Middle C:Middle C is designated C4 in scientific pitch notation because of the note's position as the fourth C key on a standard 88-key piano keyboard...

is assigned the number 60. Indeed, the mapping from pitch to real numbers defined in this manner forms the basis of the MIDI Tuning Standard
MIDI Tuning Standard
MIDI Tuning Standard is a specification of precise musical pitch agreed to by the MIDI Manufacturers Association in the MIDI protocol. MTS allows for both a bulk tuning dump message, giving a tuning for each of 128 notes, and a tuning message for individual notes as they are played.-Frequency...

, which uses the real numbers from 0 to 127 to represent the pitches C-1 to G9. 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 circular quotient space
Quotient space
In topology and related areas of mathematics, a quotient space is, intuitively speaking, the result of identifying or "gluing together" certain points of a given space. The points to be identified are specified by an equivalence relation...

that musicians call pitch class space
Pitch class space
In music theory, pitch class space is the circular space representing all the notes in a musical octave.In this space, there is no distinction between tones that are separated by an integral number of octaves...

and mathematicians call R/12Z. Points in this space can be labelled using real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

s in the range 0 ≤ x < 12. These numbers provide numerical alternatives to the letter names of elementary music theory:
0 = C, 1 = C{{music|#}}/D{{music|b}}, 2 = D, 2.5 = "D quarter tone
Quarter tone
A quarter tone , is a pitch halfway between the usual notes of a chromatic scale, an interval about half as wide as a semitone, which is half a whole tone....

sharp", 3 = D{{music|#}}/E{{music|b}},

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
Allen Forte
Allen Forte is a music theorist and musicologist. He was born in Portland, Oregon and fought in the Navy at the close of World War II before moving to the East Coast. He is now Battell Professor of Music, Emeritus at Yale University...

and Robert Morris
Robert Morris (composer)
Robert Morris is an American composer and music theorist.-Work in music theory:As a music theorist, Morris' work has bridged an important gap between the rigorously academic and the highly experimental. Born in Cheltenham, England in 1943, Morris received his musical education at the Eastman...

).
In music
Music
Music is an art form whose medium is sound and silence. Its common elements are pitch , rhythm , dynamics, and the sonic qualities of timbre and texture...

, integer notation is the translation of pitch classes and/or interval class
Interval class
In musical set theory, an interval class , also known as unordered pitch-class interval, interval distance, undirected interval, or interval mod...

es into whole numbers
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...

. Thus C=0, C#=1 ... A#=10, B=11, with "10" and "11" substituted by "t" and "e" in some sources This allows the most economical presentation of information regarding post-tonal materials.

In the integer model of pitch, all pitch classes and interval
Interval (music)
In music theory, an interval is a combination of two notes, or the ratio between their frequencies. Two-note combinations are also called dyads...

s 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
Musical analysis
Musical analysis is the attempt to answer the question how does this music work?. The method employed to answer this question, and indeed exactly what is meant by the question, differs from analyst to analyst, and according to the purpose of the analysis. According to Ian Bent , analysis is "an...

and compositional
Musical composition
Musical composition can refer to an original piece of music, the structure of a musical piece, or the process of creating a new piece of music. People who practice composition are called composers.- Musical compositions :...

tool when working with chromatic music, including twelve tone, serial
Serialism
In music, serialism is a method or technique of composition that uses a series of values to manipulate different musical elements. Serialism began primarily with Arnold Schoenberg's twelve-tone technique, though his contemporaries were also working to establish serialism as one example of...

, or otherwise atonal
Atonality
Atonality in its broadest sense describes music that lacks a tonal center, or key. Atonality in this sense usually describes compositions written from about 1908 to the present day where a hierarchy of pitches focusing on a single, central tone is not used, and the notes of the chromatic scale...

music.

Pitch classes can be notated in this way by assigning the number 0 to some note—C natural by convention{{Citation needed|date=July 2010}}—and assigning consecutive integers to consecutive semitone
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....

s; so if 0 is C natural, 1 is C sharp, 2 is D natural and so on up to 11, which is B natural. The C above this is not 12, but 0 again (12-12=0). Thus arithmetic modulo
Modular arithmetic
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus....

12 is used to represent octave
Octave
In music, an octave is the interval between one musical pitch and another with half or double its frequency. The octave relationship is a natural phenomenon that has been referred to as the "basic miracle of music", the use of which is "common in most musical systems"...

equivalence
Equivalence
Equivalence or equivalent may refer to:*In chemistry:**Equivalent **Equivalence point**Equivalent weight*In computing:**Turing equivalence *In ethics:**Moral equivalence*In history:...

. One advantage of this system is that it ignores the "spelling" of notes (B sharp, C natural and D double-flat are all 0) according to their diatonic 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 both pitches
Pitch (music)
Pitch is an auditory perceptual property that allows the ordering of sounds on a frequency-related scale.Pitches are compared as "higher" and "lower" in the sense associated with musical melodies,...

and intervals
Interval (music)
In music theory, an interval is a combination of two notes, or the ratio between their frequencies. Two-note combinations are also called dyads...

. 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{{music|#}}. (In much the same way, the term "10 degrees" can function as a label both for a temperature, and for 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 labelled "4." However, the distance between D and F{{music|#}} will still be assigned the number 4. The late music theorist David Lewin
David Lewin
David Lewin was an American music theorist, music critic and composer. Called "the most original and far-ranging theorist of his generation" , he did his most influential theoretical work on the development of transformational theory, which involves the application of mathematical group theory to...

was particularly sensitive to the confusions that this can cause{{example needed|date=March 2010}}, and both this and the above may be viewed as disadvantages.

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

Pitch class
Pitch
class
Tonal counterparts
0 C
C (musical note)
C or Do is the first note of the fixed-Do solfège scale. Its enharmonic is B.-Middle C:Middle C is designated C4 in scientific pitch notation because of the note's position as the fourth C key on a standard 88-key piano keyboard...

(also B{{music|sharp}}, D{{music|doubleflat}})
1 C{{music, D{{music (also B{{music|doublesharp}})
2 D
D (musical note)
D is a musical note a whole tone above C, and is known as Re within the solfege system.When calculated in equal temperament with a reference of A above middle C as 440 Hz, the frequency of middle D is approximately 293.665 Hz. See pitch for a discussion of historical variations in...

(also C{{music|doublesharp}}, E{{music|doubleflat}})
3 D{{music, E{{music (also F{{music|doubleflat}})
4 E
E (musical note)
E or mi is the third note of the solfège.When calculated in equal temperament with a reference of A above middle C as 440 Hz, the frequency of Middle E is approximately 329.628 Hz. See pitch for a discussion of historical variations in frequency.-Designation by octave:...

(also D{{music|doublesharp}}, F{{music|flat}})
5 F
F (musical note)
F is a musical note, the fourth above C. It is also known as fa in fixed-do solfège.When calculated in equal temperament with a reference of A above middle C as 440 Hz, the frequency of Middle F is approximately 349.228 Hz. See pitch for a discussion of historical variations in...

(also E{{music|sharp}}, G{{music|doubleflat}})
6 F{{music, G{{music (also E{{music|doublesharp}})
7 G
G (musical note)
Sol, So, or G is the fifth note of the solfège starting on C. As such it is the dominant, a perfect fifth above C.When calculated in equal temperament with a reference of A above middle C as 440 Hz, the frequency of Middle G note is approximately 391.995 Hz...

(also F{{music|doublesharp}}, A{{music|doubleflat}})
8 G{{music, A{{music
9 A
A (musical note)
La or A is the sixth note of the solfège. "A" is generally used as a standard for tuning. When the orchestra tunes, the oboe plays an "A" and the rest of the instruments tune to match that pitch. Every string instrument in the orchestra has an A string, from which each player can tune the rest of...

(also G{{music|doublesharp}}, B{{music|doubleflat}})
10, t or A A{{music, B{{music (also C{{music|doubleflat}})
11, e or B B
B (musical note)
B, also known as H, Si or Ti, is the seventh note of the solfège. It lies a chromatic semitone below C and is thus the enharmonic equivalent of C-flat....

(also A{{music|doublesharp}}, C{{music|flat}})

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
Just intonation
In music, just intonation is any musical tuning in which the frequencies of notes are related by ratios of small whole numbers. Any interval tuned in this way is called a just interval. The two notes in any just interval are members of the same harmonic series...

, 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 scale. For example, one can label the pitch classes of n-tone equal temperament
Equal temperament
An equal temperament is a musical temperament, or a system of tuning, in which every pair of adjacent notes has an identical frequency ratio. As pitch is perceived roughly as the logarithm of frequency, this means that the perceived "distance" from every note to its nearest neighbor is the same for...

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 12-fold 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-tet 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.