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${ \override Score.TimeSignature #'stencil = ##f \relative c' { \clef treble \key c \major \time 4/4 <c c'>1 } }$

Perfect octave

${ \override Score.TimeSignature #'stencil = ##f \new PianoStaff << \new Staff \relative c' { \clef treble \key c \major \time 4/4 <c' c' c'>1 \bar "|." } \new Staff \relative c' { \clef bass \key c \major \time 4/4 <c c, c, c,>1 } >> }$

All Cs from C₁ to C₇ inclusive

In music, a pitch class (p.c. or pc) is a set of all pitches that are a whole number of octaves 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 to musical set theory, a pitch class is "all pitches related to each other by octave, enharmonic equivalence, or both."^[2] Thus, using scientific pitch notation, the pitch class "C" is the set

{C_n : n is an integer} = {..., C₋₂, C₋₁, C₀, C₁, C₂, C₃, ...}.

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".^[3] 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.^[4]

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. 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 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 frequency f (measured in hertz) to a real number p using the equation

p=9+12\log _{2}{\frac {f}{440{\text{ Hz}}}}.

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 music theorists 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:

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).

${ \override Score.TimeSignature #'stencil = ##f \relative c' { \clef treble \key c \major c1 cis d dis e f |\break fis g gis a ais b \bar "||" } } \addlyrics { "0" "1" "2" "3" "4" "5" "6" "7" "8" "9" t e } \layout { \context {\Score \omit BarNumber} line-width = #100 }$

Integer notation.

In music, integer notation is the translation of pitch classes or interval classes into whole 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 and B in others^[6] (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.^[5]

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.

Pitch classes can be notated in this way by assigning the number 0 to some note and assigning consecutive integers to consecutive semitones; 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 12 is used to represent octave 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

Pitch class
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, 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

{\frac {a}{b}}=2^{n}

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, 1 ≤ ⁠p/q⁠ < 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 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 {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.

References

^ Arnold Whittall, The Cambridge Introduction to Serialism (New York: Cambridge University Press, 2008): 276. ISBN 978-0-521-68200-8 (pbk).
^ Don Michael Randel, ed. (2003). "Set theory", The Harvard Dictionary of Music, p.776. Harvard. ISBN 9780674011632.
^ Tymoczko, Dmitri (2011). A Geometry of Music: Harmony and Counterpoint in the Extended Common Practice, p.30. Oxford Studies in Music Theory. ISBN 9780199714353.
^ Müller, Meinard (2007). Information Retrieval for Music and Motion, p.60. ISBN 9783540740483. "A pitch class is defined to be the set of all pitches that share the same chroma."
^ ^a ^b ^c Whittall (2008), p.273.
^ Robert D. Morris, "Generalizing Rotational Arrays", Journal of Music Theory 32, no. 1 (Spring 1988): 75–132, citation on 83.

@@ Line 1: / Line 1: @@
+{{Short description|Set of all pitches that are a whole number of octaves apart}}
-[[Image:Perfect octave on C.png|thumb|right|Perfect octave {{audio|Perfect octave on C.mid|Play}}]]
-[[Image:Pitch class on C.png|thumb|right|All Cs from C<sub>1</sub> to C<sub>7</sub> inclusive{{audio|Pitch class on C.mid|Play}}.]]
+{{Image frame|content=<score sound="1">  {
-In [[music]], a '''pitch class''' ('''p.c.''' or '''pc''') is a [[set (music)|set]] of all [[Pitch (music)|pitches]] that are a whole number of [[octave]]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."<ref>[[Arnold Whittall]], ''The Cambridge Introduction to Serialism'' (New York: Cambridge University Press, 2008): 276. {{ISBN|978-0-521-68200-8}} (pbk).</ref> Important to [[musical set theory]], a pitch class is, "all pitches related to each other by octave, [[Enharmonic|enharmonic equivalence]], or both."<ref>Don Michael Randel, ed. (2003). "Set theory", ''The Harvard Dictionary of Music'', p.776. Harvard. {{ISBN|9780674011632}}.</ref> Thus, using [[scientific pitch notation]], the pitch class "C" is the set
+\override Score.TimeSignature #'stencil = ##f
+\relative c' {
+  \clef treble \key c \major \time 4/4
+  <c c'>1
+} }
+</score>|width=300|caption=Perfect octave}}
+{{Image frame|content=<score sound="1">  {
-:{C<sub>''n''</sub> : ''n'' is an [[integer]]} = {..., C<sub>−2</sub>, C<sub>−1</sub>, C<sub>0</sub>, C<sub>1</sub>, C<sub>2</sub>, C<sub>3</sub> ...};
+\override Score.TimeSignature #'stencil = ##f
+\new PianoStaff <<
+\new Staff \relative c' {
+  \clef treble \key c \major \time 4/4
+  <c' c' c'>1 \bar "|."
+}
+\new Staff \relative c' {
-although there is no formal upper or lower limit to this sequence, only a few of these pitches are audible to the human ear.
+  \clef bass \key c \major \time 4/4
+  <c c, c, c,>1
+} >> }
+</score>|width=300|caption=All Cs from C<sub>1</sub> to C<sub>7</sub> inclusive}}
+In [[music]], a '''pitch class''' ('''p.c.''' or '''pc''') is a [[set (music)|set]] of all [[Pitch (music)|pitches]] that are a whole number of [[octave]]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."<ref>[[Arnold Whittall]], ''The Cambridge Introduction to Serialism'' (New York: Cambridge University Press, 2008): 276. {{ISBN|978-0-521-68200-8}} (pbk).</ref> Important to [[musical set theory]], a pitch class is "all pitches related to each other by octave, [[Enharmonic|enharmonic equivalence]], or both."<ref>Don Michael Randel, ed. (2003). "Set theory", ''The Harvard Dictionary of Music'', p.776. Harvard. {{ISBN|9780674011632}}.</ref> Thus, using [[scientific pitch notation]], the pitch class "C" is the set
+:{C<sub>''n''</sub> : ''n'' is an [[integer]]} = {..., C<sub>−2</sub>, C<sub>−1</sub>, C<sub>0</sub>, C<sub>1</sub>, C<sub>2</sub>, C<sub>3</sub>, ...}.
+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 (psychophysics)|pitch-perception]] is [[Periodic function|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".<ref>[[Dmitri Tymoczko|Tymoczko, Dmitri]] (2011). ''A Geometry of Music: Harmony and Counterpoint in the Extended Common Practice'', p.30. Oxford Studies in Music Theory. {{ISBN|9780199714353}}.</ref> 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.<ref>Müller, Meinard (2007). ''Information Retrieval for Music and Motion'', p.60. {{ISBN|9783540740483}}. "A pitch class is defined to be the set of all pitches that share the same chroma."</ref>
-Note that in standard Western [[equal temperament]], distinct spellings can refer to the same sounding object: B{{music|sharp}}<sub>3</sub>, C<sub>4</sub>, and D{{music|bb}}<sub>4</sub> all refer to the same pitch, hence share the same chroma, and therefore belong to the same pitch class; a phenomenon called [[enharmonic|enharmonic equivalence]].
+In standard Western [[equal temperament]], distinct spellings can refer to the same sounding object: B{{music|sharp}}<sub>3</sub>, C<sub>4</sub>, and D{{music|bb}}<sub>4</sub> all refer to the same pitch, hence share the same chroma, and therefore belong to the same pitch class. This phenomenon is called [[Enharmonic|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]] 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:
+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 "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
-:<math>p = 9 + 12\log_2 \frac{f}{440\text{ Hz}}</math>
+:<math>p = 9 + 12\log_2 \frac{f}{440\text{ Hz}}.</math>
-This creates a linear [[pitch space]] in which octaves have size 12, [[semitone]]s (the distance between adjacent keys on the piano keyboard) have size 1, and [[middle C]] (C<sub>4</sub>) 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<sub>−1</sub> to G<sub>9</sub> (thus, middle C is 60). To represent pitch ''classes'', we need to identify or "glue together" all pitches belonging to the same pitch class&mdash;i.e.&nbsp;all numbers ''p'' and ''p''&nbsp;+&nbsp;12. The result is a cyclical [[quotient group]] that musicians call [[pitch class space]] and mathematicians call '''R'''/12'''Z'''. Points in this space can be labelled using [[real number]]s in the range 0&nbsp;≤&nbsp;''x''&nbsp;<&nbsp;12. These numbers provide numerical alternatives to the letter names of elementary music theory:
+This creates a linear [[pitch space]] in which octaves have size 12, [[semitone]]s (the distance between adjacent keys on the piano keyboard) have size 1, and [[middle C]] (C<sub>4</sub>) 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<sub>−1</sub> to G<sub>9</sub> (thus, middle C is 60). To represent pitch ''classes'', we need to identify or "glue together" all pitches belonging to the same pitch class&mdash;i.e.&nbsp;all numbers ''p'' and ''p''&nbsp;+&nbsp;12. The result is a cyclical [[quotient group]] that music theorists call [[pitch class space]] and mathematicians call '''R'''/12'''Z'''. Points in this space can be labelled using [[real number]]s in the range 0&nbsp;≤&nbsp;''x''&nbsp;<&nbsp;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{{music|t}} ([[quarter tone]] sharp), 3 = D{{music|#}}/E{{music|b}},
@@ Line 24: / Line 45: @@
 and so on. In this system, pitch classes represented by integers are classes of [[12 equal temperament|twelve-tone equal temperament]] (assuming standard concert A).
+{{Image frame|content=<score>  {
-[[Image:Integer notation.png|thumb|right|350px|Integer notation.]]
+\override Score.TimeSignature #'stencil = ##f
+\relative c' {
+  \clef treble \key c \major
+  c1 cis d dis e f |\break
+  fis g gis a ais b \bar "||"
+} }
+\addlyrics { "0" "1" "2" "3" "4" "5" "6" "7" "8" "9" t e }
+\layout { \context {\Score \omit BarNumber} line-width = #100 }
+</score>|width=|caption=Integer notation.}}
-In [[music]], '''integer notation''' is the translation of pitch classes and/or [[interval class]]es into [[integer|whole numbers]].<ref name="Whittall">Whittall (2008), p.273.</ref> Thus if C&nbsp;= 0, then C{{music|#}}&nbsp;= 1 ... A{{music|#}}&nbsp;= 10, B&nbsp;= 11, with "10" and "11" substituted by "t" and "e" in some sources,<ref name="Whittall"/> ''A'' and ''B'' in others<ref>Robert D. Morris, "Generalizing Rotational Arrays", ''Journal of Music Theory'' 32, no. 1 (Spring 1988): 75–132, citation on 83.</ref> (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.<ref name="Whittall"/>
+In [[music]], '''integer notation''' is the translation of pitch classes or [[interval class]]es into [[integer|whole numbers]].<ref name="Whittall">Whittall (2008), p.273.</ref> Thus if C&nbsp;= 0, then C{{music|#}}&nbsp;= 1 ... A{{music|#}}&nbsp;= 10, B&nbsp;= 11, with "10" and "11" substituted by "t" and "e" in some sources,<ref name="Whittall"/> ''A'' and ''B'' in others<ref>Robert D. Morris, "Generalizing Rotational Arrays", ''Journal of Music Theory'' 32, no. 1 (Spring 1988): 75–132, citation on 83.</ref> (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.<ref name="Whittall"/>
 In the integer model of pitch, all pitch classes and [[interval (music)|interval]]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 [[musical analysis|analytical]] and [[musical composition|compositional]] tool when working with chromatic music, including [[twelve tone technique|twelve tone]], [[serialism|serial]], or otherwise [[atonality|atonal]] music.
@@ Line 47: / Line 77: @@
 |-
 ! 0
-| [[C (musical note)|C]] (also B{{music|sharp}}, D{{music|doubleflat}})
+| [[C (musical note)|C]] (also B{{music|sharp}}, D{{music|doubleflat}}, A{{music|triplesharp}})
 | do
 |-
 ! 1
-| [[C♯ (musical note)|C{{music|sharp}}]], [[D♭ (musical note)|D{{music|flat}}]] (also B{{music|doublesharp}})
+| [[C♯ (musical note)|C{{music|sharp}}]], [[D♭ (musical note)|D{{music|flat}}]] (also B{{music|doublesharp}}, E{{music|tripleflat}})
 |
 |-
 ! 2
-| [[D (musical note)|D]] (also C{{music|doublesharp}}, E{{music|doubleflat}})
+| [[D (musical note)|D]] (also C{{music|doublesharp}}, E{{music|doubleflat}}, B{{music|triplesharp}}, F{{music|tripleflat}})
 | re
 |-
 ! 3
-| [[D♯ (musical note)|D{{music|sharp}}]], [[E♭ (musical note)|E{{music|flat}}]] (also F{{music|doubleflat}})
+| [[D♯ (musical note)|D{{music|sharp}}]], [[E♭ (musical note)|E{{music|flat}}]] (also F{{music|doubleflat}}, C{{music|triplesharp}})
 |
 |-
 ! 4
-| [[E (musical note)|E]] (also D{{music|doublesharp}}, F{{music|flat}})
+| [[E (musical note)|E]] (also D{{music|doublesharp}}, F{{music|flat}}, G{{music|tripleflat}})
 | mi
 |-
 ! 5
-| [[F (musical note)|F]] (also E{{music|sharp}}, G{{music|doubleflat}})
+| [[F (musical note)|F]] (also E{{music|sharp}}, G{{music|doubleflat}}, D{{music|triplesharp}})
 | fa
 |-
 ! 6
-| [[F♯ (musical note)|F{{music|sharp}}]], [[G♭ (musical note)|G{{music|flat}}]] (also E{{music|doublesharp}})
+| [[F♯ (musical note)|F{{music|sharp}}]], [[G♭ (musical note)|G{{music|flat}}]] (also E{{music|doublesharp}}, A{{music|tripleflat}})
 |
 |-
 ! 7
-| [[G (musical note)|G]] (also F{{music|doublesharp}}, A{{music|doubleflat}})
+| [[G (musical note)|G]] (also F{{music|doublesharp}}, A{{music|doubleflat}}, E{{music|triplesharp}})
 | sol
 |-
 ! 8
-| [[G♯ (musical note)|G{{music|sharp}}]], [[A♭ (musical note)|A{{music|flat}}]]
+| [[G♯ (musical note)|G{{music|sharp}}]], [[A♭ (musical note)|A{{music|flat}}]] (also F{{music|triplesharp}}, B{{music|tripleflat}})
 |
 |-
 ! 9
-| [[A (musical note)|A]] (also G{{music|doublesharp}}, B{{music|doubleflat}})
+| [[A (musical note)|A]] (also G{{music|doublesharp}}, B{{music|doubleflat}}, C{{music|tripleflat}})
 | la
 |-
 ! 10, t or A
-| [[A♯ (musical note)|A{{music|sharp}}]], [[B♭ (musical note)|B{{music|flat}}]] (also C{{music|doubleflat}})
+| [[A♯ (musical note)|A{{music|sharp}}]], [[B♭ (musical note)|B{{music|flat}}]] (also C{{music|doubleflat}}, G{{music|triplesharp}})
 |
 |-
 ! 11, e or B
-| [[B (musical note)|B]] (also A{{music|doublesharp}}, C{{music|flat}})
+| [[B (musical note)|B]] (also A{{music|doublesharp}}, C{{music|flat}}, D{{music|tripleflat}})
-| ti
+| 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, in [[just intonation]], we may express pitches in terms of positive rational numbers {{sfrac|''p''|''q''}}, expressed by reference to a 1 (often written "{{sfrac|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
@@ Line 113: / Line 143: @@
 *[[Tone row]] ([[List of tone rows and series|List]])
-==Sources==
+==References==
 {{reflist}}
 ==Further reading==
-*Purwins, Hendrik (2005). "[http://opus.kobv.de/tuberlin/volltexte/2005/1085/ Profiles of Pitch Classes: Circularity of Relative Pitch and Key—Experiments, Models, Computational Music Analysis, and Perspectives]". Ph.D. Thesis. Berlin: [[Berlin Institute of Technology|Technische Universität Berlin]].
+*Purwins, Hendrik (2005). "[https://depositonce.tu-berlin.de/items/4cb9db3c-3ff1-4849-bb62-936df7fde7b3 Profiles of Pitch Classes: Circularity of Relative Pitch and Key—Experiments, Models, Computational Music Analysis, and Perspectives]". Ph.D. Thesis. Berlin: [[Berlin Institute of Technology|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}}.
+*Tsao, Ming (2010). ''Abstract Musical Intervals: Group Theory for Composition and Analysis''. Berkeley, CA: Musurgia Universalis Press. ISBN 978-1430308355.
+*Butterfield, Sean (2023). ''[https://intmus.github.io/inttheory/ Integrated Musicianship: Theory]''. [https://intmus.github.io/inttheory/23-intro-to-post-tonal/b1-ex-mod12transnormform.html Chapter 23].
 {{Musical tuning}}

v t e Musical set theory
All-interval tetrachord All-trichord hexachord Complement Forte number Identity Interval class Interval vector Multiplication Permutation Pitch class Pitch interval Pitch-interval class Set List Similarity relation Transformation Z-relation
Diatonic set theory	Bisector Cardinality equals variety Common tone (Deep scale property) Diatonic scale Generated collection Generic and specific intervals (Myhill's property) Maximal evenness Rothenberg propriety Structure implies multiplicity

Latest revision as of 02:31, 8 December 2024

Integer notation

Disadvantages

Other ways to label pitch classes

See also

References

Further reading