Editing Pitch class (section)

==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

<math display="block">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 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: 

{{blockindent|1=0 = C, 1 = C{{music|#}}/D{{music|b}}, 2 = D, 2.5 = D{{music|t}} ([[quarter tone]] sharp), 3 = D{{music|#}}/E{{music|b}},}}

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>  {
\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 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.

Pitch classes can be notated in this way by assigning the number 0 to some note and assigning consecutive integers to consecutive [[semitone]]s; so if 0 is C natural, 1 is C{{Music|sharp}}, 2 is D{{Music|natural}} and so on up to 11, which is B{{Music|natural}}. The C above this is not 12, but 0 again (12&nbsp;−&nbsp;12&nbsp;=&nbsp;0). Thus arithmetic [[modular arithmetic|modulo]] 12 is used to represent [[octave]] [[Equivalence class (music)|equivalence]]. One advantage of this system is that it ignores the "spelling" of notes (B{{Music|sharp}}, C{{Music|natural}} and D{{Music|doubleflat}} are all 0) according to their [[diatonic functionality]].

===Disadvantages===
<!--1st disadvantage-->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,&nbsp;... 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{{music|#}} in 12-tone equal temperament, but D in 6-tone equal temperament.

<!--2nd disadvantage-->Also, the same numbers are used to represent both [[pitch (music)|pitches]] and [[interval (music)|intervals]]. For example, the number 4 serves both as a label for the pitch class E (if C&nbsp;= 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 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{{music|#}} 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").