Editing Pitch class (section)

==Other ways to label pitch classes==

{| class="wikitable" align="right"
|+Pitch class
|-
! Pitch<br />class
! Tonal counterparts
! Solfege
|-
! 0
| [[C (musical note)|C]]
| do
|-
! 1
| [[C♯ (musical note)|C{{music|sharp}}]], [[D♭ (musical note)|D{{music|flat}}]]
| 
|-
! 2
| [[D (musical note)|D]]
| re
|-
! 3
| [[D♯ (musical note)|D{{music|sharp}}]], [[E♭ (musical note)|E{{music|flat}}]]
| 
|-
! 4
| [[E (musical note)|E]]
| mi
|-
! 5
| [[F (musical note)|F]]
| fa
|-
! 6
| [[F♯ (musical note)|F{{music|sharp}}]], [[G♭ (musical note)|G{{music|flat}}]]
| 
|-
! 7
| [[G (musical note)|G]]
| sol
|-
! 8
| [[G♯ (musical note)|G{{music|sharp}}]], [[A♭ (musical note)|A{{music|flat}}]]
|
|-
! 9
| [[A (musical note)|A]]
| la
|-
! 10, t or A
| [[A♯ (musical note)|A{{music|sharp}}]], [[B♭ (musical note)|B{{music|flat}}]]
|
|-
! 11, e or B
| [[B (musical note)|B]]
| ti
|}
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

<math display="block">\frac{a}{b} = 2^n</math>

for some [[integer]] ''n''. Therefore, we can represent pitch classes in this system using ratios {{sfrac|''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&nbsp;≤&nbsp;{{sfrac|''p''|''q''}}&nbsp;<&nbsp;2.

It is also very common to label pitch classes with reference to some [[Scale (music)|scale]].  For example, one can label the pitch classes of ''n''-tone [[equal temperament]] using the integers 0 to ''n''&nbsp;−&nbsp;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.