Editing Enharmonic equivalence (section)

===Meantone===
{{Main|Meantone temperament}}
In quarter-comma meantone, there will be a discrepancy between, for example, G{{music|#}} and A{{music|b}}. If [[middle C]]'s frequency is {{mvar|f}}, the next highest C has a frequency of {{nobr| 2 {{mvar|f}} .}} The quarter-comma meantone has perfectly tuned ([[just intonation|"just"]]) [[major thirds]], which means major thirds with a frequency ratio of exactly {{nobr| {{small|{{sfrac| 5 | 4 }} }} .}} To form a just major third with the C above it, A{{music|b}} and the C above it must be in the ratio 5 to 4, so A{{music|b}} needs to have the frequency

:<math>\frac{\ 4\ }{ 5 }\ (2 f) = \frac{\ 8\ }{ 5 }\ f = 1.6\ f ~~.</math>

To form a just major third above E, however, G{{music|#}} needs to form the ratio 5 to 4 with E, which, in turn, needs to form the ratio 5 to 4 with C, making the frequency of G{{music|#}} 
:<math> \left( \frac{\ 5\ }{ 4 } \right)^2\ f ~=~ \frac{\ 25\ }{ 16 }\ f ~=~ 1.5625\ f ~.</math>

This leads to G{{music|#}} and A{{music|b}} being different pitches; G{{music|#}} is, in fact 41&nbsp;[[cent (music)|cent]]s (41% of a semitone) lower in pitch. The difference is the interval called the enharmonic [[diesis]], or a frequency ratio of {{small|{{sfrac| 128 | 125 }}}}. On a piano tuned in equal temperament, both G{{music|#}} and A{{music|b}} are played by striking the same key, so both have a frequency
:<math>\ 2^{\left(\ 8\ /\ 12\ \right)}\ f ~=~ 2^{\left(\ 2\ /\ 3\ \right)}\ f ~\approx~ 1.5874\ f ~.</math>
Such small differences in pitch can skip notice when presented as melodic intervals; however, when they are sounded as chords, especially as long-duration chords, the difference between meantone intonation and equal-tempered intonation can be quite noticeable.

Enharmonically equivalent pitches can be referred to with a single name in many situations, such as the numbers of [[integer notation]] used in [[serialism]] and [[set theory (music)|musical set theory]] and as employed by [[MIDI]].