Editing Enharmonic equivalence (section)

==Other tuning conventions==
[[File:Comparison of unisons.png|thumb|Comparison of intervals near or enharmonic with the unison]]
The standard tuning system used in Western music is [[twelve-tone equal temperament]] tuning, where the octave is divided into 12&nbsp;equal semitones. In this system, written notes that produce the same pitch, such as C{{music|#}} and D{{music|b}}, are called ''enharmonic''. In other tuning systems, such pairs of written notes do not produce an identical pitch, but can still be called "enharmonic" using the older, original sense of the word.<ref>
{{cite dictionary
 |last = Rushton |first=Julian |author-link = Julian Rushton
 |date = 2001
 |title =Enharmonic
 |dictionary = [[The New Grove Dictionary of Music and Musicians]]  |edition = 2nd
 |editor1-first = Stanley |editor1-last = Sadie   |editor1-link = Stanley Sadie
 |editor2-first = John    |editor2-last = Tyrrell |editor2-link = John Tyrrell (musicologist)
 |location = London, UK
 |publisher = Macmillan Publishers
 |isbn = 0-19-517067-9
}}
</ref>

===Pythagorean===
{{Main|Pythagorean tuning}}

In Pythagorean tuning, all pitches are generated from a series of [[Just intonation|justly tuned]] [[perfect fifth]]s, each with a frequency ratio of 3 to 2. If the first note in the series is an A{{music|b}}, the thirteenth note in the series, G{{music|#}} is ''higher'' than the seventh octave (1&nbsp;octave = frequency ratio of {{nobr|{{math| 2 to 1 {{=}} 2}} ;}} 7&nbsp;octaves is {{nobr|{{math| 2{{sup|7}} to 1 {{=}} 128}} )}} of the A{{music|b}} by a small interval called a [[Pythagorean comma]]. This interval is expressed mathematically as:

:<math>\frac{\ \hbox{twelve fifths}\ }{\ \hbox{seven octaves}\ } 
~=~ \frac{ 1 }{\ 2^7}\left(\frac{ 3 }{\ 2\ }\right)^{12}
~=~ \frac{\ 3^{12} }{\ 2^{19} }
~=~ \frac{\ 531\ 441\ }{\ 524\ 288\ }
~=~ 1.013\ 643\ 264\ \ldots
~\approx~ 23.460\ 010 \hbox{ cents} 
~.</math>

===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]].