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# Enharmonic

In music, an enharmonic is a note which is the equivalent of some other note, but spelled differently. For example, in twelve-tone equal temperament (the normal system of musical tuning in the west), the notes C sharp and D flat are enharmonically equivalent - that is, they are represented by the same key (on a musical keyboard, for example), and thus are identical in pitch, although they have different names and diatonic functionality.

This is in contrast to just intonation—in which, when singing or playing on a fretless stringed instrument such as the violin—the enharmonic equivalents actually do differ slightly in pitch. For example, consider G sharp and A flat. Call middle C's frequency x. Then high C has a frequency of 2x. Perfect major thirds need to have frequency ratios of exactly 4 to 5.

In order to form a perfect major third with the C above it, A flat and high C need to be in the ratio 4 to 5, so A flat needs to have the frequency

$\frac {2x}{\frac{5}{4}} = 1.6x$.

In order to form a perfect major third above E, however, G sharp needs to form the ratio 5 to 4 with E, which, in turn, needs to form the ratio 5 to 4 with C. Thus the frequency of G sharp is

$\left(\frac{5}{4}\right)\left(\frac{5}{4}\right)x = \left(\frac{25}{16}\right)x = 1.5625 x$

Thus, G sharp and A flat are not the same note; G sharp is, in fact 41 "cents" lower in pitch (41% of a semitone, not quite a quarter of a tone). On a piano, both would be played by striking the same key, with a frequency $2^\frac{8}{12}x = 2^\frac{2}{3} \approx 1.5874 x$ Such small differences in pitch can escape notice when presented as melodic intervals. However, when they are sounded as chords, the difference between just intonation and equal-tempered intonation is quite noticeable, even to untrained ears.

The reason that — despite, in almost all western music, Ab is exactly the same pitch as G# — we label them differently is that in tonal music notes are named for their harmonic function. This is called diatonic functionality. Also, there is one way of labeling enharmonically equivalent pitches with one and only one name, sometimes called integer notation, often used in serialism, twelve tone music, and other atonal music theory such as musical set theory.

An enharmonic is also one of the three Greek genera in music, in which the tetrachords are divided (descending) as a ditone plus two microtones. The ditone can be anywhere from 16/13 to 9/7 (3.55 to 4.35 semitones) and the microtones can be anything smaller than 1 semitone. Some examples of enharmonic genera are

1. 1/1 36/35 16/15 4/3
2. 1/1 28/27 16/15 4/3
3. 1/1 64/63 28/27 4/3
4. 1/1 49/48 28/27 4/3
5. 1/1 25/24 13/12 4/3

See also: enharmonic scale, music theory, music notation, accidental , octave equivalence, transpositional equivalence, inversional equivalence.