Enharmonic equivalence

{{Image frame|width=210|content={{center|{ \magnifyStaff #5/4 \omit Score.TimeSignature \clef F \time 2/1 fis2 s ges s }}}|caption=The notes F and G are enharmonic equivalents in 12.}} {{Image frame|width=210|content={{center|\relative c' { \magnifyStaff #5/4 \omit Score.TimeSignature \clef C \time 2/1 gisis2 s beses s}}}|caption=G and B are enharmonic equivalents; both are the same as A in 12.}} The predominant tuning system in Western music is twelve-tone equal temperament (12 ), where each octave is divided into twelve equal half-steps, or semitones; each half-step is both a chromatic semitone (a sharp or a flat) and a diatonic semitone (a minor step between two diatonic notes). The notes F and G are a whole step apart, so the note one semitone above F (F) and the note one semitone below G (G) indicate the same pitch. These written notes are enharmonic, or enharmonically equivalent. The choice of notation for a pitch can depend on its role in harmony; this notation keeps modern music compatible with earlier tuning systems, such as meantone temperaments. The choice can also depend on the note's readability in the context of the surrounding pitches. Multiple sharps or flats can produce other enharmonic equivalents; for example, F (double-sharp) is enharmonically equivalent to G. When other tuning systems were in use, prior to the adoption of 12 equal temperament|, the term enharmonic referred to notes that were very close in pitch — closer than the smallest step of a diatonic scale — but not quite identical. In a tuning system without equal half steps, F and G do not indicate the same pitch, although the two pitches would be called enharmonically equivalent. {{Image frame |width=210 |content={{center|\relative c' { \magnifyStaff #5/4 \omit Score.TimeSignature \time 2/1 1 }}}|caption=Enharmonic tritones: aug 4th = dim 5th on C.}} Sets of notes that involve pitch relationships — scales, key signatures, or intervals, for example — can also be referred to as enharmonic (e.g., in the keys of C major and D major contain identical pitches and are therefore enharmonic). Identical intervals notated with different, enharmonically equivalent, written pitches are also referred to as enharmonic. The interval of a tritone above C may be written as a diminished fifth from C to G, or as an augmented fourth (C to F). In modern , notating the C as a B leads to other enharmonically equivalent notations, an option which does not exist in most earlier notation systems. Enharmonic equivalents can be used to improve the readability of music, as when a sequence of notes is more easily read using sharps or flats. This may also reduce the number of accidentals required. ==Examples ==

At the end of the bridge section of Jerome Kern's "All the Things You Are", a G (the sharp 5th of an augmented C chord) becomes an enharmonically equivalent A (the third of an F minor chord) at the beginning of the returning Beethoven's Piano Sonata in E Minor, Op. 90, contains a passage where a B becomes an A, altering its overt musical function. The first two bars of the following passage contain a descending B major scale. Immediately following this, the Bs become As, the leading tone of B minor: Chopin's Prelude No. 15, known as the "Raindrop Prelude", features a pedal point on the note A throughout its opening section. In the middle section, these are changed to Gs as the key changes to C minor. The new key is not notated as D minor because that key signature would require a double-flat: The concluding passage of the slow movement of Schubert's final piano sonata in B (D960) contains an enharmonic change in bars 102–103, where there is a B that functions as the third of a G major triad. When the prevailing harmony changes to C major that pitch is notated as C: {{Image frame|width=210|content={{center|\relative c'' { \magnifyStaff #5/4 \omit Score.TimeSignature \set doubleSlurs = ##t 1 ()}}}|caption=G to C progression.}} ==Other tuning conventions==

In twelve-tone equal temperament tuning, the standard tuning system of Western music, an octave is divided into 12 equal semitones. Written notes that produce the same pitch, such as C and D, 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 sense of the word. Pythagorean In Pythagorean tuning, all pitches are generated from a series of justly tuned perfect fifths, each with a frequency ratio of 3 to 2. If the first note in the series is an A, the thirteenth note in the series, G is higher than the seventh octave (1 octave = frequency ratio of 7 octaves is of the A by a small interval called a Pythagorean comma. This interval is expressed mathematically as: :\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} ~. Meantone In quarter-comma meantone, there will be a discrepancy between, for example, G and A. If middle C's frequency is , the next highest C has a frequency of The quarter-comma meantone has perfectly tuned ("just") major thirds, which means major thirds with a frequency ratio of exactly To form a just major third with the C above it, A and the C above it must be in the ratio 5 to 4, so A needs to have the frequency :\frac{\ 4\ }{ 5 }\ (2 f) = \frac{\ 8\ }{ 5 }\ f = 1.6\ f ~~. To form a just major third above E, however, G 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 : \left( \frac{\ 5\ }{ 4 } \right)^2\ f ~=~ \frac{\ 25\ }{ 16 }\ f ~=~ 1.5625\ f ~. This leads to G and A being different pitches; G is, in fact 41 cents (41% of a semitone) lower in pitch. The difference is the interval called the enharmonic diesis, or a frequency ratio of . On a piano tuned in equal temperament, both G and A are played by striking the same key, so both have a frequency :\ 2^{\left(\ 8\ /\ 12\ \right)}\ f ~=~ 2^{\left(\ 2\ /\ 3\ \right)}\ f ~\approx~ 1.5874\ f ~. 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 musical set theory and as employed by MIDI. ==Enharmonic genus==

In ancient Greek music the enharmonic was one of the three Greek genera in music; in the enharmonic genus, the tetrachords are divided (in descending pitch order) as a ditone (3) plus two microtones. The ditone can be anywhere from (359.5 cents) to (435.1 cents) (3.55 to 4.35 semitones) and the microtones can be anything smaller than 1 semitone. Some examples of enharmonic genera in modern ascending pitch order are : ==Enharmonic key==

Some key signatures have an enharmonic equivalent that contains the same pitches, albeit spelled differently. In twelve-tone equal temperament, there are three pairs each of major and minor enharmonically equivalent keys: B major/C major, G minor/A minor, F major/G major, D minor/E minor, C major/D major and A minor/B minor. If a key were to use more than 7 sharps or flats it would require at least one double flat or double sharp. These key signatures are extremely rare since they have enharmonically equivalent keys with simpler, conventional key signatures. For example, G sharp major would require eight sharps (six sharps plus F double-sharp), but would almost always be replaced by the enharmonically equivalent key signature of A flat major, with four flats. ==See also==