Enharmonic equivalence

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In music, two written notes have enharmonic equivalence if they produce the same pitch but are notated differently. Similarly, written intervals, chords, or key signatures are considered enharmonic if they represent identical pitches that are notated differently. The term derives from Latin Template:Langx, in turn from Late Latin Template:Langx, from Ancient Greek Template:Langx (Template:Transliteration), from Template:Langx ('in') and Template:Langx ('harmony').

DefinitionEdit

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Template:Image frame The predominant tuning system in Western music is twelve-tone equal temperament (12 Template:Sc), where each octave is divided into twelve equivalent half steps or semitones. The notes F and G are a whole step apart, so the note one semitone above F (FTemplate:Music) and the note one semitone below G (GTemplate:Music) 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 accidentals can produce other enharmonic equivalents; for example, FTemplate:Music (double-sharp) is enharmonically equivalent to GTemplate:Music. Prior to this modern use of 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 equivalent half steps, FTemplate:Music and GTemplate:Music would not indicate the same pitch. Template:Image frame

File:Enharmonic flats.png

A musical passage notated as flats.

File:Enharmonic sharps.png

The same passage notated as sharps, requiring fewer canceling natural signs.

Sets of notes that involve pitch relationships — scales, key signatures, or intervals,<ref> Template:Cite book </ref> for example — can also be referred to as enharmonic (e.g., the keys of CTemplate:Music major and DTemplate:Music 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 GTemplate:Music, or as an augmented fourth (C to FTemplate:Music). Representing the C as a BTemplate:Music leads to other enharmonically equivalent options for notation.

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.

ExamplesEdit

At the end of the bridge section of Jerome Kern's "All the Things You Are", a GTemplate:Music (the sharp 5 of an augmented C chord) becomes an enharmonically equivalent ATemplate:Music (the third of an F minor chord) at the beginning of the returning "A" section.<ref>Kern, J. and Hammerstein, O. (1939, bars 23-25) "All the things you are", New York, T. B. Harms Co.</ref><ref>Archived at Ghostarchive Template:Cbignore and the Wayback Machine Template:Cbignore: {{#invoke:citation/CS1|citation |CitationClass=web }}Template:Cbignore</ref>

Beethoven's Piano Sonata in E Minor, Op. 90, contains a passage where a BTemplate:Music becomes an ATemplate:Music, altering its musical function. The first two bars of the following passage unfold a descending BTemplate:Music major scale. Immediately following this, the BTemplate:Musics become ATemplate:Musics, the leading tone of B minor:

File:Beethoven Sonata in E minor Op 90, first movement, bars 37-46.png

Beethoven Sonata in E Minor Op. 90, first movement, bars 37–45
File:Beethoven Sonata in E minor Op 90, first movement, bars 37-45.wav

Chopin's Prelude No. 15, known as the "Raindrop Prelude", features a pedal point on the note ATemplate:Music throughout its opening section.

File:Chopin Prelude No. 15, opening 02.png

Chopin Prelude No. 15, opening
File:Chopin Prelude No. 15, opening 01.wav

In the middle section, these are changed to GTemplate:Musics as the key changes to C-sharp minor. This is primarily a notational convenience, since D-flat minor would require many double-flats and be difficult to read:

File:Chopin Prelude No. 15, bars 28-30.png

Chopin Prelude No. 15, bars 28–29
File:Chopin Prelude Op. 28, No. 15, bars 28-29.wav

The concluding passage of the slow movement of Schubert's final piano sonata in BTemplate:Music (D960) contains a dramatic enharmonic change. In bars 102–3, a BTemplate:Music, the third of a GTemplate:Music major triad, transforms into CTemplate:Music as the prevailing harmony changes to C major:

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File:Schubert Piano Sonata D960 second movement, bars 98-107.png

Schubert Piano Sonata D960 second movement, bars 98–106
File:Schubert Piano Sonata D960 second movement, bars 98-106.wav

Other tuning conventionsEdit

File:Comparison of unisons.png

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 equal semitones. In this system, written notes that produce the same pitch, such as CTemplate:Music and DTemplate:Music, 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> Template:Cite dictionary </ref>

PythagoreanEdit

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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 ATemplate:Music, the thirteenth note in the series, GTemplate:Music is higher than the seventh octave (1 octave = frequency ratio of Template:Nobr 7 octaves is Template:Nobr of the ATemplate:Music 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>

MeantoneEdit

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} In quarter-comma meantone, there will be a discrepancy between, for example, GTemplate:Music and ATemplate:Music. If middle C's frequency is Template:Mvar, the next highest C has a frequency of Template:Nobr The quarter-comma meantone has perfectly tuned ("just") major thirds, which means major thirds with a frequency ratio of exactly Template:Nobr To form a just major third with the C above it, ATemplate:Music and the C above it must be in the ratio 5 to 4, so ATemplate:Music needs to have the frequency

To form a just major third above E, however, GTemplate: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 GTemplate:Music

<math> \left( \frac{\ 5\ }{ 4 } \right)^2\ f ~=~ \frac{\ 25\ }{ 16 }\ f ~=~ 1.5625\ f ~.</math>

This leads to GTemplate:Music and ATemplate:Music being different pitches; GTemplate:Music 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 Template:Small. On a piano tuned in equal temperament, both GTemplate:Music and ATemplate:Music 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 musical set theory and as employed by MIDI.

Enharmonic genusEdit

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In ancient Greek music the enharmonic was 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 Template:Sfrac to Template:Sfrac (3.55 to 4.35 semitones) and the microtones can be anything smaller than 1 semitone.<ref>Template:Cite journal</ref> Some examples of enharmonic genera are

Enharmonic keyEdit

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-flat major|CTemplate:Music major]], [[G-sharp minor|GTemplate:Music minor]]/[[A-flat minor|ATemplate:Music minor]], [[F-sharp major|FTemplate:Music major]]/[[G-flat major|GTemplate:Music major]], [[D-sharp minor|DTemplate:Music minor]]/[[E-flat minor|ETemplate:Music minor]], [[C-sharp major|CTemplate:Music major]]/[[D-flat major|DTemplate:Music major]] and [[A-sharp minor|ATemplate:Music minor]]/[[B-flat minor|BTemplate:Music 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 five flats.

ReferencesEdit

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External linksEdit

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