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').
Definition
Template:Image frame The predominant tuning system in Western music is [[12 tone equal temperament|twelve-tone equal temperament (12 Template:Sc)]], 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 (FTemplate:Sup) and the note one semitone below G (GTemplate:Sup) 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, FTemplate:Sup (double-sharp) is enharmonically equivalent to GTemplate:Sup.
When other tuning systems were in use, prior to the adoption of [[12 equal temperament|Template:Nobr]], 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, FTemplate:Sup and GTemplate:Sup do not indicate the same pitch, although the two pitches would be called enharmonically equivalent. Template:Image frame
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., in Template:Nobr the keys of [[C-sharp major|CTemplate:Sup major]] and [[D-flat major|DTemplate:Sup 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:Sup, or as an augmented fourth (C to FTemplate:Sup). In modern Template:Nobr, notating the C as a BTemplate:Sup 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 GTemplate:Sup (the sharp 5th of an augmented C chord) becomes an enharmonically equivalent ATemplate:Sup (the third of an F minor chord) at the beginning of the returning Template:Nobr<ref>Template:Cite book</ref><ref>Template:Cite AV media Also archived Template:Webarchive</ref>
Beethoven's Piano Sonata in E Minor, Op. 90, contains a passage where a BTemplate:Sup becomes an ATemplate:Sup, altering its overt musical function. The first two bars of the following passage contain a descending BTemplate:Sup major scale. Immediately following this, the BTemplate:Sups become ATemplate:Sups, the leading tone of B minor:
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:Sup throughout its opening section.
File:Chopin Prelude No. 15, opening 01.wav
In the middle section, these are changed to GTemplate:Sups as the key changes to [[C-sharp minor|CTemplate:Sup minor]]. The new key is not notated as [[D-flat minor|DTemplate:Sup minor]] because that key signature would require a double-flat:
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 [[B-flat major|BTemplate:Sup]] (D960) contains an enharmonic change in bars 102–103, where there is a BTemplate:Sup that functions as the third of a GTemplate:Sup major triad. When the prevailing harmony changes to C major that pitch is notated as CTemplate:Sup:
File:Schubert Piano Sonata D960 second movement, bars 98-106.wav
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 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 sense of the word.<ref> Template:Cite dictionary </ref>
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 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>
Meantone
Template:Main 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
- <math>\frac{\ 4\ }{ 5 }\ (2 f) = \frac{\ 8\ }{ 5 }\ f = 1.6\ f ~~.</math>
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 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 [[major third|ditone (Template:Sup3)]] plus two microtones. The ditone can be anywhere from Template:Small (359.5 cents) to Template:Small (435.1 cents) (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 in modern ascending pitch order are
Tonic Lower
[[microtone|Template:Mvar‑tone]]Higher
[[microtone|Template:Mvar‑tone]]( wide
gap )Ditone Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac Template:Sfrac
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-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 four flats.
See also
- Enharmonic keyboard
- Music theory
- Transpositional equivalence
- Diatonic and chromatic
- Enharmonic modulation
References
Further reading
- Eijk, Lisette D. van der (2020). "The difference between a sharp and a flat Template:Webarchive".
- Template:Cite book
- Template:Cite journal