Enharmonic equivalence

Not to be confused withAnharmonicorInharmonic.

In music, two written notes haveenharmonic equivalenceif they produce the samepitchbut arenotateddifferently. Similarly, writtenintervals,chords,orkey signaturesare consideredenharmonicif they represent identical pitches that are notated differently. The term derives from Latinenharmonicus,in turn fromLate Latinenarmonius,from Ancient Greekἐναρμόνιος(enarmónios), fromἐν('in') andἁρμονία('harmony').

Definition

The notes F♯and G♭are enharmonic equivalents in12TET.

Gand Bare enharmonic equivalents in12TET;both are the same as A♮.

The predominanttuning systemin Western music istwelve-tone equal temperament(12TET), 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 (F♯) and the note one semitone below G (G♭) indicate the same pitch. These written notes areenharmonic,orenharmonically equivalent.The choice of notation for a pitch can depend on itsrole in harmony;this notation keeps modern music compatible with earlier tuning systems, such asmeantone temperaments.The choice can also depend on the note's readability in the context of the surrounding pitches. Multipleaccidentalscan produce other enharmonic equivalents; for example, F(double-sharp) is enharmonically equivalent to G♮.Prior to this modern use of the term,enharmonicreferred to notes that werevery closein pitch — closer than the smallest step of adiatonicscale — but not quite identical. In a tuning system without equivalent half steps, F♯and G♭would not indicate the same pitch.

Enharmonic tritones: Augmented 4th = diminished 5th on C.

A musical passage notated as flats.

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

Sets of notes that involve pitch relationships — scales, key signatures, or intervals, for example — can also be referred to asenharmonic(e.g., 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 atritoneabove C may be written as a diminished fifth from C to G♭,or as an augmented fourth (C to F♯). Representing the C as a B♯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.

Examples

At the end of thebridgesection ofJerome Kern's "All the Things You Are",a G♯(the sharp 5 of an augmented C chord) becomes an enharmonically equivalent A♭(the third of an F minor chord) at the beginning of the returning "A" section.

Beethoven's PianoSonata in E Minor, Op. 90,contains a passage where a B♭becomes an A♯,altering its musical function. The first two bars of the following passage unfold a descending B♭major scale. Immediately following this, the B♭s become A♯s, theleading toneof B minor:

Beethoven Sonata in E Minor Op. 90, first movement, bars 37–45

Chopin'sPrelude No. 15,known as the "Raindrop Prelude", features apedal pointon the note A♭throughout its opening section.

Chopin Prelude No. 15, opening

In the middle section, these are changed to G♯s 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:

Chopin Prelude No. 15, bars 28–29

The concluding passage of the slow movement ofSchubert's final piano sonatain B♭(D960) contains a dramatic enharmonic change. In bars 102–3, a B♯,the third of a G♯major triad, transforms into C♮as the prevailing harmony changes to C major:

G-sharp to C progression.

Schubert Piano Sonata D960 second movement, bars 98–106

Other tuning conventions

Comparison of intervals near or enharmonic with the unison

The standard tuning system used in Western music istwelve-tone equal temperamenttuning, where the octave is divided into 12 equal semitones. In this system, written notes that produce the same pitch, such as C♯and D♭,are calledenharmonic.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.

Pythagorean

Main article:Pythagorean tuning

In Pythagorean tuning, all pitches are generated from a series ofjustly tunedperfect 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♯ishigherthan the seventh octave (1 octave = frequency ratio of2 to 1 = 2;7 octaves is2⁷to 1 = 128)of the A♭by a small interval called aPythagorean 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

Main article:Meantone temperament

In quarter-comma meantone, there will be a discrepancy between, for example, G♯and A♭.Ifmiddle C's frequency isf,the next highest C has a frequency of2f.The quarter-comma meantone has perfectly tuned ("just")major thirds,which means major thirds with a frequency ratio of exactly⁠ 5 /4⁠.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

${\frac {\ 4\ }{5}}\ (2f)={\frac {\ 8\ }{5}}\ f=1.6\ f~~.$

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 41cents(41% of a semitone) lower in pitch. The difference is the interval called the enharmonicdiesis,or a frequency ratio of⁠128/125⁠.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 ofinteger notationused inserialismandmusical set theoryand as employed byMIDI.

Enharmonic genus

Main article:Genus (music) § Enharmonic

Inancient Greek musicthe enharmonic was one of the three Greekgenerain music in which thetetrachordsare divided (descending) as aditoneplus twomicrotones.The ditone can be anywhere from⁠16/13⁠to⁠9/7⁠(3.55 to 4.35semitones) and the microtones can be anything smaller than 1 semitone.Some examples of enharmonic genera are

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

Enharmonic key

Somekey signatureshave 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♭majorandA♯minor/B♭minor.

Theoretical key

Keys that require more than 7 sharps or flats are calledtheoretical keys.They have enharmonically equivalent keys with simpler key signatures, so are rarely seen.

F flat major- (E major)
G sharp major- (A flat major)
D flat minor- (C sharp minor)
E sharp minor- (F minor)

See also

• Enharmonic keyboard
• Music theory
• Transpositionalequivalence
• Diatonic and chromatic
• Enharmonic modulation

Further reading

• Eijk, Lisette D. van der (2020). "The difference between a sharp and a flatArchived2021-03-01 at theWayback Machine".
• Mathiesen, Thomas J. (2001). "Greece, §I: Ancient". InSadie, Stanley;Tyrrell, John(eds.).The New Grove Dictionary of Music and Musicians(2nd ed.). London: Macmillan Publishers.ISBN0-19-517067-9.
• Morey, Carl (1966). "The Diatonic, Chromatic and Enharmonic Dances by Martino Pesenti".Acta Musicologica.38(2–4): 185–189.doi:10.2307/932526.JSTOR932526.

External links

• The dictionary definition ofenharmonic equivalenceat Wiktionary
• Media related toEnharmonicat Wikimedia Commons