Enharmonic equivalence

Not to be confused with Anharmonic or Inharmonic.

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 enharmonicus, in turn from Late Latin enarmonius, from Ancient Greek ἐναρμόνιος (enarmónios), from ἐν ('in') and ἁρμονία ('harmony').

Definition

The predominant tuning system in Western music is twelve-tone equal temperament tuning, in which each octave is divided into twelve equivalent half-steps. In this system, written notes such as C♯ and D♭ represent the same pitch and are considered enharmonic or enharmonically equivalent. This is not true in many other systems of tuning (see below). The choice of how to notate a pitch can depend on its role in harmony; this notational practice 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 F double-sharp is enharmonically equivalent to G natural. Prior to this modern meaning, "enharmonic" referred to notes that were very close in pitch — closer than the smallest step of a diatonic scale — but not identical. For example, in most tuning systems other than the modern 12-tone equal temperament, G♯ is not the same pitch as A♭.

A musical passage notated in flats

The same passage notated in sharps

The notes F♯ and G♭ are enharmonic equivalents.

G and B are enharmonic equivalents, both the same as A♮.

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.

Enharmonic tritones: augmented 4th = diminished 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., 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 minor sixth from C may be written as C to A♭, or as an augmented fifth (C to G♯). Representing the C as a B♯ leads to other enharmonically equivalent notational options. .

Examples in practice

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

Beethoven's Piano Sonata 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, the leading tone of B minor:

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

Chopin's Prelude No. 15, known as the "Raindrop Prelude", features a pedal point on 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 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 of Schubert's final piano sonata in 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 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 C♯ and D♭, are called enharmonic. In other tuning systems, such pairs of written notes do not produce an identical pitch.

Pythagorean

Main article: Pythagorean tuning

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 (octave = ratio of 1 to 2, seven octaves is 1 to 2⁷ = 128) of the A♭ by a small interval called a Pythagorean comma. This interval is expressed mathematically as:

${\frac {\hbox{twelve fifths}}{\hbox{seven octaves}}}={\frac {\left({\tfrac {3}{2}}\right)^{12}}{2^{7}}}={\frac {3^{12}}{2^{19}}}={\frac {531\,441}{524\,288}}=1.013\,643\,264...\approx 23.46{\hbox{ cents}}\!$

Meantone

Main article: Meantone temperament

In quarter-comma meantone, there will be a discrepancy between, for example, G♯ and A♭. If middle C's frequency is x, the next highest C has a frequency of 2x. The quarter-comma meantone has perfectly tuned ("just") major thirds, which means major thirds with a frequency ratio of exactly 4 to 5. To form a just major third with the C above it, A♭ and the C above it must be in the ratio 4 to 5, so A♭ needs to have the frequency

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

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♯

$\left({\frac {5}{4}}\right)^{2}x={\frac {25}{16}}x=1.5625x$

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 ⁠128/125⁠. On a piano tuned in equal temperament, both G♯ and A♭ are played by striking the same key, so both have a frequency

$2^{\frac {8}{12}}x=2^{\frac {2}{3}}x\approx 1.5874x$

Such small differences in pitch can escape notice when presented as melodic intervals. However, when they are sounded as 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 employed by the MIDI interface.

Enharmonic genus

Main article: Genus (music) § Enharmonic

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 ⁠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

Enharmonic key

Some key signatures have an enharmonic equivalent that contains the same pitches, albeit spelled differently. 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.

Theoretical

Keys that require more than 7 sharps or flats are called theoretical key signatures. 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
• Transpositional equivalence
• Diatonic and chromatic
• Enharmonic modulation

Further reading

• Eijk, Lisette D. van der (2020). "The difference between a sharp and a flat".
• Mathiesen, Thomas J. (2001). "Greece, §I: Ancient". In Sadie, Stanley; Tyrrell, John (eds.). The New Grove Dictionary of Music and Musicians (2nd ed.). London: Macmillan Publishers. ISBN 0-19-517067-9.
• Morey, Carl (1966). "The Diatonic, Chromatic and Enharmonic Dances by Martino Pesenti". Acta Musicologica. 38 (2–4): 185–189. doi:10.2307/932526. JSTOR 932526.

External links

• The dictionary definition of enharmonic equivalence at Wiktionary
• Media related to Enharmonic at Wikimedia Commons