Editing Enharmonic equivalence

{{Short description|Distinct pitch classes sounding the same}}
{{distinguish|Anharmonic|Inharmonic}}
{{Technical|date=September 2019}}

In music, two written notes have '''enharmonic equivalence''' if they produce the same [[pitch (music)|pitch]] but are [[musical notation|notated]] differently. Similarly, written [[Interval (music)|intervals]], [[Chord (music)|chords]], or [[key signature]]s are considered '''enharmonic''' if they represent identical pitches that are notated differently. The term derives from Latin {{langx|la|enharmonicus|label=none}}, in turn from [[Late Latin]] {{langx|la|enarmonius|label=none}}, from Ancient Greek {{Langx|grc|{{math|ἐναρμόνιος}}|label=none}} ({{transliteration|grc|enarmónios}}), from {{Langx|grc|{{math|ἐν}}|label=none}} ('in') and {{Langx|grc|{{math|ἁρμονία}}|label=none}} ('harmony').

==Definition==
{{Image frame|width=210|content={{center|<score>{ \magnifyStaff #5/4 \omit Score.TimeSignature \clef F \time 2/1 fis2 s ges s }</score>}}|caption=The notes F{{Music|#}} and G{{Music|b}} are enharmonic equivalents in [[12 equal temperament|12 {{sc|tet}}]].}}

{{Image frame|width=210|content={{center|<score>\relative c' { \magnifyStaff #5/4 \omit Score.TimeSignature \clef C \time 2/1 gisis2 s beses s}</score>}}|caption=G{{Music|x}} and B{{Music|bb}} are enharmonic equivalents in [[12 equal temperament|12 {{sc|tet}}]]; both are the same as A{{music|N}}.}}
The predominant [[musical tuning|tuning system]] in Western music is [[12 tone equal temperament|twelve-tone equal temperament]] (12 {{sc|tet}}), 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{{music|#}}) and the note one semitone below G (G{{music|b}}) indicate the same pitch. These written notes are ''enharmonic'', or ''enharmonically equivalent''. The choice of notation for a pitch can depend on its [[diatonic function|role in harmony]]; this notation keeps modern music compatible with earlier tuning systems, such as [[meantone temperament]]s. The choice can also depend on the note's readability in the context of the surrounding pitches. Multiple [[accidental (music)|accidentals]] can produce other enharmonic equivalents; for example, F{{music|x}} (double-sharp) is enharmonically equivalent to G{{music|N}}. 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, F{{music|#}} and G{{music|b}} would not indicate the same pitch.
{{Image frame |width=210 |content={{center|<score>\relative c' { \magnifyStaff #5/4 \omit Score.TimeSignature \time 2/1 <c fis>1 <c ges'>}</score>}}|caption=Enharmonic tritones: Augmented 4th = diminished 5th on C.[[File:Tritone on C.mid|Play]]}}

[[File:Enharmonic flats.png|thumb|400px|A musical passage notated as flats.]]
[[File:Enharmonic sharps.png|thumb|400px|The same passage notated as sharps, requiring fewer canceling natural signs.]]

Sets of notes that involve pitch relationships — scales, key signatures, or intervals,<ref>
{{cite book
 |last1=Benward |first1=Bruce
 |last2=Saker   |first2=Marilyn
 |year=2003
 |title=Music in Theory and Practice
 |volume=I  |page=54
 |isbn=978-0-07-294262-0
}}
</ref>
for example — can also be referred to as ''enharmonic'' (e.g., the keys of C{{music|#}} major and D{{music|b}} 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{{music|b}}, or as an augmented fourth (C to F{{music|#}}). Representing the C as a B{{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.

==Examples ==
At the end of the [[Bridge (music)|bridge]] section of [[Jerome Kern]]'s "[[All the Things You Are]]", a G{{Music|sharp}} (the sharp 5 of an augmented C chord) becomes an enharmonically equivalent A{{Music|flat}} (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 [https://ghostarchive.org/varchive/youtube/20211205/OPapxr8GvGA Ghostarchive]{{cbignore}} and the [https://web.archive.org/web/20111113215422/http://www.youtube.com/watch?v=OPapxr8GvGA&gl=US&hl=en Wayback Machine]{{cbignore}}: {{cite web| url = https://www.youtube.com/watch?v=OPapxr8GvGA| title = Ella Fitzgerald - All The Things You Are (with lyrics) | website=[[YouTube]]}}{{cbignore}}</ref>

[[Beethoven]]'s Piano [[Piano Sonata No. 27 (Beethoven)|Sonata in E Minor, Op. 90]], contains a passage where a B{{Music|flat}} becomes an A{{Music|sharp}}, altering its musical function. The first two bars of the following passage unfold a descending B{{Music|flat}} major scale. Immediately following this, the B{{Music|flat}}s become A{{Music|sharp}}s, the [[Leading-tone|leading tone]] of B minor:
[[File:Beethoven Sonata in E minor Op 90, first movement, bars 37-46.png|thumb|center|500px|Beethoven Sonata in E Minor Op. 90, first movement, bars 37–45<br>[[File:Beethoven Sonata in E minor Op 90, first movement, bars 37-45.wav]]]]
[[Chopin]]'s [[Preludes (Chopin)|Prelude No. 15]], known as the "Raindrop Prelude", features a [[pedal point]] on the note A{{Music|flat}} throughout its opening section.
[[File:Chopin Prelude No. 15, opening 02.png|thumb|center|500px|Chopin Prelude No. 15, opening<br>[[File:Chopin Prelude No. 15, opening 01.wav]]]]

In the middle section, these are changed to G{{Music|sharp}}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:

[[File:Chopin Prelude No. 15, bars 28-30.png|thumb|center|400px|Chopin Prelude No. 15, bars 28–29<br>[[File:Chopin Prelude Op. 28, No. 15, bars 28-29.wav]]]]
The concluding passage of the slow movement of [[Schubert's last sonatas|Schubert's final piano sonata]] in B{{Music|flat}} (D960) contains a dramatic enharmonic change. In bars 102–3, a B{{Music|sharp}},  the third of a G{{Music|sharp}} major triad, transforms into C{{Music|natural}} as the prevailing harmony changes to C major:

{{Image frame|width=210|content={{center|<score>\relative c'' { \magnifyStaff #5/4 \omit Score.TimeSignature \set doubleSlurs = ##t <bis dis gis>1 (<c e g!>)}</score>}}|caption=G-sharp to C progression.[[File:G sharp to C progression 01.wav]]}}

[[File:Schubert Piano Sonata D960 second movement, bars 98-107.png|thumb|center|500px|Schubert Piano Sonata D960 second movement, bars 98–106<br>[[File:Schubert Piano Sonata D960 second movement, bars 98-106.wav]]]]

==Other tuning conventions==
[[File:Comparison of unisons.png|thumb|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&nbsp;equal semitones. In this system, written notes that produce the same pitch, such as C{{music|#}} and D{{music|b}}, 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>
{{cite dictionary
 |last = Rushton |first=Julian |author-link = Julian Rushton
 |date = 2001
 |title =Enharmonic
 |dictionary = [[The New Grove Dictionary of Music and Musicians]]  |edition = 2nd
 |editor1-first = Stanley |editor1-last = Sadie   |editor1-link = Stanley Sadie
 |editor2-first = John    |editor2-last = Tyrrell |editor2-link = John Tyrrell (musicologist)
 |location = London, UK
 |publisher = Macmillan Publishers
 |isbn = 0-19-517067-9
}}
</ref>

===Pythagorean===
{{Main|Pythagorean tuning}}

In Pythagorean tuning, all pitches are generated from a series of [[Just intonation|justly tuned]] [[perfect fifth]]s, each with a frequency ratio of 3 to 2. If the first note in the series is an A{{music|b}}, the thirteenth note in the series, G{{music|#}} is ''higher'' than the seventh octave (1&nbsp;octave = frequency ratio of {{nobr|{{math| 2 to 1 {{=}} 2}} ;}} 7&nbsp;octaves is {{nobr|{{math| 2{{sup|7}} to 1 {{=}} 128}} )}} of the A{{music|b}} 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===
{{Main|Meantone temperament}}
In quarter-comma meantone, there will be a discrepancy between, for example, G{{music|#}} and A{{music|b}}. If [[middle C]]'s frequency is {{mvar|f}}, the next highest C has a frequency of {{nobr| 2 {{mvar|f}} .}} The quarter-comma meantone has perfectly tuned ([[just intonation|"just"]]) [[major thirds]], which means major thirds with a frequency ratio of exactly {{nobr| {{small|{{sfrac| 5 | 4 }} }} .}} To form a just major third with the C above it, A{{music|b}} and the C above it must be in the ratio 5 to 4, so A{{music|b}} 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, G{{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 G{{music|#}} 
:<math> \left( \frac{\ 5\ }{ 4 } \right)^2\ f ~=~ \frac{\ 25\ }{ 16 }\ f ~=~ 1.5625\ f ~.</math>

This leads to G{{music|#}} and A{{music|b}} being different pitches; G{{music|#}} is, in fact 41&nbsp;[[cent (music)|cent]]s (41% of a semitone) lower in pitch. The difference is the interval called the enharmonic [[diesis]], or a frequency ratio of {{small|{{sfrac| 128 | 125 }}}}. On a piano tuned in equal temperament, both G{{music|#}} and A{{music|b}} 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 [[set theory (music)|musical set theory]] and as employed by [[MIDI]].

==Enharmonic genus==
{{Main|Genus (music)#Enharmonic}}

In [[ancient Greek music]] the enharmonic was one of the three Greek [[genus (music)|genera]] in music in which the [[tetrachord]]s are divided (descending) as a [[major third|ditone]] plus two [[microtone]]s. The ditone can be anywhere from {{sfrac|16|13}} to {{sfrac|9|7}} (3.55 to 4.35 [[semitone]]s) and the microtones can be anything smaller than 1 semitone.<ref>{{cite journal|first=C. André|last=Barbera|title=Arithmetic and Geometric Divisions of the Tetrachord|journal=[[Journal of Music Theory]]|volume=21|issue=2|year=1977|pages=294–323|jstor=843492}}</ref> Some examples of enharmonic genera are
# {{sfrac|1|1}} {{sfrac|36|35}} {{sfrac|16|15}} {{sfrac|4|3}}
# {{sfrac|1|1}} {{sfrac|28|27}} {{sfrac|16|15}} {{sfrac|4|3}}
# {{sfrac|1|1}} {{sfrac|64|63}} {{sfrac|28|27}} {{sfrac|4|3}}
# {{sfrac|1|1}} {{sfrac|49|48}} {{sfrac|28|27}} {{sfrac|4|3}}
# {{sfrac|1|1}} {{sfrac|25|24}} {{sfrac|13|12}} {{sfrac|4|3}}

==Enharmonic key==
Some [[key signature]]s 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|C{{Music|flat}} major]], [[G-sharp minor|G{{Music|sharp}} minor]]/[[A-flat minor|A{{Music|flat}} minor]], [[F-sharp major|F{{Music|sharp}} major]]/[[G-flat major|G{{Music|flat}} major]], [[D-sharp minor|D{{Music|sharp}} minor]]/[[E-flat minor|E{{Music|flat}} minor]], [[C-sharp major|C{{Music|sharp}} major]]/[[D-flat major|D{{Music|flat}} major]] and [[A-sharp minor|A{{Music|sharp}} minor]]/[[B-flat minor|B{{Music|flat}} 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.

==See also==
* [[Enharmonic keyboard]]
* [[Music theory]]
* [[Transposition (music)|Transpositional]] equivalence
* [[Diatonic and chromatic]]
* [[Enharmonic modulation]]

==References==
{{Reflist}}

==Further reading==
* Eijk, Lisette D. van der (2020). "[https://sonid.app/blog/the-difference-between-a-sharp-and-a-flat/ The difference between a sharp and a flat] {{Webarchive|url=https://web.archive.org/web/20210301031744/https://sonid.app/blog/the-difference-between-a-sharp-and-a-flat/ |date=2021-03-01 }}". 
* {{cite book|last=Mathiesen|first=Thomas J.|year=2001|chapter=Greece, §I: Ancient|title=[[The New Grove Dictionary of Music and Musicians]]|edition=2nd|editor1-link=Stanley Sadie|editor1-first=Stanley|editor1-last=Sadie|editor2-link=John Tyrrell (musicologist)|editor2-first=John|editor2-last=Tyrrell|location=London|publisher=Macmillan Publishers|isbn=0-19-517067-9|ref=none}}
* {{cite journal|last=Morey|first=Carl|year=1966|title=The Diatonic, Chromatic and Enharmonic Dances by Martino Pesenti|journal=[[Acta Musicologica]]|volume=38|issue=2–4|pages=185–189|doi=10.2307/932526|jstor=932526|ref=none}}

==External links==
* {{Wiktionary-inline}}
* {{commons category-inline}}

{{Pitch (music)}}

[[Category:Intervals (music)]]
[[Category:Musical notes]]