# Enharmonic

In modern musical notation and tuning, an enharmonic equivalent is a note, interval, or key signature that is equivalent to some other note, interval, or key signature but "spelled", or named differently. Thus, the enharmonic spelling of a written note, interval, or chord is an alternative way to write that note, interval, or chord. For example, in twelve-tone equal temperament (the currently predominant system of musical tuning in Western music), the notes CTemplate:Music and DTemplate:Music are enharmonic (or enharmonically equivalent) notes. Namely, they are the same key on a keyboard, and thus they are identical in pitch, although they have different names and different role in harmony and chord progressions.

In other words, if two notes have the same pitch but are represented by different letter names and accidentals, they are enharmonic.[1] "Enharmonic intervals are intervals with the same sound that are spelled differently...[resulting], of course, from enharmonic tones."[2]

Prior to this modern meaning, "enharmonic" referred to relations in which there is no exact equivalence in pitch between a sharpened note such as FTemplate:Music and a flattened note such as GTemplate:Music,[3] as in enharmonic scale.

The notes FTemplate:Music and GTemplate:Music are enharmonic equivalents.
ETemplate:Music and FTemplate:Music, however, are not enharmonic equivalents, because ETemplate:Music is enharmonic with FTemplate:Music.
GTemplate:Music and BTemplate:Music are enharmonic equivalents, both the same as ATemplate:Music.
Enharmonically equivalent key signatures of BTemplate:Music and CTemplate:Music major, each followed by its respective tonic chord

Some key signatures have an enharmonic equivalent that represents a scale identical in sound but spelled differently. The number of sharps and flats of two enharmonically equivalent keys sum to twelve. For example, the key of B major, with 5 sharps, is enharmonically equivalent to the key of C-flat major with 7 flats, and 5 (sharps) + 7 (flats) = 12. Keys past 7 sharps or flats exist only theoretically and not in practice. The enharmonic keys are six pairs, three major and three minor: B major/C-flat major, G-sharp minor/A-flat minor, F-sharp major/G-flat major, D-sharp minor/E-flat minor, C-sharp major/D-flat major and A-sharp minor/B-flat minor. There are no works composed in keys that require double sharping or double flatting in the key signature, except in jest. In practice, musicians learn and practice 15 major and 15 minor keys, three more than 12 due to the enharmonic spellings.

Enharmonic tritones: A4 = d5 on C  .

For example the intervals of a minor sixth on C, on BTemplate:Music, and an augmented fifth on C are all enharmonic intervals  . The most common enharmonic intervals are the augmented fourth and diminished fifth, or tritone, for example C-FTemplate:Music = C-GTemplate:Music.[1]

Enharmonic equivalence is not to be confused with octave equivalence, nor are enharmonic intervals to be confused with inverted or compound intervals.

## Tuning enharmonics

In principle, the modern musical use of the word enharmonic to mean identical tones is correct only in equal temperament, where the octave is divided into 12 equal semitones; but even in other tuning systems enharmonic associations can be perceived by listeners and exploited by composers.[4] This is in contrast to the ancient use of the word in the context of unequal temperaments, such as quarter-comma meantone intonation, in which enharmonic notes differ slightly in pitch. It should be noted, however, that enharmonic equivalents occur in any equal temperament system, such as 19 equal temperament or 31 equal temperament, if it can be and is used as a meantone temperament. The specific equivalences define the equal temperament. 19 equal is characterized by ETemplate:Music = FTemplate:Music and 31 equal by DTemplate:Music = FTemplate:Music, for instance; in these tunings it is not true that ETemplate:Music = FTemplate:Music, which is characteristic only of 12 equal temperament.Template:Or

### Pythagorean

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In Pythagorean tuning, all pitches are generated from a series of justly tuned perfect fifths, each with a ratio of 3 to 2. If the first note in the series is an ATemplate:Music, the thirteenth note in the series, GTemplate:Music, will be higher than the seventh octave (octave = ratio of 1 to 2, seven octaves is 1 to 27 = 128) of the ATemplate:Music by a small interval called a Pythagorean comma. This interval is expressed mathematically as:

${\displaystyle {\frac {\hbox{twelve fifths}}{\hbox{seven octaves}}}=\left({\tfrac {3}{2}}\right)^{12}\!\!{\bigg /}\,2^{7}={\frac {3^{12}}{2^{19}}}={\frac {531441}{524288}}=1.0136432647705078125\!}$

### Meantone

{{#invoke:main|main}} In 1/4 comma meantone, on the other hand, consider GTemplate:Music and ATemplate:Music. Call middle C's frequency ${\displaystyle x}$. Then high C has a frequency of ${\displaystyle 2x}$. The 1/4 comma meantone has just (i.e., perfectly tuned) major thirds, which means major thirds with a frequency ratio of exactly 4 to 5.

In order to form a just major third with the C above it, ATemplate:Music and high C need to be in the ratio 4 to 5, so ATemplate:Music needs to have the frequency

${\displaystyle {\frac {8x}{5}}=1.6x.\!}$

In order 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. Thus the frequency of GTemplate:Music is

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

Thus, GTemplate:Music and ATemplate:Music are not the same note; GTemplate:Music is, in fact 41 cents lower in pitch (41% of a semitone, not quite a quarter of a tone). The difference is the interval called the enharmonic diesis, or a frequency ratio of ${\displaystyle {\frac {128}{125}}}$. 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 ${\displaystyle 2^{\frac {8}{12}}x=2^{\frac {2}{3}}x\approx 1.5874x}$. 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, even to untrained ears.

The reason that — despite the fact that in recent Western music, ATemplate:Music is exactly the same pitch as GTemplate:Music — we label them differently is that in tonal music notes are named for their harmonic function, and retain the names they had in the meantone tuning era.{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}[citation needed] }} This is called diatonic functionality. One can however label enharmonically equivalent pitches with one and only one name; for instance, the numbers of integer notation, as used in serialism and musical set theory and employed by the MIDI interface. ## Enharmonic genus {{#invoke:main|main}} In ancient Greek music the enharmonic was one of the three Greek genera in music in which the tetrachords are divided (descending) as a ditone plus two microtones. The ditone can be anywhere from 16/13 to 9/7 (3.55 to 4.35 semitones) and the microtones can be anything smaller than 1 semitone.{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}[citation needed] }} 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