Enharmonic

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In modern music, an enharmonic is a note (or key signature) which is the equivalent of some other note (or key signature), but spelled differently. For example, in twelve-tone equal temperament (the modern system of musical tuning in the west), the notes C♯ (C sharp) and D♭ (D flat) are enharmonically equivalent - that is, they are represented by the same key (on a musical keyboard, for example), and thus are identical in pitch, although they have different names and diatonic functionality.

In a given diatonic scale, an individual note name may only occur once. In the key of F for example, the major scale is: 'F, G, A, B♭, C, D, E, (F)'. Thus, the 'B♭' is called 'B♭' rather than 'A♯' as we already have a note named 'A' in the scale. The scale of F♯ major is: 'F♯, G♯, A♯, B, C♯, D♯, E♯, (F♯)'; thus we use the term 'A♯' instead of 'B♭' as we need the name 'B' to represent the 'B' note in the scale, and 'E♯' instead of 'F' as we need the name 'F' to represent the 'F♯' note in the scale.

All key signatures also have an infinite number of enharmonic key signatures that sound the same. The most common interchanges occur above 4 sharps/flats, though. For example, the key of B, with five sharps, is enharmonically equivalent to the key of C♭, with 7 flats. Interestingly enough, though, keys past 7 sharps/flats exist; they are, however, normally impractical. The key of A♭, with four flats, is equivalent to the key of G♯, with 8 sharps, the first of which is double-sharped (order of sharps: F♯♯ C♯ G♯ D♯ A♯ E♯ B♯).

In ancient Greek music, the enharmonic scale was a form of octave tuning, in which the first, second, and third notes in the octave were separated approximately by quarter tones, as were the fifth, sixth, and seventh.

1 Tuning enharmonics
2 Enharmonic genus
3 Enharmonic tetrachords in Byzantine music
4 See also
5 External links

[edit] Tuning enharmonics

The modern musical use of the word enharmonic to mean identical tones has only been correct since the adoption of Equal Temperament in the early 1900's. This is in contrast to the ancient use of the word which applies to unequal temperaments, such as 1/4 comma meantone intonation, in which enharmonic notes differ slightly in pitch. It should be noted, however, that enharmonic equivalencies 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 tuning. The specific equivalencies define the equal temperament. 19 equal is characterized by E♯ = F♭ and 31 equal by D♯♯ = F♭♭, for instance; in these tunings it is not true that E♯ = F, which is characteristic only of 12 equal temperament.

In 1/4 comma meantone, on the other hand, consider G♯ and A♭. Call middle C's frequency <math>x</math>. Then high C has a frequency of <math>2x</math>. The 1/4 comma meantone has perfect major thirds, which means major thirds with a frequency ratio of exactly 4 to 5.

In order to form a perfect major third with the C above it, A♭ and high C need to be in the ratio 4 to 5, so A♭ needs to have the frequency

In order to form a perfect 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. Thus the frequency of G♯ is

<math>\left(\frac{5}{4}\right)\left(\frac{5}{4}\right)x = \left(\frac{25}{16}\right)x = 1.5625 x</math>

Thus, G♯ and A♭ are not the same note; G♯ 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 <math>\frac{128}{125}</math>. On a piano, both would be played by striking the same key, with a frequency <math>2^\frac{8}{12}x = 2^\frac{2}{3} \approx 1.5874 x</math> 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 is quite noticeable, even to untrained ears.

The reason that — despite the fact that in recent western music, Ab is exactly the same pitch as G♯ — 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. This is called diatonic functionality. One can however label enharmonically equivalent pitches with one and only one name, sometimes called integer notation, often used in serialism and musical set theory and employed by the MIDI interface.

[edit] Enharmonic genus

Main article: Enharmonic genus

An enharmonic is also 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

[edit] Enharmonic tetrachords in Byzantine music

In Byzantine music, enharmonic describes a kind of tetrachord and the echoi that contain them. As in the Greek system, enharmonic tetrachords are distinct from diatonic and chromatic. However Byzantine enharmonic tetrachords bear no resemblance to Greek enharmonic tetrachords. Their largest division is between a whole-tone and a tone-and-a-quarter in size, and their smallest is between a quarter-tone and a semitone. These are called "improper diatonic" tetrachords in modern western usage.