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 twelvetone equal temperament (the currently predominant system of musical tuning in Western music), the notes C♯ and D♭ 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 roles 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 notes that were very close in pitch—closer than the smallest step of a diatonic scale—but not identical in pitch, such as F♯ and a flattened note such as G♭.^{[3]} as in enharmonic scale. "Enharmonic equivalence is peculiar to posttonal theory."^{[4]} "Much music since at least the 18th century, however, exploits enharmonic equivalence for purposes of modulation and this requires that enharmonic equivalents in fact be equivalent."^{[5]}
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 Cflat 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/Cflat major, Gsharp minor/Aflat minor, Fsharp major/Gflat major, Dsharp minor/Eflat minor, Csharp major/Dflat major and Asharp minor/Bflat minor. There are practically no works composed in keys that require double sharping or double flatting in the key signature. In practice, musicians learn and practice 15 major and 15 minor keys, three more than 12 due to the enharmonic spellings.
For example the intervals of a minor sixth on C, on B♯, and an augmented fifth on C are all enharmonic intervals Play . The most common enharmonic intervals are the augmented fourth and diminished fifth, or tritone, for example CF♯ = CG♭.^{[1]}
Enharmonic equivalence is not to be confused with octave equivalence, nor are enharmonic intervals to be confused with inverted or compound intervals.
Contents

Tuning enharmonics 1
 Pythagorean 1.1
 Meantone 1.2
 Enharmonic genus 2
 See also 3
 Sources 4
 Further reading 5
 External links 6
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. In other tuning systems, however, enharmonic associations can be perceived by listeners and exploited by composers.^{[6]}
Pythagorean
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 2^{7} = 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}} =\left(\tfrac32\right)^{12} \!\!\bigg/\, 2^{7} = \frac{3^{12}}{2^{19}} = \frac{531441}{524288} = 1.0136432647705078125 \approx 23.46 \hbox{cents} \!
Meantone
In 1/4 comma meantone, on the other hand, consider G♯ and A♭. Call middle C's frequency x. Then high C has a frequency of 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.
To form a just major third with the C above it, A♭ and high C must be in the ratio 4 to 5, so A♭ needs to have the frequency
 \frac {8x}{5} = 1.6 x. \!
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. Thus the frequency of G♯ is
 \left(\frac{5}{4}\right)^2x = \left(\frac{25}{16}\right)x = 1.5625 x
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 a frequency ratio of \frac{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 1.5874 x. 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 equaltempered intonation can be quite noticeable, even to untrained ears.
One can 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
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.^{[7]} 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
See also
 Enharmonic keyboard
 Enharmonic scale
 Music theory
 Music notation
 Accidental
 Octave equivalence, Transpositional equivalence, and inversional equivalence
 Diatonic and chromatic
Sources
 ^ ^{a} ^{b} Benward & Saker (2003). Music in Theory and Practice, Vol. I, p.7 & 360. ISBN 9780072942620.
 ^ Benward & Saker (2003), p.54.
 ^ Louis Charles Elson (1905) Elson's Music Dictionary, p.100. O. Ditson Company. "The relation existing between two chromatics, when, by the elevation of one and depression of the other, they are united into one".
 ^ Don Michael Randel, ed. (2003). "Set theory", The Harvard Dictionary of Music, p.776. Harvard. ISBN 9780674011632.
 ^ Don Michael Randel, "Enharmonic", The Harvard Dictionary of Music, fourth edition (Cambridge: Belknap Press of Harvard University Press, 2003), p.295. ISBN 9780674011632.
 ^ Rushton, Julian (2001). "Enharmonic", The New Grove Dictionary of Music and Musicians. Second edition, edited by Stanley Sadie and John Tyrrell. London: Macmillan Publishers. ISBN 0195170679.
 ^ C. André Barbera, "Arithmetic and Geometric Divisions of the Tetrachord", Journal of Music Theory 21, no. 2 (Autumn 1977): 294–323. Citations on 296, 299, 302–304, 307–309, 311–13, 315–19.
Further reading
 Mathiesen, Thomas J. 2001. "Greece, §I: Ancient". The New Grove Dictionary of Music and Musicians, second edition, edited by Stanley Sadie and John Tyrrell. London: Macmillan Publishers.
 Morey, Carl. 1966. "The Diatonic, Chromatic and Enharmonic Dances by Martino Pesenti". Acta Musicologica 38, nos. 2–4 (April–December): 185–89.
External links
 The dictionary definition of enharmonic at Wiktionary
