Enharmonic scale

Enharmonic scale [segment] on C.[1][2]  Play [2] Note that in this depiction C♯ and D♭ are distinct rather than equivalent as in modern notation.
Enharmonic scale on C.[3]

In music theory, an enharmonic scale is "an [imaginary] gradual progression by quarter tones" or any "[musical] scale proceeding by quarter tones".[3] The enharmonic scale uses dieses (divisions) nonexistent on most keyboards,[2] since modern standard keyboards have only half-tone dieses.

More broadly, an enharmonic scale is a scale in which (using standard notation) there is no exact equivalence between a sharpened note and the flattened note it is enharmonically related to, such as in the quarter tone scale. As an example, F♯ and G♭ are equivalent in a chromatic scale (the same sound is spelled differently), but they are different sounds in an enharmonic scale. See: musical tuning.

Musical keyboards which distinguish between enharmonic notes are called by some modern scholars enharmonic keyboards. (The enharmonic genus, a tetrachord with roots in early Greek music, is only loosely related to enharmonic scales.)

Diesis defined in quarter-comma meantone as a diminished second (m2 − A1 ≈ 117.1 − 76.0 ≈ 41.1 cents), or an interval between two enharmonically equivalent notes (from C♯ to D♭).  Play 

Consider a scale constructed through Pythagorean tuning. A Pythagorean scale can be constructed "upwards" by wrapping a chain of perfect fifths around an octave, but it can also be constructed "downwards" by wrapping a chain of perfect fourths around the same octave. By juxtaposing these two slightly different scales, it is possible to create an enharmonic scale.

The following Pythagorean scale is enharmonic:

Note Ratio Decimal Cents Difference
(cents)
C1:110
D♭256:2431.0535090.22523.460
C♯2187:20481.06787113.685
D9:81.125203.910
E♭32:271.18519294.13523.460
D♯19683:163841.20135317.595
E81:641.26563407.820
F4:31.33333498.045
G♭1024:7291.40466588.27023.460
F♯729:5121.42383611.730
G3:21.5701.955
A♭128:811.58025792.18023.460
G♯6561:40961.60181815.640
A27:161.6875905.865
B♭16:91.77778996.09023.460
A♯59049:327681.802031019.550
B243:1281.898441109.775
C′2:121200

In the above scale the following pairs of notes are said to be enharmonic:

  • C♯ and D♭
  • D♯ and E♭
  • F♯ and G♭
  • G♯ and A♭
  • A♯ and B♭

In this example, natural notes are sharpened by multiplying its frequency ratio by 256:243 (called a limma), and a natural note is flattened by multiplying its ratio by 243:256. A pair of enharmonic notes are separated by a Pythagorean comma, which is equal to 531441:524288 (about 23.46 cents).

Sources

  1. ↑  Moore, John Weeks (1875) [1854]. "Enharmonic scale". Complete Encyclopaedia of Music. New York: C. H. Ditson & Company. p. 281. . Moore cites Greek use of quarter tones until the time of Alexander the Great.
  2. 1 2 3 John Wall Callcott (1833). A Musical Grammar in Four Parts, p.109. James Loring.
  3. 1 2 Louis Charles Elson (1905). Elson's Music Dictionary, p.100. O. Ditson Company.
  • Barbieri, Patrizio (2008). Enharmonic instruments and music, 1470–1900. Latina: Il Levante Libreria Editrice.
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