Enharmonic scale

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.

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]

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)
C00001:110000
D♭00256:2431.053500090.22523.460
C♯02187:20481.067870113.685
D00009:81.1250203.910
E♭00032:271.185190294.13523.460
D♯19683:163841.201350317.595
E00081:641.265630407.820
F00004:31.333330498.045
G♭01024:7291.404660588.27023.460
F♯00729:5121.423830611.730
G00003:21.50701.955
A♭00128:811.580250792.18023.460
G♯06561:40961.601810815.640
A00027:161.68750905.865
B♭00016:91.777780996.09023.460
A♯59049:327681.802031019.550
B00243:1281.898441109.775
C′00002:121200

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. John Wall Callcott (1833). A Musical Grammar in Four Parts, p.109. James Loring.
  3. Louis Charles Elson (1905). Elson's Music Dictionary, p.100. O. Ditson Company.
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