Enharmonic scale

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Enharmonic scale [segment] on C.[1][2] About this soundPlay (help·info)[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♭). About this soundPlay (help·info)

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)
C 00001:1 1 0000
D♭ 00256:243 1.05350 0090.225 23.460
C♯ 02187:2048 1.06787 0113.685
D 00009:8 1.125 0203.910
E♭ 00032:27 1.18519 0294.135 23.460
D♯ 19683:16384 1.20135 0317.595
E 00081:64 1.26563 0407.820
F 00004:3 1.33333 0498.045
G♭ 01024:729 1.40466 0588.270 23.460
F♯ 00729:512 1.42383 0611.730
G 00003:2 1.5 0701.955
A♭ 00128:81 1.58025 0792.180 23.460
G♯ 06561:4096 1.60181 0815.640
A 00027:16 1.6875 0905.865
B♭ 00016:9 1.77778 0996.090 23.460
A♯ 59049:32768 1.80203 1019.550
B 00243:128 1.89844 1109.775
C′ 00002:1 2 1200

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. ^ Jump up to: a b c John Wall Callcott (1833). A Musical Grammar in Four Parts, p.109. James Loring.
  3. ^ Jump up to: a b Louis Charles Elson (1905). Elson's Music Dictionary, p.100. O. Ditson Company.

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