Enharmonic scale

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Enharmonic scale [segment] on C.[1][2] <phonos file="Enharmonic scale segment on C.mid">Play</phonos>[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♭). <phonos file="Enharmonic scale segment on C.mid">Play</phonos>

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 1:1 1.00000 0.00000
D♭ 256:243 1.05350 90.2250 23.4600
C♯ 2187:2048 1.06787 113.685
D 9:8 1.12500 203.910
E♭ 32:27 1.18519 294.135 23.4600
D♯ 19683:16384 1.20135 317.595
E 81:64 1.26563 407.820
F 4:3 1.33333 498.045
G♭ 1024:729 1.40466 588.270 23.4600
F♯ 729:512 1.42383 611.730
G 3:2 1.50000 701.955
A♭ 128:81 1.58025 792.180 23.4600
G♯ 6561:4096 1.60181 815.640
A 27:16 1.68750 905.865
B♭ 16:9 1.77778 996.090 23.4600
A♯ 59049:32768 1.80203 1019.55
B 243:128 1.89844 1109.78
C' 2:1 2.00000 1200.00

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 (1854). Complete Encyclopædia of Music, p.281. J. P. Jewett. Moore cites Greek use of quarter tones until the time of Alexander the Great.
  2. ↑ 2.0 2.1 2.2 John Wall Callcott (1833). A Musical Grammar in Four Parts, p.109. James Loring.
  3. ↑ 3.0 3.1 Louis Charles Elson (1905). Elson's Music Dictionary, p.100. O. Ditson Company.

External links

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