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)
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. 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.

External links

• Barbieri, Patrizio (2008). Enharmonic instruments and music, 1470–1900. Latina: Il Levante Libreria Editrice.