Enharmonic scale


In music theory, an enharmonic scale is "an gradual progression by quarter tones" or any "scale | scale proceeding by quarter tones". The enharmonic scale uses dieses nonexistent on most keyboards, since modern standard keyboards have only half-tone dieses.
More broadly, an enharmonic scale is a scale in which 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, 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.
as a diminished second, or an interval between two enharmonically equivalent notes.
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:
NoteRatioDecimalCentsDifference
C1:110
D256:2431.0535090.22523.460
C2187:20481.06787113.68523.460
D9:81.125203.910
E32:271.18519294.13523.460
D19683:163841.20135317.59523.460
E81:641.26563407.820
F4:31.33333498.045
G1024:7291.40466588.27023.460
F729:5121.42383611.73023.460
G3:21.5701.955
A128:811.58025792.18023.460
G6561:40961.60181815.64023.460
A27:161.6875905.865
B16:91.77778996.09023.460
A59049:327681.802031019.55023.460
B243:1281.898441109.775
C′2:121200

In the above scale the following pairs of notes are said to be enharmonic:
In this example, natural notes are sharpened by multiplying its frequency ratio by 256:243, 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.