Enharmonic scale

An enharmonic scale is a musical scale in which there is no exact equivalence between a sharpened note and the flattened note it is enharmonically related to.As an example, F♯ and G♭ are generally equivalent in a chromatic scale, but they would be distinguished in an enharmonic scale.

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).

The enharmonic genus is only loosely related to enharmonic scales, being a scale that has a pitch distinction too fine to accommodate with flat and sharp notation.

Musical keyboards which distinguish between enharmonic notes are called enharmonic keyboards.