# 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 Gb 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 scale is enharmonic:

Note Ratio Decimal
C 1:1 1
C# 256:243 1.053497
Db 2187:2048 1.067871
D 9:8 1.125
D# 32:27 1.185185
Eb 19683:16384 1.201354
E 81:64 1.265625
F 4:3 1.333333
F# 1024:729 1.404663
Gb 729:512 1.423828
G 3:2 1.5
G# 128:81 1.580246
Ab 6561:4096 1.601806
A 27:16 1.6875
A# 16:9 1.777777
Bb 59049:32768 1.802032
B 243:128 1.898437
C' 2:1 2

In the above scale the following pairs of notes are said to be enharmonic:

• C# and Db
• D# and Eb
• F# and Gb
• G# and Ab
• A# and Bb.

A natural note is 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.

Enharmonic scales are the third genus of musical scales.

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