Enharmonic scale
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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.