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 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.
