July 2021
We've seen that we can use a series of perfect fifths to generate a diatonic scale. We can continue the process to generate more than just seven notes, but how many do we need?
Since each pair of adjancent notes is in the ratio 3:2, after six steps in each direction we have an interval of (3⁄2)12, which is almost, but not quite, equal to seven octaves:
The notes we have at this point are:
| Name | # of fifths | Unnormalized ratio | Normalized ratio |
|---|---|---|---|
| A♭ | -6 | (3⁄2)-6 = 64⁄729 | 1024⁄729 |
| E♭ | -5 | (3⁄2)-5 = 32⁄243 | 256⁄243 |
| B♭ | -4 | (3⁄2)-4 = 16⁄81 | 128⁄81 |
| F | -3 | (3⁄2)-3 = 8⁄27 | 32⁄27 |
| C | -2 | (3⁄2)-2 = 4⁄9 | 16⁄9 |
| G | -1 | (3⁄2)-1 = 2⁄3 | 4⁄3 |
| D | 0 | (3⁄2)0 = 1 | 1 |
| A | 1 | (3⁄2)1 = 3⁄2 | 3⁄2 |
| E | 2 | (3⁄2)2 = 9⁄4 | 9⁄8 |
| B | 3 | (3⁄2)3 = 27⁄8 | 27⁄16 |
| F♯ | 4 | (3⁄2)4 = 81⁄16 | 81⁄64 |
| C♯ | 5 | (3⁄2)5 = 243⁄32 | 243⁄128 |
| G♯ | 6 | (3⁄2)6 = 729⁄64 | 729⁄512 |
In this Pythagorean tuning, A♭ and G♯ are slightly different notes, making them very dissonant when played together . Ideally, we would like them to align, creating an exact “circle of fifths” – will look at how to do that soon. For now, we are left with twelve usable notes in a “chromatic” scale (in order by frequency):
| Name | Ratio to tonic | Ratio to previous note |
|---|---|---|
| D | 1 = 1.00 | – |
| E♭ | 256⁄243 ≈ 1.05 | 1.05 / 1.00 ≈ 1.05 |
| E | 9⁄8 ≈ 1.13 | 1.13 / 1.05 ≈ 1.08 |
| F | 32⁄27 ≈ 1.19 | 1.19 / 1.13 ≈ 1.05 |
| F♯ | 81⁄64 ≈ 1.27 | 1.27 / 1.19 ≈ 1.07 |
| G | 4⁄3 ≈ 1.33 | 1.33 / 1.27 ≈ 1.04 |
| A♭ | 1024⁄729 ≈ 1.40 | 1.40 / 1.33 ≈ 1.05 |
| A | 3⁄2 ≈ 1.50 | 1.50 / 1.40 ≈ 1.07 |
| B♭ | 128⁄81 ≈ 1.58 | 1.58 / 1.50 ≈ 1.05 |
| B | 27⁄16 ≈ 1.69 | 1.69 / 1.58 ≈ 1.07 |
| C | 16⁄9 ≈ 1.78 | 1.78 / 1.69 ≈ 1.05 |
| C♯ | 243⁄128 ≈ 1.90 | 1.90 / 1.78 ≈ 1.07 |
| D | 2⁄1 = 2.00 | 2.00 / 1.90 ≈ 1.05 |
The flats and sharps fill the gaps in our diatonic scale, so the interval formed by each adjancent pair of notes is now (roughly) a half step.