June 2021
To define a scale, we would like to pick a set of notes within an octave so that the intervals between the notes are consonant. We can generate such scales using perfect fifths, after Pythagoras. Pick an arbitrary frequency as a starting point (the “tonic”), and then go up in pitch repeatedly by fifths. The first few notes in the series have the following frequency ratios relative to the tonic:
| # of fifths | Frequency ratio |
|---|---|
| 0 | (3⁄2)0 = 1 |
| 1 | (3⁄2)1 = 3⁄2 |
| 2 | (3⁄2)2 = 9⁄4 |
But, wait, 9⁄4 is greater than 2, so we need to “normalize” this note so it falls within our target range by moving its pitch down an octave to 9⁄8.
We can also find notes in the other direction, moving down by fifths instead of up. Again, we need to normalize these notes so that they are between 1 and 2 – moving them up by octaves instead of down. The first seven notes generated by this process are:
| Name | # of fifths | Unnormalized ratio | Normalized ratio |
|---|---|---|---|
| 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 |
We've also given the notes arbitrary names, which we'll explain in a moment. Let's put these normalized notes in order from lowest to highest within the octave:
| Name | Ratio to tonic | Ratio to previous note |
|---|---|---|
| D | 1 = 1.00 | – |
| E | 9⁄8 = 1.13 | 9⁄8 : 1 = 9⁄8 |
| F | 32⁄27 = 1.19 | 32⁄27 : 9⁄8 = 256⁄243 |
| G | 4⁄3 = 1.33 | 4⁄3 : 32⁄27 = 9⁄8 |
| A | 3⁄2 = 1.50 | 3⁄2 : 4⁄3 = 9⁄8 |
| B | 27⁄16 = 1.69 | 27⁄16 : 3⁄2 = 9⁄8 |
| C | 16⁄9 = 1.78 | 16⁄9 : 27⁄16 = 256⁄243 |
| D | 2⁄1 = 2.00 | 2⁄1 : 16⁄9 = 9⁄8 |
The notes in this scale are D, E, F, G, A, B, and C. The scale then lands back on D, one octave higher. So, we've simply assigned the first seven letters of the alphabet, in order from A through G, to the seven distinct notes we've created. The scale forms a loop, so we can start on any note.
We've also computed the ratio of each note relative to the previous note in the scale. Luckily1, there are only two distinct step sizes in the scale:
| Step size | Name |
|---|---|
| 9⁄8 ≈ 1.13 | Whole step |
| 256⁄243 ≈ 1.05 | Half step |
As a result, the notes in the scale are distributed somewhat evenly throughout the octave, which seems desirable. The pattern of whole (W) and half (h) steps is W h W W W h W, but since the scale repeats at every octave, so does the pattern: W h W W W h W W h W W W h W W h W W W h W ... . Any scale that follows a similar pattern of half steps separated by alternating groups of two and three whole steps is called a diatonic scale, such as the white keys on a piano.
Is it luck? There doesn't seem to be anything in the process that guarantees such an outcome mathematically, but I could be wrong. It just seems to work out nicely.↩