I've been exploring musical temperament, and I mentioned that just intonation is less common because the ratios between notes aren't constant, which prevents instruments from playing in different keys without being retuned. To get a more concrete understanding of this, let's look at the ratios between pairs of notes.
In the previous post, I found eleven ratios with low whole-number values that closely followed the polynomial curve of a twelfth root of two, the curve used by equal temperament tuning to give all notes the same frequency relationship to the note before. Those ratios were:
| 16:15 | 9:8 | 6:5 | 5:4 | 4:3 | 17:12 | 3:2 | 8:5 | 5:3 | 16:9 | 15:8 |
So for a given starting note, the next half note would have a frequency sixteen-fifteenths higher, the second half note up would have a frequency nine-eighths higher than the starting note, the third five-fourths higher than the starting note, and so on. Each ratio has a numerator and denominator lower than twenty, which helps them sound harmonious together.
But those ratios don't always have such low values when compared to each other. For example, the second half note may have a frequency nine-eighths higher than its starting key, but its frequency compared to the note below it is 135/128 higher (9/8 รท 16/15). On an instrument tuned to a particular key, changing keys would produce some weird intervals.
To see how weird, here's a table that compares the above ratios pairwise:
| 16:15 | 9:8 | 6:5 | 5:4 | 4:3 | 17:12 | 3:2 | 8:5 | 5:3 | 16:9 | 15:8 | |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 16:15 | |||||||||||
| 9:8 | 135:128 | ||||||||||
| 6:5 | 9:8 | 16:15 | |||||||||
| 5:4 | 75:64 | 10:9 | 25:24 | ||||||||
| 4:3 | 5:4 | 32:27 | 10:9 | 16:15 | |||||||
| 17:12 | 85:64 | 34:27 | 85:72 | 17:15 | 17:16 | ||||||
| 3:2 | 45:32 | 4:3 | 5:4 | 6:5 | 9:8 | 18:17 | |||||
| 8:5 | 3:2 | 64:45 | 4:3 | 32:25 | 6:5 | 96:85 | 16:15 | ||||
| 5:3 | 25:16 | 40:27 | 25:18 | 4:3 | 5:4 | 20:17 | 10:9 | 25:24 | |||
| 16:9 | 5:3 | 128:81 | 40:27 | 64:45 | 4:3 | 64:51 | 32:27 | 10:9 | 16:15 | ||
| 15:8 | 225:128 | 5:3 | 25:16 | 3:2 | 45:32 | 45:34 | 5:4 | 75:64 | 9:8 | 135:128 |
The shaded ratios in the table are those whose numerators are greater than 20, with darker colors signifying higher values.
In the key an instrument is tuned to, the frequency ratio between the key's starting note and the note two half steps up would be 9:8, but if the key changed to start on the note two half steps below what the instrument is tuned to, the frequency ratio between the new starting note and the note two half steps up would become 225:128, far less resonant than the 9:8 of the original key.