In the previous post I showed the simplest possible set of whole-number ratios for 12 musical notes, but it was hard to judge how nice it sounded compared to equal temperament's irrational-yet-familiar intervals. Perhaps, though, we can get the best of both worlds: simple whole-number ratios that approximate twelfth roots of two.
Here's the equal temperament curve we'll try to match:
Note that it doesn't matter what note we start at; the curve is exponential and looks the same if we slide the axis left or right to begin on a different note.
The simplest possible whole-number ratios we can use are 3:2, 4:3, 5:3, 5:4, 6:5, 7:4, 7:5, 7:6, 8:5, 8:7, and 9:5. Here they are placed where they best match the equal temperament curve:
Improving these intervals is tricky because we have two competing criteria. With a big enough numerator and denominator we could approximate an irrational number with arbitrary precision, so we're forced to choose between the quality of the approximation and the simplicity of the ratio. I don't have any clever metrics to suggest, so let's just start simple and see how big the ratios have to get to be within 1% of equal temperament:
The first two ratios, last two ratios, and middle ratio have changed, but all have numerators and denominators below 20. To get within 0.5% of equal temperament, we have to allow ratios up to 27:16, and some of the simpler ratios (5:3, 5:4, and 6:5) are no longer precise enough:
Such improved approximation hardly seems necessary when, using 1% precision, my ear can't tell the difference between the two sets of frequencies, except for a slight hum on certain notes when they're played simultaneously: