Here I'm applying ISO 3, the Renard series of preferred numbers, to music instead of engineering. It worked out better than I thought it would.
I mean, this is not a good instrument tuning and it's in itself a violation of ISO 16. For anyone with perfect pitch, tuning A to 450 Hz is just wrong. But what's worse is that your A at 224 Hz is going to be out of tune with it. Octaves are supposed to exactly double the frequency. The D#'s are even worse: they're 27 cents out of tune with each other, where 50 cents is the most out of tune you can be before you start being in tune with a different note.
But it sounds like a musical scale and it has a lot of nice just intonation intervals in it! Check out all those harmonically satisfying ratios, like the nice major chord of 200, 250, 300, 400 Hz.
In music theory, we're supposed to be attempting to divide up factors of 2 into rational intervals, not factors of 10. But it's really remarkable how much this almost works.
The interval between two adjacent notes in the chromatic musical scale is the 12th root of 2 (or a rational approximation to it, if you're looking for just intonation).
The intervals between Renard numbers in the R40 series approximate the 40th root of 10.
(12th root of 2) ≈ (40th root of 10)
Raise both sides to the 120th power to get:
210 ≈ 103
1024 ≈ 1000
This is the same thing as kibibytes vs. kilobytes! That's standardized in ISO 80000, by the way.
