But we still have preferred numbers [1] which never map neatly to the decimal or any other radixes due to their logarithmic nature. The ratios, sharing the same logarithmic nature, ARE one of the good reasons to select radixes if the decimal were not as pervasive as now.
The whole point of preferred numbers is fitting a rough logarithmic scale to a positional (e.g. decimal) number system.
For an alternative let me recommend just using binary logarithms written in base 12. (So that 1 corresponds to doubling, 1.7 corresponds pretty closely to tripling, and 2.4 corresponds roughly to quintupling. i.e. 0.1 (base 12) is an equal-tempered semitone.
[1] https://en.wikipedia.org/wiki/Preferred_number