I think a good way to explore these things is using a computer. Writing a little programme soon reveals the regular divisions of the octave which contain intervals near enough to the harmonics. Obviously 12 (equal temperament) and 24 (quarter tones) do it, but the surprise is that 29 gives very close to a perfect fifth and a broad range of other close intervals as well. Of course, being prime, it lacks symmetric partitions (such as augmenteds and diminisheds in the chromatic scale), but from another point of view this could be an advantage.
A few years ago I spent some time writing algorithmic composition programmes, and splitting the octave into 29 is just as easy as splitting it into 12. The 29 note split embeds an excellent approximation to the tonal scale. I tried it out by writing code to compose music modulating around the 29 key cycle in various ways. I found the effects very interesting and pleasant. I tried it on a few musicians and none of them realised, on first hearing, that they were listening to a 29 key cycle instead of the usual 12.
If I get enough time (and hopefully reduce my working hours sometime soon !) it is a direction I would very much like to take further.