1/4-1/5=x
(15)/(45)-(14)/(54)=x
5/20-4/20=1/20
so, 1/20th should be added to a 1/5 to make a 1/4 note.
lets try to do that up by adding a 1/32 note with a dot.
1/32*3/2=3/64
1/20-3/64=1/320
so we are very very close…
But it seems obvious the conversion is not easy. Probably impossible since both 2 and 5 are primes so their powers never converge. (there might coincidentally be some note that with a dot and a correct number of it enables conversion)
To a mathematician the reason notes in fractions of two are popular is obvious. That way you get the most different lengths in the most intuitive way. Let’s consider the 1/5 note. It’s one fifth of a 1/1 note, of which one fifth is 1/25. So on to 1/125 and then 625 and on to 3125.
With fractions of two, you go 1/1, 1/2, 1/4, 1/8, 1/16, 1/32, 1/64, 1/128, 1/256, 1/512… the numbers remain useable for a long time. If you want to halve any one of these, it’s no problem. From 1/256 you get 1/512 and so on.
Try halving 1/125! You get 1/250. How do you represent that in this system? That’s the same as 1/((5^3)*2). You’re mixing different primes into the system by doing something as simple as halving! It gets muddled very quickly. In the “fractions of two”-system, every note is representable as 1/(2^x) (sometimes with an added *1.5).
However, I’m beginning to see the value of dividing it into fifths… it’s a very different way of seeing music and I can hear the point your going to make that halving is not an intuitive thing to do in this system, but rather dividing it somehow in fractions of 5. This is surely a fun thing to explore. However, there’s a very easy way to accomplish this in the conventional system called tuplets! They also work for thirds and even sevenths: en.wikipedia.org/wiki/Tuplet
So, we can keep the good sides of both systems and do away with the annoying math just by using tuplets. That’s my solution anyway. What do you think?
This might also be of interest: vai.com/LittleBlackDots/tempomental.html