Why are you so obsessed with thirds? If you care about short, clean fractions on a human scale, base 6 is overall better, see this table: https://youtu.be/qID2B4MK7Y0?t=992

Plus you get all the convenience of having five digits on a hand and 5 as the highest counting digit, so you can count to 55 on two hands, and all the rest of the benefits in that (totally serious) video.