Last time I explained how Pythagorean tuning, one of the older tuning systems, arises from the fact that twelve fifths is almost the same as seven octaves. In other words, multiplying by 3/2 twelve times is almost the same as multiplying by 2 seven times:
This is a companion discussion topic for the original entry at https://johncarlosbaez.wordpress.com/2023/10/13/perfect-fifths-in-equal-tempered-scales/