This comes up every so often and in my mind, there is an answer and it has to do with how well the notes in the "temperament" combine to produce near-enough app

This comes up every so often and in my mind, there is an answer and it has to do with how well the notes in the "temperament" combine to produce near-enough approximations to simple fractions. That is, take a temperament, combine each pairs of notes together. For each pairs of notes, find a close-enough fraction to it and give it a score depending on how many of these pairs produce simple fractions. The 12 note equal temperament produces one of the best scores, assuming some (perhaps arbitrary) constraints. There are some papers getting at this idea [0]. I even wrote a small program to try and do this [1]. Farey sequences are used for best rational approximation [2] [3]. I think this even gets at why some chords sound "sour/sad" while others sound "happy/full", because they have less or more constructive interference between the notes in the range of where we can hear. Obviously this has a lot to do with culture, so it's not as clear cut but at least this approach is better than just thinking it's completely arbitrary. [0] https://www.researchgate.net/publication/267806865_Measures_of_Consonances_in_a_Goodness-of-fit_Model_for_Equal-tempered_Scales https://www.researchgate.net/publication/267806865_Measures_... [1] https://github.com/abetusk/scratch/blob/release/src/music/best-scale-subdivision.py https://github.com/abetusk/scratch/blob/release/src/music/be... [2] https://www.johndcook.com/blog/2010/10/20/best-rational-approximation/ https://www.johndcook.com/blog/2010/10/20/best-rational-appr... [3] https://en.wikipedia.org/wiki/Farey_sequence https://en.wikipedia.org/wiki/Farey_sequence