There are lots of things to love here. The interval roughness function based on the harmonic series is interesting, and I wonder if it could be used to give som

There are lots of things to love here. The interval roughness function based on the harmonic series is interesting, and I wonder if it could be used to give some kind of score to chords or tuning systems, and maybe even generate them. The article also made me realize that there are two different ways of arriving to 12 tones per octave: - The pythagorean way: you keep iterating the "pythagorean tuning algorithm" described in the article until it gives you a note that's almost exactly like one you already have (I believe this is a less convoluted way of describing the "useful coincidence" hinted at by the author). It gives you a scale made of 12 notes, with more or less complex natural relationships between them. - The logarithmic way: you test all n-tone equal (logarithmic) divisions of the octave, up to some large n where notes end up too close together. You compare how much they deviate against the most important natural intervals: pure 2nd, 3rd and 5th. You'll find that 12-tone equal temperament forms a deep local minimum. Historically, ancient scales were constructed using methods similar to the first. My interpretation is that musicians desired to freely transpose melodies without making them sound bad, so scales got more refined, and those that maintained 12 notes to the octave started approximating equal division. Eventually, logarithmic tuning satisfied that demand exactly. I'd say the more interesting coincidence is that this transition could (theoretically) be smoothly done between obvious natural and equal temperaments - precisely because the above two methods result in the same number of notes. Of course, history isn't smooth, and alternative systems can be found in all periods. Maybe the dominion of 12 will even come to an end at some point.