I like this type of research, but I think these results are not really new or surprising. In fact, there is almost a kind of "circular reasoning" at work - the Euler lattice (or Tonnetz) was constructed for the purpose of trying to represent the relationship of the basic music musical intervals. The fact that the basis of scales is the set of simple harmonic ratios of an initial pitch has been asserted by theorists for centuries, and that is what these star-convex structures correspond to. In other words, this research confirms that the established and traditional theory of scales and harmonic relationships is indeed the basis of human music of most cultures.
I agree with the feeling of "circular reasoning" in this paper. First you restrict yourself to integer ratio pitches, then you lay them out such that simple ratios occur close together, then you claim that scales contain pitches close together.
Further more, nothing is "universal" until it holds for all of the relevant set's elements. In fact, now that they've identified star convexity as a candidate for a universal, the next obvious step for an avant garde musician is to create music out of pitches picked from non-star-convex sets :)
The question is, will any of such music be considered 'beautiful'. The underlying philosophical consequences are what are (to me) most interesting in research such as this - they are another piece to add to the argument for aesthetic objectivism.
Music history has great examples of now considered beautiful music that was once (figuratively) spat on. The audience created a ruckus during the first performance of Stravinsky's Le Sacre du Printemps, Wagner looks like, in modern terminology, a "sleeper hit". Past masters who were once ignored are being re-celebrated in Indian classical music. You can tell I have my doubts that aesthetics is objective :)
Oh there are no doubt such examples; I don't consider them as an argument against aesthetic objectivism though, just as examples that our epistemology, to put it mildly, needs work :) It's an argument I've had many times and that has been fought over for 1000's of years and in countless papers and journal articles - to the point where I've given up hope of convincing anyone who knows what the argument is about, because those who do usually have thought about it so much that it's unlikely I will bring up an argument they haven't heard already, and vice versa :)
> In fact, now that they've identified star convexity as a candidate for a universal, the next obvious step for an avant garde musician is to create music out of pitches picked from non-star-convex sets :)
