The 12 tone scale of Western Tonal Harmony is a convenient, cultural map of three dimensional sonic waves in atmosphere perceived as sound. Other cultures have other types of descriptions for intervals and what Westerners call scales. Thus, the 12 tone scale and its intervals are merely higher order semantic maps of existing 3-dimensional mathematics that describe waves. You can perform any type of function upon these waves as long as you don't break a fundamental rule which defines these structures, which typically at the lowest mathematical level are defined by Euclidean geometries. If you want to use non-Euclidean descriptions, you have to make sure the waves, defined with Euclid's geometry in three dimensions and the number systems defined by those geometric rules, are redefined from the ground up using only the non-Euclidean rules. The order of The Circle of Fifths has mathematical basis in the organization of the overtone series, as do inversions. But at the same time, these seeming mathematical "truths" are a human imposition of maps upon the natural world. As an objective reader and musician, I am incredibly suspicious of any overstatement of a grand unity of anything, in physics or any other well-defined hard science. That this article makes such a claim in the incredibly soft science of music worries me, because if as valid as stated, can this effort then immediately tackle gravitational waves and string theory — because music is math, and if it is good for three dimensions, and four as posited, then why not n-dimensional arrays? It's a great consideration, but the mechanics already existed and had been stated before by others going back decades, to repackage and re-purpose for pop science seems underwhelming.

John Blake Arnold
Kerrville, Texas