Harmony Explained: Progress Towards A Scientific Theory of Music
Most music theory books are like medieval medical textbooks: they contain unjustified superstition, non-reasoning, and funny symbols glorified by Latin phrases. How does music, in particular harmony, actually work, presented as a real, scientific theory of music? 
The core to our approach is to consider not only the Physical phenomena of nature but also the Computational phenomena of any machine that must make sense of sound, such as the human brain. In particular we derive the following three fundamental phenomena of music: 
* the Major Scale, 
* the Standard Chord Dictionary, and 
* the difference in feeling between the Major and Minor Triads. 
While the Major Scale has been independently derived before by others in a similar manner [Helmholtz1863, Birkhoff1933], I believe the derivation of the Standard Chord Dictionary as well as the difference in feeling between the Major and Minor Triads to be original. 
We show to be incomplete the theory of the heretofore agreed-upon authority on this subject, 19th-century Physicist Hermann Helmholtz [Helmholtz1863]: he says notes are in "concord" because the sound playing them together is "less worse" than that of some other notes. But note that, in this theory, more notes can only penalize, some merely less than others, and so the most harmonious sound should be a single note by itself(!) and harmony would not exist as a phenomenon of music at all. 
I intend this article to be satisfying to scientists as an original contribution to science and art, yet I also intend it to be approachable by musicians and other curious members of the general public who may have long wondered at the curious properties of tonal music and been frustrated by the lack of satisfying, readable exposition on the subject. Therefore I have written in a deliberately plain and conversational style, avoiding unnecessarily formal language.