I think that starting to write a book has got to be the hardest part of doing it. I have an interest in writing about music theory, in a manner that integrates mathematics and language (psychoacoustics, psycholinguistics) in a new and comprehensive way. Others have done the same, take "A Geometry of Music" by Dmitri Tymoczko, so I may not have too much new material but I am going to give it my best attempt. A title could have been "An Algebra of Music" or "The Deep Algebraic Structure of Music" or whatever but I am not completely happy with that title -- since Geometry and Algebra have been brought together in the mathematics of the late 20th Century.
That being said, I will be using concepts from group theory, modular theory, and whatever else seems to blend into a generalized concept of music, where harmony will be the main focus, and musical scales in the same breath. Rhythm will not play into this particular treatise...perhaps once I have it written I can just keep writing or insert topics describing rhythm at that point. What is the goal of all this? To continue where Noam Chomsky and Leonard Bernstein left off, well, no, even though ideally, yes. I have my own specific take which is closer to mathematics than to grammar and linguistics. But that too could certainly change in transit.
Well, enough for a starting post. Needless to say, I will definitely speak of septachords (scales, and also 13th chords made from the same seven notes) their complements, pentachords, and hexachords, the structure of the blocks of a certain finite sporadic simple group ("M12" of Mathieu) which also plays into some of the latest Moonshine math and physics (say, Umbral and Mathieu Moonshine, mock modular forms and the like) A system of my own based on the I-Ching (Benjamin Franklin went and did likewise, that is, his own form of I-Ching and spreads, but I will not be casting fortunes or doing auguries...) Some lovely algebraic manipulations using symmetry theory (the semi direct product of "S4" and "S3" comes to mind, the full symmetries of the square and the triangle) But lest I lose my audience now (some from bewilderment and some from a sense of pretentiousness here?) I should state what the goals truly are --- A comprehensive understanding of what makes music work, in a functional way, at least, as a medium (say, a kind of colored paint) used universally in the world and exclusively in the "West", and system of 12 tones based on some seven diatonic and five remaining "accidentals", that is the "piano idea".
What else? Starting with I-Ching hexagrams, and some related matters, might form a nicer pathway into this dense forest of ideas and properties....this is where I will start, and a really basic explanation of the math involved in approximating the pitches represented in 12-tET (Twelve tone equal temperament), which are logarithmic approximations of the overtone series. A little fun with numerology-of-sorts, especially the number seven, and the first seven overtones (the fundamental and six overtones (2,3,4,5,6,7) which are the harmonics above the fundamental pitch. So there it is. Hope this works out. I think I may post a few more times on this, and concurrently get to writing the book while I go. Keeping it as simple as I can, and hopefully colorful enough to keep anyone interested in this, still interested, in this.