This question What is Mazzola's "Topos of Music" about? has already been asked, but I am dissatisfied with the response for several reasons and would like Math SE to revisit it. For starters, no one answered the question in the title.
Topos of Music is an unusual book because, despite being massive in length, its target audience is unclear. It cannot be considered to be targeted at musicians -- or if it is, it is only the subset of serious mathematicians who are also musicians. For only people with significant mathematical training can understand this book. Your average musician doesn't know what a topological space or a quotient space is let alone a presheaf and a sheaf. I doubt even many serious music theorists (eg like Frank Lerdahl) understand it. To illustrate how far off Mazzola is, consider a well respected music theorist like Lerdahl. In his book Tonal Pitch Space, Lerdahl clearly doesn't understand what a Lagrangian is at all even though he attempts to use the concepts to define musical energy functions. Given this, there is no reason to think Lerdahl knows what a presheaf is nor that he therefore understands Mazzola himself.
As a musician myself, I have no idea what the book is saying. I just feel the mathematics looks interesting, but I cannot understand it at all. So I would like a mathematician who has read or skimmed it to summarize two things:
What is the book about?
What are the top 3 theories in the book most applicable to understanding how to make better music?
Answer whatever you can.
I am uninterested in Mazzola's software projects unless they in some way directly affect my understanding of how to make pleasant music.
I would post this on Music SE, but as I said, this book is for mathematicians and only readable by the subset of them who are also musicians.
All those listed in the thread posted above
Tymoczko's Mazzola’s Counterpoint Theory
Mazzola's other books:
Several articles by Mazzola littered throughout other books