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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.

Remark

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:

  1. What is the book about?

  2. 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.

Other resources

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    $\begingroup$ I haven't looked at the book, although I do know a bit about toposes and about music, but it's very optimistic to ask for three different ways in which this work can be directly applied to making better music. That's not how this kind of thing usually works. $\endgroup$ – Kevin Carlson Jan 6 '15 at 3:14
  • $\begingroup$ There is some discussion of the book in the comments at golem.ph.utexas.edu/category/2009/05/… --- there's also a tiny bit at reddit.com/r/Musicandmathematics/comments/13k2r7/… $\endgroup$ – Gerry Myerson Jan 6 '15 at 3:20
  • $\begingroup$ @KevinCarlson do you have a suggestion for how to modify my request? I just added (2) in the hopes of getting info of that nature. $\endgroup$ – Stan Shunpike Jan 6 '15 at 3:23
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    $\begingroup$ Please stop making trivial edits to bump your question. $\endgroup$ – user98602 Jan 7 '15 at 6:46
  • $\begingroup$ Okay. Apologies. I've waited four years to ask this question so I was eager for an answer, but I will play ball by the rules. $\endgroup$ – Stan Shunpike Jan 7 '15 at 7:00
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The review in Zentralblatt is several pages long and may go some way to answering at least your first question: http://recherche.ircam.fr/equipes/repmus/mamux/ThomasOnToM.pdf

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    $\begingroup$ Maybe you should tell us what you have read, so we don't waste your time and ours telling you things you already know. It would also make your question more valuable for anyone else trying to understand the book. $\endgroup$ – Gerry Myerson Jan 6 '15 at 3:33

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