另一个运用群论的数学理论是：Guerino Mazzola 写的《The Topos of Music: Geometric Logic of Concepts, Theory, and Performance》。他展示了音乐存在于 Z12 群中，而和弦是不变子群。很有意思的读物。
Another mathematical theory using group theory is: The Topos of Music: Geometric Logic of Concepts, Theory, and Performance by Guerino Mazzola. He shows that music lives in the Z12 group and chords are invariant subgroups. Interesting read.
从类似的角度来看，还有一本《The Topos of Music》[0]，根据维基百科的说法，这本书在数学家和音乐家之间都引发了不小的争议。[0] https://www.amazon.com/Topos-Music-Geometric-Concepts-Perfor...
In a similar vein, there's the Topos of Music book[0], which, according to Wikipedia, has been somewhat controversial among mathematicians and musicians alike. [0] https://www.amazon.com/Topos-Music-Geometric-Concepts-Perfor...
…音色与和声。它开篇就展示了一种八度是不协和音的（合成）乐器，以及用它能做出怎样的音乐。然后它顺着这个思路加码，搞出了很多好玩的东西。*《The Topos of Music》，Guerino Mazzola 著。极其硬核的数学音乐理论，对我这种普通人来说太劝退了。编辑：关于看清音色的“望远镜”……你见过频谱图吗？你可以装一个…
…timbre and harmony. It starts by exhibiting a (synthetic) instrument whose octaves are dissonant, and the music that you can do with it. Then it doubles down on this idea to obtain a lot of fun. * The Topos of Music, Guerino Mazzola. Really hardcore mathematical music theory, too scary for simple people like me. EDIT: regarding telescopes for seeing timbre... have you ever seen an spectrogram? You can install…
…原话似乎是：‘那你就别指望去理解莫扎特了。’https://en.wikipedia.org/wiki/Guerino_Mazzola。批评者 Tymoczko 是《A Geometry of Music》的作者。噢，他对《Topos of Music》的书评在网上能看到。http://dmitri.mycpanel.princeton.edu/files/publications/mazz... (PDF) 摘要：本文批判了 Guerino Mazzola 对传统对位法规则的推导，认为…
…seems to be saying, then you have no business trying to understand Mozart." https://en.wikipedia.org/wiki/Guerino_Mazzola The critic, Tymoczko, is the author of A Geometry of Music. Oh, his review of Topos of Music is online. http://dmitri.mycpanel.princeton.edu/files/publications/mazz... (PDF) Abstract: This paper critiques Guerino Mazzola’s derivation of traditional counterpoint rules, arguing that…
另一个运用群论的数学理论是：Guerino Mazzola 写的《The Topos of Music: Geometric Logic of Concepts, Theory, and Performance》。他展示了音乐存在于 Z12 群中，而和弦是不变子群。很有意思的读物。
Another mathematical theory using group theory is: The Topos of Music: Geometric Logic of Concepts, Theory, and Performance by Guerino Mazzola. He shows that music lives in the Z12 group and chords are invariant subgroups. Interesting read.
从类似的角度来看，还有一本《The Topos of Music》[0]，根据维基百科的说法，这本书在数学家和音乐家之间都引发了不小的争议。[0] https://www.amazon.com/Topos-Music-Geometric-Concepts-Perfor...
In a similar vein, there's the Topos of Music book[0], which, according to Wikipedia, has been somewhat controversial among mathematicians and musicians alike. [0] https://www.amazon.com/Topos-Music-Geometric-Concepts-Perfor...
…音色与和声。它开篇就展示了一种八度是不协和音的（合成）乐器，以及用它能做出怎样的音乐。然后它顺着这个思路加码，搞出了很多好玩的东西。*《The Topos of Music》，Guerino Mazzola 著。极其硬核的数学音乐理论，对我这种普通人来说太劝退了。编辑：关于看清音色的“望远镜”……你见过频谱图吗？你可以装一个…
…timbre and harmony. It starts by exhibiting a (synthetic) instrument whose octaves are dissonant, and the music that you can do with it. Then it doubles down on this idea to obtain a lot of fun. * The Topos of Music, Guerino Mazzola. Really hardcore mathematical music theory, too scary for simple people like me. EDIT: regarding telescopes for seeing timbre... have you ever seen an spectrogram? You can install…
…原话似乎是：‘那你就别指望去理解莫扎特了。’https://en.wikipedia.org/wiki/Guerino_Mazzola。批评者 Tymoczko 是《A Geometry of Music》的作者。噢，他对《Topos of Music》的书评在网上能看到。http://dmitri.mycpanel.princeton.edu/files/publications/mazz... (PDF) 摘要：本文批判了 Guerino Mazzola 对传统对位法规则的推导，认为…
…seems to be saying, then you have no business trying to understand Mozart." https://en.wikipedia.org/wiki/Guerino_Mazzola The critic, Tymoczko, is the author of A Geometry of Music. Oh, his review of Topos of Music is online. http://dmitri.mycpanel.princeton.edu/files/publications/mazz... (PDF) Abstract: This paper critiques Guerino Mazzola’s derivation of traditional counterpoint rules, arguing that…