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1. • CommentRowNumber1.
   • CommentAuthorDrew Flieder
   • CommentTimeJul 10th 2023
   • PermaLink
   Author: Drew Flieder
   Format: MarkdownItexPresentation of the notion of a pitch-class from mathematical music theory. I plan on making more posts that relate to topics in this page, especially those involving categorical formalisms of topics from music theory. <a href="https://ncatlab.org/nlab/revision/pitch-class/1">v1</a>, <a href="https://ncatlab.org/nlab/show/pitch-class">current</a>

   Presentation of the notion of a pitch-class from mathematical music theory. I plan on making more posts that relate to topics in this page, especially those involving categorical formalisms of topics from music theory.

   v1, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeJul 10th 2023
   • PermaLink
   Author: Urs
   Format: MarkdownItexMaybe typing C\sharp gives better output than C\#

   Maybe typing

     C\sharp
   

   gives better output than

     C\#
   

3. • CommentRowNumber3.
   • CommentAuthorTodd_Trimble
   • CommentTimeJul 10th 2023
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexIn section 4, what is the meaning or significance of $Simple$?

   In section 4, what is the meaning or significance of $Simple$?

4. • CommentRowNumber4.
   • CommentAuthorDrew Flieder
   • CommentTimeJul 10th 2023
   • PermaLink
   Author: Drew Flieder
   Format: MarkdownItex@Urs: Thank you for that suggestion, it looks better now. @Todd: Good question. That is something I will unpack in another post on Mazzola's form and denotator theories. The meaning in this context is that the $\mathbb{Z}_{12}$ in $Simple(\mathbb{Z}_{12})$ is actually to be thought of as a *module* (with coefficient ring $\{ 1, 5, 7, 11\}$) from the category $Mod$ of modules with affine transformations, and not just a group. The notation $Simple(\mathbb{Z}_{12})$ means that we take the representable functor (presheaf) $Hom_{Mod}(-, \mathbb{Z}_{12})$, and then the notation \[ PC \underset{Id}{\longrightarrow} Simple (\mathbb{Z}_{12}) \] means that we name that presheaf $PC$, to indicate that the "elements" of $Hom_{Mod}(-, \mathbb{Z}_{12})$ are to be thought of as pitch-classes. A form in Mazzola is really just a module presheaf equipped with a name. There are "types constructors" other than $Simple$ as well. Specifically, there are $Limit$, $Colimit$, $Power$, and $Syn$ type constructors. These allow for the construction of more complex forms from the given module presheaves. As an example, we can define the two forms \[ Pitch \underset{Id}{\longrightarrow} Simple (\mathbb{Z}), \] the elements of which are pitches, and \[ Onset \underset{Id}{\longrightarrow} Simple (\mathbb{R}), \] the elements of which are points in time, or onsets. For the discrete diagram consisting of $Pitch$ and $Onset$, then the form \[ OnPi \underset{Id}{\longrightarrow} Limit (Onset, Pitch) \] is their product, and the idea is that the "points" in $OnPi$ provide onset and pitch information. So you can imagine how something like a melody can be encoded as a subobject of $OnPi$.

   @Urs: Thank you for that suggestion, it looks better now.

   @Todd: Good question. That is something I will unpack in another post on Mazzola’s form and denotator theories. The meaning in this context is that the $\mathbb{Z}_{12}$ in $Simple(\mathbb{Z}_{12})$ is actually to be thought of as a module (with coefficient ring $\{ 1, 5, 7, 11\}$) from the category $Mod$ of modules with affine transformations, and not just a group. The notation $Simple(\mathbb{Z}_{12})$ means that we take the representable functor (presheaf) $Hom_{Mod}(-, \mathbb{Z}_{12})$, and then the notation

   $$PC \underset{Id}{\longrightarrow} Simple (\mathbb{Z}_{12})$$

   means that we name that presheaf $PC$, to indicate that the “elements” of $Hom_{Mod}(-, \mathbb{Z}_{12})$ are to be thought of as pitch-classes. A form in Mazzola is really just a module presheaf equipped with a name.

   There are “types constructors” other than $Simple$ as well. Specifically, there are $Limit$, $Colimit$, $Power$, and $Syn$ type constructors. These allow for the construction of more complex forms from the given module presheaves. As an example, we can define the two forms

   $$Pitch \underset{Id}{\longrightarrow} Simple (\mathbb{Z}),$$

   the elements of which are pitches, and

   $$Onset \underset{Id}{\longrightarrow} Simple (\mathbb{R}),$$

   the elements of which are points in time, or onsets. For the discrete diagram consisting of $Pitch$ and $Onset$, then the form

   $$OnPi \underset{Id}{\longrightarrow} Limit (Onset, Pitch)$$

   is their product, and the idea is that the “points” in $OnPi$ provide onset and pitch information. So you can imagine how something like a melody can be encoded as a subobject of $OnPi$.

5. • CommentRowNumber5.
   • CommentAuthorTodd_Trimble
   • CommentTimeJul 11th 2023
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexWell, I've heard of Mazzola's The Topos of Music. I've never looked at it, but I'm curious about what's in it, and so I would welcome hearing more, but ideally with a succinct presentation, stripping away any notational frill. (I already suspect based on what I see here that his presentation is not all that succinct or efficient, at least not for topos theorists.) You call $\{1, 5, 7, 11\}$ a "ring". These elements are the square roots of 1 in $\mathbb{Z}_{12}$ and form a submonoid of the *multiplicative* monoid $\mathbb{Z}_{12}$. This submonoid is a group, in fact a 4-group (Viergruppe), and acts multiplicatively on $\mathbb{Z}_{12}$. But what *ring* structure do you have in mind? Please have a look at [[writing in the nLab]], if you haven't already. I myself am open to nLab articles that discuss Mazzola's work, on the condition that they can be fitted harmoniously within the nLab corpus, but this may involve extensive editing and reworking by other nLab authors.

   Well, I’ve heard of Mazzola’s The Topos of Music. I’ve never looked at it, but I’m curious about what’s in it, and so I would welcome hearing more, but ideally with a succinct presentation, stripping away any notational frill. (I already suspect based on what I see here that his presentation is not all that succinct or efficient, at least not for topos theorists.)

   You call $\{1, 5, 7, 11\}$ a “ring”. These elements are the square roots of 1 in $\mathbb{Z}_{12}$ and form a submonoid of the multiplicative monoid $\mathbb{Z}_{12}$. This submonoid is a group, in fact a 4-group (Viergruppe), and acts multiplicatively on $\mathbb{Z}_{12}$. But what ring structure do you have in mind?

   Please have a look at writing in the nLab, if you haven’t already. I myself am open to nLab articles that discuss Mazzola’s work, on the condition that they can be fitted harmoniously within the nLab corpus, but this may involve extensive editing and reworking by other nLab authors.

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeJul 11th 2023
   • (edited Jul 11th 2023)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have touched the formatting of some of the formulas, for readability, for instance giving the subtraction of pitch classes ([here](https://ncatlab.org/nlab/show/pitch-class#eq:SubtractionOfPitchClasses)) an `array`-formatting. The use of "$\underset{Id}{\to}$" for a *declaration* is most unusual, the ordinary way to typeset this would be "$\coloneqq$" (`\coloneqq`) or "$\equiv$" (`\equiv`). Above in [#4](#Comment_111271) you say: > So you can imagine how something like a melody can be encoded as a subobject of $OnPi$. It is evident that a sequence of tones can be encoded as an $\mathbb{N}$-indexed list of (real) numbers. But how do topos-theoretic notions improve on this? What's the gain? What's the aim?

   I have touched the formatting of some of the formulas, for readability, for instance giving the subtraction of pitch classes (here) an array-formatting.

   The use of “$\underset{Id}{\to}$” for a declaration is most unusual, the ordinary way to typeset this would be “$\coloneqq$” (\coloneqq) or “$\equiv$” (\equiv).

   Above in #4 you say:

   So you can imagine how something like a melody can be encoded as a subobject of $OnPi$.

   It is evident that a sequence of tones can be encoded as an $\mathbb{N}$-indexed list of (real) numbers. But how do topos-theoretic notions improve on this? What’s the gain? What’s the aim?

7. • CommentRowNumber7.
   • CommentAuthorDrew Flieder
   • CommentTimeJul 11th 2023
   • PermaLink
   Author: Drew Flieder
   Format: MarkdownItex@Todd: Thank you for welcoming discussion about Mazzola's work and referring me to [[writing in the nLab]]. I will try my best to present his work succinctly. Admittedly his work has a great deal of idiosyncracies, and a lot of stuff that would be considered "fluffy" from a mathematical perspective. This seems almost unavoidable however, since the subject of music is (historically) considerably less formal than scientific subjects, so most established musicological/music theory concepts are very fuzzy and the scope of their use is totally unregulated. So Mazzola's project seems to be to provide a "universal language" for music theory, in order that the discipline comes closer to meeting scientific standards of explicitness. All that being said, I will try to present Mazzola's work so that it fits harmoniously within the nLab corpus. If it seems unfitting and if the nLab community thinks that I should take this work elsewhere, that is fine by me. Regarding my calling $\big\{1, 5, 7, 11\big\}$ a "ring", that was a mistake. The ring should have just been $\mathbb{Z}$. The subset $\{1, 5, 7, 11\} \subset \mathbb{Z}$ is what is usually of interest in music theory, since multiplying $\mathbb{Z}_{12}$ by any of the members of $\big\{1, 5, 7, 11\big\}$ gives an automorphism of $\mathbb{Z}_{12}$ (as you suggested in \#5), and these automorphisms are often used as transformations of sets of pitch-classes. @Urs: Thank you for revising the formatting. I will take note of these standards for future posts. You're right that "$\underset{Id}{\longrightarrow}$" is unusual for declaration. The parameter, called the **identifier**, is defined by Mazzola as a functor monomorphism. In particular, whatever is the "total object" constructed from the $Type(Coordinator)$ portion (e.g. the $Limit(Onset, Pitch)$ from my comment in [#4](#Comment_111271) gives $Onset \times Pitch$), the identifier consists of a monomorphism $I \colon A \rightarrowtail Onset \times Pitch$, where $A$ is some other object in the category of module presheaves. One of the important uses of this is when you want to define the "space" of the form as a subobject of whatever is constructed from $Type(Coordinator)$. An example is something like \[ OnPi \underset{I : A \rightarrowtail Onset \times Pitch}{\longrightarrow} Limit(Onset, Pitch), \] where $A$ is, say, the subfunctor of $Onset \times Pitch$ such that the pitch coordinate values are between 0 and 127 inclusive (this would give all the midi pitch values). Regarding your comment in \#6: > It is evident that a sequence of tones can be encoded as an $\mathbb{N}$-indexed list of (real) numbers. But how do topos-theoretic notions improve on this? What’s the gain? What’s the aim? The topos-theoretic notions facilitate the organization of musical concepts in a coherent setting. They provide a consistent format for defining new forms of musical information, as well as a consistent format for accessing such information. Just about all mathematical music theory pre-Mazzola resorted to handwavy formalisms. This worked to get the field started, but the lack of any rigorous foundations for the concepts of music led to some difficulties, a couple of which are: * Making connections between musical concepts is almost always ad hoc, and thus tends to be fuzzy. * Generalization of concepts is difficult to achieve, because the precise content of the concept is difficult to fully identify. Hence what you often encounter is a theorist going a few steps in the direction of generalization, but not being able to capture the nature of a musical concept in full depth. I recall Mazzola saying somewhere (I can't remember where) that "knowledge is ordered access to information". Seen in that light, you can view the topos-theoretic foundations as providing the ordering principle for musical information.

   @Todd: Thank you for welcoming discussion about Mazzola’s work and referring me to writing in the nLab. I will try my best to present his work succinctly. Admittedly his work has a great deal of idiosyncracies, and a lot of stuff that would be considered “fluffy” from a mathematical perspective. This seems almost unavoidable however, since the subject of music is (historically) considerably less formal than scientific subjects, so most established musicological/music theory concepts are very fuzzy and the scope of their use is totally unregulated. So Mazzola’s project seems to be to provide a “universal language” for music theory, in order that the discipline comes closer to meeting scientific standards of explicitness. All that being said, I will try to present Mazzola’s work so that it fits harmoniously within the nLab corpus. If it seems unfitting and if the nLab community thinks that I should take this work elsewhere, that is fine by me.

   Regarding my calling $\big\{1, 5, 7, 11\big\}$ a “ring”, that was a mistake. The ring should have just been $\mathbb{Z}$. The subset $\{1, 5, 7, 11\} \subset \mathbb{Z}$ is what is usually of interest in music theory, since multiplying $\mathbb{Z}_{12}$ by any of the members of $\big\{1, 5, 7, 11\big\}$ gives an automorphism of $\mathbb{Z}_{12}$ (as you suggested in #5), and these automorphisms are often used as transformations of sets of pitch-classes.

   @Urs: Thank you for revising the formatting. I will take note of these standards for future posts.

   You’re right that “$\underset{Id}{\longrightarrow}$” is unusual for declaration. The parameter, called the identifier, is defined by Mazzola as a functor monomorphism. In particular, whatever is the “total object” constructed from the $Type(Coordinator)$ portion (e.g. the $Limit(Onset, Pitch)$ from my comment in #4 gives $Onset \times Pitch$), the identifier consists of a monomorphism $I \colon A \rightarrowtail Onset \times Pitch$, where $A$ is some other object in the category of module presheaves. One of the important uses of this is when you want to define the “space” of the form as a subobject of whatever is constructed from $Type(Coordinator)$. An example is something like

   $$OnPi \underset{I : A \rightarrowtail Onset \times Pitch}{\longrightarrow} Limit(Onset, Pitch),$$

   where $A$ is, say, the subfunctor of $Onset \times Pitch$ such that the pitch coordinate values are between 0 and 127 inclusive (this would give all the midi pitch values).

   Regarding your comment in #6:

   It is evident that a sequence of tones can be encoded as an $\mathbb{N}$-indexed list of (real) numbers. But how do topos-theoretic notions improve on this? What’s the gain? What’s the aim?

   The topos-theoretic notions facilitate the organization of musical concepts in a coherent setting. They provide a consistent format for defining new forms of musical information, as well as a consistent format for accessing such information. Just about all mathematical music theory pre-Mazzola resorted to handwavy formalisms. This worked to get the field started, but the lack of any rigorous foundations for the concepts of music led to some difficulties, a couple of which are:

   • Making connections between musical concepts is almost always ad hoc, and thus tends to be fuzzy.

   • Generalization of concepts is difficult to achieve, because the precise content of the concept is difficult to fully identify. Hence what you often encounter is a theorist going a few steps in the direction of generalization, but not being able to capture the nature of a musical concept in full depth.

   I recall Mazzola saying somewhere (I can’t remember where) that “knowledge is ordered access to information”. Seen in that light, you can view the topos-theoretic foundations as providing the ordering principle for musical information.

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeJul 11th 2023
   • PermaLink
   Author: Urs
   Format: MarkdownItex> The topos-theoretic notions facilitate the organization of musical concepts in a coherent setting. What's an example? A good such example would be best to highlight up-front in the entry, so that there is some motivation. Without mentioning of some real applications, the entry risks looking like a repetitive definition of $\mathbb{Z}_{12}$ in increasingly bewildering notation.

   The topos-theoretic notions facilitate the organization of musical concepts in a coherent setting.

   What’s an example? A good such example would be best to highlight up-front in the entry, so that there is some motivation. Without mentioning of some real applications, the entry risks looking like a repetitive definition of $\mathbb{Z}_{12}$ in increasingly bewildering notation.

9. • CommentRowNumber9.
   • CommentAuthorTodd_Trimble
   • CommentTimeJul 11th 2023
   • (edited Jul 11th 2023)
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexDrew: I was chatting with John Baez today and happened to mention -- it came up naturally in the course of conversation -- your addition to the nLab. Apparently lots of people interested in mathematical music theory study the 24-element group generated by translations and inversions on $\mathbb{Z}_{12}$ and its action on (for example) triads, but John perked up when I mentioned this 48-element group, which would correspond to adjoining 5 and 7 to $\{1, 11\}$ where the latter two elements correspond to the subgroup of inversions (I mean in the mathematical sense; I am aware that "inversion" in music theory usually means something else, as in EGCE being a chordal inversion of CEGC). Anyway, there may be ready-made interest in having some nLab articles on this. I know a little bit (not a lot) of music theory, but rather more of topos theory, so I may be responding a bit. Could you say quickly what is $Mod$ in this context? Is it the topos of presheaves on the 48-element group? If you are used to writing mathematics, it will help me at any rate if you write in more straightforward mathematical language like "we define $PC$ to be the presheaf blah blah" or "we abbreviate $Onset \times Pitch$" to $OnPi$", rather than $OnPi \underset{Id}{\to} Limit(Onset, Pitch)$ which is hard and confusing to read. (The stuff under Denotators is still confusing to me, like $Simple(\mathbb{Z}_{12})(n)$ -- is $n$ an object of a category you're taking presheaves on, or what is going on exactly?)

   Drew: I was chatting with John Baez today and happened to mention – it came up naturally in the course of conversation – your addition to the nLab. Apparently lots of people interested in mathematical music theory study the 24-element group generated by translations and inversions on $\mathbb{Z}_{12}$ and its action on (for example) triads, but John perked up when I mentioned this 48-element group, which would correspond to adjoining 5 and 7 to $\{1, 11\}$ where the latter two elements correspond to the subgroup of inversions (I mean in the mathematical sense; I am aware that “inversion” in music theory usually means something else, as in EGCE being a chordal inversion of CEGC).

   Anyway, there may be ready-made interest in having some nLab articles on this. I know a little bit (not a lot) of music theory, but rather more of topos theory, so I may be responding a bit. Could you say quickly what is $Mod$ in this context? Is it the topos of presheaves on the 48-element group?

   If you are used to writing mathematics, it will help me at any rate if you write in more straightforward mathematical language like “we define $PC$ to be the presheaf blah blah” or “we abbreviate $Onset \times Pitch$” to $OnPi$”, rather than $OnPi \underset{Id}{\to} Limit(Onset, Pitch)$ which is hard and confusing to read. (The stuff under Denotators is still confusing to me, like $Simple(\mathbb{Z}_{12})(n)$ – is $n$ an object of a category you’re taking presheaves on, or what is going on exactly?)

10. • CommentRowNumber10.
    • CommentAuthorDrew Flieder
    • CommentTimeJul 11th 2023
    • PermaLink
    Author: Drew Flieder
    Format: MarkdownItexClarified some notation in the *As a denotator* section. <a href="https://ncatlab.org/nlab/revision/diff/pitch-class/3">diff</a>, <a href="https://ncatlab.org/nlab/revision/pitch-class/3">v3</a>, <a href="https://ncatlab.org/nlab/show/pitch-class">current</a>

    Clarified some notation in the As a denotator section.

    diff, v3, current

11. • CommentRowNumber11.
    • CommentAuthorDrew Flieder
    • CommentTimeJul 11th 2023
    • PermaLink
    Author: Drew Flieder
    Format: MarkdownItexYes that's right, there is considerably less discussion of the 48-element group in mathematical music theory than the 24-element Dihedral group. Nonetheless many composers have used the multiplications of $5$ and $7$ to derive new pitch material from given material, and what's interesting about these operations is that they don't preserve the intervallic structure of chords. So they are used to transform the harmonic "flavor", but preserve the overall pitch-class structure. Something that is confusing about music theory is that the term "inversion" means different things in different contexts. In classical (common-practice period) music, inversion means like you said, e.g. EGCE is an inversion of CEGC. But in 20th century music theory, inversion is taken to mean the mathematical operation of multiplication by $11$. $Mod$ is the category of modules (over any ring) for objects and affine transformations for morphisms. Then $[Mod^{op}, Set]$ is the category of presheaves on $Mod$. I noticed that this is controversial though, since $Mod$ is not a small category, which is problematic regarding the existence of a subobject classifier in $[Mod^{op}, Set]$. One of the motivations for formalizing musical spaces as presheaves in $[Mod^{op}, Set]$ rather than $Mod$ was for the existence of such a subobject classifier, since e.g. a pitch-class set is a subobject of $\mathbb{Z}_{12}$. I brought this problem up to Mazzola and he had a couple responses: the first being to not think in terms of classical sets (not sure how this would affect the rest of the mathematics), the other to just take a small subcategory of $Mod$, and that way one still recovers most of the important spaces for music theory. I believe that in the history of Mazzola's theories, he first conceived of modules as being the important spaces the points of which denote musical entities. The presheaf construction came later as a result of the need to take arbitrary (co)limits and power objects of modules. I will write more in the custom of mathematical language instead of Mazzola's idiosyncratic language. The [[pitch-class]] post was perhaps premature, as it would make more sense to first have a post on e.g. Mazzola's form and denotator theories. I made a very slight edit in the *As a denotator* section, writing \[ thisPitchClass \colon 0_\mathbb{Z} \rightsquigarrow PC \underset{Id}{\longrightarrow} Simple(\mathbb{Z}_{12})(n) \] just for the time-being. Formally, the meaning of this notation is that $thisPitchClass$ *names* the element $\overline{n} \in Hom_{Mod}(0_\mathbb{Z}, \mathbb{Z}_{12})$ such that $\overline{n}(0) = n$. It's pretty cumbersome, and Mazzola often abbreviates this to just \[ thisPitchClass : 0_\mathbb{Z} \rightsquigarrow PC(n). \] Although this is pretty bloated from a mathematical view, it makes more sense in the context of music theory/musicology, where musicians speak of such entities as pitch-classes and whatnot, without the underlying spaces of these concepts being made precise.

    Yes that’s right, there is considerably less discussion of the 48-element group in mathematical music theory than the 24-element Dihedral group. Nonetheless many composers have used the multiplications of $5$ and $7$ to derive new pitch material from given material, and what’s interesting about these operations is that they don’t preserve the intervallic structure of chords. So they are used to transform the harmonic “flavor”, but preserve the overall pitch-class structure.

    Something that is confusing about music theory is that the term “inversion” means different things in different contexts. In classical (common-practice period) music, inversion means like you said, e.g. EGCE is an inversion of CEGC. But in 20th century music theory, inversion is taken to mean the mathematical operation of multiplication by $11$.

    $Mod$ is the category of modules (over any ring) for objects and affine transformations for morphisms. Then $[Mod^{op}, Set]$ is the category of presheaves on $Mod$. I noticed that this is controversial though, since $Mod$ is not a small category, which is problematic regarding the existence of a subobject classifier in $[Mod^{op}, Set]$. One of the motivations for formalizing musical spaces as presheaves in $[Mod^{op}, Set]$ rather than $Mod$ was for the existence of such a subobject classifier, since e.g. a pitch-class set is a subobject of $\mathbb{Z}_{12}$. I brought this problem up to Mazzola and he had a couple responses: the first being to not think in terms of classical sets (not sure how this would affect the rest of the mathematics), the other to just take a small subcategory of $Mod$, and that way one still recovers most of the important spaces for music theory. I believe that in the history of Mazzola’s theories, he first conceived of modules as being the important spaces the points of which denote musical entities. The presheaf construction came later as a result of the need to take arbitrary (co)limits and power objects of modules.

    I will write more in the custom of mathematical language instead of Mazzola’s idiosyncratic language. The pitch-class post was perhaps premature, as it would make more sense to first have a post on e.g. Mazzola’s form and denotator theories. I made a very slight edit in the As a denotator section, writing

    $$thisPitchClass \colon 0_\mathbb{Z} \rightsquigarrow PC \underset{Id}{\longrightarrow} Simple(\mathbb{Z}_{12})(n)$$

    just for the time-being. Formally, the meaning of this notation is that $thisPitchClass$ names the element $\overline{n} \in Hom_{Mod}(0_\mathbb{Z}, \mathbb{Z}_{12})$ such that $\overline{n}(0) = n$. It’s pretty cumbersome, and Mazzola often abbreviates this to just

    $$thisPitchClass : 0_\mathbb{Z} \rightsquigarrow PC(n).$$

    Although this is pretty bloated from a mathematical view, it makes more sense in the context of music theory/musicology, where musicians speak of such entities as pitch-classes and whatnot, without the underlying spaces of these concepts being made precise.

12. • CommentRowNumber12.
    • CommentAuthorTodd_Trimble
    • CommentTimeJul 12th 2023
    • (edited Jul 12th 2023)
    • PermaLink
    Author: Todd_Trimble
    Format: MarkdownItexOkay, thanks. Please bear with me as I continue asking questions. You said back in #7 that the ring is supposed to be $\mathbb{Z}$. Since a $\mathbb{Z}$-module is the same as an abelian group, I understand the category $Mod$ in this context to be the category whose objects are abelian groups and whose morphisms $f: A \to B$ are functions of the form $f(x) = m x + b$ where $m: A \to B$ is an ordinary homomorphism and $b$ is any element of $B$. Is this what you intend? (I somehow got the impression earlier that the 48-element group was going to be part and parcel of the site, i.e., the category that you're taking presheaves on. Could you clarify what role $\{1, 5, 7, 11\}$ was playing when you wrote #4?) Assuming the first paragraph of this comment is correct, I think using $Mod$ for this site is confusing; I think I'd write something like $Aff_\mathbb{Z}$, consistent with what is in [[affine space]]. My offhand guess is that finitely generated abelian groups would suffice for the purposes of music theory, and this is an essentially small category so that presheaves on that forms a topos. There are other tricks one can use if this feels too restrictive. For example, one can do as one does in algebraic geometry a la Demazure and Gabriel: assume a Grothendieck universe $U$ (equivalently, a strongly inaccessible cardinal) so that $U$-small sets forms an internal topos inside the topos of all sets, and then refer to e.g. $U$-small abelian groups. It's hard for me to believe that *that* wouldn't suffice for music theory foundations. Yeah, I support writing more with mathematicians in mind, since that is the main nLab audience. If you want to mention Mazzola's terminology or provide a little dictionary between his language and more ordinary language, that would of course be welcome, but in my opinion the main language should be the language that a typical nLab reader would be used to. So, for example, affine maps $n: 0 \to \mathbb{Z}_{12}$ (or if you like, maps $Aff_\mathbb{Z}(-, 0) \to Aff_\mathbb{Z}(-, \mathbb{Z}_{12})$ between the corresponding representable functors) make perfectly good sense, conceptually and notationally, and that's what I would use. I squinted at that squiggly arrow notation which seems to be Mazzola's, wondering what it was doing there.

    Okay, thanks. Please bear with me as I continue asking questions. You said back in #7 that the ring is supposed to be $\mathbb{Z}$. Since a $\mathbb{Z}$-module is the same as an abelian group, I understand the category $Mod$ in this context to be the category whose objects are abelian groups and whose morphisms $f: A \to B$ are functions of the form $f(x) = m x + b$ where $m: A \to B$ is an ordinary homomorphism and $b$ is any element of $B$. Is this what you intend?

    (I somehow got the impression earlier that the 48-element group was going to be part and parcel of the site, i.e., the category that you’re taking presheaves on. Could you clarify what role $\{1, 5, 7, 11\}$ was playing when you wrote #4?)

    Assuming the first paragraph of this comment is correct, I think using $Mod$ for this site is confusing; I think I’d write something like $Aff_\mathbb{Z}$, consistent with what is in affine space.

    My offhand guess is that finitely generated abelian groups would suffice for the purposes of music theory, and this is an essentially small category so that presheaves on that forms a topos. There are other tricks one can use if this feels too restrictive. For example, one can do as one does in algebraic geometry a la Demazure and Gabriel: assume a Grothendieck universe $U$ (equivalently, a strongly inaccessible cardinal) so that $U$-small sets forms an internal topos inside the topos of all sets, and then refer to e.g. $U$-small abelian groups. It’s hard for me to believe that that wouldn’t suffice for music theory foundations.

    Yeah, I support writing more with mathematicians in mind, since that is the main nLab audience. If you want to mention Mazzola’s terminology or provide a little dictionary between his language and more ordinary language, that would of course be welcome, but in my opinion the main language should be the language that a typical nLab reader would be used to. So, for example, affine maps $n: 0 \to \mathbb{Z}_{12}$ (or if you like, maps $Aff_\mathbb{Z}(-, 0) \to Aff_\mathbb{Z}(-, \mathbb{Z}_{12})$ between the corresponding representable functors) make perfectly good sense, conceptually and notationally, and that’s what I would use. I squinted at that squiggly arrow notation which seems to be Mazzola’s, wondering what it was doing there.

13. • CommentRowNumber13.
    • CommentAuthorDrew Flieder
    • CommentTimeJul 12th 2023
    • PermaLink
    Author: Drew Flieder
    Format: MarkdownItexYour questions are appreciated and I'm happy trying to answer them. In \#11 I tried to make something clear but now realize my wording was imprecise. In the third paragraph I say that “$Mod$ is the category of modules (over any ring)”. What I should have said is that different modules $M, N$ in $Mod$ can be over different rings $S, R$. Sometimes terminology in Topos of Music (ToM) is inconsistent, so I'm reading now in Appendix E of ToM that he is now calling the morphisms in $Mod$ *diaffine homomorphisms*. To define these requires to first define *dilinear homomorphisms*. The way that this works is the following. Given a ring homomorphism $\psi : S \rightarrow R$, and $M$ an $R$-module, we can derive an $S$-module $M_{[\psi]}$ via \[ s \cdot m = \varphi(s) \cdot m, \] $s \in S, m \in M$. A *dilinear homomorphism* is then a pair $(\psi : S \rightarrow R, g : M \rightarrow N_{[\psi]})$ where $g$ is an $S$-linear morphism. Composition of morphisms $(\varphi : T \rightarrow S, f : L \rightarrow M_{[\varphi]}), (\psi : S \rightarrow R, g : M \rightarrow N_{[\psi]})$ is defined by \[ (\psi \circ \varphi : T \rightarrow R, g \circ f : L \rightarrow N_{[\psi \circ \varphi]}). \] So for (say) an $S$-module $M$ and $R$-module $N$, $Dil(M, N)$ is the set of dilinear homomorphisms. So to answer your question from paragraph 1 of \#12, your description of a morphism in $Mod$ is true when $A$ and $B$ are equipped with the same ring. But the general case (which looks the same as what you wrote) is that a morphism is a *diaffine homomorphism*. This means that everything you wrote in paragraph 1 of \#12 is the same, except that $m : A \rightarrow B$ is in general a *dilinear homomorphism*. Just for clarity I'll rewrite it here: For an $R$-module $A$ and $S$-module $B$, a diaffine homomorphism $f : A \rightarrow B$ is of the form $f(x) = m x + b$, where $m : A \rightarrow B$ is a dilinear homomorphism and $b$ is any element of $B$. Regarding the terminology of $Aff$ instead of $Mod$, this makes sense to me. So from now on I will use that. Regarding paragraph 2 of \#12, my discussion of $\{ 1, 5, 7, 11 \}$ seems to be a case of me having confused a couple things based on readings of different texts. I will explain a bit in more detail below, but at the moment I'm wondering if there really is any significance in thinking in terms of modules instead of just groups. It could be that Mazzola had reasons for thinking in terms of modules for earlier theories, but with more recent generalizations (in terms of presheaves) the module formalism is not totally necessary. I've searched for every instance of the word "ring" in Chapter 6 of ToM, where he justifies the use of modules, but I don't see any clear role that the ring is playing. But apparently according to [this](https://www.jstor.org/stable/833391?seq=2) review of *Geometrie der Töne*, an earlier book by Mazzola, the ring structure of modules plays a role for instance in deriving intervals of equal-tempered scales. I don't have much to say about this though because I haven't read *Geometrie der Töne*. So to refer back to my mention of $\{ 1, 5, 7, 11 \}$ in \#4 as a subset of the ring $\mathbb{Z}$ that acts on $\mathbb{Z}_{12}$, this is not how such transformations are usually expressed. Instead they are expressed as group automorphisms of $\mathbb{Z}_{12}$. So the 48-element group acting on $\mathbb{Z}_{12}$ that you mentioned in \#9 is a subset of $Aff(\mathbb{Z}_{12}, \mathbb{Z}_{12})$. Unfortunately finitely generated abelian groups won't work, since e.g. a frequency value would be an element of $\mathbb{R}$. Regarding your mention of [[Grothendieck universes]], I am less familiar with this topic, but am reading about it now. It seems that it provides a trick to make a large set into a small set, so that e.g. the set of $U$-small abelian groups is small? If so, it does seem crazy to think that this wouldn't suffice for music theory foundations. However there are some "circular" constructions in Mazzola, which are like sets that contain themselves, except instead of sets they are the module presheaves. His justifications for this are speculative, but there do seem to be quite a few applications in music theory. For instance, there is a circular presheaf whose "elements" are [rhythm trees](https://support.ircam.fr/docs/om/om6-manual/co/RT1.html), a rhythm tree being a specific method for representing rhythms. These are discussed in Chapter 51 of ToM. I'm not sure though how these circular presheaf constructions relate to the size problems, but from the set theory view they would violate well-foundedness (which I know is a relation that Mazzola is critical of, in the spirit of Paul Finsler).

    Your questions are appreciated and I’m happy trying to answer them.

    In #11 I tried to make something clear but now realize my wording was imprecise. In the third paragraph I say that “$Mod$ is the category of modules (over any ring)”. What I should have said is that different modules $M, N$ in $Mod$ can be over different rings $S, R$. Sometimes terminology in Topos of Music (ToM) is inconsistent, so I’m reading now in Appendix E of ToM that he is now calling the morphisms in $Mod$ diaffine homomorphisms. To define these requires to first define dilinear homomorphisms. The way that this works is the following. Given a ring homomorphism $\psi : S \rightarrow R$, and $M$ an $R$-module, we can derive an $S$-module $M_{[\psi]}$ via

    $$s \cdot m = \varphi(s) \cdot m,$$

    $s \in S, m \in M$. A dilinear homomorphism is then a pair $(\psi : S \rightarrow R, g : M \rightarrow N_{[\psi]})$ where $g$ is an $S$-linear morphism. Composition of morphisms $(\varphi : T \rightarrow S, f : L \rightarrow M_{[\varphi]}), (\psi : S \rightarrow R, g : M \rightarrow N_{[\psi]})$ is defined by

    $$(\psi \circ \varphi : T \rightarrow R, g \circ f : L \rightarrow N_{[\psi \circ \varphi]}).$$

    So for (say) an $S$-module $M$ and $R$-module $N$, $Dil(M, N)$ is the set of dilinear homomorphisms.

    So to answer your question from paragraph 1 of #12, your description of a morphism in $Mod$ is true when $A$ and $B$ are equipped with the same ring. But the general case (which looks the same as what you wrote) is that a morphism is a diaffine homomorphism. This means that everything you wrote in paragraph 1 of #12 is the same, except that $m : A \rightarrow B$ is in general a dilinear homomorphism. Just for clarity I’ll rewrite it here: For an $R$-module $A$ and $S$-module $B$, a diaffine homomorphism $f : A \rightarrow B$ is of the form $f(x) = m x + b$, where $m : A \rightarrow B$ is a dilinear homomorphism and $b$ is any element of $B$.

    Regarding the terminology of $Aff$ instead of $Mod$, this makes sense to me. So from now on I will use that.

    Regarding paragraph 2 of #12, my discussion of $\{ 1, 5, 7, 11 \}$ seems to be a case of me having confused a couple things based on readings of different texts. I will explain a bit in more detail below, but at the moment I’m wondering if there really is any significance in thinking in terms of modules instead of just groups. It could be that Mazzola had reasons for thinking in terms of modules for earlier theories, but with more recent generalizations (in terms of presheaves) the module formalism is not totally necessary. I’ve searched for every instance of the word “ring” in Chapter 6 of ToM, where he justifies the use of modules, but I don’t see any clear role that the ring is playing. But apparently according to this review of Geometrie der Töne, an earlier book by Mazzola, the ring structure of modules plays a role for instance in deriving intervals of equal-tempered scales. I don’t have much to say about this though because I haven’t read Geometrie der Töne. So to refer back to my mention of $\{ 1, 5, 7, 11 \}$ in #4 as a subset of the ring $\mathbb{Z}$ that acts on $\mathbb{Z}_{12}$, this is not how such transformations are usually expressed. Instead they are expressed as group automorphisms of $\mathbb{Z}_{12}$. So the 48-element group acting on $\mathbb{Z}_{12}$ that you mentioned in #9 is a subset of $Aff(\mathbb{Z}_{12}, \mathbb{Z}_{12})$.

    Unfortunately finitely generated abelian groups won’t work, since e.g. a frequency value would be an element of $\mathbb{R}$. Regarding your mention of Grothendieck universes, I am less familiar with this topic, but am reading about it now. It seems that it provides a trick to make a large set into a small set, so that e.g. the set of $U$-small abelian groups is small? If so, it does seem crazy to think that this wouldn’t suffice for music theory foundations. However there are some “circular” constructions in Mazzola, which are like sets that contain themselves, except instead of sets they are the module presheaves. His justifications for this are speculative, but there do seem to be quite a few applications in music theory. For instance, there is a circular presheaf whose “elements” are rhythm trees, a rhythm tree being a specific method for representing rhythms. These are discussed in Chapter 51 of ToM. I’m not sure though how these circular presheaf constructions relate to the size problems, but from the set theory view they would violate well-foundedness (which I know is a relation that Mazzola is critical of, in the spirit of Paul Finsler).

14. • CommentRowNumber14.
    • CommentAuthorTodd_Trimble
    • CommentTimeJul 12th 2023
    • PermaLink
    Author: Todd_Trimble
    Format: MarkdownItexSo if I understand correctly, the objects of what we'd been calling $Mod$ are pairs $(R, M)$ where $R$ is a ring and $M$ is an $R$-module. This is a well-known construction, and indeed one sees frequently $Mod$ as the notation for the category where the morphisms are what he is calling "dilinear" maps. I've seen that notation used by Street, for instance. Are there papers by Mazzola on mathematical music theory in peer-reviewed mathematical journals? --- Let me try saying again what is going on with "$U$-small". In traditional set theory terms, it means we assume the existence of a cardinal $\kappa$ large enough so that the collection of sets whose size (more precisely, whose rank) is less than $\kappa$ forms a model of whatever set theory one is working with, say ZFC. What is required for this is two things: (1) if $\alpha$ is a cardinal less than $\kappa$, then also $2^\alpha \lt \kappa$, and (2) given a collection of cardinals $\{\alpha_i: \alpha_i \lt \kappa\}$ where the collection itself has fewer than $\kappa$ elements, the supremum of the collection is also less than $\kappa$. We call such a $\kappa$ "strongly inaccessible". (Assume also that $\aleph_0 \lt \kappa$.) If $V$ is a model of ZFC that has such a $\kappa$, then the "universe" of sets of rank less than $\kappa$, denoted $V_\kappa$, is also a model of ZFC. In other words, $V_\kappa$ is closed under any set-theoretic operations you care to name. Given such a $\kappa$, one may call the sets in $V_\kappa$ "$\kappa$-small". In this framework, some sets are small, and others are not; for example, $V_\kappa$ itself isn't ($\kappa$-)small, but it's still considered a set (i.e., an element of $V$). (For a set theorist, this assumption is a very mild "large cardinal hypothesis"; set theorists routinely consider much, much stronger large cardinal hypotheses without batting an eye.) This foundational set-up of "one universe" is the one adopted in Categories for the Working Mathematician, going back to the first edition. It allows one to speak of functor categories that one would like to have, in a set-theoretically respectable way. If $C$ and $D$ are categories whose object collections are (not necessarily small) sets, then the usual functor category $C^D$ is again a category whose object collection is again a set. So if $set$ is the category of all small sets (= elements of $V_\kappa$), its object collection is a set (not a small set of course), and one can form functor categories like $set^{set}$ and $set^{\left(set^{set}\right)}$ as legitimate set-theoretic constructions. All this may sound similar to the set/class distinction, but it's somewhat better than that. Particularly, with the one universe assumption, the category $Set$ of *all* sets (meaning elements of the given model $V$) can be proven to be a topos; you can't say in ordinary ZFC that the category of classes (which are really linguistic entities, not objects of the formal language) and functions between them form a topos. And under "one universe", $Set^{set^{op}}$ is also a topos. Similarly, if $aff$ is the evident category whose objects are pairs $(R, M)$ where $M$ is a small module over a small ring $R$, with affine maps as morphisms, then you still get a topos $Set^{aff^{op}}$. All this may take some getting used to, but what I found helpful in learning this stuff is to think of $V_\kappa$ as connoting a mild generalization of $V_\omega$ ("hereditarily finite sets"), in that finite sets are also closed under any set-theoretic operations you care to name, except that you don't have infinite sets like $\omega$ or $\aleph_0$ belonging to $V_\omega$. In other words, taking $\kappa$ to be $\omega$, assumptions (1) and (2) above are satisfied. So you just pump up $\omega$ to another cardinal $\kappa$ farther down the line that still satisfies (1) and (2) and so that $\omega \lt \kappa$. If anyone reading this found the discussion above confusing, then reread the discussion but mentally take $\kappa$ to be $\omega$, verify that everything should work out as advertised just fine in that case, and then relax knowing it works the same for any strongly inaccessible $\kappa$. --- Non-well-founded sets are perfectly fine to consider, but either way, with foundation or with anti-foundation, there are usually workarounds that allow constructions for whatever you want to do, including (I will boldly venture to guess) the "circular constructions" of Mazzola like the rhythm trees. I think a healthy way to consider the situation is this: the conception of well-founded sets and the cumulative hierarchy was invented to have a convenient theory of sets based solidly on (transfinite) induction and recursion. That's what traditional set theory is (to me): a gigantic elaboration of the idea of recursion. Similarly, if I understand correctly, the ill-founded sets explored by Aczel and others are good for embodying *co*recursion. But in either case, the axiomatics are founded on properties of the global membership relation $\in$, where one futzes around wondering or worrying about the existence of infinite $\in$-chains like $\ldots \in x_{n+1} \in x_n \in \ldots \in x_1 \in x_0$. Starting with Lawvere, category theorists have tended to view such considerations as spurious pseudo-problems, and category theorists have explored other options ([[structural set theory]]) that cut the Gordian knot by denying the primacy of $\in$ and the axiomatics based around that, with the understanding that mostly what mathematicians want to do with sets can be expressed directly in categorical terms, and basing their axiomatics around that instead. Anyway, I think this circularity phenomenon, however it is manifested in Mazzola's work, is probably going to be in a direction somewhat different from size considerations. I have a sneaking suspicion that the rhythm trees (whatever they are!) will turn out to be elements of a [[terminal coalgebra for an endofunctor]], which is bound up with corecursion, because generally speaking, trees are like that. It would be cool to uncover that, if true.

    So if I understand correctly, the objects of what we’d been calling $Mod$ are pairs $(R, M)$ where $R$ is a ring and $M$ is an $R$-module. This is a well-known construction, and indeed one sees frequently $Mod$ as the notation for the category where the morphisms are what he is calling “dilinear” maps. I’ve seen that notation used by Street, for instance.

    Are there papers by Mazzola on mathematical music theory in peer-reviewed mathematical journals?

    ---

    Let me try saying again what is going on with “$U$-small”. In traditional set theory terms, it means we assume the existence of a cardinal $\kappa$ large enough so that the collection of sets whose size (more precisely, whose rank) is less than $\kappa$ forms a model of whatever set theory one is working with, say ZFC. What is required for this is two things: (1) if $\alpha$ is a cardinal less than $\kappa$, then also $2^\alpha \lt \kappa$, and (2) given a collection of cardinals $\{\alpha_i: \alpha_i \lt \kappa\}$ where the collection itself has fewer than $\kappa$ elements, the supremum of the collection is also less than $\kappa$. We call such a $\kappa$ “strongly inaccessible”. (Assume also that $\aleph_0 \lt \kappa$.) If $V$ is a model of ZFC that has such a $\kappa$, then the “universe” of sets of rank less than $\kappa$, denoted $V_\kappa$, is also a model of ZFC. In other words, $V_\kappa$ is closed under any set-theoretic operations you care to name. Given such a $\kappa$, one may call the sets in $V_\kappa$ “$\kappa$-small”. In this framework, some sets are small, and others are not; for example, $V_\kappa$ itself isn’t ($\kappa$-)small, but it’s still considered a set (i.e., an element of $V$).

    (For a set theorist, this assumption is a very mild “large cardinal hypothesis”; set theorists routinely consider much, much stronger large cardinal hypotheses without batting an eye.)

    This foundational set-up of “one universe” is the one adopted in Categories for the Working Mathematician, going back to the first edition. It allows one to speak of functor categories that one would like to have, in a set-theoretically respectable way. If $C$ and $D$ are categories whose object collections are (not necessarily small) sets, then the usual functor category $C^D$ is again a category whose object collection is again a set. So if $set$ is the category of all small sets (= elements of $V_\kappa$), its object collection is a set (not a small set of course), and one can form functor categories like $set^{set}$ and $set^{\left(set^{set}\right)}$ as legitimate set-theoretic constructions.

    All this may sound similar to the set/class distinction, but it’s somewhat better than that. Particularly, with the one universe assumption, the category $Set$ of all sets (meaning elements of the given model $V$) can be proven to be a topos; you can’t say in ordinary ZFC that the category of classes (which are really linguistic entities, not objects of the formal language) and functions between them form a topos. And under “one universe”, $Set^{set^{op}}$ is also a topos. Similarly, if $aff$ is the evident category whose objects are pairs $(R, M)$ where $M$ is a small module over a small ring $R$, with affine maps as morphisms, then you still get a topos $Set^{aff^{op}}$.

    All this may take some getting used to, but what I found helpful in learning this stuff is to think of $V_\kappa$ as connoting a mild generalization of $V_\omega$ (“hereditarily finite sets”), in that finite sets are also closed under any set-theoretic operations you care to name, except that you don’t have infinite sets like $\omega$ or $\aleph_0$ belonging to $V_\omega$. In other words, taking $\kappa$ to be $\omega$, assumptions (1) and (2) above are satisfied. So you just pump up $\omega$ to another cardinal $\kappa$ farther down the line that still satisfies (1) and (2) and so that $\omega \lt \kappa$. If anyone reading this found the discussion above confusing, then reread the discussion but mentally take $\kappa$ to be $\omega$, verify that everything should work out as advertised just fine in that case, and then relax knowing it works the same for any strongly inaccessible $\kappa$.

    ---

    Non-well-founded sets are perfectly fine to consider, but either way, with foundation or with anti-foundation, there are usually workarounds that allow constructions for whatever you want to do, including (I will boldly venture to guess) the “circular constructions” of Mazzola like the rhythm trees. I think a healthy way to consider the situation is this: the conception of well-founded sets and the cumulative hierarchy was invented to have a convenient theory of sets based solidly on (transfinite) induction and recursion. That’s what traditional set theory is (to me): a gigantic elaboration of the idea of recursion. Similarly, if I understand correctly, the ill-founded sets explored by Aczel and others are good for embodying corecursion. But in either case, the axiomatics are founded on properties of the global membership relation $\in$, where one futzes around wondering or worrying about the existence of infinite $\in$-chains like $\ldots \in x_{n+1} \in x_n \in \ldots \in x_1 \in x_0$. Starting with Lawvere, category theorists have tended to view such considerations as spurious pseudo-problems, and category theorists have explored other options (structural set theory) that cut the Gordian knot by denying the primacy of $\in$ and the axiomatics based around that, with the understanding that mostly what mathematicians want to do with sets can be expressed directly in categorical terms, and basing their axiomatics around that instead.

    Anyway, I think this circularity phenomenon, however it is manifested in Mazzola’s work, is probably going to be in a direction somewhat different from size considerations. I have a sneaking suspicion that the rhythm trees (whatever they are!) will turn out to be elements of a terminal coalgebra for an endofunctor, which is bound up with corecursion, because generally speaking, trees are like that. It would be cool to uncover that, if true.

15. • CommentRowNumber15.
    • CommentAuthorDrew Flieder
    • CommentTimeJul 12th 2023
    • PermaLink
    Author: Drew Flieder
    Format: MarkdownItexThanks for all this! On your first paragraph, that is right. Objects in $Mod$ are pairs $(R, M)$ where $R$ is a ring and $M$ an $R$-module. It does seem to make more sense though to call the category $Aff$ instead of $Mod$, since in general the maps $(R, M) \rightarrow (S, N)$ of interest are the di*affine* maps, and not just the di*linear* ones. Mazzola's more recent work is mainly in journals specifically for mathematical music theory. He started his career as an algebraic geometer however, so his earlier work is in pure mathematics and is published in peer-reviewed mathematical journals. --- Your discussion on the size problems is clarifying. Admittedly I've always found treatments of size problems to be bewildering, maybe because it's hard for me to wrap my head around what's really at stake. I understand the problems with Russell's paradox, but it seems like so much blood has been shed over what seems to be (dare I say) a bureaucratic issue. Forgive me if this is arrogant. This reminds me though, have you read [[small presheaf]]? I wonder if some of what you discussed in \#14 could be brought into connection with this. When I had encountered the problem of trying to get categories of presheaves over large categories, I had looked around a lot to try to alleviate my worries. I found the small presheaf page to be somewhat therapeutic. --- Thank you also for pointing me to the [[terminal coalgebra for an endofunctor]] page! I'm going to look into that and see if rhythm trees can be expressed that way. I know that one of the reasons that Mazzola's formalism is useful is the following. Let $On \times Dur$ be the presheaf the elements of which are pairs $(o, d)$ where $o$ is an onset time and $d$ a duration. A rhythm can then be thought of as a subobject $R \hookrightarrow On \times Dur$. I would say that this is a more "concrete", or maybe "absolute", representation of a rhythm. Then if you have the presheaf $RT$ consisting of all rhythm trees, you can convert rhythm tree representations to subobjects of $On \times Dur$ via a map \[ f : RT \rightarrow \Omega^{On \times Dur} \] (where $\Omega$ is the subobject classifier). This is useful in seeing how properties of rhythm trees correspond to properties of "absolute" rhythms, which would be of interest in both theory and (musical) composition. Anyway, I'm curious to see if the coalgebra formalism works for rhythm trees, so I'm going to look into that.

    Thanks for all this!

    On your first paragraph, that is right. Objects in $Mod$ are pairs $(R, M)$ where $R$ is a ring and $M$ an $R$-module. It does seem to make more sense though to call the category $Aff$ instead of $Mod$, since in general the maps $(R, M) \rightarrow (S, N)$ of interest are the diaffine maps, and not just the dilinear ones.

    Mazzola’s more recent work is mainly in journals specifically for mathematical music theory. He started his career as an algebraic geometer however, so his earlier work is in pure mathematics and is published in peer-reviewed mathematical journals.

    ---

    Your discussion on the size problems is clarifying. Admittedly I’ve always found treatments of size problems to be bewildering, maybe because it’s hard for me to wrap my head around what’s really at stake. I understand the problems with Russell’s paradox, but it seems like so much blood has been shed over what seems to be (dare I say) a bureaucratic issue. Forgive me if this is arrogant.

    This reminds me though, have you read small presheaf? I wonder if some of what you discussed in #14 could be brought into connection with this. When I had encountered the problem of trying to get categories of presheaves over large categories, I had looked around a lot to try to alleviate my worries. I found the small presheaf page to be somewhat therapeutic.

    ---

    Thank you also for pointing me to the terminal coalgebra for an endofunctor page! I’m going to look into that and see if rhythm trees can be expressed that way. I know that one of the reasons that Mazzola’s formalism is useful is the following. Let $On \times Dur$ be the presheaf the elements of which are pairs $(o, d)$ where $o$ is an onset time and $d$ a duration. A rhythm can then be thought of as a subobject $R \hookrightarrow On \times Dur$. I would say that this is a more “concrete”, or maybe “absolute”, representation of a rhythm. Then if you have the presheaf $RT$ consisting of all rhythm trees, you can convert rhythm tree representations to subobjects of $On \times Dur$ via a map

    $$f : RT \rightarrow \Omega^{On \times Dur}$$

    (where $\Omega$ is the subobject classifier). This is useful in seeing how properties of rhythm trees correspond to properties of “absolute” rhythms, which would be of interest in both theory and (musical) composition.

    Anyway, I’m curious to see if the coalgebra formalism works for rhythm trees, so I’m going to look into that.

16. • CommentRowNumber16.
    • CommentAuthorTodd_Trimble
    • CommentTimeJul 12th 2023
    • (edited Jul 12th 2023)
    • PermaLink
    Author: Todd_Trimble
    Format: MarkdownItexI know some things about small presheaves (= colimits of small diagrams of representables). It's a slightly tricky business. For example, the small-cocompletion of a large discrete category has no terminal object! So any general statements one would like to make will have to be tested against such weird and annoying examples. The Day-Lack paper is useful, though. [Somewhere on my personal (publicly viewable) nLab page I have some material on small cocompletions, where I had wanted to investigate properties of the initial algebra of the 2-functor that sends a large category to its small cocompletion, so much of what I know of the subject came about by thinking about that problem. I think I had convinced myself at one point that that initial algebra wouldn't be a topos, but it would come damned close, being a $\Pi$W-pretopos or something. Not that I claim that any of this is important for you to know, but I may have listed some useful references there.] I don't think your attitude is arrogant; in fact, I think almost all categorists have felt somewhat hampered or annoyed by all this fuss, at one time or another. I would warn anyone, however, about the danger of being too cavalier about it. :-) Insofar as the set-up of #14 allows us to think of the morphisms of $aff$ as forming a set, the small-cocompletion of (= category of "small presheaves" on) $aff$ coincides with the usual category of presheaves, $Set^{aff^{op}}$, and as I said this will be a topos, with all the niceness that implies. I don't have too much more to say at the moment about the relation of my comment to the page [[small presheaf]], but as we've already discussed, I do get a sense that this might mean this set-up provides a viable or pragmatic approach to setting up at least one foundations for the Mazzola theory. (That of course may be premature to say, since I'm only learning a little now of this theory through this conversation.) Tell me sometime about what $Dur$ is, and better yet, how you or Mazzola would describe these rhythm trees. What sort of subobjects $R \hookrightarrow On \times Dur$ are we talking about? By the way, is there a doi for his book?

    I know some things about small presheaves (= colimits of small diagrams of representables). It’s a slightly tricky business. For example, the small-cocompletion of a large discrete category has no terminal object! So any general statements one would like to make will have to be tested against such weird and annoying examples. The Day-Lack paper is useful, though. [Somewhere on my personal (publicly viewable) nLab page I have some material on small cocompletions, where I had wanted to investigate properties of the initial algebra of the 2-functor that sends a large category to its small cocompletion, so much of what I know of the subject came about by thinking about that problem. I think I had convinced myself at one point that that initial algebra wouldn’t be a topos, but it would come damned close, being a $\Pi$W-pretopos or something. Not that I claim that any of this is important for you to know, but I may have listed some useful references there.]

    I don’t think your attitude is arrogant; in fact, I think almost all categorists have felt somewhat hampered or annoyed by all this fuss, at one time or another. I would warn anyone, however, about the danger of being too cavalier about it. :-)

    Insofar as the set-up of #14 allows us to think of the morphisms of $aff$ as forming a set, the small-cocompletion of (= category of “small presheaves” on) $aff$ coincides with the usual category of presheaves, $Set^{aff^{op}}$, and as I said this will be a topos, with all the niceness that implies. I don’t have too much more to say at the moment about the relation of my comment to the page small presheaf, but as we’ve already discussed, I do get a sense that this might mean this set-up provides a viable or pragmatic approach to setting up at least one foundations for the Mazzola theory. (That of course may be premature to say, since I’m only learning a little now of this theory through this conversation.)

    Tell me sometime about what $Dur$ is, and better yet, how you or Mazzola would describe these rhythm trees. What sort of subobjects $R \hookrightarrow On \times Dur$ are we talking about?

    By the way, is there a doi for his book?

17. • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeJul 13th 2023
    • PermaLink
    Author: Urs
    Format: MarkdownItex> By the way, is there a doi for his book? I had added the doi for "The topos of music" to the entry [here](https://ncatlab.org/nlab/show/pitch-class#Mazzola12). One finds DOI numbers by googling for the book title, finding the publisher page among the search results, and searching that for "doi" (it's typically at the top or at the bottom of the page). For a book instead of an article, if you want to see the content go to [libgen.is](http://libgen.is) and search for author name/title. E.g. [here](http://library.lol/main/2A653013266C967B86C0CC4932139E95). $\,$ By the way, the entry *[[pitch-class]]* still remains weird without some discussion of the intended application. To see the point, if it needs amplifying: At the present stage of the entry we could, without logical loss, replace the 12 musical scales with the 12 daylight hours and claim with the same right that we are working in the "Topos of Time" which will "facilitate the organization of temporal concepts in a coherent setting". This would obviously be nonsense. But why is it not nonsense when we replace "Time" by "Music". What's the substance of the matter? This should be articulated in the entry.

    By the way, is there a doi for his book?

    I had added the doi for “The topos of music” to the entry here.

    One finds DOI numbers by googling for the book title, finding the publisher page among the search results, and searching that for “doi” (it’s typically at the top or at the bottom of the page).

    For a book instead of an article, if you want to see the content go to libgen.is and search for author name/title. E.g. here.

    $\,$

    By the way, the entry pitch-class still remains weird without some discussion of the intended application.

    To see the point, if it needs amplifying: At the present stage of the entry we could, without logical loss, replace the 12 musical scales with the 12 daylight hours and claim with the same right that we are working in the “Topos of Time” which will “facilitate the organization of temporal concepts in a coherent setting”. This would obviously be nonsense. But why is it not nonsense when we replace “Time” by “Music”. What’s the substance of the matter? This should be articulated in the entry.

18. • CommentRowNumber18.
    • CommentAuthorDrew Flieder
    • CommentTimeJul 14th 2023
    • PermaLink
    Author: Drew Flieder
    Format: MarkdownItexUrs: Yes I see your point, apologies for not addressing it earlier. If there is a pitch-class entry it should be once there’s some more entries that explain the more foundational music theory concepts of Mazzola and others working in that area. But even then the pitch-class concept can probably be brought into discussion in another post, without being the sole focus. If you think the entry should be removed (at least for the time-being) that makes sense. Although would that remove this thread? That would be too bad. If that were the case, I could rename the pitch-class entry to something like "form and denotator theory", remove the content written there now, and just write a preliminary introduction to Mazzola's form and denotator theories. Then I can add to it when I organize these ideas in a more streamlined setting. What do you think?

    Urs: Yes I see your point, apologies for not addressing it earlier. If there is a pitch-class entry it should be once there’s some more entries that explain the more foundational music theory concepts of Mazzola and others working in that area. But even then the pitch-class concept can probably be brought into discussion in another post, without being the sole focus.

    If you think the entry should be removed (at least for the time-being) that makes sense. Although would that remove this thread? That would be too bad. If that were the case, I could rename the pitch-class entry to something like “form and denotator theory”, remove the content written there now, and just write a preliminary introduction to Mazzola’s form and denotator theories. Then I can add to it when I organize these ideas in a more streamlined setting. What do you think?

19. • CommentRowNumber19.
    • CommentAuthorTodd_Trimble
    • CommentTimeJul 14th 2023
    • (edited Jul 14th 2023)
    • PermaLink
    Author: Todd_Trimble
    Format: MarkdownItexHi, Drew. No, nForum threads remain, independent of what happens at the nLab. It occurred to me that [[pitch class]] might also include ideas besides Mazzola's, which is a formidable formal framework. It's nice if an nLab article can include at least one nontrivial result alongside formal definitions. John mentioned to me the other day a statement, which I've not tried to sit down to think through, that musical "triads" form a torsor over the semidirect product $\mathbb{Z}_{12} \rtimes \{1, 11\}$. If you know what he meant by this, then that could be worth developing at the nLab. If this music theory insight could be extended to incorporate the elements 5, 7 as well, then certainly that also could be interesting to some readers. John asked off-handedly whether there could be some sort of "spinorial" significance to this.

    Hi, Drew. No, nForum threads remain, independent of what happens at the nLab.

    It occurred to me that pitch class might also include ideas besides Mazzola’s, which is a formidable formal framework. It’s nice if an nLab article can include at least one nontrivial result alongside formal definitions. John mentioned to me the other day a statement, which I’ve not tried to sit down to think through, that musical “triads” form a torsor over the semidirect product $\mathbb{Z}_{12} \rtimes \{1, 11\}$. If you know what he meant by this, then that could be worth developing at the nLab. If this music theory insight could be extended to incorporate the elements 5, 7 as well, then certainly that also could be interesting to some readers. John asked off-handedly whether there could be some sort of “spinorial” significance to this.

20. • CommentRowNumber20.
    • CommentAuthorDrew Flieder
    • CommentTimeJul 14th 2023
    • (edited Jul 14th 2023)
    • PermaLink
    Author: Drew Flieder
    Format: MarkdownItexSorry, forgot to say what $Dur$ is in \#15 explicitly. It's also the presheaf $aff(-, \mathbb{R})$, as is $On$. The difference is just in the name, which is used for "semiotic" purposes, i.e. when we speak of elements in $On$ we think of onset times, and when we speak of elements in $Dur$ we think of durations. But for simplicity let me just call the product $\mathbb{R} \times \mathbb{R}$, and what I mean by this is the product of the presheaves $aff(-, \mathbb{R}) \times aff(-, \mathbb{R})$. What I mean by a subobject $R \hookrightarrow \mathbb{R} \times \mathbb{R}$ is usually (almost always) just the intuitive idea of a subset of $\mathbb{R} \times \mathbb{R}$ when treated as sets. I'll show how to derive such subobjects a couple paragraphs later, but first some preliminary information. **Preliminary Information**. For a functor $F$ in $Set^{aff^{op}}$ and a module $M$ in $aff$, then $F(M)$ is of course a set. (Mazzola uses the notation $M$[at symbol]$F$, but I'll stick to $F(M)$.) For instance, if $F$ is (say) the representable functor $aff(-, N)$, then $F(M) = aff(M, N)$. But in general $F$ need not be representable. Ok, so to get something like "subsets of $\mathbb{R} \times \mathbb{R}$", we can start by defining a "naive" subobject classifier $2$ in the following way. For a functor $F$ in $Set^{aff^{op}}$, we have the functor $2^F : aff \rightarrow Set$ defined on the modules $M$ by $2^F(M) \coloneqq 2^{F(M)}$. For instance if $F$ is the representable functor of $N$, then $2^F(0)$ will correspond to the subsets of $N$. For a case where $F$ is not one of the representable functors, say $F \coloneqq aff(-, \mathbb{Z}) \times aff(-, \mathbb{R})$, then to get subsets of $\mathbb{Z} \times \mathbb{R}$ we first get \[ F(0) \coloneqq \big\{ (z, r) \ \big\vert \ z \in aff(0, \mathbb{Z}) \wedge r \in aff(0, \mathbb{R}) \big\}, \] and then $2^F(0)$ corresponds to the subsets of $\mathbb{Z} \times \mathbb{R}$. So this is all we really need to think of a rhythm as an element $R \in 2^{\mathbb{R} \times \mathbb{R}}$, since it gives a set of onset-duration pairs. However, this naive subobject classifier $2$ can be related to the actual subobject classifier $\Omega$ in a canonical way. This is kind of cumbersome and maybe not so necessary for this discussion, but in case anyone's interested I'll write it out in the following paragraph. For a diaffine map $f : N \rightarrow M$ in $aff$, and again $F$ a functor in $Set^{aff^{op}}$, we use $F(f) : F(M) \rightarrow F(N)$ to derive the map \[ 2^F(f) : 2^F(M) \rightarrow 2^F(N) \] that sends a subset $S \subset F(M)$ to \[ S \circ f \coloneqq \big\{ s \circ f \in F(N) \ \big\vert \ s \in S \big\} \subset F(N). \] Now note that $\Omega^F$ evaluates to $\Omega^F(M) \cong S u b( aff(-, M) \times F )$ at the module $M$. So to connect the naive subobject classifier $2$ to the actual one $\Omega$, we define a natural transformation \[ \eta : 2^F \rightarrow \Omega^F \] that sends an $S \in 2^F(M)$ to $\hat{S} \in \Omega^F(M)$ defined at a module $N$ by \[ \hat{S}(N) \coloneqq \{ (a, b) \in aff(N, M) \times F(N) \ \big\vert \ b \in S \circ a \}. \] --- Ok, now on to rhythm trees. A rhythm tree is a way to encode a rhythm, which is used in the computer assisted composition programming environment [OpenMusic](http://repmus.ircam.fr/openmusic/home). I'll first talk about these informally, since they are not so crazy. A rhythm tree is a pair $R = (D, S)$ where $D \in \mathbb{Q}$ is a duration value and $S$ is a list, where each element in $S$ is either an integer or a rhythm tree. (Hence the definition of a rhythm tree is circular.) An easy example is $R = (4, (1, 1, 1, 1))$. The first coordinate $4$ expresses that the rhythm lasts 4 beats, and the list $(1, 1, 1, 1)$ expresses that these 4 beats are broken up into four segments of equal length. So $R$ is just a series of four quarter notes. (It would be easier for me to post pictures of these rhythms, but I'm not sure if that's possible on the forum, as I couldn't find any information on this.) However, since the elements in the $S$ coordinate can also be rhythm trees, we can define a rhythm tree like \[ (4, (1, 1, (2, (1, 1, 1)))), \] which consists of 2 quarter notes followed by 3 quarter-note triplets. This is because again we have 4 beats, but broken up into proportions of 1, 1, and 2, and the 2 itself is broken up into 3 segments of equal length. You can also picture this as a tree graph, where 4 is the root and it branches into (from left to right) 1, 1, and 2, and the 2 branches into 1, 1, 1. Anyway there is theoretically no limit to the nesting, so these trees can have infinite depth. Now I'm looking at Chapter 51 of ToM, where Mazzola defines rhythm trees as elements of a module presheaf, and I realize that his formalism is incorrect. So I worked out the correction which is what I'll now present. Since the circularity gets confusing (at least to me), I'm going to start out with some small steps. Given a presheaf $F$, to define lists of $F$ you can define the presheaf \[ List(F) \coloneqq Item(F) \amalg 0, \] where $Item(F)$ is defined as \[ Item(F) \coloneqq F \times List(F), \] and hence these are circular. The $0$ in the definition of $List(F)$ can be thought of as $aff(-, 0)$, so elements in $List(F)$ give a choice between taking a new item of $F$ or terminating the construction of the list. Then $Item(F)$ is a pair consisting of an element of $F$ and an element of $List(F)$, and again the latter allows you to choose a new element or terminate the list. So we can boil down the definition of a list object over $F$ as \[ List(F) \coloneqq \big( F \times List(F) \big) \amalg 0. \] Now we can define rhythm trees, which uses the list construction. The list construction is used to for the $S$ coordinate in $(D, S)$, since $S$ is a list whose elements are either integers or rhythm trees. I'll define the rhythm tree presheaf $RT$ in a moment, but first, the presheaf from which such $S$ are taken is \[ List(\mathbb{Z} \amalg RT) \coloneqq \big( ( \mathbb{Z} \amalg RT ) \times List(\mathbb{Z} \amalg RT) \big) \amalg 0 \] (where $\mathbb{Z}$ is shorthand for $aff(-, \mathbb{Z})$). Now define $DValue \coloneqq aff(-, \mathbb{Q})$. Then we define the rhythm tree presheaf as \[ RT \coloneqq DValue \times List(\mathbb{Z} \amalg RT). \] The circularity here is bewildering, but such is the nature of rhythm trees. Regarding the mapping I mentioned \[ f \colon RT \rightarrow \Omega^{On \times Dur} \] from \#15, conceiving of the general mapping rule is hard to wrap my head around, since the elements of the domain are so non-uniform. The description of such a map is probably wordy and confusing. But given any particular element $r \in RT$, it is easy to compute $f(r)$.

    Sorry, forgot to say what $Dur$ is in #15 explicitly. It’s also the presheaf $aff(-, \mathbb{R})$, as is $On$. The difference is just in the name, which is used for “semiotic” purposes, i.e. when we speak of elements in $On$ we think of onset times, and when we speak of elements in $Dur$ we think of durations. But for simplicity let me just call the product $\mathbb{R} \times \mathbb{R}$, and what I mean by this is the product of the presheaves $aff(-, \mathbb{R}) \times aff(-, \mathbb{R})$.

    What I mean by a subobject $R \hookrightarrow \mathbb{R} \times \mathbb{R}$ is usually (almost always) just the intuitive idea of a subset of $\mathbb{R} \times \mathbb{R}$ when treated as sets. I’ll show how to derive such subobjects a couple paragraphs later, but first some preliminary information.

    Preliminary Information. For a functor $F$ in $Set^{aff^{op}}$ and a module $M$ in $aff$, then $F(M)$ is of course a set. (Mazzola uses the notation $M$[at symbol]$F$, but I’ll stick to $F(M)$.) For instance, if $F$ is (say) the representable functor $aff(-, N)$, then $F(M) = aff(M, N)$. But in general $F$ need not be representable.

    Ok, so to get something like “subsets of $\mathbb{R} \times \mathbb{R}$”, we can start by defining a “naive” subobject classifier $2$ in the following way. For a functor $F$ in $Set^{aff^{op}}$, we have the functor $2^F : aff \rightarrow Set$ defined on the modules $M$ by $2^F(M) \coloneqq 2^{F(M)}$. For instance if $F$ is the representable functor of $N$, then $2^F(0)$ will correspond to the subsets of $N$. For a case where $F$ is not one of the representable functors, say $F \coloneqq aff(-, \mathbb{Z}) \times aff(-, \mathbb{R})$, then to get subsets of $\mathbb{Z} \times \mathbb{R}$ we first get

    $$F(0) \coloneqq \big\{ (z, r) \ \big\vert \ z \in aff(0, \mathbb{Z}) \wedge r \in aff(0, \mathbb{R}) \big\},$$

    and then $2^F(0)$ corresponds to the subsets of $\mathbb{Z} \times \mathbb{R}$. So this is all we really need to think of a rhythm as an element $R \in 2^{\mathbb{R} \times \mathbb{R}}$, since it gives a set of onset-duration pairs.

    However, this naive subobject classifier $2$ can be related to the actual subobject classifier $\Omega$ in a canonical way. This is kind of cumbersome and maybe not so necessary for this discussion, but in case anyone’s interested I’ll write it out in the following paragraph.

    For a diaffine map $f : N \rightarrow M$ in $aff$, and again $F$ a functor in $Set^{aff^{op}}$, we use $F(f) : F(M) \rightarrow F(N)$ to derive the map

    $$2^F(f) : 2^F(M) \rightarrow 2^F(N)$$

    that sends a subset $S \subset F(M)$ to

    $$S \circ f \coloneqq \big\{ s \circ f \in F(N) \ \big\vert \ s \in S \big\} \subset F(N).$$

    Now note that $\Omega^F$ evaluates to $\Omega^F(M) \cong S u b( aff(-, M) \times F )$ at the module $M$. So to connect the naive subobject classifier $2$ to the actual one $\Omega$, we define a natural transformation

    $$\eta : 2^F \rightarrow \Omega^F$$

    that sends an $S \in 2^F(M)$ to $\hat{S} \in \Omega^F(M)$ defined at a module $N$ by

    $$\hat{S}(N) \coloneqq \{ (a, b) \in aff(N, M) \times F(N) \ \big\vert \ b \in S \circ a \}.$$

    ---

    Ok, now on to rhythm trees. A rhythm tree is a way to encode a rhythm, which is used in the computer assisted composition programming environment OpenMusic. I’ll first talk about these informally, since they are not so crazy. A rhythm tree is a pair $R = (D, S)$ where $D \in \mathbb{Q}$ is a duration value and $S$ is a list, where each element in $S$ is either an integer or a rhythm tree. (Hence the definition of a rhythm tree is circular.) An easy example is $R = (4, (1, 1, 1, 1))$. The first coordinate $4$ expresses that the rhythm lasts 4 beats, and the list $(1, 1, 1, 1)$ expresses that these 4 beats are broken up into four segments of equal length. So $R$ is just a series of four quarter notes. (It would be easier for me to post pictures of these rhythms, but I’m not sure if that’s possible on the forum, as I couldn’t find any information on this.)

    However, since the elements in the $S$ coordinate can also be rhythm trees, we can define a rhythm tree like

    $$(4, (1, 1, (2, (1, 1, 1)))),$$

    which consists of 2 quarter notes followed by 3 quarter-note triplets. This is because again we have 4 beats, but broken up into proportions of 1, 1, and 2, and the 2 itself is broken up into 3 segments of equal length. You can also picture this as a tree graph, where 4 is the root and it branches into (from left to right) 1, 1, and 2, and the 2 branches into 1, 1, 1. Anyway there is theoretically no limit to the nesting, so these trees can have infinite depth.

    Now I’m looking at Chapter 51 of ToM, where Mazzola defines rhythm trees as elements of a module presheaf, and I realize that his formalism is incorrect. So I worked out the correction which is what I’ll now present.

    Since the circularity gets confusing (at least to me), I’m going to start out with some small steps. Given a presheaf $F$, to define lists of $F$ you can define the presheaf

    $$List(F) \coloneqq Item(F) \amalg 0,$$

    where $Item(F)$ is defined as

    $$Item(F) \coloneqq F \times List(F),$$

    and hence these are circular. The $0$ in the definition of $List(F)$ can be thought of as $aff(-, 0)$, so elements in $List(F)$ give a choice between taking a new item of $F$ or terminating the construction of the list. Then $Item(F)$ is a pair consisting of an element of $F$ and an element of $List(F)$, and again the latter allows you to choose a new element or terminate the list. So we can boil down the definition of a list object over $F$ as

    $$List(F) \coloneqq \big( F \times List(F) \big) \amalg 0.$$

    Now we can define rhythm trees, which uses the list construction. The list construction is used to for the $S$ coordinate in $(D, S)$, since $S$ is a list whose elements are either integers or rhythm trees. I’ll define the rhythm tree presheaf $RT$ in a moment, but first, the presheaf from which such $S$ are taken is

    $$List(\mathbb{Z} \amalg RT) \coloneqq \big( ( \mathbb{Z} \amalg RT ) \times List(\mathbb{Z} \amalg RT) \big) \amalg 0$$

    (where $\mathbb{Z}$ is shorthand for $aff(-, \mathbb{Z})$). Now define $DValue \coloneqq aff(-, \mathbb{Q})$. Then we define the rhythm tree presheaf as

    $$RT \coloneqq DValue \times List(\mathbb{Z} \amalg RT).$$

    The circularity here is bewildering, but such is the nature of rhythm trees.

    Regarding the mapping I mentioned

    $$f \colon RT \rightarrow \Omega^{On \times Dur}$$

    from #15, conceiving of the general mapping rule is hard to wrap my head around, since the elements of the domain are so non-uniform. The description of such a map is probably wordy and confusing. But given any particular element $r \in RT$, it is easy to compute $f(r)$.

21. • CommentRowNumber21.
    • CommentAuthorMadeleine Birchfield
    • CommentTimeJul 14th 2023
    • (edited Jul 14th 2023)
    • PermaLink
    Author: Madeleine Birchfield
    Format: MarkdownItexI wonder if there is a [[coinductive]] definition of the rhythm trees. If so that might clarify/resolve the circularity. Edit: I think one could simply define the endofunctor $\mathrm{List}(-)$ on presheaves such that every presheaf $F$ comes with the structure of an isomorphism $i(F):\mathrm{List}(F) \cong (F \times \mathrm{List}(F)) \coprod 0$.

    I wonder if there is a coinductive definition of the rhythm trees. If so that might clarify/resolve the circularity.

    Edit: I think one could simply define the endofunctor $\mathrm{List}(-)$ on presheaves such that every presheaf $F$ comes with the structure of an isomorphism $i(F):\mathrm{List}(F) \cong (F \times \mathrm{List}(F)) \coprod 0$.

22. • CommentRowNumber22.
    • CommentAuthorUrs
    • CommentTimeJul 14th 2023
    • PermaLink
    Author: Urs
    Format: MarkdownItexWhat I keep asking is to please add something to the entry to make it make sense. In its present form the entry sheds a bad light on everyone involved: The first three sections just say that a 12-periodic phenomenon is described by the cyclic group $\mathbb{Z}_{12}$. If that were in a single lead-in paragraph to something else, it might might sense. The fourth section currently *looks* crappottish, in saying that by an unspecified use of topos theory we "may denote" $\mathbb{Z}_{12}$ by some undefined and in any case bewildering notation. The fifth section doubles down on the tautologism in stating that one can "generalize" the previous discussion from $\mathbb{Z}_{12}$ to $\mathbb{Z}_n$. This again gives crackpot vibes. This leaves me baffled. I used to think there is something to topos-theoretic music theory, but this is making me suspect there is not. Why not add a sentence to the entry explaining what the deal is? So there is 12-periodicity. Okay. Now what? Just say it. Or just quote it from the literature.

    What I keep asking is to please add something to the entry to make it make sense. In its present form the entry sheds a bad light on everyone involved:

    The first three sections just say that a 12-periodic phenomenon is described by the cyclic group $\mathbb{Z}_{12}$. If that were in a single lead-in paragraph to something else, it might might sense.

    The fourth section currently looks crappottish, in saying that by an unspecified use of topos theory we “may denote” $\mathbb{Z}_{12}$ by some undefined and in any case bewildering notation.

    The fifth section doubles down on the tautologism in stating that one can “generalize” the previous discussion from $\mathbb{Z}_{12}$ to $\mathbb{Z}_n$. This again gives crackpot vibes.

    This leaves me baffled. I used to think there is something to topos-theoretic music theory, but this is making me suspect there is not.

    Why not add a sentence to the entry explaining what the deal is? So there is 12-periodicity. Okay. Now what? Just say it. Or just quote it from the literature.

23. • CommentRowNumber23.
    • CommentAuthorDrew Flieder
    • CommentTimeJul 15th 2023
    • PermaLink
    Author: Drew Flieder
    Format: MarkdownItexOk, fixing it now. I was not careful in presenting the motivation for topos theory in music. That would require much more exposition on my part. What I hope is a little clarifying is that the motivation for topos theory in music is that it provides the common space in which to talk about musical concepts *in general*. If one only wanted to talk about aspects of pitch-classes, for example, then bringing in topos theory would be absurd. But there are so many concepts in music, and the motivation for topos theory is to provide a general yet explicit format for dealing with such concepts.

    Ok, fixing it now.

    I was not careful in presenting the motivation for topos theory in music. That would require much more exposition on my part.

    What I hope is a little clarifying is that the motivation for topos theory in music is that it provides the common space in which to talk about musical concepts in general. If one only wanted to talk about aspects of pitch-classes, for example, then bringing in topos theory would be absurd. But there are so many concepts in music, and the motivation for topos theory is to provide a general yet explicit format for dealing with such concepts.

24. • CommentRowNumber24.
    • CommentAuthorDrew Flieder
    • CommentTimeJul 15th 2023
    • PermaLink
    Author: Drew Flieder
    Format: MarkdownItexRewritten with just an introduction. I provided some historical significance to the concept, with motivation for the use of integer notation. The sections on Mazzola have been removed, as there is at the moment no need to make use of his presheaf formalisms. <a href="https://ncatlab.org/nlab/revision/diff/pitch-class/4">diff</a>, <a href="https://ncatlab.org/nlab/revision/pitch-class/4">v4</a>, <a href="https://ncatlab.org/nlab/show/pitch-class">current</a>

    Rewritten with just an introduction. I provided some historical significance to the concept, with motivation for the use of integer notation.

    The sections on Mazzola have been removed, as there is at the moment no need to make use of his presheaf formalisms.

    diff, v4, current

25. • CommentRowNumber25.
    • CommentAuthorTodd_Trimble
    • CommentTimeJul 15th 2023
    • (edited Jul 15th 2023)
    • PermaLink
    Author: Todd_Trimble
    Format: MarkdownItexThe "circularity" that "defines" structures like $List(F)$ through such fixed-point equations is typically made precise through the concept of [[initial algebra of an endofunctor]] or "terminal coalgebra for an endofunctor". The trouble with relying on fixed-point equations (really isomorphisms, not equations) alone is that they will not uniquely define these structures. But initial algebras of endofunctors (for instance) do define structures uniquely (up to unique isomorphism). $List(F)$ is a paradigmatic case: it is the initial algebra for the endofunctor $\Phi_F$ that takes an object $X$ to $1 + F \times X$. (Same as your $0 \sqcup F \times X$ but in different notation.) An easy but quite wonderful theorem, due to Lambek, is that for any endofunctor $\Phi$ on a category, if an initial algebra for $\Phi$ exists, given by a morphism $\alpha: L \to \Phi(L)$, then $\alpha$ is an isomorphism (so that $L$ is indeed a "fixed point" of $\Phi$). The same goes for the dual notion of terminal coalgebra. For a presheaf topos or indeed a Grothendieck topos, initial algebras and I think also terminal coalgebras exist for any polynomial endofunctor. $List(F)$ is an example of this ($\Phi_F(X) = 1 + FX$ is polynomial. Actually, $List(-)$ itself is, according to the technical definition, polynomial as well! Without thinking about it too hard, I'll go out on a limb and conjecture that the object of rhythm trees can be described as either an initial algebra or terminal coalgebra of the endofunctor that takes $X$ to $DValue \times List(\mathbb{Z} + X)$. (Maybe the former; typically, initial algebras of (for example) polynomial endofunctors are embedded inside the terminal coalgebras, and consist of those (typically tree-like) elements of the terminal coalgebra that are "well-founded".) When Madeleine in #21 mentioned coinduction, that corresponds to taking a terminal coalgebra, as I also touched upon back in #14. The article [[continued fraction]] gives some other examples of initial algebras and terminal coalgebras, for anyone wishing to explore these notions in a classical scenario dating back centuries. :-)

    The “circularity” that “defines” structures like $List(F)$ through such fixed-point equations is typically made precise through the concept of initial algebra of an endofunctor or “terminal coalgebra for an endofunctor”. The trouble with relying on fixed-point equations (really isomorphisms, not equations) alone is that they will not uniquely define these structures. But initial algebras of endofunctors (for instance) do define structures uniquely (up to unique isomorphism).

    $List(F)$ is a paradigmatic case: it is the initial algebra for the endofunctor $\Phi_F$ that takes an object $X$ to $1 + F \times X$. (Same as your $0 \sqcup F \times X$ but in different notation.)

    An easy but quite wonderful theorem, due to Lambek, is that for any endofunctor $\Phi$ on a category, if an initial algebra for $\Phi$ exists, given by a morphism $\alpha: L \to \Phi(L)$, then $\alpha$ is an isomorphism (so that $L$ is indeed a “fixed point” of $\Phi$). The same goes for the dual notion of terminal coalgebra.

    For a presheaf topos or indeed a Grothendieck topos, initial algebras and I think also terminal coalgebras exist for any polynomial endofunctor. $List(F)$ is an example of this ($\Phi_F(X) = 1 + FX$ is polynomial. Actually, $List(-)$ itself is, according to the technical definition, polynomial as well!

    Without thinking about it too hard, I’ll go out on a limb and conjecture that the object of rhythm trees can be described as either an initial algebra or terminal coalgebra of the endofunctor that takes $X$ to $DValue \times List(\mathbb{Z} + X)$. (Maybe the former; typically, initial algebras of (for example) polynomial endofunctors are embedded inside the terminal coalgebras, and consist of those (typically tree-like) elements of the terminal coalgebra that are “well-founded”.)

    When Madeleine in #21 mentioned coinduction, that corresponds to taking a terminal coalgebra, as I also touched upon back in #14.

    The article continued fraction gives some other examples of initial algebras and terminal coalgebras, for anyone wishing to explore these notions in a classical scenario dating back centuries. :-)

26. • CommentRowNumber26.
    • CommentAuthorUrs
    • CommentTimeJul 15th 2023
    • PermaLink
    Author: Urs
    Format: MarkdownItex> Ok, fixing it now. Thanks! That looks much better.

    Ok, fixing it now.

    Thanks! That looks much better.

27. • CommentRowNumber27.
    • CommentAuthorJohn Baez
    • CommentTimeJul 15th 2023
    • (edited Jul 15th 2023)
    • PermaLink
    Author: John Baez
    Format: MarkdownItexTodd wrote: > Drew: I was chatting with John Baez today and happened to mention – it came up naturally in the course of conversation – your addition to the nLab. Apparently lots of people interested in mathematical music theory study the 24-element group generated by translations and inversions on $\mathbb{Z}_{12}$ and its action on (for example) triads, but John perked up when I mentioned this 48-element group, which would correspond to adjoining 5 and 7 to {1,11} where the latter two elements correspond to the subgroup of inversions [...] The reason I perked up was that I'd never seen anyone in music talk about multiplying pitch classes (elements of $\mathbb{Z}_{12}$) by 5 or 7. I'd be really interested to see some musical use of these extra transformations, especially if it's not just some avant-garde composer fiddling around to see what can be done with them. For a moment I thought these extra transformations might somehow be related to the circle of fifths or the seven-note major scale... but I don't really see any evidence for that.

    Todd wrote:

    Drew: I was chatting with John Baez today and happened to mention – it came up naturally in the course of conversation – your addition to the nLab. Apparently lots of people interested in mathematical music theory study the 24-element group generated by translations and inversions on $\mathbb{Z}_{12}$ and its action on (for example) triads, but John perked up when I mentioned this 48-element group, which would correspond to adjoining 5 and 7 to {1,11} where the latter two elements correspond to the subgroup of inversions […]

    The reason I perked up was that I’d never seen anyone in music talk about multiplying pitch classes (elements of $\mathbb{Z}_{12}$) by 5 or 7. I’d be really interested to see some musical use of these extra transformations, especially if it’s not just some avant-garde composer fiddling around to see what can be done with them. For a moment I thought these extra transformations might somehow be related to the circle of fifths or the seven-note major scale… but I don’t really see any evidence for that.

28. • CommentRowNumber28.
    • CommentAuthorJohn Baez
    • CommentTimeJul 15th 2023
    • (edited Jul 15th 2023)
    • PermaLink
    Author: John Baez
    Format: MarkdownItexI wrote a bunch about the 24-element group of transformations of $\mathbb{Z}_{12}$ of the form $$ x \mapsto \pm x + a \qquad a \in \mathbb{Z}_{12} $$ in [week234](https://math.ucr.edu/home/baez/week234.html) of *This Week's Finds*. This is sometimes called the **TI group**, for 'transposition' and 'inversion'. An interesting fact is that the 24-element set of major and minor triads forms a [[torsor]] for this group. If we call this set of triads $\mathrm{Tr}$, we can also look at the group of permutations $f: \mathrm{Tr} \to \mathrm{Tr}$ that commute with the action of the TI group. By some general nonsense about torsors, this must be another 24-element group isomorphic to the TI group. But since its action on $\mathsf{Tr}$ is different from the action of the TI group, it's worth thinking about separately. It's called the **PRL group** because it has 3 generators P, R, L with musically interesting meanings. I explain those. Everything I wrote is just an explanation of known work (though I'm not sure anyone had mentioned torsors before), and I give a bunch of references.

    I wrote a bunch about the 24-element group of transformations of $\mathbb{Z}_{12}$ of the form

    $$x \mapsto \pm x + a \qquad a \in \mathbb{Z}_{12}$$

    in week234 of This Week’s Finds. This is sometimes called the TI group, for ’transposition’ and ’inversion’. An interesting fact is that the 24-element set of major and minor triads forms a torsor for this group.

    If we call this set of triads $\mathrm{Tr}$, we can also look at the group of permutations $f: \mathrm{Tr} \to \mathrm{Tr}$ that commute with the action of the TI group. By some general nonsense about torsors, this must be another 24-element group isomorphic to the TI group. But since its action on $\mathsf{Tr}$ is different from the action of the TI group, it’s worth thinking about separately. It’s called the PRL group because it has 3 generators P, R, L with musically interesting meanings. I explain those.

    Everything I wrote is just an explanation of known work (though I’m not sure anyone had mentioned torsors before), and I give a bunch of references.

29. • CommentRowNumber29.
    • CommentAuthorJohn Baez
    • CommentTimeJul 15th 2023
    • PermaLink
    Author: John Baez
    Format: MarkdownItexSo, in principle going up from the 24-element TI group to the larger 48-element group of all affine transformations of $\mathbb{Z}_{12}$ could be musically interesting. But I don't see how.

    So, in principle going up from the 24-element TI group to the larger 48-element group of all affine transformations of $\mathbb{Z}_{12}$ could be musically interesting. But I don’t see how.

30. • CommentRowNumber30.
    • CommentAuthorDrew Flieder
    • CommentTimeJul 15th 2023
    • (edited Jul 16th 2023)
    • PermaLink
    Author: Drew Flieder
    Format: MarkdownItexMy general thoughts on whether or not a process/thing is musically interesting is that it is up to the skill and inventiveness of a composer. There are many composers who write awful music that is heavily based on **TI** transformations of pitch material. There are also some great composers who use such operations inventively. What has always been and still is the case is that truly inventive and skilled composers are rare. Therefore the methods and techniques that are commonplace for a certain kind of compositional thinking are often put to poor use. It is also very hard to think of a general criterion by which a musical application is interesting, which is basically a criterion for creativity. That being said, there are some interesting basic facts about multiplication by 5 and 7 (usually called the $M_5$ and $M_7$ operators). Here is the pitch-class mapping by $M_5$ * $0 \mapsto 0$ * $1 \mapsto 5$ * $2 \mapsto 10$ * $3 \mapsto 3$ * $4 \mapsto 8$ * $5 \mapsto 1$ * $6 \mapsto 6$ * $7 \mapsto 11$ * $8 \mapsto 4$ * $9 \mapsto 9$ * $10 \mapsto 2$ * $11 \mapsto 7$ Here is an interesting fact. $M_5$ acting on the set class $(037)$ of major and minor chords maps to the set class $(014)$, which is a very “atonal” sounding harmony. $(014)$ is maybe the most common building block of the music of the Second Viennese School of Schoenberg, Webern, and Berg. This is probably just a coincidence. But, since classical harmony is based on the major/minor triad, try taking a passage of classical music, say a Bach chorale, and multiply it by 5. You’ll have to revoice the harmonies a bit, but you’ll get a much more atonal kind of music that follows the same harmonic syntax as a Bach chorale! In general, apply such a mapping to any passage of tonal music and you’ll get a much different sounding result that follows the same syntax of tonal music. Note that for any of the **TI** operators, the “meaning” (if you’d like to call it that) of the harmonic syntax is the same. By this, I mean that transposing a tonal chord progression (say I-IV-V) gives you the same chord progression in another key, and inverting gives you the same thing just in a minor key (i-iv-v). In other words, **TI** *operators preserve tonal function*. On the other hand, performing $M_5$ to such a progression totally destroys the functional tonality. For instance, the tonal chord progression C-E-F transforms by $M_5$ into the chord progression (08B)-(019)-(7AB) (where "A" and "B" denote pitch-classes 10 and 11). Hence from a tonal standpoint, it is not so obvious how $M_5$ and $M_7$ could be used, since tonal music is grounded on functional tonality and that is precisely what $M_5$ destroys. On the other hand, for atonal music which is more conscious of pitch-class relations, $M_5$ and $M_7$ can be of interesting use, for instance, to preserve the structural relations of pitch-classes while transforming the harmonic "flavor". There are many pieces which use such operations, including this [string trio](https://www.youtube.com/watch?v=XwVwuPLUI2E) by Charles Wuorinen. Your connection to the circle of fourths/fifths is right. This is also related to the fact that $\{1,5,7,11\}$ are the generators of $\mathbb{Z}_{12}$. So if you have an ordered chromatic scale, which is generated by 1, and multiply it by 7 you get the circle of fifths, i.e. the elements in order generated by 7. My former professor wrote his PhD dissertation titled *Prolongation in Equal and Microtonal Temperaments*. There is extensive treatment of the multiplication operators. There's also a section titled "Constructing Generalized Diatonic Scale Cycles For Prolongation". This involves the general discussion of multiplication groups mod $n$, i.e. the groups $\mathbb{Z}_n$ under multiplication consisting of all integers coprime to $n$. These are the groups that Todd mentioned in \#5. I will need to re-read the dissertation to have more to say. But, here is an interesting fact, which you may know about but has some interesting implications for music. For any $n$ such that $n+1$ is prime, the group $(\mathbb{Z}_n, +)$ under addition is isomorphic to $(\mathbb{Z}_{n+1}, \times)$ under multiplication. So here is an interesting construction. * Choose an isomorphism $\phi : (\mathbb{Z}_{n+1}, \times) \rightarrow (\mathbb{Z}_n, +)$. * Take a generator $m \in (\mathbb{Z}_{n+1}, \times)$, and construct the sequence $\sigma = \langle 1, m, m^2, \ldots, m^{n-1} \rangle$. * Apply $\phi(\sigma) = P$, which gives you an (exhaustive) $n$-length pitch-class row, i.e. an $n$-tone row. The row $P$ has an interesting property. The idea is that many, in fact all, transpositions of $P$ are in some way "embedded" in $P$ itself, in the sense that skipping around in the order position of $P$ by some fixed amount $k \in \{1, \ldots, n\}$, and "wrapping around" when necessary, will give you a transposition of $P$. More specifically, to construct the sequence $P^k$, we do the following: For $i \in \{ 1, \ldots, n\}$, * if $1 \lt i k \leq n+1$, then $P^k_i = P_{i k \ mod \ n}$; * if $n+1 \lt i k \leq 2n + 1$, then $P^k_i = P_{i k - 1 \ mod \ n}$; * if $2n + 1 \lt i k \leq 3n + 1$, then $P^k_i = P_{i k - 2 \ mod \ n}$; * etc. Then there exists a transposition $t$ in the group **T** of transpositions of $\mathbb{Z}_n$ such that $t(P) = P^k$. In fact, there is a unique $t$ for every $P^k$, $k \in \{1, \ldots, n\}$, so the sets \[ \{ P^k \ \big\vert \ k \in \{1, \ldots, n\} \} \] and **T** are in bijection. All of this is a lot easier to picture. I wrote a program that generates such sequences in an easy-to-read format, in case anyone wants to see some of these for themselves. But here is an example with a twelve-tone row: \[ P = \langle 0,1,4,2,9,5,11,3,8,10,7,6\rangle. \] You can see, for instance, that \[ P^2 = \langle 1, 2, 5, 3, 10, 6, 0, 4, 9, 11, 8, 7 \rangle = t_1(P). \] *Anyway*, a composer, if inventive, may exploit the structure of this in a composition in a multitude of ways. It is up to their imagination to exploit such structures in non-trivial ways. Edit: Of course, the group **T** of transpositions of $\mathbb{Z}_n$ is isomorphic to $\mathbb{Z}_n$.

    My general thoughts on whether or not a process/thing is musically interesting is that it is up to the skill and inventiveness of a composer. There are many composers who write awful music that is heavily based on TI transformations of pitch material. There are also some great composers who use such operations inventively. What has always been and still is the case is that truly inventive and skilled composers are rare. Therefore the methods and techniques that are commonplace for a certain kind of compositional thinking are often put to poor use. It is also very hard to think of a general criterion by which a musical application is interesting, which is basically a criterion for creativity.

    That being said, there are some interesting basic facts about multiplication by 5 and 7 (usually called the $M_5$ and $M_7$ operators). Here is the pitch-class mapping by $M_5$

    • $0 \mapsto 0$
    • $1 \mapsto 5$
    • $2 \mapsto 10$
    • $3 \mapsto 3$
    • $4 \mapsto 8$
    • $5 \mapsto 1$
    • $6 \mapsto 6$
    • $7 \mapsto 11$
    • $8 \mapsto 4$
    • $9 \mapsto 9$
    • $10 \mapsto 2$
    • $11 \mapsto 7$

    Here is an interesting fact. $M_5$ acting on the set class $(037)$ of major and minor chords maps to the set class $(014)$, which is a very “atonal” sounding harmony. $(014)$ is maybe the most common building block of the music of the Second Viennese School of Schoenberg, Webern, and Berg. This is probably just a coincidence. But, since classical harmony is based on the major/minor triad, try taking a passage of classical music, say a Bach chorale, and multiply it by 5. You’ll have to revoice the harmonies a bit, but you’ll get a much more atonal kind of music that follows the same harmonic syntax as a Bach chorale!

    In general, apply such a mapping to any passage of tonal music and you’ll get a much different sounding result that follows the same syntax of tonal music. Note that for any of the TI operators, the “meaning” (if you’d like to call it that) of the harmonic syntax is the same. By this, I mean that transposing a tonal chord progression (say I-IV-V) gives you the same chord progression in another key, and inverting gives you the same thing just in a minor key (i-iv-v). In other words, TI operators preserve tonal function. On the other hand, performing $M_5$ to such a progression totally destroys the functional tonality. For instance, the tonal chord progression C-E-F transforms by $M_5$ into the chord progression (08B)-(019)-(7AB) (where “A” and “B” denote pitch-classes 10 and 11). Hence from a tonal standpoint, it is not so obvious how $M_5$ and $M_7$ could be used, since tonal music is grounded on functional tonality and that is precisely what $M_5$ destroys. On the other hand, for atonal music which is more conscious of pitch-class relations, $M_5$ and $M_7$ can be of interesting use, for instance, to preserve the structural relations of pitch-classes while transforming the harmonic “flavor”. There are many pieces which use such operations, including this string trio by Charles Wuorinen.

    Your connection to the circle of fourths/fifths is right. This is also related to the fact that $\{1,5,7,11\}$ are the generators of $\mathbb{Z}_{12}$. So if you have an ordered chromatic scale, which is generated by 1, and multiply it by 7 you get the circle of fifths, i.e. the elements in order generated by 7.

    My former professor wrote his PhD dissertation titled Prolongation in Equal and Microtonal Temperaments. There is extensive treatment of the multiplication operators. There’s also a section titled “Constructing Generalized Diatonic Scale Cycles For Prolongation”. This involves the general discussion of multiplication groups mod $n$, i.e. the groups $\mathbb{Z}_n$ under multiplication consisting of all integers coprime to $n$. These are the groups that Todd mentioned in #5. I will need to re-read the dissertation to have more to say. But, here is an interesting fact, which you may know about but has some interesting implications for music. For any $n$ such that $n+1$ is prime, the group $(\mathbb{Z}_n, +)$ under addition is isomorphic to $(\mathbb{Z}_{n+1}, \times)$ under multiplication. So here is an interesting construction.

    • Choose an isomorphism $\phi : (\mathbb{Z}_{n+1}, \times) \rightarrow (\mathbb{Z}_n, +)$.

    • Take a generator $m \in (\mathbb{Z}_{n+1}, \times)$, and construct the sequence $\sigma = \langle 1, m, m^2, \ldots, m^{n-1} \rangle$.

    • Apply $\phi(\sigma) = P$, which gives you an (exhaustive) $n$-length pitch-class row, i.e. an $n$-tone row.

    The row $P$ has an interesting property. The idea is that many, in fact all, transpositions of $P$ are in some way “embedded” in $P$ itself, in the sense that skipping around in the order position of $P$ by some fixed amount $k \in \{1, \ldots, n\}$, and “wrapping around” when necessary, will give you a transposition of $P$. More specifically, to construct the sequence $P^k$, we do the following: For $i \in \{ 1, \ldots, n\}$,

    • if $1 \lt i k \leq n+1$, then $P^k_i = P_{i k \ mod \ n}$;

    • if $n+1 \lt i k \leq 2n + 1$, then $P^k_i = P_{i k - 1 \ mod \ n}$;

    • if $2n + 1 \lt i k \leq 3n + 1$, then $P^k_i = P_{i k - 2 \ mod \ n}$;

    • etc.

    Then there exists a transposition $t$ in the group T of transpositions of $\mathbb{Z}_n$ such that $t(P) = P^k$. In fact, there is a unique $t$ for every $P^k$, $k \in \{1, \ldots, n\}$, so the sets

    $$\{ P^k \ \big\vert \ k \in \{1, \ldots, n\} \}$$

    and T are in bijection.

    All of this is a lot easier to picture. I wrote a program that generates such sequences in an easy-to-read format, in case anyone wants to see some of these for themselves. But here is an example with a twelve-tone row:

    $$P = \langle 0,1,4,2,9,5,11,3,8,10,7,6\rangle.$$

    You can see, for instance, that

    $$P^2 = \langle 1, 2, 5, 3, 10, 6, 0, 4, 9, 11, 8, 7 \rangle = t_1(P).$$

    Anyway, a composer, if inventive, may exploit the structure of this in a composition in a multitude of ways. It is up to their imagination to exploit such structures in non-trivial ways.

    Edit: Of course, the group T of transpositions of $\mathbb{Z}_n$ is isomorphic to $\mathbb{Z}_n$.

31. • CommentRowNumber31.
    • CommentAuthorDrew Flieder
    • CommentTimeJul 16th 2023
    • (edited Jul 16th 2023)
    • PermaLink
    Author: Drew Flieder
    Format: MarkdownItexBy the way, I just read (briefly) [week234](https://math.ucr.edu/home/baez/week234.html). I wasn't aware that the P, L, R operations could be formulated in such a way! That is very interesting, and I'll have to look into that more. Regarding some of your comments on other symmetries of $\mathbb{Z}_{12}$, I wonder if there are some important ones that have yet to be discovered yet. One thing that has been used by composers (such as [Robert Morris](https://ecmc.rochester.edu/rdm/morris.bio.html)) in many pieces is the following idea: * Choose an $n \lt 12$ (common choices are $4, 5, 6$). * Get all the $n$-element set-classes. Suppose there are $k$. * Create a $k$-length sequence of pitch-class sets that has the following properties: (1) A member of each $n$-element set-class occurs exactly once; (2) The $i$th member has $n-1$ pitch-classes in common with the $(i+1)$th member; (3) The first member also has $(n-1)$ pitch-classes in common with the last member. So each time you move to a new pitch-class set, it will have $n-1$ pitch-classes in common, and since every set-class features exactly once, you have a parsimonious cycle through the entire collection of set-classes. The composers I know who use such things use computer programs to find these cycles, but the output only gives one at a time and it takes a while to compute. I wonder if there are some groups that could help to classify such sequences, since each successive member ought to have $n-1$ elements fixed by some group action. I have no idea, but that would be of great interest...

    By the way, I just read (briefly) week234. I wasn’t aware that the P, L, R operations could be formulated in such a way! That is very interesting, and I’ll have to look into that more.

    Regarding some of your comments on other symmetries of $\mathbb{Z}_{12}$, I wonder if there are some important ones that have yet to be discovered yet. One thing that has been used by composers (such as Robert Morris) in many pieces is the following idea:

    • Choose an $n \lt 12$ (common choices are $4, 5, 6$).

    • Get all the $n$-element set-classes. Suppose there are $k$.

    • Create a $k$-length sequence of pitch-class sets that has the following properties: (1) A member of each $n$-element set-class occurs exactly once; (2) The $i$th member has $n-1$ pitch-classes in common with the $(i+1)$th member; (3) The first member also has $(n-1)$ pitch-classes in common with the last member.

    So each time you move to a new pitch-class set, it will have $n-1$ pitch-classes in common, and since every set-class features exactly once, you have a parsimonious cycle through the entire collection of set-classes.

    The composers I know who use such things use computer programs to find these cycles, but the output only gives one at a time and it takes a while to compute. I wonder if there are some groups that could help to classify such sequences, since each successive member ought to have $n-1$ elements fixed by some group action. I have no idea, but that would be of great interest…