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I just submitted the following to “The Journal of Mathematics and Music”.
“Using the traditional accidentals of western music theory, a musical space dubbed accidental space is introduced in three contexts. The first is as an algebra reminiscent of that used in quantum mechanics; this version places special significance on palindromic modes such as Dorian. The second is as a network reminiscent of that used in graph theory; this version shows clear patterns regarding chord quality clustering. The final is as a category with modes as objects and accidentals as morphisms; this version provides a singular context which encompasses both algebra and network. “
I’m fairly certain this is a mathematical something, but I’m not a professional mathematician; I’m looking forward to being proven right or wrong by others outside of me.
Just a friendly word of warning: labeling your own work a “breakthrough” is a good way to get yourself labeled as a crackpot. (-:
Thanks Mike.
I am fully aware of my choice of language and the effect that it has on a science community that is used hearing about FTL and perpetual energy “breakthroughs”.
I submit to you however that public confidence in the scope of one’s work does not a crackpot make.
In the case of this paper:
I have found Feynman-esque diagrams in music…
I have created a self-consistent algebra using Dirac’s Bra-Ket notation and able to link for the first time in history AFAIK all 7-note musical scales into one self-consistent mathematical system…
I have found a way to categorize these observations in a way that give insight not only into my theory and music but also just how endemic the CT-POV really is…
I have made pretty pictures and harmonious sounds. :)
Further, a score of 5-8 on the Crackpot Index, though admittedly for physics, allows me to allude to a breakthrough without being labeled a crackpot by those that read my paper. Those that don’t read my paper can and will say anything about it but those opinions don’t carry any weight of course.
A -5 point starting credit.
1 point for every statement that is widely agreed on to be false. NO
2 points for every statement that is clearly vacuous. NO
3 points for every statement that is logically inconsistent. TBD
5 points for each such statement that is adhered to despite careful correction. NO
5 points for using a thought experiment that contradicts the results of a widely accepted real experiment. NO
5 points for each word in all capital letters (except for those with defective keyboards). NO
5 points for each mention of “Einstien”, “Hawkins” or “Feynmann”. NONE
10 points for each claim that quantum mechanics is fundamentally misguided (without good evidence). NONE
10 points for pointing out that you have gone to school, as if this were evidence of sanity. NO
10 points for beginning the description of your theory by saying how long you have been working on it. (10 more for emphasizing that you worked on your own.) NOPE
10 points for mailing your theory to someone you don’t know personally and asking them not to tell anyone else about it, for fear that your ideas will be stolen. NEVER
10 points for offering prize money to anyone who proves and/or finds any flaws in your theory. NOPE
10 points for each new term you invent and use without properly defining it. ALL NEW TERMS DEFINED
10 points for each statement along the lines of “I’m not good at math, but my theory is conceptually right, so all I need is for someone to express it in terms of equations”. NOPE
10 points for arguing that a current well-established theory is “only a theory”, as if this were somehow a point against it. NOPE
10 points for arguing that while a current well-established theory predicts phenomena correctly, it doesn’t explain “why” they occur, or fails to provide a “mechanism”. NOT HERE
10 points for each favorable comparison of yourself to Einstein, or claim that special or general relativity are fundamentally misguided (without good evidence). NOPE
10 points for claiming that your work is on the cutting edge of a “paradigm shift”. YES
20 points for emailing me and complaining about the crackpot index. (E.g., saying that it “suppresses original thinkers” or saying that I misspelled “Einstein” in item 8.) NOPE
20 points for suggesting that you deserve a Nobel prize. NOPE
20 points for each favorable comparison of yourself to Newton or claim that classical mechanics is fundamentally misguided (without good evidence). NOPE
20 points for every use of science fiction works or myths as if they were fact. NOPE
20 points for defending yourself by bringing up (real or imagined) ridicule accorded to your past theories. NOPE
20 points for naming something after yourself. (E.g., talking about the “The Evans Field Equation” when your name happens to be Evans.) ANONYMOUS PAPER
20 points for talking about how great your theory is, but never actually explaining it. NOPE
20 points for each use of the phrase “hidebound reactionary”. LOL
20 points for each use of the phrase “self-appointed defender of the orthodoxy”. ROFLMAO
30 points for suggesting that a famous figure secretly disbelieved in a theory which he or she publicly supported. (E.g., that Feynman was a closet opponent of special relativity, as deduced by reading between the lines in his freshman physics textbooks.) NOPE
30 points for suggesting that Einstein, in his later years, was groping his way towards the ideas you now advocate. NOPE
30 points for claiming that your theories were developed by an extraterrestrial civilization (without good evidence). NOPE
30 points for allusions to a delay in your work while you spent time in an asylum, or references to the psychiatrist who tried to talk you out of your theory. NOPE
40 points for comparing those who argue against your ideas to Nazis, stormtroopers, or brownshirts. NO
40 points for claiming that the “scientific establishment” is engaged in a “conspiracy” to prevent your work from gaining its well-deserved fame, or suchlike. NOPE
40 points for comparing yourself to Galileo, suggesting that a modern-day Inquisition is hard at work on your case, and so on. NOPE
40 points for claiming that when your theory is finally appreciated, present-day science will be seen for the sham it truly is. (30 more points for fantasizing about show trials in which scientists who mocked your theories will be forced to recant.) NOPE
50 points for claiming you have a revolutionary theory but giving no concrete testable predictions. EVERYTHING IS TESTABLE
If you have the time and interest, Mike, I would of course welcome your observations on the paper and its validity.
After all, the true sign of a crackpot is not in the presentation of a work but in their defense of it, in their un-willingness to hear reason, to modify their POV in light of conflicting evidence.
let me prove to you my willingness to hear reason and change my POV; let me prove to you that my “crackpot ideas” have merit… ;)
Okay, I suggest that we stop here. We’ve had other people who want to use the nLab and nForum to advertise their work, and it’s usually turned out not too well for the advertiser, and sometimes badly for everyone.
fastlane, you’ve now put your work out there so that if people are interested, they can correspond with you. Any further advertisement (such as bragging about a “breakthrough”) is very much unwanted and discouraged.
I suggest that if someone else (whose credibility is known to the more established members of this enterprise – the Steering Committee would certainly be sufficient) feels that the material in the paper warrants incorporation into the nLab, and the author is also agreeable to this, then we could proceed along such lines. But that someone needs to be someone other than the author.
Finally, let me give my opinion that a number of discussions recently have become a bit too prolix about matters not really germane to nLab and nForum business and enquiries, and that nForum discussions work much, much better when all participants exercise self-discipline and make themselves maximally helpful in asking good questions and addressing the questions of others.
So, does anyone want to say anything about the paper instead of the choice of language in the OP?
The topic is irrefutably in line with nLab; do you judge me a “braggart” and “crackpot” before even reading the paper? ;)
“would certainly be sufficient) feels that the material in the paper warrants incorporation into the nLab, and the author is also agreeable to this, then we could proceed along such lines. But that someone needs to be someone other than the author.”
I perhaps misunderstood the purpose of this “preprints and publications” forum.
I thought it was a place to share preprints and publications, not a staging area for possible inclusion into nLab which, while I would welcome, was not the intent, tone, or request in the OP.
If I have posted in error, I apologize.
All I want is to have professional mathematicians, of which I am not, look at my amateur work and see if what I think “breaks through” actually does…. that’s all I asked for and I’m sorry if I didn’t make that clearer or if you read my “proven right or wrong” as a challenge and not the request for help and external mathematical validation that it actually is.
So, does anyone want to say anything about the paper …?
Yes, that should be the question of the thread. If anyone has anything to say on it, then I suppose they will say it.
Whether the paper is a breakthrough and in line with the nLab may now be left for others to judge, but repeated declarations from you that this is the case are really not helping your cause.
I thought it was a place to share preprints and publications
I think usually people have used this category to bring to attention the preprints and publications of others here that seem relevant to nLab interests. One can mention one’s own submissions and publications here, certainly, but if you want to say anything more about it than the brief abstract, then it should not take the form of “I think this is a mathematical breakthrough”. Please don’t get defensive about this; we are just trying to say that this is frowned upon here.
Problem solved: Breakthrough —> Something.
Now it is my hope that everyone on this thread will spend as much time on my paper as they have on my OP and maybe help me find out what that “something” is.
I’ve updated my categorical music research paper.
I would be interested in particular in your assessment if I used QUIVER correctly as it is a reference that I make directly from the n-lab.
Also, In section 3.2 I make ample use of Leinster’s 2014 “Basic Category Theory” to establish my objects and relationships as a category; critique in that area would be most useful and welcome.
A quiver is nothing more and nothing less than a directed graph, with loops allowed and allowing multiple edges all with the same source and target. Usually though when people use the word “quiver”, they have in mind certain resonances having to do with representation theory. We used it in the nLab I think mostly because “directed graph” has different denotations for different groups of mathematicians, whereas “quiver” at least has just one denotation, even if the connotations differ.
Figure 1 certainly has an underlying quiver. But if for your purposes there is an important distinction between solid arrows and dashed arrows (or if colors are important), then it looks like you’re dealing not with a mere quiver, but a quiver with some extra structure (namely, the data that tells you which arrows are dotted or dashed, etc.).
There are multiple issues in how mathematical language is being used in section 3.2. For example, “category” is defined incorrectly. For one thing, it is simply not true that there must be one or more morphisms going from and object $A$ to an object $B$. The word “relationship” is question-raising (what do you mean by a “relationship”?), and should be avoided. Saying that the morphisms “commute” as part of the definition is “not even wrong” – as presented it’s close to meaningless (I’m sorry if that sounds harsh, but it’s true): what you want instead is to introduce a composition function as part of the data.
As a general rule of thumb, when you define a mathematical concept, you introduce (1) sets of the sorts you will be discussing (e.g., a set $O$ of objects, a set $M$ of morphisms), (2) some structure that connects these sets in some way, in the form of specified functions and relations (e.g., a function $id: O \to M$, functions $dom: M \to O$, $cod: M \to O$, etc.), and (3) some conditions which the data introduced in (1) and (2) must satisfy (e.g., identity axiom, associativity axiom). If any of these steps is skipped or is unclear, then the definition will be incomplete and confusing. For example, saying “morphisms commute” is, grammatically, of type (3), whereas you haven’t even introduced the composition data of type (2) for the reader to know what that phrase might be talking about. (Try going back to Leinster’s book, and see if you can view his definition of category as fitting into this general scheme.)
There may be some interesting ideas in your paper, but I believe a lot of clean-up will be required to bring this up to mathematical standards.
Thanks a million for taking the time to comment. Thanks for your insight on quivers and the mis-use of “commute” in the definitions. I hope I can trouble you for a little extra time though….
I tried to follow Leinster p. 10, Definition 1.1.1 step by step in defining my category but you point out three concerns I’d like to address:
1) I’m at a lost as to how to better define my objects. I’ve listed them in table 1, I’ve given a bra-ket construction of these objects in section 1.1, a matrix definition in 1.5, and then I define them again in section 3.3 when I discuss identities. Can you give me any advice on how I might more clearly define my objects?
2) Also, you say
it is simply not true that there must be one or more morphisms going from and object A A to an object B B.
but Leinster’s defintion say
for each A,B in ob( A A ), a collection A A (A,B) of maps or arrows or morphisms from A to B
How am I misreading this definition and/or how can I bring my definition and usage more in line with convention?
3) Finally, the statement that “all triangles and squares commute” is the topic of section 3.4. Was your original concern exclusively the mis-use of the word “commute” as part of the definition or that the procedure outlining how “triangles and squares commute” in 3.4 is inadequate as well?
Thanks again!
The collection might be empty…
Okay, I hadn’t commented yet on “accidental categories”; so far I only commented on how you presented the general notion of category. Let me turn to how you present accidental categories.
One thing that sticks out is notation. You write ${|\natural|}_A$ for (apparently) the identity morphism $A \to A$ on a mode $A$. Why the absolute value symbolism? (I’ll come back to this in a moment.) And it seems you have for any pair of modes $A, B$ a flat “operator” (not morphism?) ${|\flat|}: A \to B$. Should that be ${|\flat|}_{A, B}: A \to B$, to avoid ambiguity? (You don’t want the same symbol ${|\flat|}$ for two different morphisms $A \to B$, $C \to D$.)
Ordinarily, in category theory, the composite of $f: A \to B$ and $g: B \to C$ is written by a simple juxtaposition $g f: A \to C$ (or $g \circ f$). So ordinarily you would write the composite of ${|\natural|}_A: A \to A$ and ${|\flat|}_{A, B}: A \to B$ by the juxtaposition ${|\flat|}_{A, B}{|\natural|}_A: A \to B$. You have instead ${|\flat \natural_A|}$ which strictly speaking makes little sense, as the bare notations $\flat$, $\natural$ were never introduced to apply an absolute value to.
I think it would be simpler and notationally less confusing just to drop the absolute value notation, unless there is really a compelling reason for it. So you could write simply $\flat_{A, B} \natural_A$ for the composite. Similarly, if I’m parsing you write, you have a morphism $\sharp_{A, B}: B \to A$ for any pair of modes $A, B$.
“The accidental operators between modes A, B, and C commute; thus accidental operators compose naturally and universally.” What does that mean?? “Composing naturally and universally” is just not a phrase in my lexicon. Does this mean (I’m dropping the absolute value notation) for example $\flat_{B, C} \flat_{A, B} = \flat_{A, C}: A \to C$? Do we have $\sharp_{A, B}\flat_{A, B} = \natural_A$?
Relying on handwaving (and idiosyncratic) phrases like the above should be avoided. If you mean to convey equational axioms, then write those out precisely.
Can there be, in an accidental category, other morphisms besides the accidental ones? If so, would the natural morphisms $\natural_A$ behave like identities with respect to other morphisms?
(By the way: I have just a rudimentary grasp of music theory. I am aware of things like the canonical modes and relationships in the circle of fifths and I know basic musical notation, e.g., for the piano. Musically, I am a piano student, for about seven years now. So I don’t think your article would be totally beyond my ken, although I haven’t tried hard to grasp what you’re really driving at; right now I’m commenting just at the formal level of mathematical presentation.)
As always, let me start off by thanking you for taking the time to review my work. I don’t have anybody with any categorical knowledge to discuss this work with and therefore your perspective is overwhelmingly useful.
Why the absolute value symbolism?
This is a throwback to my physics days where operators are annotated with the absolute value sign to distinguish them from other objects in the theory. Thus |b3| is an operator but |b3> is a mode. The other reason is that it does have an absolute value (a vacuum state more accurately) and thus it can actually be mapped to a real number. But your point is well met and I think it best if I just drop the notation in text and keep the absolute value sign to reflect a mapping from an accidental to a real number and thus bring it more in line with math convention as you state. Let me know how that reads to you or if you have another suggestion for how to conventionally annotate these symbols
And it seems you have for any pair of modes A,B A, B a flat “operator” (not morphism?)
I see no difference between operator, morphism, map, path, or arrow; they all take one object and turn it into the same or another The sense of the word is that Ionian is in one “shape” to which when we apply the b7 operator (for example) and it changes “shape” to Myxolydian; hence the operator b7 is the morphism. Again, this is a throwback to my physics days and how we view and talk about operators.
Should that be |♭|A,B:A→B {|\flat|}_{A, B}: A \to B, to avoid ambiguity? (You don’t want the same symbol |♭| {|\flat|} for two different morphisms A→B A \to B, C→D C \to D.)
This point is a critical part of my paper and personal understanding so I’m glad you brought this up. Let’s use figure one, the Major Scale, as a reference.
In terms of notation, the “juxtaposition” of operators you mention comes about because I think of them as paths. Thus the composite b3b7 reflects a path whereby you take a step in the b7 direction first and another step in the b3 direction; hence a statement like b3b7 stands for a composite path. As for my use of subscripts, I’ll need to give that more thought. The same operator can connect two different modes… so the b7 operation connecting Ionian and Myxolydian in figure one is the same operator connection Lydian and Lydian Dominant. Figure three shows this and other mapping redundancies. As such, the b7 operation doesn’t carry any of these modes as a subscript. I’ll have to give this more thought though…
AFAIK, we have only one object, the mode, so we are dealing with a monoidal category… so really A -> A is the only morphism (what I call the free natural operator). In figure one we see this insofar that every mode leads directly to another mode with a single arrow. Take away all color and identifying marks and then any transition between two modes looks the same. But the morphisms, the arrows, are invertable and thus we are dealing with a group category. To me, this means that now there is a proper A-> B transition where A is in the “sharp” category and B is in the opposite “flat” category. This create an identity based on this adjunction, what I call the identity natural operator.
The extra structure on the quiver that you mentioned in #12 comes, as best I an tell, from the Z12 parent group. We have six generators in the form of the six arrows in figure 1 (what I may be mistakenly referring to as my quiver). These are the six generators (as I see it) of a free Z7 group. In effect, my belief is that there is a map from Z12 to Z7 that reflects a chromatic scale (12-tone equal temperament) “symmetry breaking down” to a heptatonic (7-tone unequal temperament) scale. This map picks two intervals in Z12 to create a 2-generator free group; hence the major scale uses two intervals from Z12 ($e^1$, a half-step H, and $e^2$, a whole-step W) to create the major scale as a two generator free group WWHWWWH. Thus my thesis that these six arrows comprise a “universal map” from which any heptatonic scale can be built (which again encouraged the “quiver” designation as in a finite set of arrows).
Bear in mind that that the nature of the paper is I’m just discovering the nature of this structure and I see no evidence of this discovery or this structure in the literature. Thus I’m stumbling about, not quite blindly, but certainly out of my ken as a physicist trying to mathematically explain these structures I see in musical notation.
“The accidental operators between modes A, B, and C commute; thus accidental operators compose naturally and universally.”
Yes, as pointed out in your last, this will be confusing and I’m re-writing this up to be more in line with Leinster’s definition and less “paraphrasing” on my part.
Can there be, in an accidental category, other morphisms besides the accidental ones?
As near as I can tell, the answer is “no”. Referring to a musical mode or scale by accidentals is necessary and sufficient (or so I claim) to map out all heptatonic scales and modes. There is no need to introduce any other object or morphism to accomplish this. In effect, the Accidental Category reflects the fact that accidentals are the only morphism in this space.
If so, would the natural morphisms ♮A \natural_A behave like identities with respect to other morphisms?
HOWEVER, if we consider the mapping of an accidental to a real number, to a musical interval like a perfect fifth, then we can see the identity still behaves like an identity since it will map to unison, the “do nothing” music interval consisting of zero half-steps. I “think” this is the correct answer though I get the sense that this mapping takes us out of the Accidental Category and into the category of real numbers and thus is a functor; not sure how that affects my model to be honest.
although I haven’t tried hard to grasp what you’re really driving at; right now I’m commenting just at the formal level of mathematical presentation
Absolutely! What I’m presenting is a mathematical model derived from musical notation. In fact, you don’t have to know any music theory to appreciate my model since it relys entirely on structure, not form. As such, to reference my work all you need is my table 1 that list the scales I use or an outside reference by way of a book of scales and modes (Here’s the one I used) and then see how my structure put them together. As such, I would LOVE for you to understand the musical ramifications of what I propose but I am equally THANKFUL for your formal mathematical presentation… which makes me more formal in turn!
Random thought which just entered my brain: is there a rough analogy between the operators $\flat$, $\sharp$ and ladder operators in physics?
Anyway, I’m glad you mentioned why that “absolute value” notation. If I were reading more carefully I might have picked up on that. So modes are like states and accidentals are like operators or observables?
Random thought which just entered my brain: is there a rough analogy between the operators ♭ \flat, ♯ \sharp and ladder operators in physics?
From section 1.1:
“Accidental operators thus raise and lower scale degrees in our modal bra-kets much like ladder operators raise and lower quantum numbers in a physics bra-ket (Shankar, 1988, eqn. 12.5.3). “
.
So modes are like states and accidentals are like operators or observables?
From the introduction
“Chapter one casts Accidental Space as an algebra acting on the scale degrees of a heptatonic mode; here we get accustomed to thinking of modes as states and accidentals as operators”
The observables are given by the operator vacuum state as outlined in section 1.2 and reflect musical intervals we can hear (i.e., observe) like a perfect fifth or major second.
Not so random.
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