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February 26, 2010
A Look at the Mathematical Origins of Western Musical Scales
Posted by Simon Willerton
I want to explain a little of the background behind Tom Fiore’s musical post last year. In this post the aim is to explain to a numerate audience some of the origins of the Western seven note musical scale. I will try to assume no formal knowledge of music except perhaps a vague notion of what a piano keyboard looks like. I won’t get very far in the historical development, only up to about the middle ages.
There are two aspects of music relevant to this discussion: melody and harmony. Melody involves the consecutive playing of notes, like in any tune you can hum or whistle; harmony involves the simultaneous playing of notes like in a chord or multipart singing. In terms of Western music, it appears that harmony did not make an appearance until the middle ages; in this post I will not get on to harmony and how it had a significant effect on the precise pitch of each note. What I will explain here is the origin of the seven note Western musical scale in terms of the consonance of the octave and the consonance of the fifth, which in turn have their origins in the physics of the vibrating string.
I should add at this point that I learnt much of this stuff from the following great book which is freely available as a pdf.
- Dave Benson, Music: A Mathematical Offering, Cambridge University Press, 2006.
I also learnt a lot from talking to various friends and there is lots and lots of information on the internet one interesting looking document is
- Peter A. Frazer, The Development of Musical Tuning Systems.
February 25, 2010
3000 and One Things to Think About
Posted by Urs Schreiber
In a burst of activity, Zoran Škoda a few minutes ago went beyond the Lab entry of nominal count 3000 with beginning a discussion of crystals.
Other activity we have seen recently:
Peter Arndt just expanded the section on Chow groups at motivic cohomology.
Jacques Distler has added to the software an impressive built-in SVG-graphics editor and everyone is now drawing nice diagrams for their Lab entries, such as Andrew Stacey on his page on Frölicher spaces.
David Corfield is, as you know, pushing us to look into the internal logic in an -topos. While Mike Shulman of course is developing 2-categorical logic all along.
We found (Zoran and I did, at least) that we need to be pushing to get to more genuine physics on the Lab, but also that we still need to get a bit more abstract structure in place first to do it right . Sort of as a reminder for where we are headed, now an POV on BV-BRST formalism in terms of derived higher geometry.
But the physics is gradually coming, hopefully? There is now lots to think about at multisymplectic geometry. I am hoping for somebody to start talking about the references on multisymplectic quantization mentioned at the very end!
Or: today we figured out what exactly quantum information theorists mean by quantum channels and noticed that these obviously form a nice category.
Well, it takes its time. I have been spending a bit of time preparing notes that ought to eventually be useful in seminars on -rings and the -Grothendieck construction.
And then there is notable activity on the personal webs.
André Joyal is explaining distributors and factorization systems.
Todd Trimble is beginning to explain Buildings for category theorists.
And you haven’t even contributed yet! (Unless you have, of course).
(Infinity, 1)-logic
Posted by David Corfield
We’re having a chat over here about what an -logic might look like. The issue is that if we can extract a (1)-logic from ordinary toposes, shouldn’t there be an -logic to be extracted from -toposes. This post originated as an ordinary comment, but as things have gone a little quiet at the Café (10 days without a post!), I thought I’d promote it.
Won’t there be a sense in which this internal logic to an -topos will have to be interpretable as a ‘logic of space’? If is an especially nice topos which being well-pointed allows us to understand 1-logic internally and externally, we might hope that the well-pointed -topos of -groupoids, , does the same for -logic, given -groupoids are models for spaces.
Had I not known anything about logic beyond that it is a language used to formulate statements in a domain and to represent valid reasoning, then had I been asked to extract such a thing given the definition of a topos, that would seem to be a tricky task. But give me and I can make a start.
February 15, 2010
Higher Structures in Göttingen IV
Posted by Urs Schreiber
guest post by Christoph Wockel
Dear all,
we cordially invite you to participate the CRCG Workshop “Higher Structures in Topology and Geometry IV”, which will take place June 2-4 at the Göttingen Mathematics Institute (Germany). Main speakers will be
- Ezra Getzler (Northwestern)
- Birgit Richter (Hamburg)
- Urs Schreiber (Utrecht)
- Christoph Schweigert (Hamburg)
In addition, there will be a minor amount of talks given by PhD students and postdocs. For additional information you may consult the
Best wishes,
Christoph Wockel (on behalf of the organisers Giorgio Trentinaglia and Chenchang Zhu)
February 14, 2010
Algorithmic Thermodynamics
Posted by John Baez
Mike Stay and I would love comments on this paper:
- John Baez and Mike Stay, Algorithmic thermodynamics.
February 12, 2010
Intrinsic Naturalness
Posted by Urs Schreiber
Lately there have been remarkable developments in higher category theory.
What used to have a touch of alchemy to it – in its mystery, its grand hopes, its plethora of recipes tried out in hard and lonely work in long nights – is becoming chemistry: what used to be hypothesis and conjectures have become theorems; what used to be a philosopher’s stone seemingly out of reach has become a tangible jewel that you can touch, hold against the light, marvel at – and finally use to cut through the glass roof that has been impeding progress for so long: higher category theory.
Or so some think. Discussion with colleagues reveals that the perception of and feelings about what has been achieved – and is being achieved as we speak – varies. While parts of the community are storming ahead with the new technology that has become available, in other parts reservation and scepticism towards this activity is being felt.
Is this really the philosopher’s stone that the search was after? Isn’t its shiny appearance a cheap trick achieved by taking that old pebble called homotopy theory, and polishing it a bit? Is what superficially looks impressive rather intrinsically a kludge geared to serve a purpose and good enough to impress the mundane, but far from being the natural god-given structure that the inner circle of researchers in higher category theory knows – and rightly knows! – is the true goal of the search?
This are, in impressionistic paraphrase, questions being asked behind the scenes. As I have realized in long private discussions recently. And these are good and important questions. If anything as important as a first working implementation of higher category theory is being claimed – explicitly or not – , the claim deserves to be carefully scrutinized.
But careful scrutiny requires an effort to obtain a clear picture of the situation to the same extent that it requires a critical mindset. Imagine the alchemist producing finally the philosopher’s stone – and then discarding it onto the heap of failed attempts for not recognizing it.
Recently I was standing in an alchemist friend’s laboratory and we were looking at that stone. My colleague pointed at it and exclaimed: “Look, it is not natural. This cannot be the answer.” To which I replied: “But wait, you are looking at it as we use it to cut through that glass. Pick it up instead and hold it against the sunlight, so that you see its intrinsic colors, not the reflection of the workbench.” I picked up the stone and held it against the light in three different angles, and we were bathed in its light.
My friend agreed that he hadn’t looked at the stone in this light, and that this did make a difference. He then asked me to share the view from these three angles here on the blog. Which is what I now do.
February 10, 2010
Grothendieck Said: “Stop”
Posted by John Baez
Recently a team of mathematicians have been trying to republish the famous series of papers called “SGA”, or “Séminaire de Géométrie Algébrique”. This was largely the work of Alexander Grothendieck, who had refused to let Springer Verlag republish the original version.
February 6, 2010
This Week’s Finds in Mathematical Physics (Week 293)
Posted by John Baez
In week293 of This Week’s Finds, catch up on recent papers and books about -categories. Hear about last weekend’s Conference on the Mathematics of Environmental Sustainability and Green Technology at Harvey Mudd College. And learn how to think of networks of resistors as chain complexes which are also morphisms in a category.
February 5, 2010
Sheaves Do Not Belong to Algebraic Geometry
Posted by Tom Leinster
…and here’s a proof.
They are, of course, very useful in algebraic geometry (as is the equals sign). Also, human beings discovered them while developing algebraic geometry, which is why many of them still make the association.
But as we’ll see, sheaves are an inevitable consequence of general ideas that have nothing to do with algebraic geometry. In fact, sheaves (and various related notions) arise automatically from two completely general categorical constructions, together with one almost imperceptibly small topological observation.
February 3, 2010
Quantum Physics and Logic at Oxford
Posted by John Baez
I’m trying to cut back on jetting about, with some success — but I couldn’t resist going to this:
- Quantum Physics and Logic (QPL7), Saturday/Sunday May 29-30, 2010, Oxford. Organized by Bob Coecke, Prakash Panangaden, and Peter Selinger. Before the workshop itself there will be a school from May 24th to 28th.
It’s the seventh of the QPL series. But the meaning of the abbreviation “QPL” has changed. For the first four workshops of this name, it meant “Quantum Programming Languages”. Now it means “Quantum Physics and Logic”. That’s because the scope has broadened to cover everything about “the interaction between modern computer science logic, quantum computation and information, models of spatio-temporal causality, and quantum foundations.”
February 2, 2010
Derived Synthetic Differential Geometry
Posted by Urs Schreiber
Ordinary synthetic differential geometry – at least the well adapted models – is concerned with 0-truncated generalized spaces that are modeled on smooth loci: the formal duals of finitely generated -rings.
Under derived synthetic differential geometry I suppose we should want to understand the study of the notions of space that are induced from the geometry (in the sense of geometry for structured -toposes) that is the geometric envelope of the pregeometry constituted by , with one of its familiar site structures.
This would seem to be an excellent candidate for the ambient geometry in which most of fundamental physics, as presently conceived, takes place.
Maybe we can chat a bit about it here.