Tue 1:30-3:20 room Smith 111, continued Spring 2009 as Music 571, Music room 212, Mondays 1:30-3:20.Note on the continuation:
In Spring, we will concentrate more on individual research projects, which will be presented to the seminar in various stages of completion for discussion and feedback.
If you would like to jump in at this stage, some knowledge of music theory is a good idea (at least for example, if you play an instrument and read music). Of course more music theory is better. In math, a BA in math or the equivalent, or a keen interest and some talent and background, will see you through.
Math majors and graduate students etc are welcome. The areas of mathematics that are currently most active in music theory include:
Actions of wreath product groups. These are the music-theory "UTTs", which construe the "neo-Riemannian" groups (the music theory Riemann) and "contextual transformations."
Directed graphs with arrows labelled in a monoid and nodes labelled in a set acted on by the monoid. These are called "Lewin nets". They are called simply "Nets" if we relax the restriction that the diagram must (in the category theory sense) commute, and allow multiple arrows (or labels) between any ordered pair of nodes.
Theory of Sturmian morphisms on words. This is used as a model for scale theory.
Grothendieck topologies are at the root of the voluminous music theory work by Guerino Mazzola and his associates (the Topos of Music book).
March 30, Mon, 1:30-3:20, first meeting to discuss how the seminar will progress this term.
Mon April 6 1:30-2:30 Music rm 212, and Th April 9 1:30-2:30 Music rm 114: Drew on contextual groups
Generalized Contextual Groups, Thomas M. Fiore and Ramon Satyendra, MTO 11/3 (Aug 2005)
Two papers by Drew Nobile (attached as VoiceLeading and UTT)
NB in general these involve wreath-product group actions on the set of all major and minor triads. The action is "contextual" in that for each operation, the action on an element depends on some characteristic of that element (whether the triad is major or minor).
There is a separate page below we can use for notes and comments about this.
For the Future:
Here is some interesting work in mathematics of cognition:
See also Michael Leyton and my article about it, Chloe's Friends
More on wreath products in the future. See
Wreath Products of Groups and Semigroups, J D P Meldrum, Longman 1995.
April 13 Mon 12:30-1:20 Music 212, then Th 1:30-2:20: Fiore/Satyendra Generalized Contextual Groups, Thomas M. Fiore and Ramon Satyendra, MTO 11/3 (Aug 2005) paper discussion. See also Lewin GMIT duality of "T"s and "P"s for nonAbelian groups, Theorems 3.4.1-2, 3.4.4-6, 3.61-4, 184.108.40.206 - 4.1.5, 4.1.6, 4.1.7.
For a concise summary about UTTs see Douthett and Hook, UTTs and the Twelve-Tone Music of Webern, Perspectives of New Music 46/1 (2008), 91ff. The pdf is attached below.
On Thursday, Drew led a great discussion of SIMPs and COMMs (GMIT's Ts and Ps), and diagrammed all the relations among the three related groups IVLS and its Ts and Ps, and the mixed Abelian groups half SIMP half COMM in both ways. Ada asked some great questions about duality. (Apparently the T vs P dual groups that arise in music theory and their kind of duality are not well known, or known at all, in regular group theory.) The anti-isomorphism between T and IVLS that arises from the GMIT definitions of IVLS and transposition only plays when these are non-Abelian groups. Yet all three are isomorphic (e.g. to D24 in the Fiore/Satyendra-article case); the anti-isomorphism only plays when you have all three groups acting on the same set and so have to keep the names of the elements consistent across all the groups. Remarkably, Lewin explains all this very clearly already at the end of Ch 1 of GMIT. There is a very clear little proof at the end of the Fiore/Satyendra that just nails down the co-extensivity of COMM and P for a given GIS.
How are IVLS and T related anti-isomorphically?
INT<x, Tj(Ti(x))> = ij
Ts: Ti Tj
x Ti(x) Tj(Ti(x))
ints: -- i --> -- j ----->
------------ ij ---------->
You recall that I was trying to remember something from standard group theory that reminded me of this. On p 158 of my edition of D&F, exercise 8 for section 5.2 on direct products (and ex 7 too) show that a direct product group that is a homogeneous "power" group, i.e. G to the nth, is isomorphic to any such group formed by any permutation of the indices of its components Gi. It turns out that the resulting indices involve the inverses of the pemutation (member of Sn). "Any permutation of the 'position vectors' ... which fixes their coefficients is the same as the inverse permutation on the coefficients" (fixing each position vector), and the reason for this is that the permutation element acts on the position vectors as a left group action. (The "position vectors" are the orthogonal projection elements of the direct product group....)
Similarly, we could use right group action notation for the transpositions, which would eliminate the anti-isomorphism -- the group of transpositions would be just isomorphic to the group of intervals, as Lewin notes casually in GMIT 1.11.4. This might be more intuitive. You have to choose whether you prefer T or P to be the anti-isomorphic group to IVLS.
However, switching to a right group action for the direct product group permutations can break the homomorphism there. There seems to be a certain, dare I say, duality between swapping hands -- right vs left action -- and homo/antihomomorphism. The other dualities we saw in music theory are the Riemannian mode-duality, where the action depends on the mode of the object acted on, and the T-group vs P-group duality, where they are pairs of anti-isomorphic groups in a GIS; whenever there is one, there is its dual, in that particular set of relations among the groups IVLS, T, and P..
April 20 12:30 rm 212 and April 23 1:30 rm 114:
Professor Perkins gives us an intro to category theory, topoi, and Grothendieck topologies
(D&F Appendix II is a brief introduction via hom-sets, but Patrick does not use this approach.) The standard book is Mac Lane, Categories for the Working Mathematician; this is a hard read. Here is my fly-over, or drive-by:
A category has objects and arrows. When the arrows compose into chains, the chain compositions must be associative. There must be an identity arrow for each object, that composes as an identity arrow with the other arrows. The arrows are called morphisms between objects in a category. A morphism between categories is called a functor. A morphism between functors is called a natural transformation (which is where it gets interesting). A hom-set hom(a, b) is the set of all morphisms from one object to another. Functors may be full, faithful, and/or forgetful (see refs).
The opposite category reverses the arrows in a way that is consistent with the original category (yet another duality of the anti kind), so that arrows (f-opp compose g-opp) = (g compose f)-opp and any sentence about C translates into a dual sentence about C-opp (e.g. f-opp is monic IFF f is epi). Functors are covariant or contravariant; a contravariant functor is a morphism of arrows that composes arrows in the anti fashion (as in IVLS and T). A hom-functor leaves one of the argument-slots blank, so that hom(a, _) ranges over all sets of functors from a to each of the other objects in that category. The (set-valued) contravariant hom-functor hom(_, b) takes C-opp to Set (p 34 Mac Lane) as a composition on the right (handedness again), such that for arrows f and g, g* compose-on-the-right f = f compose-on-the-left g. This is the familiar anti-morphism situation from GMIT. ... and so on. But this has taken us to the point of Mazzola's basic and ubiquitous "@" notation in Topos. See my remarks on the Topos of Music subpage to this site, linked below.
From Prof Perkins (See three articles attached to this site):
My goal is to understand Mazzola's book the Topos of Music. The subject of category theory is vast but I want to focus on the parts he uses (as best as I can tell).
Here are some things to read on categories. There's also a paper on abelian groups because I think the category of modules over the ring of integers will be important.
Wikipedia has a decent article on rings: http://en.wikipedia.org/wiki/Ring_(mathematics)
On Monday, I'd like to give some examples of categories. I'll talk about sets, vector spaces and modules over a ring. Mazzola uses the category of contravariant functors on the category of modules over a ring. (Gulp!) Given time, I'll introduce functors. He only seems to use some very special cases.
plus the tutorial by GM in the Mazzolaexamples attached file.
For Mon April 27 and Th April 30:
Be sure to read my comments above, plus the attached file cat101.pdf. When this file bores you -- though the pushouts of specifications excite me -- try this link to Mac Lane's book on Sheaves, pp. 31 ff on characteristic functions and subobject classifiers (and yes, presheaves are important, if you read on). This is tough stuff.
Prof. Perkins will lead us through Mazzola's Topos of Music constructions this week. The attached file Mazzolaexamples.pdf is also good to look at at this time.
I also attached two more files, my Cool Tools article JMMRahn-as published.pdf , which does connect Lewin nets and Nets with categories etc, and the PNM article on the same subject by Mazzola and Andreatta (mazzolapnm.pdf).
There is some music-specific math for Reti-type motivic analysis at http://www.brocku.ca/mathematics/people/buteau/
Mon April 27 John leads through cat101 and other cat theory basics.
There is some interesting advanced stuff here: http://www.staff.ncl.ac.uk/peter.jorgensen/index.html
There is some music-specific math for Reti-type motivic analysis at http://www.brocku.ca/mathematics/people/butea
Thursday April 30 Read Rahn Cool Tools article (JMMRahn-as published.pdf attached below). Look at Mazzola/Andreatta on K-nets attached as mazzolapnm.pdf.
During May we have been independently of TOM doing the basic math underlying TOM. As of May 14 we have done Z Modules, category theory with limits etc, and covered independently the point made the the PNM article by Mazzola and Andreatta, that a Lewin-net of the Knet kind is an element of the limit of the category. I'll try to summarize here.
We note that the usual module morphisms are in TOM tweaked by composition with translations (transpositions) so that the revised module morphisms now parody the Tn/TnI group. This is necessary to avoid subgroup black holes, so to speak, as the usual morphisms (like those in the Hall article) cannot get you out of a set of elements forming a subgroup of Zn. This fixes that problem at the expense of breaking the module category so that it probably no longer has finite limits, as needed for a topos. Mazzola then uses the customary "gadget" of set valued contravariant hom functors (presheaves) over the new module category; such as these always form a category of functors such that, as the Yoneda lemma states, the natural transformations among these hom functors are the same as (NB not just isomorphic to) the original hom sets. This drastically compresses and oversimplifies just for a reminder of the trajectory so far.
We should all read M/A PNM pp 101-109 mazzolapnm.pdf at this point to pin down Mazzola's use of the Yoneda construction via presheaves. I have attached to this webpage pp 744-749 from Eduard Cech, Topological Spaces, on presheaves and sheaves, the topological definitions. (This is the book I found in a used bookstore in 1967....)
We continue next week to examine suboject classifiers in this topos (pp 103-104 PNM) , and how this fits together into a Grothendieck topology.
This term will be a chance for you to fill out gaps in your music/mathematical knowledge and to pursue research projects in related areas of interest to you. We will discuss how to arrange all this at our first meeting on Jan 6, room Smith 111. So far, people have expressed interested in:
1. studying the algebra (group theory through category theory) that is used most often in music theory
2. Klumpenhouwer Networks and Lewin Nets
3. mathematical models of abstract voice leading and topological spaces of connections among pc set types
but we will be open to whatever people are interested in. This may end up with some sessions being tutorials by me or another seminar member, and some sessions having members of the seminar present ongoing readings and research in some particular area.
I have ordered the usual algebra textbook for those who need it, Dummit and Foote Abstract Algebra; we'll use this for any review of the math. The most relevant journals are Perspectives of New Music, Journal of Music Theory, Music Theory Spectrum, and Journal of Mathematics and Music. The most important music theory books for this topic are Lewin, GMIT, and Mazzola, The Topos of Music.
Articles of interest:
Oren Kolman, Transfer Principles for GIS, PNM 42/1 (2004)
Rachel Hall, see separate page below http://www.sju.edu/~rhall/Contextual/contextual.pdf
Dan Vuza, review of GMIT, PNM 26/1 (1988)
Dan Vuza, Supplementary Sets and Regular Complementary Unending Canons (Part One), PNM 29/2; part 2, PNM 30/1, part 3, PNM 30/2.
Moreno Andreatta, Methodes algebriques (PhD thesis, Paris) http://faculty.washington.edu/jrahn/TesiMoreno.pdf
Rahn articles: Cool Tools... Vol 1 no 1, Journal of Mathematics and Music;
Approaching Musical Actions, PNM 45/2 http://faculty.washington.edu/jrahn/RahnActions.pdf;
The Swerve and the Flow, PNM 42/1 http://faculty.washington.edu/jrahn/SwervePNM.pdf
Chloe's Friends (A Symposium About Music and Mathematics). Perspectives of New Music 41/2 (Summer 2003): 8-30. A drunken conversation about semi-direct products, splitting group extensions, Wreath products, related to Michael Leyton's books, plus (in an appendix) applications to the Mazzola Topos (see page)
Reminder: For Jan 26, we'll do more D&F, more on GMIT GIS and my questions in that page, and start discussing Kolman. We will do the Hall article later, as I think we'll have more than enough to do this week.
For Feb 3: Discuss Kolman (see page), and further review the relevant group theory (mostly in Ch 4). Start discussing the Hall article. Our next new reading will be my Musical Actions article, which seems relevant (see above).
For Feb 10: Finish Kolman, start Hall. For Hall, see p. 35 D&F, but to really dig in we should read D&F Chapter 11, which we can't quite do yet (need to get more done of Ch 1-6.) And Chapter 10, also; see the sections on exact sequences etc. in Ch 10. We may just have to rely on Drew to breeze us through this for the time being as the Hall relies on a lot of the more advanced algebra (but still well within D&F).
For Feb 17: focus on Lewin nets in GMIT, Ch 9 et al.
For Feb 24: Rahn presents Actions.
March 3: Drew on Hall's article. Doug on game theory.
March 10: Orit on Stockhausen Klavierstueck 10. Note the deep Z6 structure and its GIS. Ben on a Tango!