Music 575, Seminar and Theory, Autumn 2004

Music and Mathematics

Mondays 3:30-5:20, room 212

John Rahn



VERSION DATED 22 November 2004




In this seminar we will learn some common basic mathematics used in music theory, mostly group theory and other algebra. We will then read some recent  research from Europe and the USA in the area of mathematical models of music and musical performance. This is a hot, emerging research area in music theory.


This syllabus is adapted from a public lecture series I taught in Barcelona, Spain, during 2003-2004.

The pacing and content of it are subject to revision as we learn what we already know and what we need to learn. I am not including a week-by-week schedule in advance here for that reason. Instead, I will be posting updated versions of this syllabus on the web each week so you know what we covered, where to find it in the text or other books, and what we’ll do next week; all the details for the next week may not be up until shortly before the class.



By The Week (Though we will stay flexible)


Week 1 10/04/04: introduction and discussion; sets, relations and graph theory. Arrows everywhere. Reflexivity, transitivity, symmetry. Orderings, linear, lattice, partial; trees and rhizomes. Similarity relations vs. equivalence; equivalence classes and partitions. The axiom of choice and the continuum hypothesis. Cantor’s Paradise. Modelling—pasting math objects onto the real world: triumphs and caveats. (references: Dummit and Foote pp. 1-12, and see the references in the Additional Remarks of this syllabus on models, and on structure, algebra etc (Suppes and Gordon, Mates, Hunter, van Heijenoort, Fraenkel, Bar-Hillel, and Levy.)



Week 2 10/11/04: Mappings: domain, codomain, range; Image=range, preimage, fiber of f over b element of range; into (injective, distinct images have distinct preimages), onto (surjective, every element of codomain has a preimage so codomain=range), 1-1 (bijective, both inj and surj).. Composition of mappings. The integers mod 12 as a model of pitch classes. (NB: The material covered so far can be referenced in pp. 1-12 of Dummit and Foote. The rest of this week’s material is mostly in D&F pp. 13-21.)


Groups. Def: A group is a carrier set G and binary operation * G X G à G such that * is associative, there exists an identity element in G such that a*e=a (and e*a=a), and for each a in G there is some inverse of a, -a in G, such that a*-a=-a*a=e. If a*b=b*a for all elements of G then the group is abelian or commutative.


Semigroup (associative only), monoid (semigroup with an identity). Semigroups are important as they are the basis (for example) of algebraic machine theory (computer science) and other applications. Monoids are important as they are the structure of  more abstract constructions, such as categories. The theories of semigroups and monoids are not simpler than the theory of groups.


A subgroup is a subset of the group carrier set which satisfies the conditions for a group, e.g. {0 3  6 9}, +mod12 is a subgroup of Z12, +mod12, but {0  1 3 6 9 11} is not, because e.g. 1+3=4 which is not in that subset of the carrier.


N.B. Def of binary operation means no result of the operation can be outside the group, i.e. closure under the operation. A subgroup also must be closed with respect to inverses; each element in it must be pairable with its inverse element in the subgroup. In a group, the identity is unique and the inverses of each element are unique. A well-known result states that H is a subgroup of G IFF H is not empty and for all x,y in H, x-y is an element of H (the composition of x and the inverse of y; this condenses the criteria into a single test). For finite H, this amounts to being closed under the binary operation.


Z12 is a group. Z12 models the pitch classes. Subgroups and cosets of Z12 and their pitch-class interpretations. (Tease: Generators and cyclic groups with implications for other ets. E.g. in prime moduli every interval size is a cyclic generator, a very different group structure from Z12 or D12 with very different musical implications. Ets 7, 53, and so on; scale theory.)


For next week, also read Basic Atonal Theory pp 1-39.



Week 3 10/18/04:


Permutation (permutation group, symmetric group)


A permutation is any bijection from a set A to itself. The set of all such permutations of A is a group under the binary operation composition of mappings, called the “symmetric group.” If A is finite with n elements, e.g. the integers from 0 to n-1, this group is called the (non-Abelian) symmetric group of degree n, Sn. The order of Sn is n! since there are n! different possible orderings of the n things in A. Sn is isomorphic to every group of all permutations whose carrier set A has n elements. Any subgroup of an Sn is called a permutation group. Cyclic decomposition of permutations. (D&F 29ff) NB mappings that are not permutations do not have cyclic decompositions. 


Homomorphism: If (G, *) and (H, @) are groups, a map Q: GàH such that Q(x*y)=Q(x)@Q(y) for all x, y in G is a homomorphism from G to H.


Informally, we might remember this as: the image of the combination equals the combination of the images (under the homomorphism mapping).


If the homomorphism map Q is also a bijection, then Q is called an isomorphism, and G and H are called isomorphic, written G~=H. Two groups that are isomorphic are (group-)structurally identical, so they are the same group to within rewriting the names of the operations and elements. (D&F p. 37)


(Homomorphism itself, or more generally “morphism,”  is important because it is a map between algebraic structures such as groups which preserves some, but not necessarily all, structure from one to the other. It is the basic tool for relating algebraic entities. For example, it lies behind the idea of group representation, and is basic to the definition of categories.)


NB Isomorphism is equivalence for groups (and other algebraic entities) as far as group structure is concerned. However, two isomorphic structures may display important musical differences when interpreted in music. We show that the cycle-of-fifths transform of a musical structure (tune) is isomorphic to its pre-image, but clearly quite different in musical effect.


Remarkably (Cayley’s Theorem), every group is isomorphic to a permutation group. This means that no matter what the carrier set and binary operation of the original group are, the structure of that group is replicated in a group whose set has mappings (permutations) as its elements, and composition of mappings as its group operation. This evacuates any ontology below this level, for group theory. (D&G 122)



Group action. A group action of a group G  (with binary operation *) on set A is a map from G X A to A, written as g(a) for all g in G and a in A, such that universally in G and A, g1(g2(a))=(g1*g2)(a) and e(a)=a (e the identity in G).


The notion of group action is to ensure that combination of group elements by the group binary operation is consistent with application of each group element to each member of the set A. For example, a computer program in a functional language (as described by John Backus) is a string of embedded function applications, such as f(g(h(i(j(k(x)))))), where x is the input datum and the output is the result of evaluating the expression. If the language is a group action on the data set X, we can replace any part of the string of functions with a single function which is their composite. For example, if f composed with g = w, then f(g(h(i(j(k(x)))))) = w(h(i(j(k(x))))).


Then we can prove (see D&F p. 42, also pp. 114 ff for a more advanced treatment):

For each fixed g in G there is a map sg: A à A, sg(a)=g(a).

This map sg is a permutation of A.

The map from G to SA (the group of all permutations of the carrier set A) defined by gàsg is  a homomorphism. This is called the permutation representation associated with the group action.

(For the “symmetric group” of permutations see D&F pp. 29ff. NB  in general, a “group representation” is a homomorphic image of a group.)


The group of pitch-class transpositions Tn(x) àx+n acting on the set (or group) of integers mod 12.


Pitch-class interval defined informally in terms of transpositional operations acting on Z12. This will be relevant when we later study Lewin’s Generalized Interval Systems (GIS). Informally, the name of each pc is the interval that some reference pc must be transposed to get the pc named --  an action with reference to an arbitrary base pc. Thus, the pc named zero is named after the number of unit intervals one must transpose the base pc (itself) to get the pc (itself), namely 0 units.  


Read BAT Ch 3 and Ch 4.


Week 4: 10/25/04


First, let’s review some BAT constructions if we need to. If not, skip to [end of review of BAT] below. (yes, this is like FORTRAN…)


Ordered interval and unordered interval for pitches and for pitch classes. The interval content of  a set of pc. The Tn common-tone theorem. Tn preserves interval so Tn is a isometry group; there are others to study later. If the ordered interval between two pcs x and y is i<x,y>=y-x, then

T(i<x,y>)(x)=T(y-x) (x)= x+(y-x)=x, proving that transposing x by the interval between x and y gets you y. The unordered pc interval i(x,y) is defined as the smaller of i<x,y> and i<y,x>. i<x,y>+i<y,x>=0.


Note that although the classical notion of distance is symmetric, i.e. the same in both directions, here

ordered interval is primary, unordered interval (distance) is secondary and defined in terms of ordered interval. All scientifically measured quantities such as interval need to be operationally oriented, related to what you do to get the measurement. Moving through an interval to measure it is always in one direction and conceivably one could get two different quantities depending on direction – as we do for pitch class interval. (Real life examples: the distance between Paris and BCN for some airline might vary according to the route one must take in either direction.) This is Lewin’s transformational outlook.


Types of sets of pc are equivalence classes of sets of pc under some canonical operation such as Tn. Tn types. The normal form of  a set of pc. The representative form of a type of pc set. (For this sort of material see Rahn BAT CH 1-3.)


TnI (inversion), the other isometry of the line. Tn and TnI are the only two isometries of the line, so we can’t hope to find any other musical transformations that preserve (unordered) interval structure. Only Tn preserves ordered interval.


TnI(x)= -x+n. TnI preserves unordered pc interval, the classic notion of distance, but not ordered pc interval as

i<TnI(x), TnI(y)>=i<(-x+n), (-y+n)>=(-y+n)-(-x+n)=x-y=i<y,x>.

One could say mapping each pc to its TnI image converts each ordered pc interval to the same distance “in the opposite direction,” that is, from y to x instead of x to y. Also,

i<x,y>+i<TnI(x), TnI(y)>=0 in all cases (y-x+x-y=0). But since i(x,y) is always the smaller of the two possible ordered intervals it remains unchanged under TnI.


The “n” in TnI is called the “inversional index” of TnI. Each pair of pcs related by any TnI add up to the index, n. To show this, x+TnI(x)=x+(-x+n)=n for any x, n. The axis of symmetry for TnI is n/2, one half of the index.


Tn and I do not commute, since T-nI=ITn (ITn(x) = I(x+n)=-x-n=T-nI) as is true for the dihedral group in general; in music theory, TnI is the conventional order: first invert, then transpose. Tn/TnI equivalence classes of pc sets. (See BAT.)


[end of review of BAT]


The dihedral group D24 of Tn and TnI, which we notate “Tn/TnI.” (D&F pp 23ff)  Historically, the dihedral groups are visualized in group theory as groups of symmetries of rigid objects (in our case, planar polygons), where a symmetry is (informally) any rigid motion in n+1-space of the n-dimensional polygon which covers the original polygon, permuting the vertices among themselves by this rigid motion. This is exactly the group of rotations and reflections of the polygon. The clock diagram models such a polygon.


Generators and relations of a group. Group presentations: D2n = <r,s | r**n=s**2=1, r*s=s*(r**-1)>, fitting what we have said about Tn/TnI, where r=T1 and s=I. (D&F pp 26-7) Cyclic groups.


The permutation representation associated with the group action of the dihedral group of Tn and TnI on the set of all pitch classes. Cyclic decompositions of TTOS.


Recall from last week that a permutation is any bijection from a set A to itself. 


Examples from Tn/TnI:  we take advantage of the permutation representation associated with the group action of the dihedral group D24=Tn/TnI on the set of integers modulo 12 which model the pitch classes. Each operation in Tn/TnI is represented by the permutation of A which is its image under the homomorphism, as defined above under group action. D24 is thus represented as a permutation group.



T5*T1 = T5(T1(x)) = T6(x)


T1 = 0 1 2 3 4 5 6 7 8 9 10 11   e.g. 0 maps to 1

        1 2 3 4 5 6 7 8 9 10 11 0


T5= 0 1 2 3 4 5   6  7 8 9 10 11  e.g. then 1 maps to 6

       5 6 7 8 9 10 11 0 1 2  3   4


T1*T5 =   0 1 2 3 4   5   6  7 8 9 10 11 e.g. in the composite, 0 maps to 6

                 6 7 8 9 10 11 0  1 2 3  4   5



Orbits (D&G 116-117). Let G be a group acting on a nonempty set A. Then


(1)   The set {ga | g element of G} is called the orbit of G containing a. (This is the set of all elements of the set A which are images of some one element of A, a, under all operations in group G, G acting on A.)


(2)   The action of G on A is called transitive if there is only one orbit, that is, for any two elements of A a and b, there is some g in G such that a=g(b).


(3)   The set of all orbits partitions A: the orbits do not intersect, and their union is A. So each orbit is an equivalence class – the class of all elements of A equivalent to some one element a under the action. (The number of elements in each class equals the index of the stabilizer of a in G, |G : Ga|; for the moment we will not pursue definitions of normality, kernel, stabilizer, centralizer, and so on; see D&F pp. 114 ff.)


Let us recall from last week some things about cyclic groups.



Cyclic groups. (D&G 55ff) A group G is cyclic IFF G can be generated by a single element, that is, for at least one g element of G, G={gn | n an integer}. The cyclic group generated by an element g is notated <g>. Any two cyclic groups of the same order are isomorphic. Every subgroup of a cyclic group is cyclic.


Cyclic notation. Each element of Sn (each permutation) has a unique cyclic decomposition whose product is Sn. (D&G p. 117ff)


And let us tie this to the idea of orbits, above:


The sets of numbers that appear in the individual cycles of the cyclic decomposition of some permutation s are the orbits of the cyclic subgroup generated by s, <s>.



Examples from Tn/TnI


T2 (0 2 4 6 8 10) (1 3 5 7 9 11) where of the two whole-tone sets, the evens are in one equivalence class, and all the odds in the other. The two cycles contain the orbits of <T2>, that is, the two sets of pc obtained by letting <T2>={T0 T2 T4 T6 T8 T10} act on the integers mod 12. The “multiplication” of the two cycles (within Sn) yields Z12. Also, their union is Z12, and the two have no intersection, so they partition Z12, as noted above (because the orbits of the cyclic group generated by T2, <T2>, must partition Z12).


T9 (9 6 3 0) (10 7 4 1) (11 8 5 2) and these are the three orbits of <T9> acting on Z12.


T5I (0 5) (1 4) (2 3) (11 6) (10 7) (9 6)  which are also the orbits of <T5I> acting on Z12.


T6I (0 6) (1 5) (2 4) (3) (11 7) (10 8) (9) which are the orbits of <T6I> acting on Z12.



The complete list of the permutation group representing Tn/TnI in cyclic notation is:


T0 (0)(1) etc

T1 (0 1 2 3 4 5 6 7 8 9 10 11); note that <T1> =Tn; this is true only for a transposition whose number does not divide the n of Zn, 12: 1, 5, 7, 11

T2 (0 2 4 6 8 10) (1 3 5 7 9 11)   2 cycles of 6

T3 (0 3 6 9) (1 4 7 10) (2 5 8 11) 3 cycles of 4

T4 (0 4 8) (1 5 9) (2 6 10) (3 7 11) 4 cycles of 3

T5 (0 5 10 3 8 1 6 11 4 9 2 7) <T5>=Tn

T6 (0 6) (1 7) (2 8) (3 9) (4 10) (5 11) 6 cycles of 2

T7 (0 7 2 9 4 11 6 1 8 3 10 5) retrograde cycle of T5 showing that T5 and T7 are each other’s inverse

T8 (0 8 4) (1 9 5) (2 10 6) (3 11 7) retrograde cycles of T4

T9 (0 9 6 3) (1 10 7 4) (2 11 8 5) retrograde cycles of T3

T10 (0 10 8 6 4 2) (1 11 9 7 5 3) retrograde cycle of T2

T11 (0 11 10 9 8 7 6 5 4 3 2 1) retrograde cycle of T1



For the TnI, an even inversional index n means 5 2-cycles and two 1-cycles, odd index means 6 2-cycles, and in all cases each cycle sums to the inversional index n.


T0I (0) (6) (1 11) (2 10) (3 9) (4 8) (5 7)

T1I (0 1) (2 11) (3 10) (4 9) (5 8) (6 7)

T2I (0 2) (1) (3 11) (4 10) (5 9) (6 8) (7)

T5I (0 5) (1 4) (2 3) (11 6) (10 7) (9 6) 

T6I (0 6) (1 5) (2 4) (3) (11 7) (10 8) (9)

T7I (0 7) (1 6) (2 5) (3 4) (11 8) (10 9)

T8I (0 8) (1 7) (2 6) (3 5) (4) (11 9) (10)

T9I (0 9) (1 8) (2 7) (3 6) (4 5) (11 10)

T10I (0 10) (1 9) (2 8) (3 7) (4 6) (5) (11)

T11I (0 11) (1 10) (2 9) (3 8) (4 7) (5 6)


We can easily see that any TnI is its own inverse since all cycles in any TnI are 2-cycles (construing the singletons e.g. (1) as (1 1).) Thus performing a TnI on itself will simply flip the 2-cycles back to their original position – a set of binary switches, as it were. Among the Tn, only T6 has this kind of cyclic structure, 6 2-cycles, and is its own inverse. Note that each of the 6 2-cycles in T6 sums to a different even number and thus will also be found in just one of the 6 even TnI cyclic decompositions, e.g. (1 7) in T6 is found in T8I.


From the cyclic decompositions above we can predict all common-tone behavior of pc sets under any of the 24 operations in Tn/TnI. For example, if a pc set contains a complete cycle of one of the 24 operations, that cyclic content will appear in the image of that set under that operation.





T4 {0 1 2 5 9} will contain as a subset {1 5 9} which is a cycle of T4.

T9I {0 1 4 5 9} will contain the union of the cyclic subsets {9 0} and  {1 4}



We can also manipulate the order resulting from an operation on some original  ordering. This gives us complete control over our use of the group operations acting on any subset or ordering of some subset of pcs.


Example: Webern Symphony row retrogrades under T6 (maps into itself under RT6) because the members of the T6 cycles appear in retrograde symmetrical positions.


Exercise 1: make three different rows that retrograde-invert under T1I, i.e. each maps into itself under RT1I.


Exercise 2: Make two different rows whose order positions permute in this pattern:

4  3 10 1 0 6 5 9 11 7 2 8

hint – draw a pattern of cycles of arrows showing this permutation from the original 0 1 2 ...


Note that so far, we have restricted our operations to isometries, either the translation group of musical transpositions Tn or the dihedral group D24=Tn/TnI.


The degree of symmetry of a pc set (or other entity) is defined as the number of operations that map the set into itself. This is relative to some group of operations. The degree of symmetry of {0 4 8} is 3 in Tn, but 6 in Tn/TnI (T0, T4, T8, T0I, T4I, T8I). A set will map into itself under an operation IFF its content completely includes only some subset of the cycles of that operation, i.e. complete orbits of <s> in the permutation group representing the parent group.



Week 5: 11/1/04


I am away at a conference this day. The seminar should meet with out me. 


1. Here is an assignment for you all as a group to work out together this week:

Do the Webern analysis (Op 27, 2d movement, on reserve). Work out how what we have been talking about in seminar can apply to this piece. Get as far as you can. You can present it all to me Nov 8.


2. I suggest that you do the Lewin reading, but, if you wish, postpone discussion of it until the following week – or, go ahead with discussion and you can bring up any opacities with me Nov 8.



1. Combinatoriality, all-combinatoriality, and their relation to orbits and cyclic decompositions


Analysis of Webern Piano Variations second movement as 5 2s and 2 1s.


Applying pc set types in analysis. Two approaches to using invariance in the same way: how Schoenberg liked to keep the hexachord content invariant (while varying the order inside the hexachords), two 6s, and Webern liked to keep the dyad content invariant (while varying the order inside the dyads), six 2s. Generalization of content/order dichotomy to working with sets less than 12 pc (manipulating the order resulting from an operation by some TTO), using cycles.

Even if the pc content is not invariant, the resulting set will have the same structure of intervals (measures) if the TTO group consists solely of isometries, or (weaker) the same group-induced structure to within isomorphism if the TTOs include M7.


So the basic syntactical rhetoric is still: Keep it the same in some ways (structure) and different in others (order).



2. Lewin GMIT Ch 1, 2, and 3. Read before the seminar and discuss in seminar. (CH 1 is a review of group theory, most of which we have covered, adding other things as we went.) 



Week 6: 11/8/2004

This week we discussed the Webern Op 27 analysis, then went over Lewin Chapters 1 and 2. The worm at the heart of music theory for all possible universes. Preliminary discussion of Lewin’s def of GIS.



Week 7: 11/15/2004:  more group theory

Be sure to meet with me during the next week or so to talk about the topic of your term paper!



First a little more basic group theory used in Lewin Ch 3:


The quotient group G/H: the set of fibers over elements of H a subgroup of G can be a group with the binary operation defined by XaXb=Xab (the combination of the fibers over a and b is the fiber over the combination (in H) of a and b).


The kernel of a homorphism is the fiber over the identity of H


If a homomorphism of G onto H has kernel K, then G/K (G mod K) is a  group.


If N is a subset of G, for any fixed g element of G define the left coset of N in G as

{gN={gn | n element of G} (right coset, Ng)


If G is a group with kernel K then G/K is a group whose elements are the left cosets of K in G with binary operation uKvK=(uv)K (The combination in the coset group of the u coset with the v coset is defined as the uv coset where u and v are combined in G.)


The cosets of G partition G.


uKvK=(uv)K IFF gn-g is an element of K for all g in G and all n in N (K being the kernel of a homomorphism mapping G into N)


gn-g is the conjugate of n by g


g “normalizes” N if gN-g=N (if N maps into itself under conjugation by g)


N subgroup of G is “normal” IFF every g in G normalizes N (N maps into itself under conjugation by every g element of G)


These statements are equivalent:


1. N is a normal subgroup of G

2. The normaliser of N in G is G (set of elements of G that normalize N=G)

3. gN=Ng

4. left cosets form a group as above


Moreover, N is a normal subgroup of G IFF N is the kernel of some homomorphism from G


The natural projection of G onto G/N is defined as (Greek) p(g)=gN (each element of G is mapped into the left coset it forms with N)


NB the pitch classes are the quotient group Z/Z12. Illustrations of all this.



The Mother of All Music Groups: Summary of groups in basic music theory.


There are many different musical groups: we will learn the deep meaning of the numbers 576 and 2304.


The group of Tn with 12 elements { T0 T1 … T11}, * with composition of mappings as the binary operation.


The dihedral group of Tn and TnI, of order 24.


The 24 X 24 direct product group of  row TTOs, a double dihedral group of order 576 (!) acting on order-number, pc-number pairs, that includes R and r. This can be written as a group of operations <onop, pcop>  where each onop is an order-number operation from rn/rnR and each pcop is a pc operation from Tn/TnI, acting on a set A of elements of form <on, pc> -- an ordered pair of an order number and a pc number. The group action then maps A into A and each element of A, <on, pc>, into <onop(on), pcop(pc)>.


Note that Tn/TnI is isomorphic to rn/rnR, with one slight adjustment in our thinking so that a retrograde r0R = T0I; the “normal” musical retrograde, playing it backwards, would then correspond to the operation r11R. THIS REDEFINES THE MODELLING OF MUSICAL RETROGRADE.


Define an operation on pcs M7(x) = 7x (the circle-of-fifths transform).

The Klein 4-group {Identity, I, M7, M7I} acting on pcs.


The Klein 4-group of {identity, R, I, RI} with composition of mappings. For its group action, see the double dihedral above.


The affine group on pcs, of order 48, that includes Tn, TnI, and TnM7. NB not all isometries any more.


The Mother of All TTO-Groups including all of the above, a 48 X 48 direct product group acting on order number, pc-number pairs. The group is{onops, pcops} taking each onop from the group that includes rn, rnR/, and rnM7 and each pcop from the group that includes Tn, TnI, TnM7. This big group has 2.304 operations.



musical illustration:

Take a tune with 12 notes and order numbers Z12. We want to partition of the tune into 4 instrumental parts (flute, violin, cello, and tuba). Use the homomorphism M3 on the order numbers. The kernel  of this homomorphism is {0 4 8} (fiber over 0). The cosets are the translates of the kernel. The set of cosets partitions the tune into 4 set of notes, one for each instrument, with order numbers {0 4 8}, {1, 5, 9, {2, 6, 10}, {3, 7, 11}. This means each part plays every 3rd note in the tune.





Week 8: 11/22/2004

Discuss Lewin Ch 7, 8, 9, 10 (Networks).


Finish reviewing and discussing Lewin Ch 2 and 3. Read Lewin Ch 7, 8, 9 and 10. Start discussion of  7, 8, 9 as time permits.

There are some problems in the formal ideas in this book, and extensions to them:


1.      GIS: Oren Kolman has recently shown (Kolman 2003) that every GIS can be rewritten as a group, so that all group theory applies directly (“transfers”) to GIS. Among other things, this points up a possible flaw in the definition of GIS; a more intuitive definition would restrict a group of intervals to some cyclic group of one generator (my assertion). (See Kolman 2003.)


2.      Definitions in Ch 9: There is a problem here which prevents having more than one arrow-label between any two nodes. Lewin defines an arrow in his node-arrow def (p 193) as an ordered pair of points, then maps ARROW into SGP, so each ordered pair of nodes has exactly one transformation in the semigroup that labels the arrow (one arrow). This probably originates in Lewin’s work with groups of intervals, which are constrained to work this way. Of course in most groups, such as D24, you need multiple arrows. There are various alternatives which would work for networks with multiple arrow(-labels) for a given ordered pair of nodes. Multiple arrows (or labels on an arrow, depending on the definitional system) in digraphs are standard, and it is hard to see what is accomplished by not allowing more than one relationship between any two nodes in the model. You also need multiple arrows for groups applied to graphs, category theory, etc.


3.      With this change, a Lewin network is formally a commutative diagram in some musical category – a directed graph with arrows labeled in a monoid, such that the composition of paths in the underlying category is associative and so on (definition of category and of commutative diagram.) Lewin says the labels are in a  semigroup but his definition of node-arrow system makes every graph reflexive, providing the identities that augment a semigroup to a monoid. So it is possible to use category theory to explore Lewin networks, much as GIS turned out to be groups: group theory transfers into GIS theory, and category theory transfers into Lewin network theory.


4.      I made this connection in my paper, “The Swerve and the Flow: Music’s Relation to Mathematics,” delivered at IRCAM in October 2003 and subsequently published in PNM 42/1; I think I was the first to say this. I expanded on this idea in a talk at the ICMC, Miami, Nov 2 2004, called “Musical Acts”; in this talk I expanded into the relation of Lewin nets to the fundamental group of a topological space, and to homotopy classes, and adding category theory as a solution to part of a set of criteria for a general music theory. Later in this seminar I’ll give a talk about all this.


Week 8: 11/29/2004  More on Lewin Networks.


Week 10: 12/6/2004

I will present and we can discuss the material from my ICMC presentation, “Musical Acts.” This will reference the ideas in my earlier paper “The Swerve and the Flow” (PNM  42/1), Lewin networks, category theory, and some ideas from topology, in the service of an exploration of  a possibly more adequate, or improved, music theory in general.


To prepare, read “The Swerve and the Flow,” and look at the short appendix in Dummit and Foote on Category Theory (Appendix III, pp. 877-884).








Basic Texts


The most important books for you to read in connection with this seminar are:


Rahn, John, Basic Atonal Theory

Lewin, David, Generalized Musical Intervals and Transformations.


Both are out of print so please be careful with the copies on reserve on the Music Library.


For mathematics, the best source is probably:

Dummit and Foote, Abstract Algebra, available as a text in the Bookstore and on reserve in the Math Library. This is our basic reference for most of the mathematics.


The most mathematically advanced book in music theory is the newly published:

Mazzola, Guerino, The Topos of Music (see notes in bibliography)


Moreno Andreatta has written the first reasonably comprehensive history of mathematical music theory in the 20th century in “Methodes algebraiqes….” This also contains some exposition of material also being taught in this seminar that is a little more mathematical than what is found in BAT.


See the bibliography below for details.



Euromamuth, USmumath, and the new journal


In the recent past few years, there has been a renaissance of mathematics/music research in Europe. Its centers have been the MaMuth seminars run for some years now by Guerino Mazzola (Zurich), and a series of meetings at IRCAM in Paris, organized by  Moreno Andreatta. Recently, Moreno and I co-organized a conference on American “Set Theory” at IRCAM in October 2003, and I gave papers at other meetings there, and served on the jury for Moreno’s PhD defense in Paris. I had many long talks there with interesting people such as Moreno Andreatta, Guerino Mazzola, Thomas Noll (Berlin, an associate of Mazzola’s), and Oren Kolman (London).


The most startling European development is Mazzola’s mammoth new book (1400 pp) called the Topos of Music. This book re-invents all music theory on a formal underlying basis of Grothendieck Topologies, which are a kind of topology of sheaves of categories. These are quite recherché entities based on ground-breaking research done in Paris in the 1960s. Most mathematicians are not yet fully conversant with this subject. The math involved is primarily group theory, category theory, and topology. Mazzola in particular relies heavily on category theory and the Yoneda perspective. For a fuller description of the Mazzola book, see

You can order this directly from the publisher at


Meanwhile, in the USA, Robert Peck has been organizing music and mathematics conferences as sub-conferences of meetings of the American Mathematical Society. These get-togethers have sparked a new interest and solidarity among Americans who do this kind of research. Thomas Noll, Moreno Andreatta, and Guerino Mazzola have attended some meetings in the USA, which has helped introduce the Euro strand here. PNM has recently published a number of papers in this area, as part of Call for Papers in music and mathematics which I put up after David Lewin died. See especially “Chloe” and “The Swerve” by myself, and the papers by Robert Peck and Oren Kolman in the recent issues of PNM. JMT has also been publishing more heavily in this area, with David Clampitt as editor. Most recently, Thomas Noll is heading up an effort to found a new, international journal for this kind of research. The project now has an Editorial Board and a publisher so it should materialize some time in early 2005. Guerino Mazzola and myself have been invited to write two “visionary articles” for its first issue.



Annotated Select Bibliography  * means on reserve


Andreatta, Moreno. Methodes algebriques en musique et musicologies du XX siecle: Aspects theoriques, analytiques, et compositionels. Unpublished doctoral thesis, 2003. An interesting doctoral thesis in French, on the history of algebraic methods in music theory, including a French take on the American tradition, with some original work on canons.


Birkhoff, Garrett, and MacLane, Saunders. A Survey of Modern Algebra. NY: MacMillan, 1977. This is a classic undergraduate textbook for math majors, much less comprehensive and advanced than Dummit and Foote (see below).


Callender, Clifton. Formalized Accelerando: An Extension of Rhythmic Techniques in Nancarrow's Acceleration Canons. PNM 39/1 (Winter 2001): 188-210. Calculus applied to tempo.


Carey, Norman, and Clampitt, David. Self-Similar Pitch Structures, Their Duals, and Rhythmic Analogs. PNM 34/2 (Summer 1996): 62-87. Scale theory using continued fractions produces infinitely recursive fractal rhythmic sequences.


*Dummit and Foote, Abstract Algebra, Second Edition. NY: Wiley, 1999. A comprehensive graduate textbook for math majors, including treatment of groups, rings, fields, modules and vector spaces, Galois theory, introduction to homological algebra and representation theory. This is the textbook for this course, and is on reserve in the mathematics library.


Haralick, Robert. The Language of Mathematics. Unpublished paper, University of Washington College of Engineering, 1991. Basic intro to graph theory etc.


Haralick, Robert. A Consistent Labeling Theoretic Approach to Music. In the proceedings of the Music and Science symposium, University of Washington, February 1991. CCISM, Music DN-10, University of Washington. Graph theory underlying finite topology and geometry as a model for musical structure, based on his well-known work in robot vision.


Hook, Julian. 2002. Uniform Triadic Transformations. JMT 46,1&2 (2002): 127-52.


Kolman, Oren. Transfer Principles for Generalized Interval Systems. PNM 41/2 (2003).


*Lewin, David. 1987. Generalized Musical Intervals and Transformations. New Haven: Yale University Press. A canonical work which every serious student of music should read. (See related articles by Lewin, Klumpenhouwer, and others, and subsequent analytical case book by Lewin (1993).)


Lewin, David. 1993. Musical form and transformation: 4 analytic essays. New Haven: Yale University Press. Analytic essays using Lewin nets.


*Michael Leyton. 2001.  A Generative Theory of Shape. Berlin: Springer Verlag. A wreath-product (group) theory of shape in general, derived from design software but applied very widely.


Mac Lane, Saunders. Categories for the Working Mathematician, Second Edition. NY: Springer Verlag, 1978. The standard text, graduate level.


Mazzola, Guerino. 1985. Gruppen und Kategorien in der Musik : Entwurf einer mathematischen Musiktheorie. Berlin : Heldermann.


Mazzola, Guerino, and Zahorka, Oliver. 1993. Geometry and Logic of Musical Performance. SNSF Report, University of Zurich.


*Mazzola, Guerino. 2002. The Topos of Music. Basel: Birkhauser Verlag. Monstrous book, over 1300 pages, re-inventing music theory from foundations up, using very advanced research mathematics – algebraic geometry and Grothendieck topologies.


Mazzola, Noll, and Lluis-Puebla, editors. 2004. Perspectives in Mathematical and Computational Music Theory. Osnabruck: EPOS. A recent collection of essays by Europeans, at an advanced level.


Morris, Robert. 1995. Compositional Spaces and Other Territories. PNM 33 (1995): 328-59.


*Morris, Robert. 1987. Composition with pitch-classes : a theory of compositional design. New Haven : Yale University Press. Extends Rahn 1980 in many ways, covering serial compositional techniques.


*Rahn, John. 1980. Basic Atonal Theory. NY: Schirmer Music Books. A classic basic text. (The exercises and optional sections contain information on the group theory of Z12.)


Rahn, John.  1995. Some Remarks on Network Models for Music. In Musical Transformations and Intuitions: A Festschrift for David Lewin, ed. Raphael Atlas and Michael Cherlin. Pendragon Press.


Rahn, John. 2003. Chloe’s Friends: A Symposium about Music and Mathematics. PNM 41/2.


Rahn, John. 2003. The Swerve and the Flow: Music’s Relation to Mathematics. Paper read at Colloque Autour de la Set-Theory, IRCAM, Paris, Oct 15. 2003. Published in PNM 42/1.


Reiner, David. Enumeration in Music Theory. American Mathematical Monthly, January 1995: 51-54.


Rumelhart, D. E. and J. L. McClelland. Parallel Distributed Processing: Explorations in the Microstructure of Cognition. In two volumes. Cambridge, MA: MIT Press. (See the "Nets" bibliography on my web site for related work in Computer Music Journal, and by Gjerdingen and others.)


Suppes, Patrick.  1972. Axiomatic Set Theory. NY: Dover. Classic undergraduate text at Stanford.


Vuza, Dan. Supplementary Sets and Regular Complementary Unending Canons. Serialized in four parts in PNM 29/2, 30/1, 30/2, and 31/1. Lots of group theory applied to canons.




additional remarks:

 on “Neo-Riemannian Theory”


There has been a remarkable efflorescence of theory based on groups of tonal transformations, originating in Lewin’s book, propelled further by Brian Hyer,  Rick Cohn, and John Clough’s group at SUNY Buffalo (including Jack Douthett and David Clampitt), and energized by a conference at Buffalo in 1993. Much of this is collected in a special issue of JMT (Spring 1998, vol. 42/2), with a good introductory essay by Rick Cohn. The most recent development is Julian Hook’s excellent treatment using wreath products, in the most recent issue of JMT.



additional remarks:

 Scale Theory


John Clough also re-energized tonal scale theory (somnolent since the Renaissance) in work since the 1980s, initially focussing on properties of the diatonic scale and variants of this. Jack Douthett, a mathematician, entered into a lengthy and fruitful collaboration with Clough, and Clough’s students David Clampitt and Norman Carey (now teaching at Eastman and Yale) wrote a brilliant paper in MTS which jolted this line of research further along. You will see plenty along these lines in the journals.


Scale theory and Neo-Riemannian theory together have transformed the music theory of tonality into a modern mathematical study.


additional remarks:

Similarity measures


The spate of articles on measuring similarity among pc sets originated in three articles published simultaneously in PNM 18 (1980), by Morris, Rahn, and Lewin. Morris generalized and extended a measure from Forte based on interval vectors, and interval-based similarity measures have remained a major thread in the subsequent discussion (including Erickson’s). Rahn generalized and extended Morris’s ideas to measures of subset-type content is sizes greater than 2 (interval). Lewin reformulated some of Rahn’s notions with different mathematics. Later, Mark Hoover closed some loopholes in Rahn’s ATMEMB measure, and Marcus Castren (Finland) wrote a good monograph surveying the similarity literature to his time and proposing a measure which improves Rahn. The most recent entry is Quinn (PNM 39/2). Some articles on formal voice-leading also lead from the interval-based similarity thinking.


additional remarks:

structure, algebra, category theory, and geometry


Suppes, Patrick.  1972. Axiomatic Set Theory, an undergraduate textbook, is a good place to start; Gordon, Charles.  1967. Gordon, Introduction to Mathematical Structures is even more elementary.


Further work in this line needs elementary logic (first-order predicate calculus); see any good text, such as Benson Mates, Elementary Logic (Oxford). Slightly more advanced is a text such as Geoffrey Hunter, Metalogic: An Introduction to the Metatheory of Standard First-Order Logic (UC Press). From Frege to Goedel: A Source Book in Mathematical logic, 1879-1931 (ed. Jean van Heijenoort) is an excellent compilation of mind-boggling breakthroughs, including Goedel's most famous contributions (just the original research papers, translated into English when necessary). You can end up with Alonzo Church, Introduction to Mathematical Logic, Vol. 1 (Princeton, 1956). Vol. 2 was never published because its scope reaches beyond current research.


At some point in this series you can enjoy Foundations of Set Theory, by Fraenkel, Bar-Hillel, and Levy (North Holland, 1973). Axiomatic Set Theory by Bernays and Fraenkel (North Holland, 1968) is a good advanced practical construct to supplement Suppes.


The interesting structure of almost-not-structure is the subject of point-set topology, which could prove quite fruitful for music theory; e.g. Topological Spaces, by Eduard Cech (Interscience/Wiley 1966), though this is a somewhat quirky treatment of it; this connects with Haralick's approach, and with Mazzola’s recent book.


For algebraic structures (group theory, fields, category theory, etc.) see any good text on algebra, such as Birkhoff and MacLane, Algebra (introductory); Serge Lang, Algebra (advanced). I particularly recommend Dummit and Foote, Abstract Algebra, Second Edition, NY: Wiley, 1999, which is advanced enough to contain most relevant basic algebra (graduate level text), with a short appendix introducing category theory.


Mac Lane, Categories for the Working Mathematician, is a standard introduction to category theory.


Guerino Mazzola’s new book, The Topos of Music, relies heavily on not only category theory but on theory of sheaves, algebraic geometry, and Grothendieck topologies. There are many, highly condensed but lucid appendices in this book about the relevant mathematics. This is very advanced research-level mathematics. 



additional remarks:

on models


A classic collection of papers from 1960 called Logic, Methodology and Philosophy of Science (ed. Nagel, Suppes, and Tarski) snapshots some ferment in model theory, which you should follow up by reading Tarski (various works, use the library software). There is plenty of recent work in this area which is too profuse and diverse to detail. See issues of the journal Synthese from the 60s-80s for one. Various authors have explored alternate universe theory (at times called discourse theory) as an extension to model theory. Much work has a computer-science connection, as in UW Professor Robert Haralick's highly abstract relational models for robot vision, applicable to music as in "A Consistent Labelling Theoretic Approach to Music" and its references (see Network bibliography).


For a very simple, non-mathematical, musical application see Rahn, John (1979) "Aspects of Musical Explanation." A simple, small model of a formalized music theory for tonal music is in "Logic, Set Theory, Music Theory."  College Music Symposium 19, no. 1 (Spring): 114-27. See also my "Some Remarks on Network Models for Music" for a sketch of a synthesis of relations, grammars, neural nets, and Lewin nets.


additional remarks:

on explanation


The locus classicus on "scientific" explanation is Karl Popper, The Logic of Scientific Discovery (Harper, 1965). See also the short, undergraduate text Philosophy of Natural Science, by Carl Hempel (Prentice-Hall, 1966). The sociology of it is pioneered in the misnamed The Structure of Scientific Revolutions by Thomas Kuhn. Milton Babbitt pioneered the "scientific" model of explanation for music (various articles). Ben Boretz in Meta-Variations, The Logic of What?, and various articles, and others such as myself ("How do You DU", "Aspects of Musical Explanation, " "Notes on Methodology in Music Theory") examined Babbitt's assertion skeptically or refigured it. More recently, Brown and Dempster misread all this, in the course of asserting again a cruder version of a quasi-Popperian model for explanation for music. (The whole exchange is in Journal of Music Theory Vol 33/1, 1989). 


There are plenty of alternatives to the Popperian model for explanation, especially for an art. For a somewhat faded snapshot of some, see my "New Research Paradigms,"  Spectrum 12/1 (Spring): 84-94, its bibliography; see also my article “Aspects of Musical Explanation” (PNM 17/2, 1979), and the general bibliography above.