Music 575, Seminar and Theory, Autumn 2004

Music
and Mathematics

Mondays
3:30-5:20, room 212

John
Rahn

**Syllabus**

VERSION
DATED 22 November 2004

http://faculty.washington.edu/jrahn/5752004.htm

** **

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.

Examples:

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 3^{rd} 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).

PAPERS
ARE DUE FRIDAY DEC 10

** **

** **

**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 20^{th} 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 http://www.ifi.unizh.ch/groups/mml/publications/special4.php4

You can order this directly
from the publisher at http://www.birkhauser.ch/books/math/5731.htm

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.

http://www.ircam.fr/equipes/repmus/moreno/TheseMoreno.pdf

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.

** **