Music 575, Seminar and Theory, Autumn 2005
Music
and Mathematics
Mondays
John
Rahn
Syllabus
VERSION
DATED 22 September 2005
http://faculty.washington.edu/jrahn/575A2005.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
This syllabus is adapted from a
public lecture series I taught in
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 10/03/05: 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
Week 2 10/10/05: 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 10/17/05:
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: 10/24/05
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
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=
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: 10/31/05
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: 11/7/2005
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: 11/14/2005: 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: 11/21/2005
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,
Week 8: 11/28/2005 More on
Lewin Networks.
Week 10: 12/5/2005
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 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
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
You can order this directly
from the publisher at http://www.birkhauser.ch/books/math/5731.htm
Meanwhile, in the
Annotated Select Bibliography * means
on reserve
Andreatta,
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,
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,
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.
Lewin, David. 1993. Musical
form and transformation: 4 analytic
essays.
*Michael
Leyton. 2001. A Generative Theory of Shape.
Mazzola, Guerino. 1985. Gruppen und Kategorien in der Musik : Entwurf einer
mathematischen Musiktheorie.
Mazzola,
Guerino, and Zahorka, Oliver. 1993.
Geometry and Logic of Musical Performance. SNSF Report,
*Mazzola, Guerino. 2002. The
Topos of Music.
Mazzola,
Noll, and Lluis-Puebla, editors. 2004.
Perspectives in Mathematical and Computational Music Theory.
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.
*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.
Suppes,
Patrick. 1972. Axiomatic Set Theory. NY:
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 (
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 (
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.
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.