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