Also available at http://math.ucr.edu/home/baez/week234.html
June 12, 2006
This Week's Finds in Mathematical Physics (Week 234)
John Baez
Today I'd like to talk about the math of music - including
torsors, orbifolds, and maybe even Mathieu groups. But first,
some movies of the n-body problem:
1) Cris Moore, The 3-body (and n-body) problem,
http://www.santafe.edu/~moore/gallery.html
In 1993 Cris Moore discovered solutions of the gravitational
n-body problem where the particles' paths lie in a plane and
trace out braids in spacetime! I spoke about these in "week181".
More recently, Moore and Michael Nauenberg have found solutions
with cubic symmetry and vanishing angular momentum, and made
movies of these. For the mathematical details, try this:
2) Cristopher Moore and Michael Nauenberg, New periodic orbits
for the n-body problem, available at math.DS/0511219
Next, math and music.
Some of you have been in this situation. A stranger at a party
asks what you do. You reluctantly admit you're a mathematician,
expecting one of the standard responses: "Oh! I hate math!" or
"Oh! I was pretty good at math until...."
But instead, after a strained moment they say: "Oh! Do you play
an instrument too? Isn't music really mathematical?"
I guess it's like meeting a Martian and asking them if they like
Arizona: an attempt to humanize something alien and threatening.
You may not have much in common, but at least you can chat about
red rocks.
Of course there *is* something mathematical about music, and lots
of mathematicians play music. I rarely think about music in a
mathematical way. But I know they have something in common: the
transcendent beauty of pure form.
Indeed, in the Middle Ages, music was part of a "quadrivium" of
mathematical arts: arithmetic, geometry, music, and astronomy.
These were studied after the "trivium" of grammar, rhetoric and
logic. This is why mathematicians scorn a result as "trivial"
when it's easy to see using straightforward logic. When a
result seems more profound, they should call it "quadrivial"!
Try saying it sometime: "Cool! That's quadrivial!" It might
catch on.
There are also modern applications of math to music theory. I had
never heard of "neo-Riemannian theory" until Tom Fiore explained it
to me while I was visiting Chicago. Tom is a postdoc who works on
categorified algebraic theories, double categories and the like -
but he's also into music theory:
3) Thomas M. Fiore, Music and mathematics, available at
http://www.math.uchicago.edu/~fiore/1/music.html
4) Thomas M. Fiore and Ramon Satyendra, Generalized contextual
groups, Music Theory Online 11 (2005), available at
http://mto.societymusictheory.org/issues/mto.05.11.3/toc.11.3.html
The first of these is a very nice gentle introduction, suitable
both for musicians who don't know group theory and mathematicians
who don't know a triad from a tritone!
When Tom first mentioned "neo-Riemannian theory", I thought this
was some bizarre application of differential geometry to music.
But no - we're not talking about the 19th-century mathematician
Bernhard Riemann, we're talking about the 19th-century music
theorist Hugo Riemann!
Based on the work on Euler - yes, *the* Euler - Hugo Riemann
introduced diagrams called "tone nets" to study the network of
relations between similar chords. You can see his original
setup here:
5) Joe Monzo, Tonnetz: the tonal lattice invented by Riemann,
Tonalsoft: the Encyclopedian of Microtonal Music Theory,
http://www.tonalsoft.com/enc/t/tonnetz.aspx
6) Paul Dysart, Tonnetz: musics, harmony and donuts,
http://members2.boo.net/~knuth/
Apparently Riemann's ideas have caught on in a big way. Monzo
says that "use of lattices is endemic on internet tuning lists",
as if they were some sort of infectious disease.
Dysart seems more gung-ho about it all. The "donuts" he mentions
arise when you curl up tone nets by identifying notes that differ
by an octave. He has some nice pictures of them!
In neo-Riemannian theory, people like Lewin and Hyer started
extending Riemann's ideas by using *group theory* to systematize
operations on chords. The best easy introduction to this is
Fiore's paper "Music and mathematics". Here you can read about
math lurking in the music of Elvis and the Beatles! Or, if
you're more of a highbrow sort, see what he has to say about
Hindemith and Liszt's "Transcendental Etudes". And if you
like doughnuts and music, you'll love the section where he
explains how Beethoven's Ninth traces out a systematic path in
a torus-shaped tone net! This amazing fact was discovered by
Cohn, Douthett, and Steinbach.
(If I weren't so darn honest, I'd add that Liszt wrote the
"Transcendental Etudes" as a sequel to his popular "Algebraic
Etudes", and explain how Mozart's "eine kleine Nachtmusik"
tours a tone net shaped like a Klein bottle. But alas....)
Let me explain a bit about group theory and music - just
enough to reach something really cool Tom told me.
If you're a musician, you'll know the notes in an octave go
like this, climbing up:
C, C#, D, D#, E, F, F#, G, G#, A, A#, B
until you're back to C. If you're a mathematician, you might
be happier to call these notes
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11
and say that we're working in the group of integers mod 12,
otherwise known as Z/12. Let's be mathematicians today.
The group Z/12 has been an intrinsic feature of Western music
ever since pianos were built to have "equal temperament"
tuning, which makes all the notes equally spaced in a certain
logarithmic sense: each note vibrates at a frequency of 2^{1/12}
times the note directly below it.
Only 7 of the 12 notes are used in any major or minor key -
for example, C,D,E,F,G,A,B is C major and A,B,C,D,E,F,G is
A minor. So, as long as Western composers stuck to writing
pieces in a single fixed key, the Z/12 symmetry was "spontaneously
broken" by their choice of key, only visible in the freedom to
change keys.
But, as composers gradually started changing keys ever more
frequently within a given piece, the inherent Z/12 symmetry
became more visible. In the late 1800s this manifested itself
in trend called "chromaticism". Roughly speaking, music is
"chromatic" when it freely uses all 12 notes, but still within
the context of an - often changing - key. I guess Wagner and
Richard Strauss are often mentioned as pinnacles of chromaticism.
Chromaticism then led to full-fledged "twelve-tone music"
starting with Schoenberg in the early 1900s. This is music
that fully exploits the Z/12 symmetry and doesn't seek to
privilege a certain 7-element subset of notes defining a key.
People found Schoenberg's music disturbing and dissonant at
the time, but I find it very beautiful.
Now comes the really exciting thing Tom told me: two other
symmetry groups lurking in music, and a relationship between them.
First, the transposition-inversion group. This acts as
permutations of the set Z/12. It's generated by two
especially nice permutations. The first is "transposition".
This raises each note a step:
x |-> x + 1
Musicians would call this a half-step, just like physicists
measure spin in multiples of 1/2, but we're being mathematicians!
The second is "inversion". This turns notes upside down:
x |-> -x
The relevance of this to music is a bit less obvious: composers
like Bach and Schoenberg used it explicitly, but we'll see it
playing a subtler role, relating major and minor chords.
The transposition-inversion group has 24 elements. Mathematicians
call it the 24-element "dihedral group", since it consists of the
symmetries of a regular 12-sided polygon where you're allowed
to rotate the polygon (transposition) and also flip it over
(inversion). I hope you see that this geometrical picture is
just a way of visualizing the 12 notes.
So, the transposition-inversion group obviously on the 12-element
set of notes. But, it also acts on the 24-element set of "triads"!
Triads are among the most basic chords in music. Mathematically
they are certain 3-element subsets of Z/12. They come in two
kinds, major and minor. There are 12 major triads, namely
{0,4,7} C major triad: {C,E,G}
and everything you can get from this by transposition. If you
invert these, you get the 12 minor triads, namely
{0,-4,-7} = {5,8,0} F minor triad: {F,Ab,C}
and everything you can get from *this* by transposition.
(Note that {0,-4,-7} = {5,8,0} because we're working mod 12
and the order doesn't matter. I've also included the way musicians
talk about these triads, in case you care.)
Major triads sound happy; when you invert them they sound sad,
just like an upside-down smile looks sad. There could be some
profound truth lurking here. A smile has a positive second
derivative:
. .
. .
. .
which says that things are "looking up", while a frown has negative
second derivative:
. .
. .
. .
which says that things are "looking down". An upside-down smile
is a frown.
(On the other hand, a backwards smile is still a smile, and a
backwards frown is still a frown. So, if you're a company and
the second derivative of your profits is positive, you can say
business is looking up - and you could still say this if time
were reversed!)
But never mind. We had this transposition-inversion group acting
on our set of notes, namely Z/12. Since tranposition and inversion
act on notes, they also act on triads. For example, transposition
does this:
{0,4,7} |-> {1,5,8} C major triad |-> C# major triad
while inversion does this:
{0,4,7} |-> {5,8,0} C major triad |-> F minor triad
So, we've got this 24-element transposition-inversion group
acting on the 24-element set of triads!
But here's really cool part: there's *another* important
24-element group acting on the same set! It's easy to define
mathematically, but it also has a musical meaning.
Mathematically, it's just the "centralizer" of the transposition-
inversion group. In other words, it consists of all ways of
permuting triads that *commute* with transposition and inversion!
Musically, it's called the "PLR" group, because it's generated
by 3 famous transformations.
To describe these transformations, I'll need to talk about the
"bottom", "middle" and "top" note of a triad. If you know a
wee bit of music theory this should be obvious as long as you
know I'm talking about triads in root position. If you're a
mathematician who has never studied music theory and you think
of triads as 3-element subsets of Z/12, it might be less obvious,
since Z/12 doesn't have a nice ordering - it only has a *cyclic*
ordering. But this is enough. The point is that major triads
are sets of the form
{n,n+4,n+7},
while minor triads are of the form
{n,n+3,n+7}.
So, we can call the note n the "bottom", the note n+3 or n+4 the
"middle", and n+7 the "top". Musicians call them the "root",
"third" and "fifth", but let's be simple-minded mathematicians.
Okay, what are the transformations P, L, and R? They stand
for "parallel", "leading tone change", and "relative" - but
what *are* they?
Each of these transformations keeps exactly 2 of the notes
in our triad the same. Also, each changes major triads into
minor triads and vice versa. These features make these
transformations musically interesting.
The transformation "P" keeps the top and bottom notes the same.
I've now said enough for you to figure out what it does...
at least in principle. For example:
P: {0,4,7} |-> {0,3,7} C major triad |-> C minor triad
P: {0,3,7} |-> {0,4,7} C minor triad |-> C major triad
The tranformation "L" turns the middle and top note into the bottom
and middle note when you start with a MAJOR triad. It turns the
bottom and middle note into the middle and top note when you start
with a MINOR triad. For example:
L: {0,4,7} |-> {4,7,11} C major triad |-> E minor triad
L: {0,3,7} |-> {8,0,3} C minor triad |-> G# major triad
The transformation "R" works the other way around. It turns the
middle and top note into the bottom and middle note when you start
with a MINOR triad. And it turns the bottom and middle note into
the middle and top note when you start with a MAJOR triad:
R: {0,4,7} |-> {9,0,4} C major triad |-> A minor triad
R: {0,3,7} |-> {3,7,10} C minor triad |-> D# major triad
Can you see why the transformations P, L, and R commute with
transposition and inversion? It should be easy to see that they
commute with transposition. Commuting with inversion means that
if I switch the words "top" and "bottom" and also the words "major"
and "minor" in my descriptions above, these transformations don't
change!
You should be left wondering why P, L, and R generate the group
of *all* transformations of triads that commute with transposition
and inversion - and why this group, like the transposition-inversion
group itself, has exactly 24 elements!
It turns out some of this has a simple explanation, which has very
little to do with the details of triads or even the 12-note scale.
Imagine a scale with n equally spaced notes. Transpositions
and inversions will generate a group with 2n elements. Let's
call this group G. If you take any "sufficiently generic" chord
in our scale, G will act on it to give a set S consisting of 2n
different chords. Then it's a mathematical fact that the group of
permutations of S that commute with all transformations in G
will be isomorphic to G! So, it too will have 2n elements.
To explain *why* this is true, I need a bit more math.
First of all, I need to define my terms. I'm defining a chord
to be "sufficiently generic" if no element of G maps it to itself.
We then say G acts *freely* on S. By the way we've set things up,
G also acts *transitively* on S. A nonempty set on which G
acts both freely and transitively is called a "G-torsor". You can
read about torsors here:
7) John Baez, Torsors made easy,
http://math.ucr.edu/home/baez/torsors.html
They're philosophically very interesting, since they're related
to gauge symmetries in physics... but right now the only fact we
need is that any G-torsor is isomorphic to G. So, we can identify
S with G, with G acting by left multiplication.
Then, it's a well-known fact that any permutation of G that
commutes with left multiplication by all elements of G must be
given by *right* multiplication by some element of G. And
these right multiplications form a group of transformations
that is isomorphic to G... just as we were trying to show!
In other words: the group of permutations of G has a subgroup
isomorphic to G, namely the left translations. It also has
another subgroup isomorphic to G, namely the right translations.
Each of these subgroups is the "centralizer" of the other. That
is, each one consists of all permutations that commute with every
permutation in the other one! Fiore and Satyendra call them
"dual groups".
In our application to music, the first copy of G is our good old
transposition-inversion group, while the second copy is a
generalization of the PLR group. Fiore and Satyendra call it the
"generalized contextual group".
All this is indeed very general. I don't know a similarly
general explanation of why the operations P, L, and R succeed
in generating all transformations that commute with transposition
and inversion.
I asked Tom Fiore if he and Ramon Satyendra were the first to
show that the PRL group was the centralizer of the transposition-
inversion group. His reply was packed with information, so
I'll quote it:
The initial insight about the duality between the T/I group and
the PLR group was at least 20 year ago. Dual groups in the musical
sense were introduced in David Lewin's seminal 1987 book "Generalized
Musical Intervals and Transformation Theory." This book stimulated
interest in neo-Riemannian theory, since Lewin recalled the
transformations P,L, and R as objects of study.
Major-minor duality was a concern of Hugo Riemann, a theorist from
the second half of the 19th century. Given his interest in duality,
Riemann may have had some intuition about a duality between T/I and
PLR, though it wasn't until after his death that this duality was
formulated in algebraic terms. An algebraic proof of the duality of
T/I and PLR was in the thesis of Julian Hook in 2002.
Ramon and I were the first to prove that the "generalized contextual
group" is dual to the T/I group acting on a set generated by an
arbitrary pitch-class segment satisfying the tritone condition.
(The tritone condition says that the inital pitch-class segment
contains an interval other than a tritone and unison.) Our
theorem has the PLR group and major/minor triads as a special case,
since the generalized contextual group becomes the PLR group when one
takes the generating pitch class segment to be the three pitches of a
major chord. The advantage of our generalization is that one can now
apply the PLR insight to passages that are not triadic. There was a
general move toward this in practice for the past decade (Childs and
Gollin considered seventh chords rather than triads, Lewin analyzed
instances of a non-diatonic phrase in a piano work of Schoenberg, we
analyzed Hindemith, and so on). Most music does not consist entirely
of triads (e.g. late 19th century chromatic music), so the restriction
of PLR to triads was not conclusive.
We did a literature review of recent neo-Riemannian theory in Part
5 of our article "Generalized Contextual Groups", since there have
been a lot of insights in the past 10 years. One of the main
thinkers is Rick Cohn, who came up with (among other things) a
nice tiling of the plane which one navigates using P,L, and R
(Richard Cohn, Neo-Riemannian operations, parsimonious trichords,
and their Tonnetz representations, Journal of Music Theory, 1997).
It is quite geometric.
You read more about these matters here... I'll list these references
in the order Tom mentions them:
8) David Lewin, Generalized Musical Intervals and Transformations,
Yale University Press, New Haven, Connecticut, 1987.
9) Julian Hook, Uniform Triadic Transformations, Ph.D. thesis, Indiana
University, 2002.
10) Adrian P. Childs, Moving beyond neo-Riemannian triads: exploring
a transformational model for seventh chords, Journal of Music
Theory 42/2 (1998): 191-193.
11) Edward Gollin, Some aspects of three-dimensional Tonnetze,
Journal of Music Theory 42/2 (1998): 195-206.
12) Richard Cohn, Neo-Riemannian operations, parsimonious
trichords, and their "Tonnetz" representations, Journal of
Music Theory 41/1 (1997), 1-66.
13) David Lewin, Transformational considerations in Schoenberg's
Opus 23, Number 3, preprint.
In fact, the notion of "torsor" pervades the work of David Lewin,
but not under this name - Lewin calls it a "general interval system".
Stephen Lavelle noticed the connection to torsors in 2005:
14) Stephen Lavelle, Some formalizations in musical set theory,
June 3, 2005, available at http://www.maths.tcd.ie/~icecube/lewin.pdf
and http://www.maths.tcd.ie/~icecube/lewin.ps
Unfortunately the music theorists seem not to have set up
an "arXiv", so some of their work is a bit hard to find.
For example, all of Volume 42 Issue 2 of the Journal of Music
Theory is dedicated to neo-Riemannian theory, but I don't
think it's available online. Luckily, the music theorists have
set up some free online journals, like this:
15) Music Theory Online, http://mto.societymusictheory.org/
and this one has links to others. The Society for Music Theory
also has online resources including a nice bibliography on the
basics of music theory:
16) Society for Music Theory, Fundamentals of music theory,
selected bibliography, http://societymusictheory.org/index.php?pid=37
Now let me turn up the math level a notch....
If you're the right sort of mathematician, you'll have noticed by
now that we're doing some fun stuff starting with the abelian
group A = Z/12. First we're considering certain "affine
transformations" of A. These are the transformations that
preserve all these operations:
(x,y) |-> cx + (1-c)y
where c is an integer. For A = Z/n, the group of affine
transformations has the transposition-inversion group as a
subgroup. The whole affine group has 48 elements, but for
now we only keep this subgroup with 24 elements. Call it G.
Then, we're saying that we can take any "sufficiently generic"
subset of A, hit it with all elements of G, and get a G-torsor,
say S. G is then seen as a subgroup of the group of permutations
of S, and the centralizer of this subgroup is again isomorphic to
G.
You may be more familiar with affine transformations on a vector
space, where we get to use any real number for c. Then
cx + (1-c)y
describes the line through x and y, so you can say that affine
transformations are those that preserve lines. Vector spaces are
R-modules for R the reals, while abelian groups are R-modules for
R the integers. The concept of "affine transformations" of an
R-module works pretty much the same way whenever R is any
commutative ring. And, indeed, everything I just said in the last
paragraph works if we let A be an R-module for any commutative ring
R.
So, there's some very simple nice abstract stuff going on here:
we're taking an abelian group A, looking at a subgroup G of its
affine transformations, and seeing that sufficiently generic
subsets of A give rise to G-torsors!
These are nice examples of G-torsors, since nobody is likely to
accidentally confuse them with the group G. If you read my webpage
on torsors, you'll see it's often easy to mix up a G-torsor with
the group G itself.
In fact, I just committed this sin myself! The set of notes is
not naturally an abelian group until we pick an origin - a place
for the chromatic scale to start. It's really just an A-torsor,
where A is the abelian group generated by transposition.
So, there lots of torsors lurking in music....
The pretty math I've just described only captures a microscopic
portion of what makes music interesting. It doesn't, for example,
have anything to say about what makes some intervals more dissonant
than others. As Pythagoras noticed, simple frequency ratios like
3/2 or 4/3 make for less dissonant chords than gnarly fractions
like 1259/723. The equal tempered tuning system, where the basic
frequency ratio is 2^{1/12}, would have made Pythagoras roll in
his grave! Advocates of other tuning systems say these irrational
frequency ratios are driving us crazy, making wars break out and
plants wilt - but there's an unavoidable conflict between the desire
for simple ratios and the desire for evenly spaced notes, built into
the fabric of mathematics and music. Every tuning system is thus a
compromise. I would like to understand this better; there's bound
to be a lot of nice number theory here.
To study different tuning systems in a unified way, one first step
is replace the group Z/12 by a continuous circle. Points on this
circle are "frequencies modulo octaves", since for many - though
certainly not all - purposes it's good to consider two notes
"the same" if they differ by an octave. Mathematically this circle
is R+/2, namely the multiplicative group of positive real numbers
modulo doubling. As a group, it's isomorphic to the usual circle
group, U(1).
This "pitch class circle" plays a major role in the work of Dmitri
Tymoczko, a composer and music theorist from Princeton, who emailed
me after I left a grumpy comment on the discussion page for this
fascinating but slightly obscure article:
17) Wikipedia, Musical set theory,
http://en.wikipedia.org/wiki/Musical_set_theory
He's recently been working on voice leading and orbifolds. They're
related topics, because if you have a choir of n indistinguishable
angels, each singing a note, the set of possibilities is:
T^n / S_n
where T^n is the n-torus - the product of n copies of the pitch
class circle - and S_n is the permutation group, acting on n-tuples
of notes in the obvious way. This quotient is not usually a manifold,
because it has singularities at certain points where more than one
voice sings the same note. But, it's an *orbifold*. This kind of
slightly singular quotient space is precisely what orbifolds were
invented to deal with.
Tymoczko is coming out with an article about this in Science
magazine. For now, you can learn more about the geometry of
music by playing with his "ChordGeometries" software:
18) Dmitri Tymoczko, ChordGeometries,
http://music.princeton.edu/~dmitri/ChordGeometries.html
As for "voice leading", let me just quote his explanation,
suitable for mathematicians, of this musical concept:
BTW, if you're writing on neo-Riemannian theory in music, it
might be helpful to keep the following basic distinction in
mind. There are chord progressions, which are essentially
functions from unordered chords to unordered chords (e.g. the
chord progression (function) that takes C major to E minor).
Then there are voice leadings, which are mappings from the notes
of one chord to the notes of the other E.g. "take the C in a C
major triad and move it down by semitone to the B." This voice
leading can be written: (C, E, G)->(B, E, G).
This distinction is constantly getting blurred by neo-Riemannian
music theorists. But to really understand "neo-Riemannian
chord progressions" you have to be quite clear about it.
To form a generalized neo-Riemannian chord progression, start
with an ordered pair of chords, say (C major, E minor). Then
apply all the transpositions and inversions to this pairs,
producing (D major, F# minor), (C minor, Ab major), etc. The
result is a function that commutes with the isometries of the
pitch class circle. As a result, it identifies pairs of chords
that can be linked by exactly similar collections of voice
leading motions.
For example, I can transform C major to E minor by moving C down
by semitone to B.
Similarly, I can transform D major to F# minor by moving D down
by semitone to C#.
Similarly, I can transform C minor to Ab major by moving G up to
Ab.
This last voice leading, (C, Eb, G)->(C, Eb, Ab) is just an
inversion (reflection) of the voice leading (C, E, G)->(B, E, G).
As a result it moves one note up by semitone, rather than moving
one note down by semitone.
More generally: if you give me *any* voice leading between C
major and E minor, I can give you an exactly analogous voice
leading between D major and F# minor, or C minor and Ab major,
etc. So "neo-Riemannian" progressions identify a class of
*harmonic* progressions (functions between unordered collections
of points on the circle) that are interesting from a *voice
leading* perspective. (They identify pairs of chord progressions
that can be linked by the same voice leadings, to within rotation
and reflection.)
You can learn more about this here:
19) Dmitri Tymoczko, Scale theory, serial theory, and voice leading,
available at http://music.princeton.edu/~dmitri/scalesarrays.pdf
I'd like to conclude tonight's performance with a "chromatic fantasy" -
some wild ideas that you shouldn't take too seriously, at least as
far as music theory goes. In this rousing finale, I'll list some
famous subgroups of the permutations of a 12-element set. They may
not be relevant to music, but I can't resist mentioning them and
hoping somebody dreams up an application.
So far I've only mentioned two: the cyclic or "transposition" group,
Z/12, and the dihedral or "transposition/inversion" group with 24
elements. These are motivated by thinking of Z/12 as a discrete
analogue of a circle and considering either just its rotations, or
rotations together with reflections. But, mathematically, it's
nice to loosen up this rigid geometry and consider *projective*
transformations of a circle, now viewed as a line together with a
point at infinity - a "projective line".
Indeed, the group Z/11 becomes a field with 11 elements if we multiply
as well as add mod 11. If we throw in a point at infinity, we get a
projective line with 12 elements. It looks just like our circle of 12
notes. But now we see that the group PGL(2,Z/11) acts on this projective
line in a natural way. This group consists of invertible 2x2 matrices
with entries in Z/11, mod scalars. People call it PGL(2,11) for short.
So, PGL(2,11) acts on our 12-element set of notes. And, it's a
general fact for any field F that PGL(2,F) acts on the corresponding
projective line in a "triply transitive" way. In other words, given
any ordered triple of distinct points on the projective line, we can
find a group element that maps it to any *other* ordered triple of
distinct points.
Even better, the action is "sharply" triply transitive, meaning
there's *exactly one* group element that does the job!
This lets us count the elements in PGL(2,11). Since we can find
exactly one group element that maps our favorite ordered triple of
distinct elements to any other, we just need to count such triples,
and there are
12 x 11 x 10 = 1320
of them - so this is the size of PGL(2,11).
This may be too much symmetry for music, since this group carries
*any* three-note chord to any other, not just in the sense of
chord progressions but in the sense of voice leadings. Still,
it's cute.
We might go further and look for a quadruply transitive group of
permutations of our 12-element set of notes - in other words, one
that maps any ordered 4-tuple of distinct notes to any other.
But if we do, we'll run smack dab into MATHIEU GROUPS!
Here's an utterly staggering fact about reality. Apart from the
group of *all* permutations of an n-element set and the group of
*even* permutations of an n-element set, there are only FOUR
groups of permutations that are k-tuply transitive for k > 3.
Here they are:
* The Mathieu group M_{11}. This is a quadruply transitive group
of permuations of an 11-element set - and sharply so! It has
11 x 10 x 9 x 8 = 7920
elements.
* The Mathieu group M_{12}. This is a quintuply transitive group
of permutations of a 12-element set - and sharply so! It has
12 x 11 x 10 x 9 x 8 = 95,040
elements.
* The Mathieu group M_{23}. This is a quadruply transitive group
of permutations of a 23-element set - but not sharply so. It has
23 x 22 x 21 x 20 x 48 = 10,200,960
elements. As you can see, 48 group elements carry any distinct
ordered 4-tuple to any other.
* The Mathieu group M_{24}. This is a quintuply transitive group
of permutations of a 24-element set - but not sharply so. It has
24 x 23 x 22 x 21 x 20 x 48 = 244,823,040
elements. As you can see, 48 group elements carry any distinct
ordered 4-tuple to any other.
These groups all arise as symmetries of certain discrete geometries
called Steiner systems. An "S(L,M,N) Steiner system" is a set of N
"points" together with a collection of "lines", such that each line
contains M points, and *any* set of L points lies on a unique line.
The symmetry group of a Steiner system consists of all permutations
of the set of points that map lines to lines. It turns out that:
* There is a unique S(5,6,12) Steiner system, and the Mathieu group
M_{12} is its symmetry group. The stabilizer group of any point
is isomorphic to M_{11}.
* There is a unique S(5,8,24) Steiner system, and the Mathieu group
M_{24} is its symmetry group. The stabilizer group of any point
is isomorphic to M_{23}.
So, the group M_{12} could be related to music if there were a
musically interesting way of taking the chromatic scale and choosing
6-note chords such that any 5 notes lie in a unique chord. I can't
imagine such a way - most of these chords would need to be wretchedly
dissonant. Another way to put the problem is that such a big group
of permutations would impose more symmetry on the set of chords than
I can imagine my ears hearing. It's like those grand unified theories
that posit symmetries interchanging particles that look completely
different. They could be true, but they've got their work cut out
for them.
Luckily, the Mathieu groups appear naturally in other contexts -
wherever the numbers 12 and 24 cast their magic spell over mathematics!
For example, M_{24} is related to the 24-dimensional Leech lattice,
and M_{12} can be nicely described in terms of 12 equal-sized balls
rolling around the surface of another ball of the same size. See
"week20" for more on this - and the book by Conway and Sloane cited
there for even more.
For a pretty explanation of M_{24}, also try this:
20) Steven H. Cullinane, Geometry of the 4 x 4 square,
http://finitegeometry.org/sc/16/geometry.html
For explanations of both M_{24} and M_{12}, try this:
21) Peter J. Cameron, Projective and Polar Spaces, QMW Math Notes
13, 1991. Also available at http://www.maths.qmul.ac.uk/~pjc/pps/
Chapter 9: The geometry of the Mathieu groups, available at
http://www.maths.qmul.ac.uk/~pjc/pps/pps9.pdf
It would be fun to dream up more relations between incidence
geometry and music theory. Could Klein's quartic curve play a
role? Remember from "week214", "week215" and "week219" that this
3-holed torus can be nicely tiled by 24 regular heptagons. Its
orientation-preserving symmetries form the group PSL(2,7), which
consists of all 2x2 matrices with determinant 1 having entries in
Z/7, modulo scalars. This group has 24 x 7 = 168 elements. Since
there are 7 notes in a major or minor scale, and 24 of these scales,
it's hard to resist wanting to think of each heptagon as a scale!
Indeed, after I mentioned this idea to Dmitri Tymoczko, he said
that David Lewin and Bob Peck have written about related topics.
Alas, the heptagonal tiling of Klein's quartic has a total of 56
vertices, not a multiple of 12, so there's no great way to think
of the vertices as notes. But, it has 84 = 7 x 12 edges, so
maybe the edges are labelled by notes and each note labels 7 edges.
Unlike some groups I mentioned earlier, PSL(2,7) is not a subgroup
of the permutations of a 12-element set. And while PSL(2,7) has
lots of 12-element subgroups, these are not cyclic groups but
instead copies of A_4. These facts put some further limitations
on any crazy ideas you might try.
On the bright side, mathematically if not musically, there is a
fascinating way to embed PSL(2,7) into the Mathieu group M_{24},
which can be described by getting M_{24} to act on the set of
24 heptagons in the Klein quartic:
22) David Richter, How to make the Mathieu group M24,
http://homepages.wmich.edu/~drichter/mathieu.htm
He works in the Poincare dual picture, where the Klein quartic
is tiled by 56 triangles, but that's no big deal.
By the way, in "week79" I explained how PSL(2,F) acts on the
projective line over the field F; the same thing works for PGL(2,F).
I also passed on some interesting facts mentioned by Bertram Kostant,
which relate PSL(2,5), PSL(2,7) and PSL(2,11) to the symmetry groups
of the tetrahedron, cube/octahedron and dodecahedron/icosahedron.
Kostant put these together to give a nice description of the buckyball!
Kepler would be pleased. But, he'd be happier if we could find
the music of the spheres lurking in here, too.
-----------------------------------------------------------------------
Quote of the Week:
A guiding principle in modern mathematics is this lesson: Whenever you
have to do with a structure-endowed entity S, try to determine its group
of automorphisms, the group of those element-wise transformations which
leave all structural relations undisturbed. You can expect to gain a
deep insight into the constitution of S in this way. - Hermann Weyl
-----------------------------------------------------------------------
Addenda: This week's issue provoked more discussion than any in recent
history! You can read a lot on sci.math.research - just search for
"Week 234". Here are some comments from Dave Rusin, David Corfield, Mike
Stay, Dmitri Tymoczko, Cris Moore, Robert Israel, Noam Elkies, Stephen
Lavelle and Steve Lubin.
Dave Rusin explained the logic behind having 12 notes in the chromatic
scale. David Corfield mentioned a book on topos theory in music, and
a paper by Noam Elkies on Mathieu groups. Mike Stay pointed out William
Sethares' work on how the timbre of an instrument affects which scales
sound good. Dmitri Tymoczko had more comments on this issue. Cris Moore
mentioned an interesting microtonal composer named Easley Blackwood.
Robert Israel pointed out an unusual fact about Riemann and Einstein.
Noam Elkies explained what David Lewin was trying to do with PSL(2,7) in
music theory. And Stephen Lavelle gave some more references on torsors
and topoi in music, and said more about the origin of the 12-note scale.
So, here we go! Dave Rusin wrote:
You wrote:
...there's an unavoidable conflict between the desire for simple
ratios and the desire for evenly spaced notes, built into the
fabric of mathematics and music. Every tuning system is thus a
compromise. I would like to understand this better; there's
bound to be a lot of nice number theory here.
Sure there is. You want to choose a number N of intervals into which
to divide the octave, so that there are two tones in the scale that,
like C and G, have frequencies very nearly in a 3:2 ratio. (This
also gives a bonus pair like G and the next C up, which are then in
a 4:3 ratio.) But that just means you want 2^{n/N} to be nearly 3/2,
i.e. n/N is a good rational approximation to log_2(3/2). Use
continued fractions or Farey sequences as you like. You'll find
that a five-note octave is not a bad choice (roughly giving you
just the black keys on a piano, and roughly corresponding to ancient
Oriental musical sounds) but a 12-note octave is a really good choice.
So it's not just happenstance that we have a firmly-entrenched system
of 12-notes-per-octave. I'm sure you've seen this "7 - 12" magic
before, e.g. the circle-of-fifths in music takes you through 7
octaves, or the simple arithmetic that 2^{19} ~ 3^{12} (i.e.
524288 ~ 531441). Long ago I programmed an old PC to play a
41-tone scale because the next continued-fractions approximant
calls for such a scale.
Of course you could argue that music consists of more than just
(musical) fourths and fifths and so the REAL number theory comes
about by choosing numbers of tones which allow lots of sets of
notes to be in (or nearly in) simple Pythagorean harmonies. How,
exactly, you balance the conflicting goals is a matter of personal
choice.
What with the musicians in your family and all that, I'm guessing
you probably knew all this already and simply withheld the comments
because of space limitations, but just in case, I thought I'd
complete your train of thought for you. This stuff is pretty
classic and it's all over the web. I get more hits on my web
page about this than any of my math pages!
dave
Here's Dave's web page:
23) Dave Rusin, Mathematics and music,
http://www.math.niu.edu/~rusin/uses-math/music/
David Corfield wrote:
Hi,
Next you need to wade through all 1300 pages of The Topos of
Music. This is "topos" in the category theoretic sense.
Check out the table of contents!
24) Guerino Mazzola, The Topos of Music: Geometric Logic of
Concepts, Theory and Performance, Birkhauser, Berlin, 2002.
Preface and contents available at
http://www.encyclospace.org/tom/tom_preface_toc.pdf
Guerino Mazzola, homepage,
http://www.ifi.unizh.ch/staff/mazzola/mazzola.html
Colin McLarty reviewed it for MathSciNet. I like the part:
Symmetries within scores, and structural relations between
scores, drive the mathematics up to sheaves, and very briefly
to toposes and Grothendieck topologies. The author candidly
states he is unsure whether this musicological perspective
can use topos cohomology (p. 436).
Did you ever hear about Conway's M_{13}?
25) John H. Conway, Noam D. Elkies, Jeremy L. Martin,
The Mathieu group M_{12} and its pseudogroup extension M_{13},
available as math.GR/0508630.
I can't remember whether it was this that Alexander Borovik
mentioned to me as a sign that the simple sporadic groups are
just islands sticking up above the water.
Best, David
Needless to say, David doesn't write me emails with numbered references;
I often polish the emails I get, with the permission of the authors,
trying not to violate the spirit of the thing.
My student Mike Stay wrote:
Music really does sound better if the piano is tuned to the
particular key, i.e. the Pythagorean intervals.
Start with a frequency for C. At each step, multiply by 2
(up an octave) or by 2/3 (down a fourth). Go down a
fourth unless it will take you out of the octave; in the
latter case, multiply by 2 first.
*2 C'
*2/3 G
*2/3 D
*2 D'
*2/3 A
*2/3 E
*2 E'
*2/3 B
*2/3 F#
*2/3 C#
etc.
Classical music was written for a particular key because the
keys sounded different! Using the tuning above induces a
"distance" on the keys--how in tune they are. Pieces would
use the dissonant tunings of other keys for effect. My friend
is an organist and piano tuner; he says that with the logarithmic
tuning all keys sound "equally bad."
But the timbre of the instrument--the harmonics and overtones--
apparently have a great deal to do with whether a particular
chord is consonant or not. This is a really cool paper that
illustrates how to choose nearly any collection of frequencies
as a scale and then come up with a timbre for which it sounds
natural and right:
26) William Sethares, Relating tuning and timbre,
http://eceserv0.ece.wisc.edu/~sethares/consemi.html
Sethares' home page has a bunch of MP3's on it for people who
want to listen:
27) William Sethares, MP3 Download Central,
http://eceserv0.ece.wisc.edu/~sethares/otherperson/all_mp3s.html
I like "Truth on a Bus", played in a 19-note scale.
There are some tracks from the CD mentioned above here:
28) William Sethares, Tuning, Timbre, Spectrum, Scale,
2nd edition, Springer Verlag, Berlin, 2004. Author's guide
available at http://eceserv0.ece.wisc.edu/~sethares/ttss.html
Sound examples available at
http://eceserv0.ece.wisc.edu/~sethares/html/soundexamples.html
The first several tracks play a tune on a typical 12-tone
instrument. Then they change its timbre by adjusting the
harmonics. Now if played in 12 divisions of a perfect octave
(twice the frequency), it sounds perfectly awful; but if played
in 12 divisions of 2.1, it sounds "right" again.
Mike Stay
http://math.ucr.edu/~mike
In response to my comment "Every tuning system is thus a compromise.
I would like to understand this better...", Dmitri Tymoczko wrote:
William Sethares' "Tuning, Timbre, Spectrum, Scale" is the best
book about this. He has a convincing demonstration that "pure
ratios" are not in themselves important: what's important is
that the overtones of two simultaneously-sounding notes match.
Since harmonic tones have partials that are integer multiples
of the fundamental, you get pure ratios.
However, for inharmonic tones, such as bell-sounds, the overtones
are not integer multiples of the lowest tone. Hence, to get the
partials to match you often need to use non-integer ratios.
Sethares' book comes with a CD demonstrating this. It has to
be heard to be believed.
It seems that a bunch of the music on Sethares' CD is available online,
as Mike pointed out above. I find most of this music interesting but
unpleasant, not because of the tuning systems, but because it lacks
soul. I haven't listened to "Truth on a Bus" yet.
Cris Moore wrote:
By the way, you should check out the music of Easley Blackwood.
He wrote a series of microtonal etudes, where the number of tones
per octave ranges from 13 to 24. Some of them (17, I think) are
quite beautiful.
Cris
Here's an interesting syllabus and list of references that gives a
feel for what mathematically sophisticated music theorists need to
know these days:
29) John Rahn, Music 575: Music and Mathematics, November 2004,
syllabus available at http://faculty.washington.edu/jrahn/5752004.htm
Rahn makes some interesting comments on David Lewin's book Generalized
Musical Intervals and Transformations, which defines a concept of
"generalized interval system", or GIS. As far as I can tell without
having read the book, a generalized interval system is a G-torsor for
some group G, where quite likely we might wish to restrict G to be
abelian or even cyclic. Thus, concretely, a generalized interval
system is a set S of "pitch classes" on which some group G acts,
and such that for any two elements s,s' in S there is a unique g
in G with gs = s'. In this situation we say g is the "difference
in pitch" between s and s'.
A subtle feature of G-torsors is that they are isomorphic to G,
but not in a canonical way, because they don't have a god-given
"identity element". I explain the importance of this in my webpage
"Torsors made easy". However, as in physics and mathematics, some
people in music theory seem willing to ignore this subtlety and
identity any G-torsor with G.
Rahn has the following comments on Lewin's book. I find them
interesting because it shows music theorists grappling with ideas
like category theory.
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 D_{24},
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.
I don't know what a Lewin network is, except from the above.
Unfortunately, Lewin's book is out of print. Lewin died in
2003, and Milton Babbitt said that a fair portion of his writing
remains unpublished:
30) Ken Gewertz, Composer, music theorist David Lewin dies at 69,
Harvard University Gazette,
http://www.news.harvard.edu/gazette/2003/05.15/13-lewinobit.html
Robert Israel pointed out an interesting connection between Einstein
and Riemann:
John Baez wrote:
>When Tom first mentioned "neo-Riemannian theory", I thought this
>was some bizarre application of differential geometry to music.
>But no - we're not talking about the 19th-century mathematician
>Bernhard Riemann, we're talking about the 19th-century music
>theorist Hugo Riemann!
Interestingly, both Bernhard Riemann and Hugo Riemann had a
connection with Einstein. But in Hugo's case, it was not Albert
but his cousin Alfred, the musicologist, who edited Hugo Riemann's
"Musik Lexicon".
Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
Noam Elkies explained what David Lewin was doing with PSL(2,7):
John Baez wrote:
> Today I'd like to talk about the math of music - including
> torsors, orbifolds, and maybe even Mathieu groups. [...]
Lots of interesting points there (though curiously none about
Fourier analysis, which seem to be a natural connection in the context
of mathematical physics). For now I'll just address Mathieu groups
and the like. First, though, a trivial etymological point:
> [...] in the Middle Ages, music was part of a "quadrivium" of
> mathematical arts: arithmetic, geometry, music, and astronomy.
> These were studied after the "trivium" of grammar, rhetoric and logic.
> This is why mathematicians scorn a result as "trivial" when it's easy
> to see using straightforward logic. When a result seems more profound,
> they should call it "quadrivial"!
>
> Try saying it sometime: "Cool! That's quadrivial!" It might catch on.
I hope not; while the usual sense "commonplace, ordinary, unimportant"
of "trivial" does come from a Latin word "trivium", it's not the academic
trivium but a fork in the road -- the meeting of "tres viae", three ways.
Apparently it was common for people to make gossip and other trivial
small talk at these three-way crossings. See for instance
, which to its credit
includes also the technical meaning
2b: relating to or being the mathematically simplest case;
specifically: characterized by having all variables equal to zero
-- it would surely be too much to ask a general dictionary to mention
identity elements or maps involving an arbitrary group. To its
*dis*credit, the same dictionary glosses "trivium" as "crossroads",
which should of course be where at least *four* ways meet, that is,
a quadrivium! In fact the www.m-w.com entry for "quadrivium" also
glosses that Latin word as "crossroads"...
[Given the TWFiMP context, perhaps this is the place to digress further
and suggest that a Feynman diagram is a network of trivialities, in the
original meaning of "3-way meetings" :-)]
Back to groups and music:
> I'd like to conclude tonight's performance with a "chromatic fantasy" -
> some wild ideas that you shouldn't take too seriously, at least as
> far as music theory goes. In this rousing finale, I'll list some
> famous subgroups of the permutations of a 12-element set. They may
> not be relevant to music, but I can't resist mentioning them and
> hoping somebody dreams up an application. [...]
>
> * The Mathieu group M_{12}. This is a quintuply transitive group
> of permutations of a 12-element set - and sharply so! [...]
>
> These groups all arise as symmetries of certain discrete geometries
> called Steiner systems. An "S(L,M,N) Steiner system" is a set of N
> "points" together with a collection of "lines", such that each line
> contains M points, and *any* set of L points lies on a unique line.
> The symmetry group of a Steiner system consists of all permutations
> of the set of points that map lines to lines. It turns out that:
>
> * There is a unique S(5,6,12) Steiner system, and the Mathieu group
> M_{12} is its symmetry group. The stabilizer group of any point
> is isomorphic to M_{11}. [...]
>
> So, the group M_{12} could be related to music if there were a
> musically interesting way of taking the chromatic scale and choosing
> 6-note chords such that any 5 notes lie in a unique chord. I can't
> imagine such a way - most of these chords would need to be wretchedly
> dissonant.
This from the man who finds Schoenberg's music so beautiful?...
(Actually not all of Schoenberg's music is 12-tone, and some of it
is not even atonal; perhaps you had heard only his tonal music and/or
early explorations outside tonality such as the Op.19 piano pieces.)
> Another way to put the problem is that such a big group
> of permutations would impose more symmetry on the set of chords than
> I can imagine my ears hearing.
Curiously the real problem here is that M_{12}, big as it is,
is *missing* an important kind of symmetry: a 12-cycle.
While one wouldn't need all the symmetries of the Steiner system
to be evident in the music, it would be nice for any transposition
of any Steiner chord to be again a Steiner chord. But, since
M_{12} is a simple group, it contains only even permutations,
whereas a 12-cycle is odd. It follows that M_{12} does not contain
the triply-transitive group PGL_2(Z/11Z) -- though it *does*
contain the index-2 subgroup PSL_2(Z/11Z), in two different ways
that are switched by an outer automorphism of M_{12}!
While M12 does not contain any 12-cycles, it does contain double 6-cycles.
We can choose one of them (doesn't matter which, because they're
all conjugate in M12), and then choose an action of M12 on Z/12Z
that makes our double 6-cycle act by translation by 2. This gives
a Steiner (5,6,12) system on Z/12Z that's invariant under all
even translations. Now there are Binom(12,5)/Binom(6,5) = 132
hexads in the system, in 66 complementary pairs, and it turns out
that none of the hexads is taken to itself or its complement
by a nontrivial element of M12. So, under the standard identification
of Z/12Z with the chromatic pitch classes, we get 11 six-note chords
that, together with their whole-tone transpositions and complements of
whole-tone transpositions (all different), form a (5,6,12) Steiner system
and thus contain every 5-note chord in just one way.
One problem with this is that our musical training does not prepare us
to distinguish whole-tone transpositions from non-whole-tone
transpositions. So, let's form a double Steiner system by allowing
arbitrary transpositions. This ruins most of the M12 symmetry
(most of which wasn't audible anyway), but gives us symmetry
under the full group of musical transpositions, inversion,
and complements. It so happens that this group still acts freely,
so we get 11 six-note chords that, together with their
transpositions and complements of transpositions, contain
every 5-note chord in exactly two ways.
Back in 1991 I figured out what these 11 chords are.
Written as subsets of Z/12Z rather than of {C,C#,D,...},
they are:
{0,1,2,3,4,6},
{0,1,2,3,5,7},
{0,1,2,3,6,7},
{0,1,2,4,5,8},
{0,1,3,4,6,9},
{0,1,3,5,7,9},
{0,1,2,4,5,9},
{0,1,2,4,7,8},
{0,1,2,5,6,8},
{0,1,2,4,7,9},
{0,1,3,5,6,8}.
Each five-note chord can be regarded as a link between two
of these chords (not necessarily different ones; for instance,
the two completions of a symmetrical chord such as {0,1,2,3,4}
must be each others inversion, here {0,1,2,3,4,6} and {-2,0,1,2,3,4}).
The resulting graph is here, with some cryptic labeling to indicate
things like whether one of the chords linked by an edge must be inverted
to get a 5-note overlap:
http://www.math.harvard.edu/~elkies/m12.pdf
> It would be fun to dream up more relations between incidence geometry
> and music theory. Could Klein's quartic curve play a role? [...]
> Its orientation-preserving symmetries form the group PSL(2,7), which
> consists of all 2x2 matrices with determinant 1 having entries in Z/7,
> modulo scalars. This group has 24 x 7 = 168 elements. Since there are
> 7 notes in a major or minor scale, and 24 of these scales, it's hard
> to resist wanting to think of each heptagon as a scale!
>
> Indeed, after I mentioned this idea to Dmitri Tymoczko, he said
> that David Lewin and Bob Peck have written about related topics.
Right. In fact this group, call it G168, also has
a doubly-transitive action on 7 objects, which includes
a 7-cycle and thus gives a natural way for this group
to act on a 7-note diatonic scale. David Lewin noted
this possibility in one of his last papers.
The key (no pun intended) is that this group G168 is also
isomorphic with the invertible 3x3 matrices over Z/2Z --
one of those remarkable isomorphisms between small matrix groups.
So, G168 permutes the 2^3-1 = 7 nonzero vectors in a 3-dimensional
vector space V over Z/2Z. The existence of a 7-cycle is automatic
because 7 is prime, though it is true in general that for any
finite field k and any positive integer n the group of nxn matrices
over k contains elements that cyclically permute the nonzero vectors
of k^n. The reason is that k has a degree-n extension k'; any finite
field has cyclic unit group, and multiplication by a generator is
a cyclic permutation of the nonzero elements that acts linearly over k',
so a fortiori over k!
But back to G168. This group also permutes the 7 codimension-1
subspaces (planes) of V, each of which contains three nonzero vectors.
Any two distinct nonzero vectors are contained in a unique such plane,
so we get a (2,3,7) Steiner system, a.k.a. projective plane of order 2,
whose group of symmetries is G168.
If we use a 7-cycle to identify the nonzero vectors of V with
a diatonic scale, then the Steiner system gives us a distinguished
collection of 7 three-note chords, which form a single orbit under
translation (or "diatonic transposition" in music-theory lingo).
It is well-known that these can be chosen to be either {n,n+1,n+3}
or {n,n-1,n-3}. Using the first one yields the chords
{CDF}, {DEG}, {EFA}, {FGB}, {GAC}, {ABD}, {BCE},
and either this or its inversions is what David Lewin
suggested using.
> Unlike some groups I mentioned earlier, PSL(2,7) is not a subgroup
> of the permutations of a 12-element set.
Not a transitive subgroup, you mean. Since the group acts on 7 objects,
it can certainly act on 12 while keeping five of them unmoved.
That's basically what we did above, with the five untouched objects
being the "black keys" C#, D#, F#, G#, A#. Or, use the action on
the 8 points of the projective line over Z/7Z and leave four of the 12
untouched.
--Noam D. Elkies
P.S.
> (If I weren't so darn honest, I'd add that Liszt wrote the
> "Transcendental Etudes" as a sequel to his popular "Algebraic Etudes",
> and explain how Mozart's "eine kleine Nachtmusik" tours a tone net
> shaped like a Klein bottle. But alas....)
Ha. Here's another kind of transcendental etude:
math.harvard.edu/~elkies/stego.pdf
which sounds like this:
math.harvard.edu/~elkies/stego.mid
(an automatic realization that has all the notes but makes
no attempt at interpretations). I premiered the piece here
last year on March 14; for more hints see
math.harvard.edu/~elkies/stego.hints
:-)
I used Noam Elkies' remark to correct my claim that PSL(2,7) isn't a
subgroup of permutations of a 12-element set. Interestingly, the
triples (n,n+1,n+3) and (n,n-1,n-3) in Z/7 are also famous as two
equivalent ways of defining the octonion multiplication tables! If
we call the seven unit imaginary octonions A, B, C, D, E, F, and G,
then we can define octonion multiplication using the chords Elkies
lists by saying that
CD = F, DE = G, EF = A, FG = B, GA = C, AB = D, BC = E
along with knowing they anticommute and square to -1.
Unfortunately, these chords don't seem particularly fundamental to
music! Alas, if only they were triads....
Here are some more nice references and comments from Stephen Lavelle:
I may have been the first to say torsor, but there's a lot of stuff in
existence about group actions as they apply to music theory. In
particular, Fripertinger's page here:
31) Harald Fripertinger, Mathematical music theory,
http://www.uni-graz.at/~fripert/index_11.html
has a lot of pretty damned useful articles.
There're also some pretty cool, and possibly musical, applications of
proper Topos Theory to music by Mazzola's school - see, for instance:
32) Thomas Noll, The topos of triads, available at
http://www.cs.tu-berlin.de/~noll/ToposOfTriads.pdf
John Baez wrote:
> Apparently Riemann's ideas have caught on in a big way. Monzo
> says that "use of lattices is endemic on internet tuning lists",
> as if they were some sort of infectious disease.
>
> Dysart seems more gung-ho about it all. The "donuts" he mentions
> arise when you curl up tone nets by identifying notes that differ
> by an octave. He has some nice pictures of them!
More general "nets", which look quite like commutative diagrams, are
put into use by Lewin. Actually, they are commutative diagrams, and
one can conceptually quite reasonably formulate a category of musically
meaningful limits - see:
33) Guerino Mazzola and Moreno Andreatta, From a categorical point of
view: K-nets as limit denotators, available at
recherche.ircam.fr/equipes/repmus/mamux/documents/mazzola-andreatta.pdf
> In neo-Riemannian theory, people like Lewin and Hyer started
> extending Riemann's ideas by using *group theory* to systematize
> operations on chords. The best easy introduction to this is
> Fiore's paper "Music and mathematics". Here you can read about
> math lurking in the music of Elvis and the Beatles! Or, if
> you're more of a highbrow sort, see what he has to say about
> Hindemith and Liszt's "Transcendental Etudes". And if you
> like doughnuts and music, you'll love the section where he
> explains how Beethoven's Ninth traces out a systematic path in
> a torus-shaped tone net! This amazing fact was discovered by
> Cohn, Douthett, and Steinbach.
Mazzola has a nice argument as well (in his book "The Topos of Music"),
where he shows an inconsistency in Riemann's logic of harmonic
functions by showing that his construction would allow one to fix an
orientation on a Moebius strip : ) (if you take the seven notes in
the major scale, and connect any three points with a triangle when they
form a major, minor, or diminished chord, you get a Moebius strip - this
was first come up with by Schoenberg).
> (If I weren't so darn honest, I'd add that Liszt wrote the
> "Transcendental Etudes" as a sequel to his popular "Algebraic
> Etudes", and explain how Mozart's "eine kleine Nachtmusik"
> tours a tone net shaped like a Klein bottle. But alas....)
Encore! Encore!
> C, C#, D, D#, E, F, F#, G, G#, A, A#, B
>
> until you're back to C. If you're a mathematician, you might
> be happier to call these notes
>
> 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11
>
> and say that we're working in the group of integers mod 12,
> otherwise known as Z/12. Let's be mathematicians today.
Here's a question for you, that you might want to know the answer to:
*Historically*, why are there twelve notes in the scale? And why are
seven white and five black?
The answer is that one that ties in lots of stuff about continued
fractions, but goes along these lines: one is looking at the octave,
and divides it up by looking at the first n fifths (in our scale
c,g,d,a,e,b,...) - this divides up the scale.
Pythagoras et al. thought that one should try to keep the variety of
intervals between consecutive notes as small as possible - in the end,
deciding that the fewer different intervals present the better. Scales
generated by fifths that have only two intervals present between
side-by-side notes are called Pythagorean. None have just one interval,
and the first three Pythagorean scales have 5,7, and 12 notes. 12 was
thought pretty much enough, I'm guessing, and it can have nicely
embedded into it the two smaller scales (as white and black notes).
I should have a reference for the original article where I read this
(some Irish maths society bulletin I think), but I've said enough that
the material should be findable online. Ah yes, here it is:
34) Maria Jose Garmendia Rodriguez and Juan Antonio Navarro Gonzalez,
Musical scales, IMS Bulletin 35 (Christmas 1995), 24.
for all the good it'll do you.
Steve Lubin wrote:
Dear John,
I just came across your charming page on the web about donuts and music
and related matters, and I'd like to contribute a bit of info. Your statement:
And if you like doughnuts and music, you'll love the section where he
[Fiore] explains how Beethoven's Ninth traces out a systematic path in a
torus-shaped tone net! This amazing fact was discovered by Cohn,
Douthett, and Steinbach.
isn't quite accurate; or at least, doesn't do justice to priority of
discovery. In 1974 I wrote a dissertation in musicology at NYU entitled
Techniques for the Analysis of Development in Middle-Period Beethoven,
in which I laid out four ideas:
The 18th-century European composers behaved as though they inhabited a
communally shared abstract tonal space that existed independently of
individual works and served as a substrate for them.
This space evolved in the course of the century, with respect to its
internal features and the ways in which it encouraged or discouraged
composers' choices in navigating through it.
A suitable graphic representation of this space could be used as a map
for describing composers' itineraries in particular pieces; and these
intineraries in themselves constitute an important contribution to the
effect and beauty of their works (in coordination with other factors:
rhythmic, textural, and so on).
The tonal space inhabited by the late-18th-century composers had the
form of a torus, for reasons arising from the internal geometry of the
key-relationships they favored.
Recently I've been gratified to discover that some neo-Riemannian
theorists have been acknowledging my contribution in their publications:
e.g., Cohn, Lerdahl and Gollin.
I wonder what Bernhard Riemann might have thought of this! My
understanding of what his namesake Otto did is as follows: Otto
described two-dimensional tone nets that extended out indefinitely in
all directions like wallpaper -in this I believe he followed out earlier
work of Gottfried Weber. The initial insight of taking the patterns on
this flat surface and giving them closure by covering a torus with them
was mine--I'm not aware of any prior precedents.
BTW, I'm a physics groupie, always starved for laypersons' accounts of
the latest stuff. (The collapse of Scientific American as a classy
journal was a catastrophe for me.) I've decided to set aside the rest
of August for reading your diaries.
Thanks and best,
Steve
http://www.stevenlubin.com/
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