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. So 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://www.math.uchicago.edu/~fiore/1/music.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 Encyclopedia 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:

.          .
 .        .
    .  .

    .  .
 .        .
.          .

(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.

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 forming the group G consisting of all "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:

Tⁿ/S_n

where Tⁿ 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 2×2 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 × 11 × 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 M₁₁. This is a quadruply transitive group of permuations of an 11-element set - and sharply so! It has

  11 × 10 × 9 × 8 = 7920

  elements.

• The Mathieu group M₁₂. This is a quintuply transitive group of permutations of a 12-element set - and sharply so! It has

  12 × 11 × 10 × 9 × 8 = 95,040

  elements.

• The Mathieu group M₂₃. This is a quadruply transitive group of permutations of a 23-element set - but not sharply so. It has

  23 × 22 × 21 × 20 × 48 = 10,200,960

  elements. As you can see, 48 group elements carry any distinct ordered 4-tuple to any other.

• The Mathieu group M₂₄. This is a quintuply transitive group of permutations of a 24-element set - but not sharply so. It has

  24 × 23 × 22 × 21 × 20 × 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 M₁₂ is its symmetry group. The stabilizer group of any point is isomorphic to M₁₁.

• There is a unique S(5,8,24) Steiner system, and the Mathieu group M₂₄ is its symmetry group. The stabilizer group of any point is isomorphic to M₂₃.

So, the group M₁₂ 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, M₂₄ is related to the 24-dimensional Leech lattice, and M₁₂ 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 M₂₄, also try this:

20) Steven H. Cullinane, Geometry of the 4 × 4 square, http://finitegeometry.org/sc/16/geometry.html

For explanations of both M₂₄ and M₁₂, 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 × 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 × 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 transitive 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 A₄. 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 M₂₄, which can be described by getting M₂₄ 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 Poincaré 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.

Addenda: This week's issue provoked more discussion than any in recent history! You can read a lot on sci.math.research. 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 log₂(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 2¹⁹ ~ 3¹² (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

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 M₁₃?

25) John H. Conway, Noam D. Elkies, Jeremy L. Martin, The Mathieu group M₁₂ and its pseudogroup extension M₁₃, 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

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

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.

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

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 D₂₄, you need multiple arrows. There are various alternatives which would work for networks with multiple arrow(-labels) for a given ordered pair of nodes. Multiple arrows (or labels on an arrow, depending on the definitional system) in digraphs are standard, and it is hard to see what is accomplished by not allowing more than one relationship between any two nodes in the model. You also need multiple arrows for groups applied to graphs, category theory, etc.

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

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

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
Department of Mathematics
University of British Columbia

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 http://www.m-w.com/dictionary/trivial, 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 discredit, 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 M12.  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
  M12 is its symmetry group.  The stabilizer group of any point 
  is isomorphic to M11. [...]
 
  So, the group M12 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 M₁₂, 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 M₁₂ is a simple group, it contains only even permutations, whereas a 12-cycle is odd. It follows that M₁₂ does not contain the triply-transitive group PGL₂(Z/11Z) -- though it does contain the index-2 subgroup PSL₂(Z/11Z), in two different ways that are switched by an outer automorphism of M₁₂!

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 3×3 matrices over Z/2Z -- one of those remarkable isomorphisms between small matrix groups. So, G168 permutes the 2³-1 = 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 n×n matrices over k contains elements that cyclically permute the nonzero vectors of kⁿ. 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:

http://math.harvard.edu/~elkies/stego.pdf

which sounds like this:

http://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

http://math.harvard.edu/~elkies/stego.hints

:-)

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.

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/

© 2006 John Baez
baez@math.removethis.ucr.andthis.edu