This Week's Finds in Mathematical Physics (Week 234)

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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 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, G is just the good old
transposition-inversion group, otherwise known as a "dihedral
group".

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 its group G of 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.

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

-----------------------------------------------------------------------
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mathematics and physics, as well as some of my research papers, can be
obtained at

http://math.ucr.edu/home/baez/

For a table of contents of all the issues of This Week's Finds, try

http://math.ucr.edu/home/baez/twfcontents.html

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If you just want the latest issue, go to

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jhnr...@yahoo.co.uk

unread,
Jun 13, 2006, 11:18:55 PM6/13/06
to

John Baez wrote:
>
> 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

How about topoi? Being a category theory buff, you may be
interested in a work called "The Topos of Music", discussed
on sci.math a couple of months ago:

http://groups.google.com/group/sci.math/browse_frm/thread/74260e4d010f4b5/b0161928089c6c1b?lnk=st&q=&rnum=33#b0161928089c6c1b


Cheers

John R Ramsden

Robert Israel

unread,
Jun 14, 2006, 3:00:11 AM6/14/06
to
In article <e6la3o$911$1...@glue.ucr.edu>,
John Baez <ba...@math.removethis.ucr.andthis.edu> 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 isr...@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada

Paul Danaher

unread,
Jun 14, 2006, 3:00:22 AM6/14/06
to
From the point of view of a practical musician, this approach provides an
interesting framework for chromatic and serial analysis, but lacks an
important dimension - human auditory sensitivity. Belkin
http://www.musique.umontreal.ca/personnel/Belkin/bk.H/harm.PDF points out
that "octave equivalence" is an unjustifiable assumption - "In the extreme
registers, pitch discrimination is very inexact and dependant on many
factors, including orchestration, duration etc" (p. 6). The "equal
temperament" assumption is also very artificial once you get away from
keyboard instruments, but this is covered by the "frequency modulo octave"
circle. However, the auditory consideration suggests that this should be
regarded not as a circle but as a helix tapering as it rises and descends.

John Baez

unread,
Jun 14, 2006, 9:05:44 AM6/14/06
to
In article <1150232891.9...@u72g2000cwu.googlegroups.com>,
<jhnr...@yahoo.co.uk> wrote:

>How about topoi? Being a category theory buff, you may be
>interested in a work called "The Topos of Music", discussed
>on sci.math a couple of months ago:
>
>http://groups.google.com/group/sci.math/browse_frm/thread/74260e4d010f4b5/b0161928089c6c1b?lnk=st&q=&rnum=33#b0161928089c6c1b

Good point! I'll have to take a look. As David Corfield
pointed out, the preface and table of contents of this book
are available online.

Here's what David said, along with comments from some other
folks... for fans of category theory, check out John Rahn's
syllabus for his "Music and Mathematics" course.

Hmm, right now they're playing "the most unwanted music" on
the radio - it's got a rap beat, sung by an opera singer,
accompanied by a tuba, with a bunch of atonal electronic effects
thrown in. I sort of like it!

.....................................................................

Addenda: Here are some comments from Dave Rusin, David Corfield, Mike
Stay, Dmitri Tymoczko and Cris Moore. 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

McLarty reviewed it for MathSciNet:

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).

I never quite get Colin when he's being ironic, but I believe
this may be a case. I guess he has to be a little careful as
they're both in the Grothendieck Circle.

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 divide by 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'
/3 G
/3 D
*2 D'
/3 A
/3 E
*2 E'
/3 B
/3 F#
/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

tc...@lsa.umich.edu

unread,
Jun 14, 2006, 7:47:09 PM6/14/06
to
In article <e6p4t8$9uj$1...@glue.ucr.edu>,
John Baez <ba...@math.removethis.ucr.andthis.edu> wrote:

>Dave Rusin wrote:
> 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.

There have been some serious musical compositions in other scales. For
example, I once attended a concert in which one of the pieces was based
on the 19-tone equal tempered scale. They specially tuned two pianos for
the concert. Some notes existed on both pianos, but others existed only
on one of them; between the two of them, the pianos covered all the notes.

As I recall, the 19-tone scale realizes the major third and perfect fifth
slightly better than the 12-tone scale. Of course you can get even better
approximations with something like the 41-tone scale, but 19 is small
enough that it's quite feasible to make instruments for it.

The concert I attended had a pre-concert talk which explained and
demonstrated some of the other properties of the 19-tone scale. One
amusing thing you can do is to try to play "approximations" to familiar
tunes on the new scale; sometimes the performer can even do tricks like
modulate to a new key in such a way that an untrained listener feels as
though there has been no point at which the key changed.
--
Tim Chow tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences

eb...@lfa221051.richmond.edu

unread,
Jun 14, 2006, 9:07:28 PM6/14/06
to
In article <e6p4t8$9uj$1...@glue.ucr.edu>,
John Baez <ba...@math.removethis.ucr.andthis.edu> wrote:
>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 divide by 3 (down a fourth).

This should be "divide by 3/2" or "multiply by 2/3", right? If not,
I don't understand the "take you out of the octave" bit below.
The first /3 would take you to the G below the initial C, and the
next /3 would take you to the D that's over an octave below that.
Or am I missing something?

> Go down a
> fourth unless it will take you out of the octave; in the
> latter case, multiply by 2 first.
>
> *2 C'
> /3 G
> /3 D
> *2 D'
> /3 A
> /3 E
> *2 E'
> /3 B
> /3 F#
> /3 C#
>
> etc.

[...]

>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.

I'm certainly not an expert on this, so someone please tell me if I'm wrong,
but isn't this true even for pianos? I've been told that the
overtones for piano strings aren't harmonic multiples of the fundamentals,
because the strings have stiffness as well as tension, and that piano
tuners have to "stretch" the tuning (make the upper registers sharp
and the lower registers flat) in order for the piano to sound like it's
in tune.

After all the scholarly works being cited in this thread, I'm embarrassed
to go so lowbrow, but wikipedia has a bit to say about this:

http://en.wikipedia.org/wiki/Piano_tuning
http://en.wikipedia.org/wiki/Stretched_tuning

-Ted

--
[E-mail me at na...@domain.edu, as opposed to na...@machine.domain.edu.]

Tom Roberts

unread,
Jun 15, 2006, 12:39:22 PM6/15/06
to
Paul Danaher wrote:
> The "equal
> temperament" assumption is also very artificial once you get away from
> keyboard instruments, [...]

It doesn't even apply exactly to a properly tuned piano. Pianos are
"stretch tuned" because the harmonics of lower strings are not exactly
integer multiples of their fundamental frequencies, they are slightly
sharp. Octaves are stretched slightly to reduce the dissonance between
the actual harmonics of the lower notes and the fundamentals of the
higher notes. This is, of course, due to the fact that steel piano
strings have stiffness, especially the lower ones that are wrapped.

This obviously implies that the "octave equivalence" is not exact for
real pianos. There is clearly an _audible_ equivalence, but it is
different from the mathematical equivalence discussed here.

Tom Roberts

J. B. Wood

unread,
Jun 15, 2006, 12:39:57 PM6/15/06
to
In article <44908007$0$24622$b45e...@senator-bedfellow.mit.edu>,
tc...@lsa.umich.edu wrote:

> As I recall, the 19-tone scale realizes the major third and perfect fifth
> slightly better than the 12-tone scale. Of course you can get even better
> approximations with something like the 41-tone scale, but 19 is small
> enough that it's quite feasible to make instruments for it.
>
> The concert I attended had a pre-concert talk which explained and
> demonstrated some of the other properties of the 19-tone scale.

Hello, and a 19-tone equally-tempered (19-TET) scale has a fifth that
comes in at about 695 cents (based on 1200 * log2(2) = 1200 cents for an
octave). The fifth in 12-TET is 700 cents and a just (pure) fifth is 1200
* log2(3/2) or about 702 cents. OTOH, a minor third in 19-TET at 315.8
cents is very close to the just value of 1200 * log2(6/5) or 315.6 cents.
Sincerely

John Wood (Code 5550) e-mail: wo...@itd.nrl.navy.mil
Naval Research Laboratory
4555 Overlook Avenue, SW
Washington, DC 20375-5337

John Baez

unread,
Jun 15, 2006, 8:28:47 PM6/15/06
to
In article <e6pm1j$9mf$1...@bigbang.richmond.edu>,
<eb...@lfa221051.richmond.edu> wrote:

>In article <e6p4t8$9uj$1...@glue.ucr.edu>,
>John Baez <ba...@math.removethis.ucr.andthis.edu> wrote:

>>My student Mike Stay wrote:

>> Start with a frequency for C. At each step, multiply by 2
>> (up an octave) or divide by 3 (down a fourth).

>This should be "divide by 3/2" or "multiply by 2/3", right?

Right - I'll fix that on my website.

Dmitri Tymoczko pointed out another mistake, namely that
the 24-element "transposition-inversion" group is not the
whole affine group of Z/12. Remember, the transposition-
inversion group is generated by the operations

x |-> x + 1

and

x |-> -x

The whole affine group of Z/12 is generated by these and the
operation

x |-> 5x

It has 48 elements!

For a detailed study of affine transformations of Z/12 and
their application to music, see this paper mentioned by Stephen
Lavelle:

Thomas Noll, The topos of triads, available at
http://www.cs.tu-berlin.de/~noll/ToposOfTriads.pdf

Mathematicians will enjoy this, since he considers Lawevere-Tierney
topologies on the topos Set^T, where T is the 8-element monoid of
*not necessarily invertible* affine maps f: Z/12 -> Z/12 which
preserve a given triad.

>I'm certainly not an expert on this, so someone please tell me if I'm wrong,
>but isn't this true even for pianos? I've been told that the
>overtones for piano strings aren't harmonic multiples of the fundamentals,
>because the strings have stiffness as well as tension, and that piano
>tuners have to "stretch" the tuning (make the upper registers sharp
>and the lower registers flat) in order for the piano to sound like it's
>in tune.

Yes it's true: the tuning is "stretched" so that the lowest notes on
the piano are about 30 cents flat - where there are 1200 cents per octave -
and the highest notes are about 30 cents sharp.

Other fun physics facts:

The tension on a grand piano string is about 30 tons! The length of
the strings does not double as one goes down an octave, since this
would be a bit inconvenient; instead a ratio between 1.88 and 1.94 is
used, and a change in the mass density of the strings is used to get
the rest of the effect. The piano hammer hits the key at about 1/7
the way down the string, to minimize the effect of the 7th harmonic,
which sounds out of tune.

I know the tuning on a piano is stretched because I have an electronic
piano, a Yamaha Clavinova, which goes to great lengths to sound like
the real thing. For example, it deliberately simulates the dull
thumping sound made by the hammers as they hit the piano strings!
And, it has settings for stretched and unstretched tuning, with the
default being stretched.

It may seem silly to go to elaborate lengths to simulate what could
be considered *defects* of the traditional piano - but if you like
how a piano sounds, you like those things.

Why not just use a real piano? Well, I prefer an electronic piano
because 1) it's cheaper than a comparable-sounding real one, 2) it
never goes out of tune, and 3) I can save the music I play, or feed
it into my computer using MIDI.

It also has settings for different tuning systems: Pythagorean, meantone,
Werkmeister, Kirnberger, and so on. A lot of these systems were used
for pianos before the equal-tempered system became standard. At first
I wondered if my ear was good enough to detect the difference between
these systems and equal temperament. The answer turned out to be: YES.
All the other systems sound horribly out of tune! At first. But then,
if I play in one long enough, I get used to it... and then equal temperament
sounds horribly out of tune! If I switch between these systems too often,
they ALL sound horribly out of tune - and I wonder if I've permanently
destroyed my ability to enjoy music. Luckily, the effect goes away.

I now see why people get so emotional about tuning systems. For a
well-balanced introduction, see:

http://www.kirnberger.fsnet.co.uk/TempsI.html
http://home.no.net/wimkamp/instruments/Harpsichord/harpsichord.html

The root of all evil is the fact that if you go up a fifth, you
multiply the frequency of a note by 3/2. No amount of doing this
is the same as going up an integral number of octaves (a power of 2).
So, we compromise and say that 12 fifths is almost 7 octaves:

(3/2)^{12} ~ 2^7

129.746 ~ 128

The difference is called the "Pythagorean comma". It's like a
lump in the carpet: you can push it around, maybe try to hide it
under a sofa, but you can't get rid of it. The different tuning
systems shove this lump in different places.

Now perhaps you can see why the Pythagoreans drowned the guy
who proved the existence of irrational numbers: it's the
irrationality of

ln(3/2)/ln(2)

that creates the nasty lump in the carpet. (The irrationality
of sqrt(2) is related to the "diabolus in musica", but that's
another matter.)

robert bristow-johnson

unread,
Jun 17, 2006, 8:43:59 AM6/17/06
to
in article e6la3o$911$1...@glue.ucr.edu, John Baez at

ba...@math.removethis.ucr.andthis.edu wrote on 06/13/2006 08:10:

> 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.

that's a sorta "best fit" compromise we come to because we don't like
keyboards with 19 or 31 notes per octave.



> 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.

that, i think, is more of an historical accident or coincidence. they could
have chosen a subset of those 7 (e.g. the "pentatonic scale", the same 5
notes picked out of the minor set is often used as the "rock scale" or
"blues scale", sometimes with the dim5 note added, that lead guitarists like
to riff with). there are other modes (that can also be played on the "white
keys) as well as variations of the minor (such as "harmonic minor" vs.
"melodic minor", the former can be played on white keys if you're in the key
of A).

> 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.

i am not sure what this is about. of course, with equally tempered
instruments, any of these keys have the same relative frequency ratios
between notes. any qualitative difference sensed is purely a function of
absolute pitch and i am not sure that those of us without perfect pitch (but
with a good sense of relative pitch) would know the difference between
pieces played a semitone or two different in pitch. i think they would
evoke the same feeling that the music aims to make.

unfretted string instruments and human voice will likely not be done in
equal temperament (if not accompanied by piano or some other keyboard)
because the temperament that people will naturally drift to will likely be
what is often called "just intonation".

horns are different. if a musical piece is transposed to a neighboring key,
because some note intervals in horns are more like just intonation
(particularly different notes with identical valve fingering) rather than
equal temperament, different keys will have different frequency ratios
between corresponding pairs of notes and that might give the piece a
different feel for folks with "golden ears".


> 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,

i thought we called those "12-tone" songs. kinda a genre of modern
composition.

________

i'm still trying to deal with some of the mathematical concepts presented,
but i am still convinced that any mathematical theory of music scales built
upon the 12 note/octave equal tempered scale is, "occidentocentric".

here is my mathematical and perceptual spin on this musical scale thing. áit
ain't historical but i think what happened historically regarding the Equal
Temperament is that, using their ears, these early musicians and keyboard
instrument designers were trying to accomplish the same thing but without
the any explicit use of logarithms. of course the math behind it is pretty
simple for s.p.r. folk.

in the musical domain, the term "interval" might correspond to "frequency
ratio" (of fundamental frequencies of two tones) in the mathematical domain.
the octave is the most primitive interval (and the least dissonant), hearing
the same tune ("Yankee Doodle" or whatever) played twice at an octave
interval will qualitatively sound the same except one is higher in pitch.
but it will be in the same *key*. ámost often, when men and women sing some
melody together, they will sing the same notes, but one octave apart. áeven
if one begins a phrase and the other (gender) completes the phrase, they
will usually be an octave apart (or possibly unison), but not a fifth or any
other interval because that would be a key change and "it wouldn't sound
right". áthe non-octave interval change in the melody would *change* the
melody. ábut that is not so if it were an interval shift of an octave.

physically, the frequency ratio for an octave is exactly 2 (going up) or 1/2
(going down). áwhy is that? áwhy not a frequency ratio of 10 or "e" or pi or
something like that to be the most primitive (other than unison) musical
interval?

(sorry for being so fundamentalist about this.) áthe reason is clearly that
a physical measure of the perceptual quality of dissonance vs. harmonicity
of two notes has to do with how well the quasi-periodic waveforms of the two
notes "mesh together" (at least within tolerances of sensing pitch). áif one
note's waveform completes exactly two cycles in exactly the time that the
other completes one cycle, there won't be very much dissonance between those
two notes. áthat means, allowing for a 0.35% slop (or whatever is our
tolerance in pitch error) in frequency, notes that are the most harmonious
(in the simplest sense) are those with frequency ratios that are the
simplest, that is ratios with the smallest integers.

so probably unison 1/1 is, by definition, the least dissonant. áthen
probably the octave, 2/1, comes next. ánow if you restrict your intervals to
be between unison and an octave: á1 <= M/N <= 2, what would be the least
dissonant intervals ( M, N are integers)? átheir frequency ratios would be
(in order of increasing dissonance):


á á á á | á á| á á| á á| á á| á á| á á| á á| á á| á á| á á| á á| á á|

á á á á1/1 á á á á á á á á á á á á á á á á á á á á á á á á á á á á 2/1
á á á á á á á á á á á á á á á á á á á á á 3/2
á á á á á á á á á á á á á á á á 4/3 á á á á á á á á5/3
á á á á á á á á á á á á á 5/4 á á á á á á á á á á á á á7/4
á á á á á á á á á á á á6/5 á á á á á7/5 á á á á 8/5 á á á 9/5
á á á á á á á á á á 7/6 á á á á á á á á á á á á á á á á á 11/6
á á á á á á á á á á8/7 á á á 9/7 á á 10/7 á á11/7 á á12/7 á13/7
á á á á á á á á á9/8 á á á á á á 11/8 á á á á á 13/8 á á á á15/8
á á á á á á á á10/9 á 11/9 á á á á á á13/9 á14/9 á á á 16/9 á17/9
á á á á á á á 11/10 á á á á á13/10 á á á á á á á á 17/10 á á á19/10
á á á á...

á á á á | á á| á á| á á| á á| á á| á á| á á| á á| á á| á á| á á| á á|
cents: á0 á 100 á200 á300 á400 á500 á600 á700 á800 á900 1000 1100 1200

á á á á á á á cents = 1200*log2(M/N) = 100*semitones

if you read this with a mono-spaced font, i tried to place the horizontal
position of the intervals to be strictly increasing in value. ánow if you
computed the base 2 log of all of these ratios and plotted those values
along a number line from 0 to 1, although there would be a few "orphans",
there would be some noticeable clustering at some pitch intervals. ánow if
you divided the interval from log2(1/1) to log2(2/1) into equal parts, you
would see a very good fit if it were divided into 12 equal parts (it would
be at least as good if it were divided into 24 equal parts, but a 24
key/octave piano might be pretty hard to play).

this is why we in the "west" ended up with the 12 note/octave equal tempered
scale. ái think 31 notes/octave hits these intervals better (when i was a
freshman, i ran a Fortran program testing this, i should redo it in MATLAB),
but such a discrete-pitch instrument, might be unwieldy to play.

if i remember right, the intervals they selected for the "just intonation"
scale are

á á á á1/1 á á á 9/8 á á á5/4 á 4/3 á á á 3/2 á á á5/3 á á á15/8 á 2/1

cents: á0 á á á á204 á á á386 á 498 á á á 702 á á á884 á á á1088 á1200

this might be the scale that a violinist, cellist, trombonist, or some other
continuous-pitch instrumentalist (or a vocalist) would *naturally* settle
on. áa keyboard could be designed (or tuned) to hit those intervals spot on
(with mean-tone temperament for the "sharps" or accidentals), but it would
be good for that key only. in terms of log frequency (which is closely
coupled to perceived pitch quasi-periodic tones), people could tell there
was very close to twice as much distance between 1/1 and 9/8 or between 9/8
andá5/4 as there was between 5/4 and 4/3. the fact that there was very
nearly two "units" of pitch spacing between the pairs (1/1):(9/8),
(9/8):(5/4), (4/3):(3/2), (3/2):(5/3), (15/8):(5/3), than there is between
(5/4):(4/3) and (15/8):(2/1) leads directly, in this compromise to dividing
the octave up into 12 equally spaced intervals. but, if we had 50 fingers
on our hands, it might have come out to a higher number that fits these nice
harmonic intervals even better. the 12 note equally tempered scale
represents a compromise so that the scale sounds equally out of tune for
whatever key you play in. ánote how close to multiples of 100 the cents is
for each just interval.

perhaps you can come up with some interesting discrete group relationship
with the fact that, in this occidentocentric scale, the pitch differences
between adjacent notes of a major (sometimes called "Ionian") or minor
(sometimes called "Aeolian") scales (or one of 5 other "modes" such as
"Dorian" or "Phrygian" or "Lydian" or "Mixo-lydian" or "Locrian") can be
picked out of this circle (depending on where you start):


h w

w w

w h
w

Ionian (major): w w h w w w h
Dorian: w h w w w h w
Phrygian: h w w w h w w
Lydian: w w w h w w h
Mixo-lydian: w w h w w h w
Aeolian (minor): w h w w h w w
Locrian: h w w h w w w


w = "whole-step" = 2 semitones = 200 cents = freq ratio of 2^(2/12)
h = "half-step" = 1 semitones = 100 cents = freq ratio of 2^(1/12)


doesn't matter which way you go around the circle. there is symmetry about
the "main diagonal".

anyway, if you compare the scales for Ionian and Mixo-lydian, you will see
that they are identical except at one place which can be fixed with one
small half-step adjustment (one sharp), and then you can change the key to
"G major" repeat and get another sharp. this is where the key signatures
and the "circle of fifths" comes from. (and then the mathematical fact that
(3/2)^12 is very nearly the same as 2^7 = 128.)

what we popularly call "major" or "minor" are just two, and if you bring in
the orient or other musical traditions, of a zillion other scales. like the
SI system of units, there is nothing special or universal about these
western scales. they're not like Planck units.


--

r b-j r...@audioimagination.com

"Imagination is more important than knowledge."

John Baez

unread,
Jun 17, 2006, 12:17:06 PM6/17/06
to
In article <C0B8F249.155F3%r...@audioimagination.com>,
robert bristow-johnson <r...@audioimagination.com> wrote:

> John Baez wrote:

>> 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.

>that's a sorta "best fit" compromise we come to because we don't like
>keyboards with 19 or 31 notes per octave.

>> 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.

>that, i think, is more of an historical accident or coincidence.

Maybe!

I wasn't trying to pass judgement on questions like that.

I hope you didn't think that just because I was using math to talk
about Western music, that I thought these features were "natural"
or "optimal".

>> 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.

>i am not sure what this is about.

I was just joking about the analogy to spontaneous symmetry
breaking in physics:

When water freezes, the translation and rotation symmetry
get spontaneously broken by how the water molecules pick out
a specific crystal lattice. An ice crystal does not exhibit symmetry
under arbitrary translations and rotations: only those that map
the lattice to itself. The symmetry is still there, but
it's only visible in our freedom to translate or rotate the crystal.

Similarl, playing music in C major, say, breaks the Z/12 symmetry
of the equal-tempered scale by picking out a 7-element subset
of notes that you play more. A composition in C major doesn't
exhibit any sort of Z/12 symmetry. The symmetry is only visible
in our freedom to transpose the whole composition to another key.

Spontaneous symmetry breaking happens at low temperatures;
symmetry gets restored at high temperatures.

When we heat up ice to 273 kelvin, it melts and the rotation and
translation symmetry is restored: liquid water favors no lattice.
If we could heat up some water to about 1 or 2 quadrillion kelvin,
the symmetry between the electromagnetic and weak would (we
believe) get restored: the Higgs field no longer picks out a
specific direction. Similarly, when we "heat up" a piece of music
by throwing in lots of accidentals and shifts of key, the Z/12
symmetry gets restored.

That's all I was trying to say; don't take it too seriously,
it's just a fun idea.

>> 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,

>i thought we called those "12-tone" songs. kinda a genre of modern
>composition.

Here I was talking about late-1800s chromaticism - guys named Richard
are the ones to listen to for this, like Strauss and Wagner. They
kept changing keys and throwing in accidentals. Schoenberg
started out doing that, but then pushed it to 12-tone music, where
the whole notion of "key" evaporates. A good little intro:

http://en.wikipedia.org/wiki/Chromatic_scale

If you like my little physics analogy, you can think of diatonic
music (Bach, Mozart) as the "solid" phase of Western music,
chromaticism as the "liquid" phase, and 12-tone music as the
"gas" phase. It was fun and exciting to gradually turn up the
temperature, but once the gas phase was reached there wasn't
much fun left in this direction - I suppose when people start hacking
apart pianos or playing lots of radios on stage it's the "plasma"
phase. :-)

>i'm still trying to deal with some of the mathematical concepts presented,
>but i am still convinced that any mathematical theory of music scales built
>upon the 12 note/octave equal tempered scale is, "occidentocentric".

Of course! I'm terrified that you think I don't believe that.
That's why I kept saying "Western" all the time.

>physically, the frequency ratio for an octave is exactly 2 (going up) or 1/2

>(going down). why is that? why not a frequency ratio of 10 or "e" or pi or


>something like that to be the most primitive (other than unison) musical
>interval?

You answer your question, but a related, compatible answer is that
if you pluck a string, you'll typically excite mainly its fundamental
mode, then some of the mode that wiggles twice as fast, then a bit of
the mode that wiggles three times as fast, and so on. For a while,
each new mode sounds nice next to the previous mode, so we get intervals
that are famous in Western music:

1 fundamental
2 = 2/1 x 1 and 2/1 is an octave
3 = 3/2 x 2 and 3/2 is a fifth
4 = 4/3 x 3 and 4/3 is a major fourth
5 = 5/4 x 4 and 5/4 is a major third
6 = 6/5 x 5 and 6/5 is a minor third

At the next step, we get an interval of 7/6, which is not part
of Western music; it sounds "bad". So, when people build pianos,
they make the hammer hit the string at a point 1/7 of the way down
the string, so this mode isn't excited! This is called the "seventh
harmonic problem". People building brass instruments also need
to deal with it, apparently.

Here's a picture:

http://hyperphysics.phy-astr.gsu.edu/HBASE/music/harmon.html#c1

I guess 8/7 is also "bad", but when we get to 9/8 that's a
"major second", i.e. a whole tone step.

I like your analysis here, even though on my screen half the
spaces look like lower-case a's with acute accents on them!

I'll fix that:

> 1/1 2/1
> 3/2
> 4/3 5/3
> 7/4
> 6/5 7/5 8/5 9/5
> 7/6 11/6
> 8/7 9/7 10/7 11/7 12/7 13/7
> 9/8 11/8 13/8 15/8
> 10/9 11/9 13/9 14/9 16/9 17/9
> 11/10 13/10 17/10 19/10

> 1/1 9/8 5/4 4/3 3/2 5/3 15/8 2/1

As you can see, the intervals that got picked involve a lot of fractions
that show up naturally as ratios of harmonics: 2/1, 3/2, 4/3, 5/4. All
the denominators are powers of 2, or simply 3.

It's interesting to compare this "just intonation" with "Pythagorean
tuning":

http://www.medieval.org/emfaq/harmony/pyth2.html

which is even heavily based on powers of 2 and 3.

>in terms of log frequency (which is closely
>coupled to perceived pitch quasi-periodic tones), people could tell there
>was very close to twice as much distance between 1/1 and 9/8 or between 9/8

>and 5/4 as there was between 5/4 and 4/3. the fact that there was very


>nearly two "units" of pitch spacing between the pairs (1/1):(9/8),
>(9/8):(5/4), (4/3):(3/2), (3/2):(5/3), (15/8):(5/3), than there is between
>(5/4):(4/3) and (15/8):(2/1) leads directly, in this compromise to dividing
>the octave up into 12 equally spaced intervals.

Right. But the desire to "fill in" these holes may have arisen
when people tried to trasnpose the above 7-note scale; e.g. take
a melody and play it a fifth or third higher.

>but, if we had 50 fingers
>on our hands, it might have come out to a higher number that fits these nice
>harmonic intervals even better.

Dave Rusin suggest that the next really good number after 12 notes is
41 notes:

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.

>perhaps you can come up with some interesting discrete group relationship


>with the fact that, in this occidentocentric scale, the pitch differences
>between adjacent notes of a major (sometimes called "Ionian") or minor
>(sometimes called "Aeolian") scales (or one of 5 other "modes" such as
>"Dorian" or "Phrygian" or "Lydian" or "Mixo-lydian" or "Locrian") can be
>picked out of this circle (depending on where you start):
>
>
> h w
>
> w w
>
> w h
> w
>
>
>
>Ionian (major): w w h w w w h
>Dorian: w h w w w h w
>Phrygian: h w w w h w w
>Lydian: w w w h w w h
>Mixo-lydian: w w h w w h w
>Aeolian (minor): w h w w h w w
>Locrian: h w w h w w w

Thanks for taking the time to draw that! Yes, I should think
about the math of this. I know people already have. Here's
something interesting, mildly related:

http://en.wikipedia.org/wiki/Rothenberg_propriety

>what we popularly call "major" or "minor" are just two, and if you bring in
>the orient or other musical traditions, of a zillion other scales. like the
>SI system of units, there is nothing special or universal about these
>western scales. they're not like Planck units.

Yup. Again, I hope you don't think I don't think this!


robert bristow-johnson

unread,
Jun 17, 2006, 6:54:39 PM6/17/06
to

John Baez wrote:
..

> >what we popularly call "major" or "minor" are just two, and if you bring in
> >the orient or other musical traditions, of a zillion other scales. like the
> >SI system of units, there is nothing special or universal about these
> >western scales. they're not like Planck units.
>
> Yup. Again, I hope you don't think I don't think this!

John, i confess that i dunno what i think you think. i am still
decoding stuff in your introductory post and trying to break it down
into mathematical language that i understand. but, you're right that i
was wondering if in this mathematical analysis that i have not yet
decoded, there was some implication that this 12 note per octave
equal-tempered scale was somehow mathematically more natural than other
tuning systems. and i understand your disclaimer above.

actually i *do* think that there is some kinda mathematical
*serendipity* in the 12-note per octave equal-tempered scale.
equal-tempered scales are equally good or equally bad in all keys
(which makes them useful for discretely keyed instruments) and the
12-note equal-tempered scale is, IMO, the one that lies at a sorta
equilibrium of two different opposing pressures, one to have intervals
as harmonic as possible and the other to ergomonically have as few
buttons to push as possible. it really *is* fortuitous that (3/2)^12
is very nearly equal to 2^7. but like numbers such as 0.30282212 (that
is sqrt(4*pi*alpha) which is the number i think you physikers should be
putting up on your walls rather than 137.03599911), i really just view
that so far as one of them sorta unexplained numerical gifts from
nature that we don't yet understand.

bestest,

r b-j

John Baez

unread,
Jun 17, 2006, 10:47:35 PM6/17/06
to
In article <1150569266.9...@g10g2000cwb.googlegroups.com>,
robert bristow-johnson <r...@audioimagination.com> wrote:

>John Baez wrote:

>> >what we popularly call "major" or "minor" are just two, and if you bring in
>> >the orient or other musical traditions, of a zillion other scales.
>> >like the SI system of units, there is nothing special or universal
>> >about these western scales. they're not like Planck units.

>> Yup. Again, I hope you don't think I don't think this!

>John, i confess that i dunno what i think you think. i am still
>decoding stuff in your introductory post and trying to break it down
>into mathematical language that i understand. but, you're right that i
>was wondering if in this mathematical analysis that i have not yet
>decoded, there was some implication that this 12 note per octave
>equal-tempered scale was somehow mathematically more natural than other
>tuning systems.

No! It's got its charms, and its charms can be studied mathematically.
But so do lots of other scales.

The main charms I discussed were its Z/12 transposition group
and its 24-element transposition-inversion group. But if we had an
equal-tempered 93-note scale, it would have a Z/93 transposition
group and a 186-element transposition-inversion group! As I
explained - using too much math jargon for you to easily digest -
a bunch of stuff works for ANY equal-tempered scale, including
the superficially shocking fact that the operations on sufficiently
generic chords which *commute* with transposition and inversion
themselves form a group *isomorphic* to the transposition-inversion group.

All the far-out weird stuff at the end of "week234" - about groups
that act on 7-element or 12-element or 24-element sets, like
PSL(2,11) and M_{12} and M_{24} and PSL(2,7) - is stuff that
turns out *not* to have much to do with Western music, much as
I wish it did. I had to work out some stuff to see that this
sad fact is true, so that's what I did - with a huge boost from
Noam Elkies, who (as we see from his post) had studied these
issues before. Luckily, these groups have plenty of interests
besides music.

>actually i *do* think that there is some kinda mathematical
>*serendipity* in the 12-note per octave equal-tempered scale.
>equal-tempered scales are equally good or equally bad in all keys
>(which makes them useful for discretely keyed instruments) and the
>12-note equal-tempered scale is, IMO, the one that lies at a sorta
>equilibrium of two different opposing pressures, one to have intervals

>as harmonic as possible and the other to ergonomically have as few


>buttons to push as possible. it really *is* fortuitous that (3/2)^12
>is very nearly equal to 2^7.

Yes. As Dave Rusin explains:

http://www.math.niu.edu/~rusin/uses-math/music/12

the continued fraction expansion of log(3)/log(2) shows that
the best rational approximations to this number are:

1/1, 2/1, 3/2, 8/5, 19/12, 65/41, 84/53, 485/306, 1054/667, ....

The 19/12 here means that going up 12 fifths is darn close
to going up 19-12 = 7 octaves:

(3/2)^{12} ~ 2^7

129.746 ~ 128

which makes the 12-note-per-octave equal-tempered scale nice.

If we didn't want so many notes, the 8/5 up there says that
the pentatonic scale would be our next best bet, since going
up 5 fifths is close to going up 8-5 = 3 octaves:

(3/2)^5 ~ 2^3

7.59375 ~ 8

Not quite so impressive, but perhaps this explains the
popularity of pentatonic scales!

If we had lots of fingers, maybe we'd go for the 41-note
scale that Rusin likes, since the 65/41 up there says that
going up 41 fifths is close to going up 65-41 = 24 octaves:

(3/2)^{41} ~ 2^{24}

16,585,998 ~ 16,777,216

But, the 12-note scale is not bad, and it has the added
charm of 12 being a highly divisible number, etc. etc.

But, proponents of other scales can marshal arguments for
their preferences, too. The great thing is, we don't need
to choose - we can listen to all the music we like.

Phillip Helbig---remove CLOTHES to reply

unread,
Jun 18, 2006, 9:07:23 PM6/18/06
to
bristow-johnson <r...@audioimagination.com> writes:

> of course, with equally tempered
> instruments, any of these keys have the same relative frequency ratios
> between notes. any qualitative difference sensed is purely a function of
> absolute pitch and i am not sure that those of us without perfect pitch (but
> with a good sense of relative pitch) would know the difference between
> pieces played a semitone or two different in pitch. i think they would
> evoke the same feeling that the music aims to make.

If I play an D, tell you it is D, then play something else and you can
tell me what it is, then you have good relative pitch. If I play
something and don't tell you what it is, but you can tell me what it is,
then you have good absolute pitch. However, this is just a fancy name
for a good memory, i.e. you know what that note is because you have
heard it before. "Absolute pitch" sounds mysterious, but it's really
just a good memory. (Of course, there is nothing physiological about it
in the sense that one can somehow, through some in-born ability or
whatever, recognise particular notes, since the pitch of the notes---one
could see this as a basic offset---is more or less arbitrary. These
days, a' is set to 440 Hz, but that is a relatively recent development.
I have some CDs of baroque music with a' set to 415 or 392 Hz, since
those were more common tunings at the time the music was written. It's
not the same as just transposing down to a lower key, since tuning the
instruments to a lower pitch changes the colour of the sound (it makes
the high overtones less prominent).)

Phillip Helbig---remove CLOTHES to reply

unread,
Jun 18, 2006, 9:07:23 PM6/18/06
to
In article <e71dcg$fp6$1...@glue.ucr.edu>,
ba...@math.removethis.ucr.andthis.edu (John Baez) writes:

> >what we popularly call "major" or "minor" are just two, and if you bring in
> >the orient or other musical traditions, of a zillion other scales. like the
> >SI system of units, there is nothing special or universal about these
> >western scales. they're not like Planck units.

Actually, one doesn't have to go that far afield. The Doric mode is
common in, for example, mediaeval music and heavy-metal music.

Michael Jørgensen

unread,
Jun 19, 2006, 4:45:48 PM6/19/06
to
"Tom Roberts" <tjrobe...@sbcglobal.net> wrote in message
news:vAUjg.24200$VE1....@newssvr14.news.prodigy.com...

> Pianos are
> "stretch tuned" because the harmonics of lower strings are not exactly
> integer multiples of their fundamental frequencies, they are slightly
> sharp.

This is new to me!

I learnt that the harmonics are determined from the wave length, and the
latter must obviously be the total length divided by some integer.

So I guess you're saying that the stretching force of the string is
dependent on the frequency?

I'm just curious to learn exactly where the simple-minded theory breaks
down.

-Michael.

Ken S. Tucker

unread,
Jun 19, 2006, 4:50:03 PM6/19/06
to
John Baez wrote:
> In article <C0B8F249.155F3%r...@audioimagination.com>,
> robert bristow-johnson <r...@audioimagination.com> wrote:

>
> > John Baez wrote:
>
> >> 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.
>
> >that's a sorta "best fit" compromise we come to because we don't like
> >keyboards with 19 or 31 notes per octave.
>
> >> 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.
>
> >that, i think, is more of an historical accident or coincidence.
>
> Maybe!
>
> I wasn't trying to pass judgement on questions like that.
>
> I hope you didn't think that just because I was using math to talk
> about Western music, that I thought these features were "natural"
> or "optimal".
>
> >> 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.
>
> >i am not sure what this is about.
>
> I was just joking about the analogy to spontaneous symmetry
> breaking in physics:
>
> When water freezes, the translation and rotation symmetry
> get spontaneously broken by how the water molecules pick out
> a specific crystal lattice. An ice crystal does not exhibit symmetry
> under arbitrary translations and rotations: only those that map
> the lattice to itself. The symmetry is still there, but
> it's only visible in our freedom to translate or rotate the crystal.
>
> Similarl, playing music in C major, say, breaks the Z/12 symmetry
> of the equal-tempered scale by picking out a 7-element subset
> of notes that you play more. A composition in C major doesn't
> exhibit any sort of Z/12 symmetry. The symmetry is only visible
> in our freedom to transpose the whole composition to another key.
>
> Spontaneous symmetry breaking happens at low temperatures;
> symmetry gets restored at high temperatures.
>
> When we heat up ice to 273 kelvin, it melts and the rotation and
> translation symmetry is restored: liquid water favors no lattice.

> If we could heat up some water to about 1 or 2 quadrillion kelvin,
> the symmetry between the electromagnetic and weak would (we
> believe) get restored: the Higgs field no longer picks out a
> specific direction. Similarly, when we "heat up" a piece of music
> by throwing in lots of accidentals and shifts of key, the Z/12
> symmetry gets restored.
>
> That's all I was trying to say; don't take it too seriously,
> it's just a fun idea.
>
> >> 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,
>
> ...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.
>
> >what we popularly call "major" or "minor" are just two, and if you bring in
> >the orient or other musical traditions, of a zillion other scales. like the
> >SI system of units, there is nothing special or universal about these
> >western scales. they're not like Planck units.
>
> Yup. Again, I hope you don't think I don't think this!

While the discussion feels about subjective music,
the transmission of information given by,

http://en.wikipedia.org/wiki/DTMF

seems to favor anti-harmonics. Subjectivity aside,
that seems to favor prime numbers in ratio.
At night bugs are screaming out various frequency's
- you know - for mating, and it's practically a chorus.

We set-up EAR's,

http://earco.travisktucker.com/

to listen to natural sounds.
Ken

Gene Ward Smith

unread,
Jun 19, 2006, 4:50:08 PM6/19/06
to
tc...@lsa.umich.edu wrote:

> There have been some serious musical compositions in other scales. For
> example, I once attended a concert in which one of the pieces was based
> on the 19-tone equal tempered scale.

Some compositions in it can be found linked to the Wikipedia article on
it:

http://en.wikipedia.org/wiki/19_equal_temperament

> As I recall, the 19-tone scale realizes the major third and perfect fifth
> slightly better than the 12-tone scale.

No, the fifth is flatter, that is characteristic of meantone tunings.
It makes up for it with a better (if flat) major third and a nearly
just minor third.

Gene Ward Smith

unread,
Jun 19, 2006, 4:50:13 PM6/19/06
to
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.

For one thing they can be generalized to higher dimensions, involving
higher "prime limits":

http://www.xenharmony.org/sevlat.htm

or

http://66.98.148.43/~xenharmo/sevlat.htm

(whichever works for you.) Moreover, they can be put into normed vector
spaces with something other than the L2 norm, which sometimes has
advantages.

> 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.

This group theory has a parallel treatment in terms of just intonation
and the Tonnetz and its higher dimensional analogues, which I think
makes
for a stronger analogy between what is going on and symmetry breaking
in physics; at least, that is how I've thought of it for the last few
decades. You can read a little about it on the same page above, but I
should elaborate the connection. The group of the hexagon, of
isometries of the Tonnetz lattice fixing a note which we can identify
as the
lattice (a lattice being, among other things, a group) identity. We can
then lift this to a broken symmetry on pitches, rather than pitch
classes, by requiring that the
approximate pitch measurement provided by 3-equal temperament (<3 5 7|
as a homomorphic map) be preserved; and then extend it by including
inversion to get the full group of the hexagon. Similarly, we can take
the 3D lattice (an A3 ~ D3 type of lattice for the lattice buffs) and
consider the symmetries preserving 4-equal temperament, which leads to
the group of the cube of order 48.

One result of all of this is this little ditty:

http://tunesmithy.netfirms.com/tunes/tunes.htm#hexany_phrase

This was all written up in a paper of twenty years ago, which the
editors where I sent it declined to publish on the grounds that it (and
I'm quoting) was "too mathematical".

> It would be fun to dream up more relations between incidence
> geometry and music theory. Could Klein's quartic curve play a
> role?

The Klein Quadric does. Rational points on it correspond to rank two
temperaments of the seven-limit. The Klein Quadric is a Grassmann
manifold, and in general the regular temperaments in any prime limit p
may be classified by rational points on a Grassmannian, which is a
projective variety, but in some sense also a generalization of
projective space.

Gene Ward Smith

unread,
Jun 19, 2006, 4:50:22 PM6/19/06
to
John Baez wrote:

> At the next step, we get an interval of 7/6, which is not part
> of Western music; it sounds "bad".

It does not sound bad, and it is a part of Western music. However, the
traditional name for it in Western musical theory is "augmented
second". An augmented second was very close to a 7/6 ratio in the range
of more or less optimal meantone tunings in the vicinity of the
traditional 1/4 comma meantone. In other words the meantone tuning
which makes it pure, which has a fifth of 696.3 cents, is very close to
the fifth which makes the major third pure, which is 696.6 cents, and
was the de facto standard tuning in the West for a considerable period.

It has a darker sound than the just minor third, but is generally about
as consonant. In 12 equal temperament, the two are both pretty far out
of tune, and conflated in a compromise pitch of 300 cents.

> > 1/1 2/1
> > 3/2
> > 4/3 5/3
> > 7/4
> > 6/5 7/5 8/5 9/5
> > 7/6 11/6
> > 8/7 9/7 10/7 11/7 12/7 13/7
> > 9/8 11/8 13/8 15/8
> > 10/9 11/9 13/9 14/9 16/9 17/9
> > 11/10 13/10 17/10 19/10
>
> > 1/1 9/8 5/4 4/3 3/2 5/3 15/8 2/1

This is the Stern-Brocot tree between 1 and 2, incidentally.

> Dave Rusin suggest that the next really good number after 12 notes is
> 41 notes:

I'd say it was 19. 41 derives from the denominators of the continued
fraction for log2(3), but this is an extremely limited perspecive on
equal tunings.

On Sloane's OEIS, there are a number of integer sequences relating the
Riemann zeta function on the critical line which I think are of
theoretical interest.

Highest maximum values picks 19 as the next after 12:

http://www.research.att.com/~njas/sequences/A117536

Integrating between two zeros also picks 19 after 12:

http://www.research.att.com/~njas/sequences/A117538

Increasingly large relative spacing between the zeros of the Riemann
zeta function also picks 19 after 12, though this seems less reliable
as an indicator (it doesn't pick 41 at all!)

http://www.research.att.com/~njas/sequences/A117537

> http://en.wikipedia.org/wiki/Rothenberg_propriety

Wow, I just wrote that last week! The net is fast.

Dik T. Winter

unread,
Jun 19, 2006, 4:50:45 PM6/19/06
to
In article <C0B8F249.155F3%r...@audioimagination.com> robert bristow-johnson <r...@audioimagination.com> writes:
> in article e6la3o$911$1...@glue.ucr.edu, John Baez at
> ba...@math.removethis.ucr.andthis.edu wrote on 06/13/2006 08:10:
.
> > 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.
>
> that's a sorta "best fit" compromise we come to because we don't like
> keyboards with 19 or 31 notes per octave.

Because *some* people don't like keyboards with 19 or 31 notes per octave.
Ever heard about the Huygens-Fokker 31-tone organ? There has even been
music composed for it, and (at least some) is available on CD.

> > 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.
>
> that, i think, is more of an historical accident or coincidence.

It is more than an historical accident or coincidence in Western music.
In Western music it dates from long ago.

> they could
> have chosen a subset of those 7 (e.g. the "pentatonic scale", the same 5
> notes picked out of the minor set is often used as the "rock scale" or
> "blues scale", sometimes with the dim5 note added, that lead guitarists like
> to riff with).

Yes, other scales are possible, and exist. But before the equal-tempered
scale the distances between notes was not the same, and there was a
distinction between C major and D major that did go beyond a slightly
higher pitch.

> > 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.
>
> i am not sure what this is about.

Mathematics.

> physically, the frequency ratio for an octave is exactly 2 (going up) or 1/2

> (going down). why is that? why not a frequency ratio of 10 or "e" or pi or


> something like that to be the most primitive (other than unison) musical
> interval?

Look up Pythagoras. He found already that two tones sounded nice together
as a chord when the ratio of the frequencies was equal to the ratio of two
small integers. 10 fits (but the difference in pitches is quite large).
pi and e might fit when you approximate them (but even the 22/7 approximation
of pi would sound horrible to some).

> this is why we in the "west" ended up with the 12 note/octave equal tempered
> scale.

Before Bach there were piano's that were not well-tempered.

> i think 31 notes/octave hits these intervals better (when i was a
> freshman, i ran a Fortran program testing this, i should redo it in MATLAB),
> but such a discrete-pitch instrument, might be unwieldy to play.

Go to Teylers museum in Haarlem (the Netherlands) where a 31
notes/octave instrument is played on a fairly regular basis.

> if i remember right, the intervals they selected for the "just intonation"
> scale are

The "they selected" is just a wrong terminology. "They found" would be
better. In Western music you had the ratios 2/1 (octave), 3/2 (fifth),
4/3 (fourth), 5/4 (major third) and 6/5 (minor third). (See where the
major and minor scales in Western music come from?)
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/

Paul Danaher

unread,
Jun 19, 2006, 4:50:51 PM6/19/06
to
robert bristow-johnson wrote:
..

> i
> really just view that so far as one of them sorta unexplained
> numerical gifts from nature that we don't yet understand.

This takes us back to the question why music and physics and mathematics can
have a sensuous satisfaction, doesn't it? I remember reading a comment about
the proof of the five-colour theorem that expressed disappointment on the
lines of "If this is the answer, it can't have been a good question!". Then
there's the Feynman "lost lecture" about the motion of planets around the
sun, which is ultimately unsatisfying ...

eb...@lfa221051.richmond.edu

unread,
Jun 20, 2006, 9:55:40 AM6/20/06
to
In article <44925e33$0$38663$edfa...@dread12.news.tele.dk>,

Michael Jørgensen <ccc5...@vip.cybercity.dk> wrote:
>"Tom Roberts" <tjrobe...@sbcglobal.net> wrote in message
>news:vAUjg.24200$VE1....@newssvr14.news.prodigy.com...
>
>> Pianos are
>> "stretch tuned" because the harmonics of lower strings are not exactly
>> integer multiples of their fundamental frequencies, they are slightly
>> sharp.
>
>This is new to me!
>
>I learnt that the harmonics are determined from the wave length, and the
>latter must obviously be the total length divided by some integer.
>
>So I guess you're saying that the stretching force of the string is
>dependent on the frequency?

It's more that "stretching" or "tension" isn't sufficient to describe
the behavior of the string. You also need something like "stiffness"
to have an adequate model of the internal forces in the string.
Tension is, of course, the property of a string that characterizes its
desire not to be stretched; stiffness characterizes its desire not to
be bent.

If a string has tension but negligible stiffness, then you get the familiar
wave equation

(d^2 y / dt^2) = c^2 (d^2 y / dx^2),

with c^2 = T/lambda = tension / (mass per unit length). The
solutions, of course, are waves with constant speed c. For constant
wave speed, the frequencies are inversely related to the wavelengths.
The wavelengths of standing waves on the string are lambda = 2L/n with
L the length of the string, so you get nice harmonics: frequency is
proportional to n.

Stiffness adds another term to the wave equation -- a d^4 y / dx^4
term if I'm not mistaken. Now the wave speed is frequency-dependent.
So even though the wavelengths on the string are still 2L/n, the
frequencies are not proportional to n. And your ear doesn't care
about the wavelength along the string; it cares about frequency.

David W.Cantrell

unread,
Jun 20, 2006, 9:30:20 PM6/20/06
to
Dik.W...@cwi.nl (Dik T. Winter) wrote:
> In article <C0B8F249.155F3%r...@audioimagination.com> robert
> bristow-johnson <r...@audioimagination.com> writes:
[snip]

> > this is why we in the "west" ended up with the 12 note/octave equal
> > tempered scale.
>
> Before Bach there were piano's that were not well-tempered.

Perhaps something got lost in "translation". Of course, before Bach there
were keyboard instruments (Klavier, in the _general_ sense) which were not
well-tempered. That was your point. But the piano _per se_ did not exist
before Bach. It was invented during his lifetime (spec. in 1709) by
Cristofori, who originally called it "gravicembalo col pian e forte". IIRC,
it is a matter of some speculation whether Bach ever actually encountered a
pianoforte.

David

Matt Noonan

unread,
Jun 20, 2006, 9:30:20 PM6/20/06
to

eb...@lfa221051.richmond.edu wrote:
> In article <44925e33$0$38663$edfa...@dread12.news.tele.dk>,
> Michael Jørgensen <ccc5...@vip.cybercity.dk> wrote:
> >"Tom Roberts" <tjrobe...@sbcglobal.net> wrote in message
> >news:vAUjg.24200$VE1....@newssvr14.news.prodigy.com...
> >
> >> Pianos are
> >> "stretch tuned" because the harmonics of lower strings are not exactly
> >> integer multiples of their fundamental frequencies, they are slightly
> >> sharp.
> >
> >This is new to me!
> >
> >I learnt that the harmonics are determined from the wave length, and the
> >latter must obviously be the total length divided by some integer.
> >
> >So I guess you're saying that the stretching force of the string is
> >dependent on the frequency?
>
> It's more that "stretching" or "tension" isn't sufficient to describe
> the behavior of the string. You also need something like "stiffness"
> to have an adequate model of the internal forces in the string.
<snip>

> Stiffness adds another term to the wave equation -- a d^4 y / dx^4
> term if I'm not mistaken. Now the wave speed is frequency-dependent.
> So even though the wavelengths on the string are still 2L/n, the
> frequencies are not proportional to n. And your ear doesn't care
> about the wavelength along the string; it cares about frequency.

Here's something I've been wondering: the derivation of the wave
equation uses the low-amplitude assumption to say that the force on the
string is proportional to the displacement, acting at right angles to
the rest position of the string. In reality, the force acts in the
direction of the string's curvature vector. For a low note, it seems
that the amplitude would be relatively high. So are the errors in the
right-angle approximation large enough to be audible on the low notes
of a piano?

Matt

robert bristow-johnson

unread,
Jun 21, 2006, 9:36:28 AM6/21/06
to
Dik T. Winter wrote:
> In article <C0B8F249.155F3%r...@audioimagination.com> robert bristow-johnson <r...@audioimagination.com> writes:
..

> > that's a sorta "best fit" compromise we come to because we don't like
> > keyboards with 19 or 31 notes per octave.
>
> Because *some* people don't like keyboards with 19 or 31 notes per octave.
> Ever heard about the Huygens-Fokker 31-tone organ?

no, can't say that i have.

> There has even been
> music composed for it, and (at least some) is available on CD.
>
> > > 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.
> >
> > that, i think, is more of an historical accident or coincidence.
>
> It is more than an historical accident or coincidence in Western music.
> In Western music it dates from long ago.
>
> > they could
> > have chosen a subset of those 7 (e.g. the "pentatonic scale", the same 5
> > notes picked out of the minor set is often used as the "rock scale" or
> > "blues scale", sometimes with the dim5 note added, that lead guitarists like
> > to riff with).
>
> Yes, other scales are possible, and exist. But before the equal-tempered
> scale the distances between notes was not the same, and there was a
> distinction between C major and D major that did go beyond a slightly
> higher pitch.

of course. i think that was some impetus for the equal tempered scale.

>
> > physically, the frequency ratio for an octave is exactly 2 (going up) or 1/2
> > (going down). why is that? why not a frequency ratio of 10 or "e" or pi or
> > something like that to be the most primitive (other than unison) musical
> > interval?
>
> Look up Pythagoras.

i know about Pythagorean tuning.

> He found already that two tones sounded nice together
> as a chord when the ratio of the frequencies was equal to the ratio of two
> small integers. 10 fits (but the difference in pitches is quite large).
> pi and e might fit when you approximate them (but even the 22/7 approximation
> of pi would sound horrible to some).

my question "why" was rhetorical.

> > this is why we in the "west" ended up with the 12 note/octave equal tempered
> > scale.
>
> Before Bach there were piano's that were not well-tempered.

probably with mean-tone tuning for the sharps. and if you transposed a
piece of music up or down a half step or whole step, the music would
sound qualitatively different (not just a little higher or lower).

> > i think 31 notes/octave hits these intervals better (when i was a
> > freshman, i ran a Fortran program testing this, i should redo it in MATLAB),
> > but such a discrete-pitch instrument, might be unwieldy to play.
>
> Go to Teylers museum in Haarlem (the Netherlands) where a 31
> notes/octave instrument is played on a fairly regular basis.

i've heard of the 31 (and 19) note scales. just didn't know about the
instruments.

> > if i remember right, the intervals they selected for the "just intonation"
> > scale are
>
> The "they selected" is just a wrong terminology. "They found" would be
> better. In Western music you had the ratios 2/1 (octave), 3/2 (fifth),
> 4/3 (fourth), 5/4 (major third) and 6/5 (minor third). (See where the
> major and minor scales in Western music come from?)

i thinkl it *is* the correct terminology. there were all sorts of
intervals that they "found" and you have alluded to some of them. why
use 5/4 for the major third instead of (9/8)^2 (the pythagorean major
third)? it's a selection. they liked it better. why use 15/8 for a
major 7th instead of 13/7 or 11/6? probably because they were starting
to realize that the simpler ratios moved the note farther from 2/1 and
they wanted to maintain that "half step" relationship. they *selected*
15/8 and *rejected* 13/7. it's a selection (perhaps for some very good
reasons), but no natural order forced musicians to use 15/8 over 13/7.

i still think the choices of just-intoned intervals, that eventually
led to the 12-note equal tempered scale has to do with the fact that in
perceived pitch (closely related to log of frequency), there was very
nearly twice the measure of pitch between some intervals than others so
that with this selection of just-tuned intevals:

1/1 9/8 5/4 4/3 3/2 5/3 15/8 2/1

or for the minor scale:

1/1 9/8 6/5 4/3 3/2 5/3 7/4 2/1

there are seven adjacent intervals of which five adjacent intervals
have about twice the spacing in log frequency as the remain two
adjacent intervals. that's convenient and it's a choice and it will
lead to the compromise of the 12-note equal tempered scale.

r b-j

J. B. Wood

unread,
Jun 21, 2006, 9:36:28 AM6/21/06
to
In article <20060620125902.679$5...@newsreader.com>, "David W.Cantrell"
<DWCan...@sigmaxi.org> wrote:

Hello, and I would just like to add that authoritative references like the
Oxford and Harvard music dictionaries refer to this category of scale
tunings as "irregular" temperaments (temperings that generally preserve
some just 3rds and just 5ths while eliminating the wolf fifth present in
meantone tunings). Calling them "well" draws the inevitable linkage to
J.S. Bach's "Well-Tempered Clavier" (WTC) pieces. There is considerable
disagreement among scholars as to whether Bach intended the WTC as a
demonstration of the advantages of equal-temperament (a tune when
transposed to another key sounds the same) or an irregular temperament
(all keys usable but a tune has a distinct sound in a particular key).
Perhaps Bach intended his WTC to showcase the advantages/disadvantages of
both types of tempering. Sincerely,

eb...@lfa221051.richmond.edu

unread,
Jun 21, 2006, 7:42:00 PM6/21/06
to
In article <1150827532....@i40g2000cwc.googlegroups.com>,
Matt Noonan <matt....@gmail.com> wrote:

>Here's something I've been wondering: the derivation of the wave
>equation uses the low-amplitude assumption to say that the force on the
>string is proportional to the displacement, acting at right angles to
>the rest position of the string. In reality, the force acts in the
>direction of the string's curvature vector. For a low note, it seems
>that the amplitude would be relatively high. So are the errors in the
>right-angle approximation large enough to be audible on the low notes
>of a piano?

I'm not quite sure why there'd be a pitch-dependence. This effect is
certainly amplitude-dependent, though. To try to measure it, you
should play your piano really really loud.

I think it's likely to be quite small, though. Here's why. (I'll
admit that this is a heuristic argument, not a real calculation. Feel
free to rip it to shreds.)

In the usual approximations people always make to derive things about
waves on a string, the dispersion relation is

omega^2 = c^2 k^2.

That is, the wave equation has solutions that look like
A exp(i k x - i omega t) with the above relation between
frequency and wavenumber.

Now suppose you turn up the amplitude to the point where nonlinear
amplitude-dependent effects start to matter. Presumably, the
dispersion relation becomes a perturbed version of the above:

omega^2 = c^2 k^2 [1 + (something)*A + (something)*A^2 + ... ]

On dimensional grounds, the first something must be proportional to k,
to cancel out the dimension of A. But the dispersion relation must
be an even function of k -- the string doesn't care if waves are propagating
left or right -- so the leading correction must be the A^2 term.
In fact, the dispersion relation must be

omega^2 = c^2 k^2 [1 + c (kA)^2]

where c is a dimensionless constant. k is essentially the reciprocal
wavelength, so I claim that changes in the frequency due to
amplitude-dependent effects must scale like the square of amplitude /
wavelength.

A/lambda must be at most a few times 10^-3, right? (I
haven't looked closely at a vibrating piano string for a while, but
that seems right). So amplitude-dependent corrections ought to shift
the frequency by a part in 10^5. That's about 100 times smaller than,
for instance, the differences in temperament people have been
talking about in this thread.

Gene Ward Smith

unread,
Jun 21, 2006, 7:42:01 PM6/21/06
to
[ Mod. note: It may be time to wrap up or steer back on topic,
i.e. physics. -ik ]

J. B. Wood wrote:

> There is considerable
> disagreement among scholars as to whether Bach intended the WTC as a
> demonstration of the advantages of equal-temperament (a tune when
> transposed to another key sounds the same) or an irregular temperament
> (all keys usable but a tune has a distinct sound in a particular key).

Can you cite a scholar who claims Bach was using equal temperament?
Because while there is quite a lot of arguing going on, all of it I
know about is over which circulating temperament he used, eg the
brouhaha over Brad Lehman's recent proposal. I know of no source which
says equal temperament was being used on keyboard instruments during
this period.

> Perhaps Bach intended his WTC to showcase the advantages/disadvantages of
> both types of tempering. Sincerely,

A much more reasonable suggestion, I think, is that it is intended to
compare meantone with circulating temperaments. We don't know what Bach
meant by "well tempered", however.

One striking fact about WTC I in particular is that in any given key,
the harmonic compass is small enough that it can be played very
successfully in meantone. If a performer was willing to dig out his
tuning wrench (and harpsichords then were eminently retunable, and
often retuned) before playing a particular prelude and fugue, it could
be played that way. Whether Bach meant anything by this fact is unknown.

tadchem

unread,
Jun 22, 2006, 2:42:32 AM6/22/06
to
John Baez wrote:
> 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 -

There are many types of musical scales, each suited to its own purpose.

A good place to start:
http://en.wikipedia.org/wiki/Musical_scales

Harmony and chords work well in the *chromatic scale* due to the simple
ratios of the frequencies. The other links embedded within the article
are also quite interesting to a true student of the mathematics of
music.

Tom Davidson
Richmond, VA

David W.Cantrell

unread,
Jun 22, 2006, 2:43:04 AM6/22/06
to
In article <20060620125902.679$5...@newsreader.com>, "David W.Cantrell"
<DWCan...@sigmaxi.org> wrote:

I didn't remember correctly, as pointed out by Gene. My retraction appeared
yesterday in sci.physics and sci.math, but not here. Let me correct that
now:

"Gene Ward Smith" <genewa...@gmail.com> wrote:
> Silbermann took up making early fortepianos, and Silbermann of course
> knew Bach. The story as I understand it is that Silbermann showed Bach
> one of his early pianos, and Bach didn't like it, but liked later
> versions and even promoted them. Do you have a cite for the
> controversy, by any chance?

My response:

I apologize. My memory just needed some refreshing, let's say. I have no
reason to suppose that <http://en.wikipedia.org/wiki/Gottfried_Silbermann>
is inaccurate. So Bach must have seen one, and he liked Silbermann's later
instruments better. But I doubt that he liked them well enough to buy one
for himself. There were quite a few instruments inventoried at the time of
his death (including IIRC more than one Lautenwerk) but no pianoforte.

David

John Baez

unread,
Jun 22, 2006, 2:43:15 AM6/22/06
to
In article <e74j35$b4o$2...@online.de>,
Phillip Helbig wrote:

>If I play an D, tell you it is D, then play something else and you can
>tell me what it is, then you have good relative pitch. If I play
>something and don't tell you what it is, but you can tell me what it is,
>then you have good absolute pitch.

There's an interesting effect called the "tritone paradox" which
is evidence that many people have some form of absolute pitch
without knowing it:

http://en.wikipedia.org/wiki/Tritone_paradox
http://psy.ucsd.edu/~ddeutsch/psychology/deutsch_research6.html

To understand this effect you first need to understand a
"Shepard tone". A Shepard tone is a superposition of sine
waves separated by octaves - ideally, of all frequencies:

http://en.wikipedia.org/wiki/Shepard_tone

Digression: a perpetually rising Shepard tone sounds like it's
getting higher and higher... but mysteriously never getting too
high to hear! It's the auditory equivalent of the spiral on a
turning barbershop pole. It's also a bit like this illusion:

http://asa.aip.org/gif/demo27b.gif

It's fun to listen to a rising Shepard tone and try to detect
how one is being fooled:

http://www.cs.ubc.ca/nest/imager/contributions/flinn/Illusions/ST/st.html
http://www.netalive.org/tinkering/shepard-effect/

The fast you play it, the easier it is to catch the "tradeoff"
where one rising notes fades out and the rising note an octave
below fades in. I find the first of the above websites to be a
bit nicer. Click on "APPLET: Shepard's Tones", then click "Start"
and also click on "Display" to see what's going on.

Anyway: the "tritone paradox" consists of playing two Shepard's
tones separated by half an octave. Neither is "higher" than the
other in any objective sense, but it turns out that people often
have a strong, fixed opinion about which one sounds higher - which
depends on where in the scale the notes in question lie! This
suggests they have a certain amount of absolute pitch ability.

You can try it here:

http://www.cs.ubc.ca/nest/imager/contributions/flinn/Illusions/ST/st.html

.........................................................................

In article <e74j6s$b4o$3...@online.de>,
Phillip Helbig:

Note that it was Robert Bristow-Johnson, not I, who said that
major or minor are just two of a zillion scales.

John Baez

unread,
Jun 22, 2006, 2:43:21 AM6/22/06
to
Dik.W...@cwi.nl (Dik T. Winter) wrote:

> In article <C0B8F249.155F3%r...@audioimagination.com> robert
> bristow-johnson <r...@audioimagination.com> writes:

> > this is why we in the "west" ended up with the 12 note/octave equal
> > tempered scale.

> Before Bach there were piano's that were not well-tempered.

I'm sure everyone here knows this, but it's worth pointing
out that Bach's "well-tempered" tuning system is probably
different from the "equal-tempered" tuning system.

In the currently popular "equal-tempered" tuning system, each note
on the piano vibrates 2^{1/12} times as fast as the one below it...
at least if we ignore the slight "stretching" of this system that
people use on actual pianos.

There were a number of "well-tempered" systems in Bach's day,
none of which was the "equal-tempered" system:

http://en.wikipedia.org/wiki/Well_temperament

Nobody really knows which system Bach used. This page:

http://www.kirnberger.fsnet.co.uk/Temps3.htm

argues that it was the Werckmeister III system. They also explain
how this system works.

The advantage of non-equal-tempered systems, of course, is
that different keys have different personalities, making for
more interesting music.

J. B. Wood

unread,
Jun 22, 2006, 5:38:26 PM6/22/06
to
In article <1150910239.2...@p79g2000cwp.googlegroups.com>, Gene
Ward Smith <genewa...@gmail.com> wrote:

> Can you cite a scholar who claims Bach was using equal temperament?

Rasche. But we don't know if JSB favored equal temperament (ET) over
other extant tuning systems.

> Because while there is quite a lot of arguing going on, all of it I
> know about is over which circulating temperament he used, eg the
> brouhaha over Brad Lehman's recent proposal. I know of no source which
> says equal temperament was being used on keyboard instruments during
> this period.

AFAIK while equal temperament was known during JSB's time (used on lutes)
its use on keyboard instruments is questionable. I think the practicality
of unlimited modulation (desirable for transposing pieces to accomodate
differing human voice ranges) into any of 12 keys would be recognized but
in the case of keyboard instruments (harpsichord, clavichord, etc) the
tuning procedures for ET had not been worked out.

> A much more reasonable suggestion, I think, is that it is intended to
> compare meantone with circulating temperaments. We don't know what Bach
> meant by "well tempered", however.

That is reasonable but we also should keep in mind that ET certainly
circulates and for all practical purposes is indistinguishable from a 1/11
syntonic comma meantone. The centuries-old Pythagorean tuning is in fact
a "0" comma meantone. In all 12-tone meantone tunings we adjust 11 just
fifths (pitch ratio 3/2) identically by some fraction of a syntonic
comma. The remaining fifth is the "wolf". When we temper by 1/11 comma
this remaining fifth has the same width as the other 11.

> One striking fact about WTC I in particular is that in any given key,
> the harmonic compass is small enough that it can be played very
> successfully in meantone. If a performer was willing to dig out his
> tuning wrench (and harpsichords then were eminently retunable, and
> often retuned) before playing a particular prelude and fugue, it could
> be played that way. Whether Bach meant anything by this fact is unknown.

An examination of the circle of 5ths reveals that a meantone tuning
permits modulation into six adjacent keys without crossing the wolf. I
would not want to hear the WTC played in a key where the wolf was
crossed. I would think because of this JSB would have favored either an
irregular temperament (such as proposed by Vallotti/Young, Barnes, or
Kellner) or ET for his WTC. But as you point out his goal might have been
to show the modulation limits of meantone.
Unless additional historical evidence surfaces, we'll continue
speculating. Sincerely,

Hans Aberg

unread,
Jun 22, 2006, 5:38:42 PM6/22/06
to
In article <e7d3cv$j40$1...@glue.ucr.edu>,
ba...@math.removethis.ucr.andthis.edu (John Baez) wrote:

> In the currently popular "equal-tempered" tuning system, each note
> on the piano vibrates 2^{1/12} times as fast as the one below it...
> at least if we ignore the slight "stretching" of this system that
> people use on actual pianos.

And the fact that the pitch changes depending on how hard the key is
struck. The scale stretch can in fact be quite considerable; it is most
conservative on concert grand pianos, which typically have no-stretch zone
in the middle octaves.

> There were a number of "well-tempered" systems in Bach's day,
> none of which was the "equal-tempered" system:
>
> http://en.wikipedia.org/wiki/Well_temperament

And 12-TET came into use rather late, in the 19th century:
   http://en.wikipedia.org/wiki/Equal_temperament

One should note that the scale considerable affects the chord texture, and
in this way, one can end up using hight partials, like 19. A program where
one can experiment with this is Scala:
  http://www.xs4all.nl/~huygensf/scala/
Check for example out the rational (Just) chords:
  m               10:12:15
  quasi-equal m   16:19:24
Then, in a MIDI church organ, the former minor, which is classical Just
intonation, gets a high partial, which the quasi-equal minor. Hence, the
latter might be preferred, if this partial is viewed as disturbing (but in
another musical context, it might be viewed as favorable), leading to a
scale which involves the partial 19.

Using 12-TET chords usually produce brighter chords. One can note that in
modern music, instruments with ability to produce more partials are
preferred. A theory I have is that the reason for this is that such
instruments have better capability to compete with the bright 12-TET
chords.

It is otherwise unclear what exact scale orchestral instruments are plying
in, as it depends on how the musician bends the pitch. An instrument
"tuned in 12-TET" is in reality merely an instrument that facilitates the
playing in 12-TET if the musician so wills. See for example the tone
offsets of a flute:
  http://www.landellflutes.com/-Products/NCS.htm
Especially on older flute scales, the compensation the musician has to do
on each note in order to play 12-TET scale is considerable.

I can bend some +- 30-40 cents on the flute. So it seems possible to play
some Just intonations. The style of playing would then to use 12-TET
merely as a method of facilitating scale key modulation, while playing in
some Just intonation, once the modulation has been done. Another
complication is that the formal scale pitches are not actually played, but
merely used as a reference, making it hard to measure these pitches down
by means of a physical measurement apparatus. Musicians are not
very conscious in this formal way about these physical pitches during
performance, but use instead intuition to make it sound right, making
it difficult to get hard facts about what pitches are used as a reference.

--
Hans Aberg

eb...@lfa221051.richmond.edu

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Jun 22, 2006, 5:38:47 PM6/22/06
to
In article <e7bogm$qi1$1...@bigbang.richmond.edu>,

<eb...@lfa221051.richmond.edu> wrote:
>In article <1150827532....@i40g2000cwc.googlegroups.com>,
>Matt Noonan <matt....@gmail.com> wrote:
>
>>Here's something I've been wondering: the derivation of the wave
>>equation uses the low-amplitude assumption to say that the force on the
>>string is proportional to the displacement, acting at right angles to
>>the rest position of the string. In reality, the force acts in the
>>direction of the string's curvature vector. For a low note, it seems
>>that the amplitude would be relatively high. So are the errors in the
>>right-angle approximation large enough to be audible on the low notes
>>of a piano?
>
>I'm not quite sure why there'd be a pitch-dependence. This effect is
>certainly amplitude-dependent, though. To try to measure it, you
>should play your piano really really loud.
>
>I think it's likely to be quite small, though. Here's why. (I'll
>admit that this is a heuristic argument, not a real calculation. Feel
>free to rip it to shreds.)

I realized later that there's a simpler argument for the same conclusion.
There must be some function giving the frequency of waves on a string in
terms of wavenumber k and amplitude A:

omega=omega(k,A).

We want to assume that the amplitude dependence is weak but not
quite negligible, so let's do a Taylor expansion in A. The
frequency must be an even function of A, for the simple reason that
negative values of A describe equivalent physical solutions to positive
values -- the string doesn't care if it's waving up or down. So
to leading order, things must look like

omega = omega_0 (1 + q A^2)

where omega_0=ck is the usual relationship with no amplitude-dependence.
On dimensional grounds, q must be of order 1/L^2 where L is either the
length of the string or the wavelength of the waves, because
these are the only quantities in the problem with dimensions of length.
So the corrections must go like (A/L)^2.

Hendrik van Hees

unread,
Jun 22, 2006, 11:22:39 PM6/22/06
to
David W.Cantrell wrote:

> Perhaps something got lost in "translation". Of course, before Bach
> there were keyboard instruments (Klavier, in the _general_ sense)
> which were not well-tempered. That was your point. But the piano _per
> se_ did not exist before Bach. It was invented during his lifetime
> (spec. in 1709) by Cristofori, who originally called it "gravicembalo
> col pian e forte". IIRC, it is a matter of some speculation whether
> Bach ever actually encountered a pianoforte.

As far as I know, he infact tried some pianofortes, and he was not
pleased at all.

--
Hendrik van Hees Texas A&M University
Phone: +1 979/845-1411 Cyclotron Institute, MS-3366
Fax: +1 979/845-1899 College Station, TX 77843-3366
http://theory.gsi.de/~vanhees/faq mailto:he...@comp.tamu.edu

J Jensen

unread,
Jun 30, 2006, 5:34:49 PM6/30/06
to
John Baez wrote:
> In article <e6pm1j$9mf$1...@bigbang.richmond.edu>,
> <eb...@lfa221051.richmond.edu> wrote:
>
> >In article <e6p4t8$9uj$1...@glue.ucr.edu>,
> >John Baez <ba...@math.removethis.ucr.andthis.edu> wrote:
>
> >>My student Mike Stay wrote:
>
> >> Start with a frequency for C. At each step, multiply by 2
> >> (up an octave) or divide by 3 (down a fourth).
>
> >This should be "divide by 3/2" or "multiply by 2/3", right?
>
> Right - I'll fix that on my website.
[snip]
>
> For a detailed study of affine transformations of Z/12 and
> their application to music, see this paper mentioned by Stephen
> Lavelle:
>
> Thomas Noll, The topos of triads, available at
> http://www.cs.tu-berlin.de/~noll/ToposOfTriads.pdf
>
> Mathematicians will enjoy this, since he considers Lawevere-Tierney
> topologies on the topos Set^T, where T is the 8-element monoid of
> *not necessarily invertible* affine maps f: Z/12 -> Z/12 which
> preserve a given triad.
>
[snip]
>> I now see why people get so emotional about tuning systems. For a
> well-balanced introduction, see:
>
> http://www.kirnberger.fsnet.co.uk/TempsI.html
> http://home.no.net/wimkamp/instruments/Harpsichord/harpsichord.html
>
> The root of all evil is the fact that if you go up a fifth, you
> multiply the frequency of a note by 3/2. No amount of doing this
> is the same as going up an integral number of octaves (a power of 2).
> So, we compromise and say that 12 fifths is almost 7 octaves:

>
> (3/2)^{12} ~ 2^7
>
> 129.746 ~ 128
>
> The difference is called the "Pythagorean comma". It's like a
> lump in the carpet: you can push it around, maybe try to hide it
> under a sofa, but you can't get rid of it. The different tuning
> systems shove this lump in different places.
>
> Now perhaps you can see why the Pythagoreans drowned the guy
> who proved the existence of irrational numbers: it's the
> irrationality of
>
> ln(3/2)/ln(2)
>
> that creates the nasty lump in the carpet. (The irrationality
> of sqrt(2) is related to the "diabolus in musica", but that's
> another matter.)

Sorry to jump into this thread a little late. I spent a fair bit of
time investigating tuning and temperament, not from the point of view
of irrational numbers and continued fractions, but from the physical
point of view of harmonics and what I call acoustical landmarks.
Various different temperaments are designed to achive various
acoustical goals ( pure 5ths = pure 3rd harmonics, pure major 3rds =
pure 5th harmonic, pure minor thirds...etc).

I wrote up what I have here:
http://home.austin.rr.com/jmjensen/TEMPER/Temperament.html
It is very much a work in progress, and I welcome input to tie up any
lose ends.

Another remark is that (I believe) the math and physics have a lot of
applications to music theory (classical tonal music), and temperaments
are by no means the most important one. However, I am deeply skeptical
about the usefulness of the "tonnetz" diagrams mentioned in the
original post. Yes, they are mathematical and elegant in some ways, but
I just don't see it saying anything about *music*.

--Jeff

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