Google Groups no longer supports new Usenet posts or subscriptions. Historical content remains viewable.

Dismiss

139 views

Skip to first unread message

Jun 13, 2006, 8:10:22 AM6/13/06

to

Also available at http://math.ucr.edu/home/baez/week234.html

June 12, 2006

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

John Baez

Today I'd like to talk about the math of music - including

torsors, orbifolds, and maybe even Mathieu groups. But first,

some movies of the n-body problem:

1) Cris Moore, The 3-body (and n-body) problem,

http://www.santafe.edu/~moore/gallery.html

In 1993 Cris Moore discovered solutions of the gravitational

n-body problem where the particles' paths lie in a plane and

trace out braids in spacetime! I spoke about these in "week181".

More recently, Moore and Michael Nauenberg have found solutions

with cubic symmetry and vanishing angular momentum, and made

movies of these. For the mathematical details, try this:

2) Cristopher Moore and Michael Nauenberg, New periodic orbits

for the n-body problem, available at math.DS/0511219

Next, math and music.

Some of you have been in this situation. A stranger at a party

asks what you do. You reluctantly admit you're a mathematician,

expecting one of the standard responses: "Oh! I hate math!" or

"Oh! I was pretty good at math until...."

But instead, after a strained moment they say: "Oh! Do you play

an instrument too? Isn't music really mathematical?"

I guess it's like meeting a Martian and asking them if they like

Arizona: an attempt to humanize something alien and threatening.

You may not have much in common, but at least you can chat about

red rocks.

Of course there *is* something mathematical about music, and lots

of mathematicians play music. I rarely think about music in a

mathematical way. But I know they have something in common: the

transcendent beauty of pure form.

Indeed, in the Middle Ages, music was part of a "quadrivium" of

mathematical arts: arithmetic, geometry, music, and astronomy.

These were studied after the "trivium" of grammar, rhetoric and

logic. This is why mathematicians scorn a result as "trivial"

when it's easy to see using straightforward logic. When a

result seems more profound, they should call it "quadrivial"!

Try saying it sometime: "Cool! That's quadrivial!" It might

catch on.

There are also modern applications of math to music theory. I had

never heard of "neo-Riemannian theory" until Tom Fiore explained it

to me while I was visiting Chicago. Tom is a postdoc who works on

categorified algebraic theories, double categories and the like -

but he's also into music theory:

3) Thomas M. Fiore, Music and mathematics, available at

http://www.math.uchicago.edu/~fiore/1/music.html

4) Thomas M. Fiore and Ramon Satyendra, Generalized contextual

groups, Music Theory Online 11 (2005), available at

http://mto.societymusictheory.org/issues/mto.05.11.3/toc.11.3.html

The first of these is a very nice gentle introduction, suitable

both for musicians who don't know group theory and mathematicians

who don't know a triad from a tritone!

When Tom first mentioned "neo-Riemannian theory", I thought this

was some bizarre application of differential geometry to music.

But no - we're not talking about the 19th-century mathematician

Bernhard Riemann, we're talking about the 19th-century music

theorist Hugo Riemann!

Based on the work on Euler - yes, *the* Euler - Hugo Riemann

introduced diagrams called "tone nets" to study the network of

relations between similar chords. You can see his original

setup here:

5) Joe Monzo, Tonnetz: the tonal lattice invented by Riemann,

Tonalsoft: the Encyclopedian of Microtonal Music Theory,

http://www.tonalsoft.com/enc/t/tonnetz.aspx

6) Paul Dysart, Tonnetz: musics, harmony and donuts,

http://members2.boo.net/~knuth/

Apparently Riemann's ideas have caught on in a big way. Monzo

says that "use of lattices is endemic on internet tuning lists",

as if they were some sort of infectious disease.

Dysart seems more gung-ho about it all. The "donuts" he mentions

arise when you curl up tone nets by identifying notes that differ

by an octave. He has some nice pictures of them!

In neo-Riemannian theory, people like Lewin and Hyer started

extending Riemann's ideas by using *group theory* to systematize

operations on chords. The best easy introduction to this is

Fiore's paper "Music and mathematics". Here you can read about

math lurking in the music of Elvis and the Beatles! Or, if

you're more of a highbrow sort, see what he has to say about

Hindemith and Liszt's "Transcendental Etudes". And if you

like doughnuts and music, you'll love the section where he

explains how Beethoven's Ninth traces out a systematic path in

a torus-shaped tone net! This amazing fact was discovered by

Cohn, Douthett, and Steinbach.

(If I weren't so darn honest, I'd add that Liszt wrote the

"Transcendental Etudes" as a sequel to his popular "Algebraic

Etudes", and explain how Mozart's "eine kleine Nachtmusik"

tours a tone net shaped like a Klein bottle. But alas....)

Let me explain a bit about group theory and music - just

enough to reach something really cool Tom told me.

If you're a musician, you'll know the notes in an octave go

like this, climbing up:

C, C#, D, D#, E, F, F#, G, G#, A, A#, B

until you're back to C. If you're a mathematician, you might

be happier to call these notes

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11

and say that we're working in the group of integers mod 12,

otherwise known as Z/12. Let's be mathematicians today.

The group Z/12 has been an intrinsic feature of Western music

ever since pianos were built to have "equal temperament"

tuning, which makes all the notes equally spaced in a certain

logarithmic sense: each note vibrates at a frequency of 2^{1/12}

times the note directly below it.

Only 7 of the 12 notes are used in any major or minor key -

for example, C,D,E,F,G,A,B is C major and A,B,C,D,E,F,G is

A minor. So, as long as Western composers stuck to writing

pieces in a single fixed key, the Z/12 symmetry was "spontaneously

broken" by their choice of key, only visible in the freedom to

change keys.

But, as composers gradually started changing keys ever more

frequently within a given piece, the inherent Z/12 symmetry

became more visible. In the late 1800s this manifested itself

in trend called "chromaticism". Roughly speaking, music is

"chromatic" when it freely uses all 12 notes, but still within

the context of an - often changing - key. I guess Wagner and

Richard Strauss are often mentioned as pinnacles of chromaticism.

Chromaticism then led to full-fledged "twelve-tone music"

starting with Schoenberg in the early 1900s. This is music

that fully exploits the Z/12 symmetry and doesn't seek to

privilege a certain 7-element subset of notes defining a key.

People found Schoenberg's music disturbing and dissonant at

the time, but I find it very beautiful.

Now comes the really exciting thing Tom told me: two other

symmetry groups lurking in music, and a relationship between them.

First, the transposition-inversion group. This acts as

permutations of the set Z/12. It's generated by two

especially nice permutations. The first is "transposition".

This raises each note a step:

x |-> x + 1

Musicians would call this a half-step, just like physicists

measure spin in multiples of 1/2, but we're being mathematicians!

The second is "inversion". This turns notes upside down:

x |-> -x

The relevance of this to music is a bit less obvious: composers

like Bach and Schoenberg used it explicitly, but we'll see it

playing a subtler role, relating major and minor chords.

The transposition-inversion group has 24 elements. Mathematicians

call it the 24-element "dihedral group", since it consists of the

symmetries of a regular 12-sided polygon where you're allowed

to rotate the polygon (transposition) and also flip it over

(inversion). I hope you see that this geometrical picture is

just a way of visualizing the 12 notes.

So, the transposition-inversion group obviously on the 12-element

set of notes. But, it also acts on the 24-element set of "triads"!

Triads are among the most basic chords in music. Mathematically

they are certain 3-element subsets of Z/12. They come in two

kinds, major and minor. There are 12 major triads, namely

{0,4,7} C major triad: {C,E,G}

and everything you can get from this by transposition. If you

invert these, you get the 12 minor triads, namely

{0,-4,-7} = {5,8,0} F minor triad: {F,Ab,C}

and everything you can get from *this* by transposition.

(Note that {0,-4,-7} = {5,8,0} because we're working mod 12

and the order doesn't matter. I've also included the way musicians

talk about these triads, in case you care.)

Major triads sound happy; when you invert them they sound sad,

just like an upside-down smile looks sad. There could be some

profound truth lurking here. A smile has a positive second

derivative:

. .

. .

. .

which says that things are "looking up", while a frown has negative

second derivative:

. .

. .

. .

which says that things are "looking down". An upside-down smile

is a frown.

(On the other hand, a backwards smile is still a smile, and a

backwards frown is still a frown. So, if you're a company and

the second derivative of your profits is positive, you can say

business is looking up - and you could still say this if time

were reversed!)

But never mind. We had this transposition-inversion group acting

on our set of notes, namely Z/12. Since tranposition and inversion

act on notes, they also act on triads. For example, transposition

does this:

{0,4,7} |-> {1,5,8} C major triad |-> C# major triad

while inversion does this:

{0,4,7} |-> {5,8,0} C major triad |-> F minor triad

So, we've got this 24-element transposition-inversion group

acting on the 24-element set of triads!

But here's really cool part: there's *another* important

24-element group acting on the same set! It's easy to define

mathematically, but it also has a musical meaning.

Mathematically, it's just the "centralizer" of the transposition-

inversion group. In other words, it consists of all ways of

permuting triads that *commute* with transposition and inversion!

Musically, it's called the "PLR" group, because it's generated

by 3 famous transformations.

To describe these transformations, I'll need to talk about the

"bottom", "middle" and "top" note of a triad. If you know a

wee bit of music theory this should be obvious as long as you

know I'm talking about triads in root position. If you're a

mathematician who has never studied music theory and you think

of triads as 3-element subsets of Z/12, it might be less obvious,

since Z/12 doesn't have a nice ordering - it only has a *cyclic*

ordering. But this is enough. The point is that major triads

are sets of the form

{n,n+4,n+7},

while minor triads are of the form

{n,n+3,n+7}.

So, we can call the note n the "bottom", the note n+3 or n+4 the

"middle", and n+7 the "top". Musicians call them the "root",

"third" and "fifth", but let's be simple-minded mathematicians.

Okay, what are the transformations P, L, and R? They stand

for "parallel", "leading tone change", and "relative" - but

what *are* they?

Each of these transformations keeps exactly 2 of the notes

in our triad the same. Also, each changes major triads into

minor triads and vice versa. These features make these

transformations musically interesting.

The transformation "P" keeps the top and bottom notes the same.

I've now said enough for you to figure out what it does...

at least in principle. For example:

P: {0,4,7} |-> {0,3,7} C major triad |-> C minor triad

P: {0,3,7} |-> {0,4,7} C minor triad |-> C major triad

The tranformation "L" turns the middle and top note into the bottom

and middle note when you start with a MAJOR triad. It turns the

bottom and middle note into the middle and top note when you start

with a MINOR triad. For example:

L: {0,4,7} |-> {4,7,11} C major triad |-> E minor triad

L: {0,3,7} |-> {8,0,3} C minor triad |-> G# major triad

The transformation "R" works the other way around. It turns the

middle and top note into the bottom and middle note when you start

with a MINOR triad. And it turns the bottom and middle note into

the middle and top note when you start with a MAJOR triad:

R: {0,4,7} |-> {9,0,4} C major triad |-> A minor triad

R: {0,3,7} |-> {3,7,10} C minor triad |-> D# major triad

Can you see why the transformations P, L, and R commute with

transposition and inversion? It should be easy to see that they

commute with transposition. Commuting with inversion means that

if I switch the words "top" and "bottom" and also the words "major"

and "minor" in my descriptions above, these transformations don't

change!

You should be left wondering why P, L, and R generate the group

of *all* transformations of triads that commute with transposition

and inversion - and why this group, like the transposition-inversion

group itself, has exactly 24 elements!

It turns out some of this has a simple explanation, which has very

little to do with the details of triads or even the 12-note scale.

Imagine a scale with n equally spaced notes. Transpositions

and inversions will generate a group with 2n elements. Let's

call this group G. If you take any "sufficiently generic" chord

in our scale, G will act on it to give a set S consisting of 2n

different chords. Then it's a mathematical fact that the group of

permutations of S that commute with all transformations in G

will be isomorphic to G! So, it too will have 2n elements.

To explain *why* this is true, I need a bit more math.

First of all, I need to define my terms. I'm defining a chord

to be "sufficiently generic" if no element of G maps it to itself.

We then say G acts *freely* on S. By the way we've set things up,

G also acts *transitively* on S. A nonempty set on which G

acts both freely and transitively is called a "G-torsor". You can

read about torsors here:

7) John Baez, Torsors made easy,

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

They're philosophically very interesting, since they're related

to gauge symmetries in physics... but right now the only fact we

need is that any G-torsor is isomorphic to G. So, we can identify

S with G, with G acting by left multiplication.

Then, it's a well-known fact that any permutation of G that

commutes with left multiplication by all elements of G must be

given by *right* multiplication by some element of G. And

these right multiplications form a group of transformations

that is isomorphic to G... just as we were trying to show!

In other words: the group of permutations of G has a subgroup

isomorphic to G, namely the left translations. It also has

another subgroup isomorphic to G, namely the right translations.

Each of these subgroups is the "centralizer" of the other. That

is, each one consists of all permutations that commute with every

permutation in the other one! Fiore and Satyendra call them

"dual groups".

In our application to music, the first copy of G is our good old

transposition-inversion group, while the second copy is a

generalization of the PLR group. Fiore and Satyendra call it the

"generalized contextual group".

All this is indeed very general. I don't know a similarly

general explanation of why the operations P, L, and R succeed

in generating all transformations that commute with transposition

and inversion.

I asked Tom Fiore if he and Ramon Satyendra were the first to

show that the PRL group was the centralizer of the transposition-

inversion group. His reply was packed with information, so

I'll quote it:

The initial insight about the duality between the T/I group and

the PLR group was at least 20 year ago. Dual groups in the musical

sense were introduced in David Lewin's seminal 1987 book "Generalized

Musical Intervals and Transformation Theory." This book stimulated

interest in neo-Riemannian theory, since Lewin recalled the

transformations P,L, and R as objects of study.

Major-minor duality was a concern of Hugo Riemann, a theorist from

the second half of the 19th century. Given his interest in duality,

Riemann may have had some intuition about a duality between T/I and

PLR, though it wasn't until after his death that this duality was

formulated in algebraic terms. An algebraic proof of the duality of

T/I and PLR was in the thesis of Julian Hook in 2002.

Ramon and I were the first to prove that the "generalized contextual

group" is dual to the T/I group acting on a set generated by an

arbitrary pitch-class segment satisfying the tritone condition.

(The tritone condition says that the inital pitch-class segment

contains an interval other than a tritone and unison.) Our

theorem has the PLR group and major/minor triads as a special case,

since the generalized contextual group becomes the PLR group when one

takes the generating pitch class segment to be the three pitches of a

major chord. The advantage of our generalization is that one can now

apply the PLR insight to passages that are not triadic. There was a

general move toward this in practice for the past decade (Childs and

Gollin considered seventh chords rather than triads, Lewin analyzed

instances of a non-diatonic phrase in a piano work of Schoenberg, we

analyzed Hindemith, and so on). Most music does not consist entirely

of triads (e.g. late 19th century chromatic music), so the restriction

of PLR to triads was not conclusive.

We did a literature review of recent neo-Riemannian theory in Part

5 of our article "Generalized Contextual Groups", since there have

been a lot of insights in the past 10 years. One of the main

thinkers is Rick Cohn, who came up with (among other things) a

nice tiling of the plane which one navigates using P,L, and R

(Richard Cohn, Neo-Riemannian operations, parsimonious trichords,

and their Tonnetz representations, Journal of Music Theory, 1997).

It is quite geometric.

You read more about these matters here... I'll list these references

in the order Tom mentions them:

8) David Lewin, Generalized Musical Intervals and Transformations,

Yale University Press, New Haven, Connecticut, 1987.

9) Julian Hook, Uniform Triadic Transformations, Ph.D. thesis, Indiana

University, 2002.

10) Adrian P. Childs, Moving beyond neo-Riemannian triads: exploring

a transformational model for seventh chords, Journal of Music

Theory 42/2 (1998): 191-193.

11) Edward Gollin, Some aspects of three-dimensional Tonnetze,

Journal of Music Theory 42/2 (1998): 195-206.

12) Richard Cohn, Neo-Riemannian operations, parsimonious

trichords, and their "Tonnetz" representations, Journal of

Music Theory 41/1 (1997), 1-66.

13) David Lewin, Transformational considerations in Schoenberg's

Opus 23, Number 3, preprint.

In fact, the notion of "torsor" pervades the work of David Lewin,

but not under this name - Lewin calls it a "general interval system".

Stephen Lavelle noticed the connection to torsors in 2005:

14) Stephen Lavelle, Some formalizations in musical set theory,

June 3, 2005, available at http://www.maths.tcd.ie/~icecube/lewin.pdf

and http://www.maths.tcd.ie/~icecube/lewin.ps

Unfortunately the music theorists seem not to have set up

an "arXiv", so some of their work is a bit hard to find.

For example, all of Volume 42 Issue 2 of the Journal of Music

Theory is dedicated to neo-Riemannian theory, but I don't

think it's available online. Luckily, the music theorists have

set up some free online journals, like this:

15) Music Theory Online, http://mto.societymusictheory.org/

and this one has links to others. The Society for Music Theory

also has online resources including a nice bibliography on the

basics of music theory:

16) Society for Music Theory, Fundamentals of music theory,

selected bibliography, http://societymusictheory.org/index.php?pid=37

Now let me turn up the math level a notch....

If you're the right sort of mathematician, you'll have noticed by

now that we're doing some fun stuff starting with the abelian

group A = Z/12. First we're 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

-----------------------------------------------------------------------

Previous issues of "This Week's Finds" and other expository articles on

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

A simple jumping-off point to the old issues is available at

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

If you just want the latest issue, go to

http://math.ucr.edu/home/baez/this.week.html

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:

Cheers

John R Ramsden

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:

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

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.

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.

Jun 14, 2006, 9:05:44 AM6/14/06

to

In article <1150232891.9...@u72g2000cwu.googlegroups.com>,

<jhnr...@yahoo.co.uk> wrote:

<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

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.

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

Jun 14, 2006, 9:07:28 PM6/14/06

to

>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).

>

> 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.]

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, [...]> The "equal

> temperament" assumption is also very artificial once you get away from

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

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:

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

Jun 15, 2006, 8:28:47 PM6/15/06

to

In article <e6pm1j$9mf$1...@bigbang.richmond.edu>,

<eb...@lfa221051.richmond.edu> wrote:

<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.)

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:

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."

Jun 17, 2006, 12:17:06 PM6/17/06

to

In article <C0B8F249.155F3%r...@audioimagination.com>,

robert bristow-johnson <r...@audioimagination.com> wrote:

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!

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

Jun 17, 2006, 10:47:35 PM6/17/06

to

In article <1150569266.9...@g10g2000cwb.googlegroups.com>,

robert bristow-johnson <r...@audioimagination.com> wrote:

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.

Jun 18, 2006, 9:07:23 PM6/18/06

to

In article <C0B8F249.155F3%r...@audioimagination.com>, robert

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).)

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:

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.

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...

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.

Jun 19, 2006, 4:50:03 PM6/19/06

to

> 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.

>

> 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.

>

> ...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.

>

>

> > 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.

>

> >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.

>

> >> 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".

>

>

> 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.

>

> >> 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.>

> 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

> 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,

>

> >> 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,

>

> 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.

>

> >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

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.

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.

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.

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.

> 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/

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.

..

> 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 ...

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?

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.

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]> 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.

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

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.

> 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

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:

..> 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

Jun 21, 2006, 9:36:28 AM6/21/06

to

In article <20060620125902.679$5...@newsreader.com>, "David W.Cantrell"

<DWCan...@sigmaxi.org> wrote:

<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,

Jun 21, 2006, 7:42:00 PM6/21/06

to

In article <1150827532....@i40g2000cwc.googlegroups.com>,

Matt Noonan <matt....@gmail.com> wrote:

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.

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 ]

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.

Jun 22, 2006, 2:42:32 AM6/22/06