Algebra and Geometry

Hi there,

I'm glad Gene is promoting the algebraic
approach because it is a source of insights.
However as there is no mathematician here
to promote the geometrical approach I'd
like to put in a word for that too.

Much of Erv Wilson's research seems
to be geometrically based from what I've
seen of it, so one can't doubt its value
for generting new musical results and
scales.

Geometrically what we have is a lattice
embedded in multi-dimensional space.

A particular musical interval then such
as 5/4 can be represented as the point
{2, 0, 1) but that is just one of its points.

You can also get to it as e.g.
2^0.386...

So geometrically it is also represented by
{0,386...., 0, 0)
in the Tenney lattice considered as a
geometrical structure.

More generally, if we join those two
points, then any point along that line
can also represent 5/4. So more generally
every interval is a line in the geometrical
space of the lattice.

If we restrict our language to modules
with integer exponents or even vector
spaces with rational exponents, we
can't even make this remark. But
it is true geometrically. That means
we may be able to get neater proofs
geometrically of some results, or
get insights into things by seeing
things geometrically.

So for instance - well the hexanies
and dekanies could be derived
algebraically - but the regularity
of the structure and their symmetry
is perhaps best studied geometrically
using geometrical transformations
of the Tenney lattice.

So also Penrose tilings
can be viewed as a slice of
a five dimensional lattice
and those types of constructions
are most easily studied
geometrically - and the construction
has musical significance all the way.

They can also be constructed by
using n-grid duals which I won't
go into here but it is a geometrical
construction again, but one that
remains in 2D all the way.

Anyway just look at all Erv Wilson's
results. They just speak for themselves.

The algebraic approach is undoubtedly useful
and since the most high power mathematician
currently posting to this forum is an algebraicist
and as he has frequently shown the power of his
approach with interesting and surprising
results, then it is natural that it has come to
take centre stage in the mathematical
treatment of the subject here.

However, anyone who feels
they prefer to study the subject
geometrically is in excellent
company. We really need to foster
both approaches.

I think if one looks at a broader context,
outside thsi current forum, or even within
this current forum since it has
started, one would find that the
geometrical approach has produced
as many new results as the
algebraic one. It is also more
accessible to many newbies.

Robert

hi Robert,

--- In tuning@yahoogroups.com, "Robert Walker" <robertwalker@n...>
wrote:

> Hi there,
>
> I'm glad Gene is promoting the algebraic
> approach because it is a source of insights.
> However as there is no mathematician here
> to promote the geometrical approach I'd
> like to put in a word for that too.
>
> Much of Erv Wilson's research seems
> to be geometrically based from what I've
> seen of it, so one can't doubt its value
> for generting new musical results and
> scales.

hear, hear! i'm all for tuning geometry!

> Geometrically what we have is a lattice
> embedded in multi-dimensional space.
>
> A particular musical interval then such
> as 5/4 can be represented as the point
> {2, 0, 1) but that is just one of its points.

i see you do this a lot, Robert ... you have to
watch those signs! the monzo for 5/4 is [-2 0, 1> .

>
> You can also get to it as e.g.
> 2^0.386...
>
> So geometrically it is also represented by
> {0,386...., 0, 0)
> in the Tenney lattice considered as a
> geometrical structure.

on this one you accidentally replaced a decimal-point
with a comma-mark. should be {0.386...., 0, 0)

but in any case, that's not the correct number.
5/4 = 2^0.321928095 = 3^0.203114014 = 5^0.138646884.

> More generally, if we join those two
> points, then any point along that line
> can also represent 5/4. So more generally
> every interval is a line in the geometrical
> space of the lattice.

if you're including prime-factor 2, then
every interval in the 3-dimensional 5-limit lattice
is a plane, not a line.

it would be a line on an 8ve-equivalent lattice
which uses only 3 and 5 for its axes.

> If we restrict our language to modules
> with integer exponents or even vector
> spaces with rational exponents, we
> can't even make this remark. But
> it is true geometrically. That means
> we may be able to get neater proofs
> geometrically of some results, or
> get insights into things by seeing
> things geometrically.

excellent points.

> Anyway just look at all Erv Wilson's
> results. They just speak for themselves.

which i suppose is why Erv doesn't say much
in words when he publishes his lattices.

> The algebraic approach is undoubtedly useful
> and since the most high power mathematician
> currently posting to this forum is an algebraicist
> and as he has frequently shown the power of his
> approach with interesting and surprising
> results, then it is natural that it has come to
> take centre stage in the mathematical
> treatment of the subject here.
>
> However, anyone who feels
> they prefer to study the subject
> geometrically is in excellent
> company. We really need to foster
> both approaches.
>
> I think if one looks at a broader context,
> outside thsi current forum, or even within
> this current forum since it has
> started, one would find that the
> geometrical approach has produced
> as many new results as the
> algebraic one. It is also more
> accessible to many newbies.

thanks for all of this, Robert.
i believe that seeing tuning-math geometrically
*definitely* is easier for the newbie.

anyway, aren't algebra and geometry just
two sides of the same coin? that was Descarte's
whole point when he created the coordinate system.

-monz

Hi Monz,

Yes I do have to watch out for missed minus signs and
things.

> > on this one you accidentally replaced a decimal-point
> > with a comma-mark. should be {0.386...., 0, 0)

> but in any case, that's not the correct number.
> 5/4 = 2^0.321928095 = 3^0.203114014 = 5^0.138646884.

Yes quite right. I was thinking of cents, but
of course these are fractional octaves (or tritaves etc), so you
need to think of e.g. 100 units to an octave instead
of 1200.

So your 0.3219 is 0.386 * 1000/1200

And yes, it is a plane of course, a line under octave equivalence,
thanks.

Robert

--- In tuning@yahoogroups.com, "Robert Walker" <robertwalker@n...> wrote:

> So for instance - well the hexanies
> and dekanies could be derived
> algebraically - but the regularity
> of the structure and their symmetry
> is perhaps best studied geometrically
> using geometrical transformations
> of the Tenney lattice.

Strangely enough, it was me who discovered the hexany geometrically
and Erv who discovered it algebraically. I think both points of view
are important. Of course, no one much liked my presentation of the
geometrical appoach, since it was algebraic.

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:

> anyway, aren't algebra and geometry just
> two sides of the same coin?

They tend to be. That's why there is an undergraduate subject called
analytic geometry, which actually uses algebra, not analysis. Then
there is geometric algbebra, which does all of the stuff about wedge
products, etc in a way which seems useful to physicists, but not very
much to us. Then there is a dark and sinister subject called algbraic
geometry, which is just analytic geometry after it has grown shark's
teeth and adopted some very technical and high-powered concepts.
Objects such as lattices can be seen in algebraic terms or even in
graph-theory terms.
And so forth...