Conductors for intervals

Here's something which may turn out interesting for music, though it
probably won't.

Given a positive rational number N/D in lowest terms, we can define a
corresponding elliptic curve by the equation

y^2 = x (x-N+D) (x-N)

Elliptic with integer coefficients have something called a
"conductor", an integer whose definition I won't give because it is
too technical. But the smallest conductor I've found is 15, for 81/80.
Another small one is 21, for 49/48.

I was led to this because this is a little like how Fermat's Last
Theorem gets proven; you come up with an elliptic curve with conductor
2, which is impossible (the smallest is 11.) I'll try to get hold of
some better software for computing conductors and see if 81/80 is
really the smallest conductor which arises like this.

On 7/29/06, Gene Ward Smith <genewardsmith@coolgoose.com> wrote:
> Elliptic with integer coefficients have something called a
> "conductor", an integer whose definition I won't give because it is
> too technical.

Too technical for tuning-math?? That's a new one! Can you recommend
any books on the subject?

Keenan

Wild. -C.

At 04:21 PM 7/29/2006, you wrote:
>Here's something which may turn out interesting for music, though it
>probably won't.
>
>Given a positive rational number N/D in lowest terms, we can define a
>corresponding elliptic curve by the equation
>
>y^2 = x (x-N+D) (x-N)
>
>Elliptic with integer coefficients have something called a
>"conductor", an integer whose definition I won't give because it is
>too technical. But the smallest conductor I've found is 15, for 81/80.
>Another small one is 21, for 49/48.
>
>I was led to this because this is a little like how Fermat's Last
>Theorem gets proven; you come up with an elliptic curve with conductor
>2, which is impossible (the smallest is 11.) I'll try to get hold of
>some better software for computing conductors and see if 81/80 is
>really the smallest conductor which arises like this.

--- In tuning-math@yahoogroups.com, "Keenan Pepper" <keenanpepper@...>
wrote:
>
> On 7/29/06, Gene Ward Smith <genewardsmith@...> wrote:
> > Elliptic with integer coefficients have something called a
> > "conductor", an integer whose definition I won't give because it is
> > too technical.
>
> Too technical for tuning-math?? That's a new one! Can you recommend
> any books on the subject?

Elliptic Curves by Knapp discusses it a little. A book now available
online is Cremona's Algorithms for Modular Elliptic Curves:

http://www.maths.nottingham.ac.uk/personal/jec/book/amec.html

Now that we know all elliptic curves are modular, this takes on a new
aspect. The book actually explains, more or less, how to compute the
conductor and I may have to mess with that if I can't find a usable
program.

A definition of conductor from another point of view may be found in
the following Wikipedia article, which is one of mine:

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

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" wrote:
> --- In tuning-math@yahoogroups.com, "Keenan Pepper" wrote:
> > On 7/29/06, Gene Ward Smith wrote:
> > > Elliptic with integer coefficients have something called
a "conductor", an integer whose definition I won't give because it
is too technical.
> >
> > Too technical for tuning-math?? That's a new one! Can you
recommend any books on the subject?
>
> Elliptic Curves by Knapp discusses it a little. A book now
available online is Cremona's Algorithms for Modular Elliptic Curves:
> http://www.maths.nottingham.ac.uk/personal/jec/book/amec.html
>
> Now that we know all elliptic curves are modular, this takes on a
new aspect. The book actually explains, more or less, how to compute
the conductor and I may have to mess with that if I can't find a
usable program.
>
> A definition of conductor from another point of view may be found
in the following Wikipedia article, which is one of mine:
> http://en.wikipedia.org/wiki/Classical_modular_curve

Hi Gene,

The term Conductor is mentioned on wikipedia's Modularity Theorem
page, as well as on the Classical Modular Curve page, but not itself
clearly defined; nor does it appear on the disambiguation page for
Conductor.

It might be neat to rectify , especially, the last point.

As to how useful a "conductor" might be in making music ...! I
suspect that the largeish integers involved in even the simplest
cases might take them out of the range of most JI enthusiasts. But
who knows - stranger things have happened. Did you have any speific
ideas on how you yourself might use a conductor?

Regards,
Yahya

--- In tuning-math@yahoogroups.com, "yahya_melb" <yahya@...> wrote:

> As to how useful a "conductor" might be in making music ...!

I've concluded that this business isn't going anywhere.