Various Conjectures

Here is a list of some various conjectures relating to music theory. The proofs are pretty trivial, so I'll omit them and leave them as an exercise for the reader.

1.) No two ratios can have the same Tenney Height and the same Weil Height.
2.) No two ratios can have the same Tenney Height and the same Arithmetic (N+D) Height.
3.) No two ratios can have the same Arithmetic Height and the same Weil Height.
4.) For p_n(m)>=2^K, and Q being the set of all ratios with Tenney Height K, the following is true: 2^(m)>=2*|Q|. If p_n(m)=2^K, then 2^(m)=2*|Q|.
5.) If H is the set of all ratios with Arithmetic Height K, then K>=2*|H|+1. If K is a prime other than 2, then K=2*|H|+1.
6.) (N*D)*(N/D)^k is a height function iff -1<k<infinity.

Conditions:
Ratios P and 1/P are considered to be equivalent.
p_n(m) is the primorial function, where m is an integer.
|T| is the cardinality of a set T.
N>=D.

Ryan Avella

On Thu, Sep 27, 2012 at 1:41 AM, Ryan Avella <domeofatonement@yahoo.com>
wrote:
>
> Here is a list of some various conjectures relating to music theory. The
> proofs are pretty trivial, so I'll omit them and leave them as an exercise
> for the reader.

Er, are they conjectures are proofs? If it's proven then it's a
theorem, not a conjecture. A conjecture is something you just think is
true, but you haven't proven it yet (and it might still be wrong).

-Mike

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Thu, Sep 27, 2012 at 1:41 AM, Ryan Avella <domeofatonement@...>
> wrote:
> >
> > Here is a list of some various conjectures relating to music theory. The
> > proofs are pretty trivial, so I'll omit them and leave them as an exercise
> > for the reader.
>
> Er, are they conjectures are proofs? If it's proven then it's a
> theorem, not a conjecture. A conjecture is something you just think is
> true, but you haven't proven it yet (and it might still be wrong).
>
> -Mike
>

Yeah, I guess all of them are pretty much theorems then with the possible exception of #4. That one I'm a little doubtful about.

Ryan