Two questions for Gene

1. When you're done enumerating 12-tone blocks, how about looking at
5-tone? There are some nice japanese scales that are pelograndpa :)

2. Can you explain, in as non-technical a manner as possible, the
proof that there are 21.5 :) commas in the 5-limit with epimericity <
1/2?

.5:) (1/1) - undefined
1.5:) 2/1 - exo
2.5:) 3/2 - exo
3.5:) 4/3 - exo
4.5:) 5/4 - exo
5.5:) 6/5 - exo
6.5:) 9/8 - exo
7.5:) 10/9 - exo
8.5:) 16/15 - ?
9.5:) 25/24 - ?
10.5:) 27/25 - ?
11.5:) 32/27 - exo
12.5:) 81/80
13.5:) 128/125
14.5:) 135/128 - ?
15.5:) 250/243
16.5:) 256/243
17.5:) 648/625
18.5:) 2048/2025
19.5:) 3125/3072
20.5:) 15625/15552
21.5:) 32768/32805

I think for 5-limit linear we can just cover these in a few pages,
and separately make brief mention of anything else of "historical
importance" (531441/524288 and probably that's it?) . . .

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> 1. When you're done enumerating 12-tone blocks, how about looking
at
> 5-tone? There are some nice japanese scales that are pelograndpa :)

Hadn't thought about doing that; 7 and 10 was a little on my mind.

> 2. Can you explain, in as non-technical a manner as possible, the
> proof that there are 21.5 :) commas in the 5-limit with epimericity
<
> 1/2?

I'm afraid I don't have a proof, and proving it would probably be
difficult. The proof is a proof that the number of p-limit commas
with epimericity e < 1 is finite. In practice, you get to a point
where it seems obvious no futher commas are going to show up; if they
did, it would be very odd from a number theoretic point of view.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

> > 2. Can you explain, in as non-technical a manner as possible, the
> > proof that there are 21.5 :) commas in the 5-limit with
epimericity
> <
> > 1/2?
>
> I'm afraid I don't have a proof, and proving it would probably be
> difficult. The proof is a proof that the number of p-limit commas
> with epimericity e < 1 is finite. In practice, you get to a point
> where it seems obvious no futher commas are going to show up; if
they
> did, it would be very odd from a number theoretic point of view.

Hmm . . . that severely reduces the appeal of using epimericity to
define our badness contour. In this case, we should probably use
something that crosses zero error at some finite complexity.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

> The proof is a proof that the number of p-limit commas
> with epimericity e < 1 is finite.

There are an infinite number of commas with e < 1, but for any
positive d, there are a finite number of commas with e < 1-d --
correct?

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:

> Hmm . . . that severely reduces the appeal of using epimericity to
> define our badness contour. In this case, we should probably use
> something that crosses zero error at some finite complexity.

A curious remark for a physics major to make. We know the list is
compelte, we just can't prove it, and if there were one more comma we
missed it would be of no concievable musical use anyway, as Dave
would be quick to point out.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> wrote:
>
> > The proof is a proof that the number of p-limit commas
> > with epimericity e < 1 is finite.
>
> There are an infinite number of commas with e < 1, but for any
> positive d, there are a finite number of commas with e < 1-d --
> correct?

That's what I was trying to say. There are a finite number of commas
(in some prime limit) with comma < e < 1.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> > Hmm . . . that severely reduces the appeal of using epimericity
to
> > define our badness contour. In this case, we should probably use
> > something that crosses zero error at some finite complexity.
>
> A curious remark for a physics major to make. We know the list is
> compelte, we just can't prove it, and if there were one more comma
we
> missed it would be of no concievable musical use anyway, as Dave
> would be quick to point out.

Then why not define our badness contour to exclude anything of "no
conceivable musical use"?

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:

> Then why not define our badness contour to exclude anything of "no
> conceivable musical use"?

Fine, but then it would be worth mentioning that you really didn't
need to.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> > Then why not define our badness contour to exclude anything
of "no
> > conceivable musical use"?
>
> Fine, but then it would be worth mentioning that you really didn't
> need to.

If there is some other comma out there satisfying the epimericity
requirement, then you *did* really need to.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:

> If there is some other comma out there satisfying the epimericity
> requirement, then you *did* really need to.

Yes, but there won't be. We all knew Catalan's conjecture was true
even before it was proven. There are tons of number theoretic
conjectures about which there is essentially no doubt about their
truth but which have not been proven.