Paul's New Paper

Hi!

Paul's new paper was in my mailbox at 4th of February so it has been
enriching my life for almost a month now. May I say it is a pleasure
to read. But it is way too short! :)

For one thing the paper doesn't explain how one derives the
generators of the temperament from the vanishing commas. I have no
idea how to do that and some kind of explanation would've been nice.
It is pretty straightforward (at least with a computer) to find the
optimal (for example TOP) generators when you know how the
generators are mapped to primes. But I want to know how to get the
generator mappings from the comma list.

I understand that the math involved might've been too complicated to
be included in the paper.

P.S. In the paper there is a list of 5-prime-limit ratios of
Harmonic Distance less than 6. I think the ratios 9/4, 5/2, 15/2 and
27/1 are missing from that list.

Kalle

--- In tuning@yahoogroups.com, "Kalle Aho" <kalleaho@m...> wrote:

> Hi!
>
> Paul's new paper was in my mailbox at 4th of February so it has
been
> enriching my life for almost a month now.

Glad to hear it.

> May I say it is a pleasure
> to read.

Why thank you!

>But it is way too short! :)

John Chalmers was very generous for allowing me that many pages in
his journal. But yes, many things had to be omitted.

> For one thing the paper doesn't explain how one derives the
> generators of the temperament from the vanishing commas. I have no
> idea how to do that

Did you follow the tuning-math list last year or the year before
that? This came up a number of times.

> and some kind of explanation would've been
> nice.

I'll try to include one in part 2 or 3 of the paper. (You can see
that the paper you have says "Part 1".)

> It is pretty straightforward (at least with a computer) to find the
> optimal (for example TOP) generators when you know how the
> generators are mapped to primes. But I want to know how to get the
> generator mappings from the comma list.
>
> I understand that the math involved might've been too complicated
to
> be included in the paper.

Yes, that's essentially what I decided -- I stuck to the real simple
parts of the math involved.

> P.S. In the paper there is a list of 5-prime-limit ratios of
> Harmonic Distance less than 6. I think the ratios 9/4, 5/2, 15/2
and
> 27/1 are missing from that list.

Yikes! I can't believe no one noticed this before! Ouch, ouch,
ouch . . . how did I let this happen? Well, if you fill in the
missing lines, the point I'm making about TOP Meantone remains true,
right?

--- In tuning@yahoogroups.com, "Kalle Aho" <kalleaho@m...> wrote:

> But I want to know how to get the
> generator mappings from the comma list.

Kalle, you may or may not have noticed some patterns relating the
commas (in vector form) to the generator part of the mappings in
Table 1 in the paper. Here's the pattern. Take the comma in vector
form, ignore the first entry (the power of 2), and determine the GCD
(greatest common divisor) of the last two entries. The generator part
of the mapping of 2 is always 0. The generator part of the mapping of
3 is the power of 5 in the comma vector, divided by the GCD you
found. And the generator part of the mapping of 5 is minus the power
of 3 in the comma vector, again divided by that GCD. That why we see
that for

Meantone 81:80 or [-4 4 -1>, the GCD of the last two entries is 1, so

the generator part of the mapping is 0, -1, -4; for

Vishnu [23 6 -14>, the GCD of the last two entries is 2, so

the generator part of the mapping is 0, -7, -3.

Get it? It always works this way.

The GCD mentioned above also tells you how many periods there are per
octave.

Things basically work the same way in the 7-limit case, but instead
of starting with the comma in vector form, you start with the bicomma
in bivector form (i.e., a wedge product of any two basis commas).
Since this is a bit more complicated, we should continue this
discussion on the tuning-math list . . .

>> P.S. In the paper there is a list of 5-prime-limit ratios of
>> Harmonic Distance less than 6. I think the ratios 9/4, 5/2, 15/2
>> and 27/1 are missing from that list.
>
>Yikes! I can't believe no one noticed this before! Ouch, ouch,
>ouch . . . how did I let this happen? Well, if you fill in the
>missing lines, the point I'm making about TOP Meantone remains true,
>right?

Paul, until I get a chance to scan this, can you keep track of
essential corrections?

-Carl

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
>
> --- In tuning@yahoogroups.com, "Kalle Aho" <kalleaho@m...> wrote:
>
> > But I want to know how to get the
> > generator mappings from the comma list.
>
> Kalle, you may or may not have noticed some patterns relating the
> commas (in vector form) to the generator part of the mappings in
> Table 1 in the paper. Here's the pattern. Take the comma in vector
> form, ignore the first entry (the power of 2), and determine the
GCD
> (greatest common divisor) of the last two entries. The generator
part
> of the mapping of 2 is always 0. The generator part of the mapping
of
> 3 is the power of 5 in the comma vector, divided by the GCD you
> found. And the generator part of the mapping of 5 is minus the
power
> of 3 in the comma vector, again divided by that GCD. That why we
see
> that for
>
> Meantone 81:80 or [-4 4 -1>, the GCD of the last two entries is 1,
so
>
> the generator part of the mapping is 0, -1, -4; for
>
> Vishnu [23 6 -14>, the GCD of the last two entries is 2, so
>
> the generator part of the mapping is 0, -7, -3.
>
> Get it? It always works this way.

Got it, thanks!

> The GCD mentioned above also tells you how many periods there are
per
> octave.

Aha!

> Things basically work the same way in the 7-limit case, but
instead
> of starting with the comma in vector form, you start with the
bicomma
> in bivector form (i.e., a wedge product of any two basis commas).
> Since this is a bit more complicated, we should continue this
> discussion on the tuning-math list . . .

Then let's do that!

Hi Paul,

> > For one thing the paper doesn't explain how one derives the
> > generators of the temperament from the vanishing commas. I have
no
> > idea how to do that
>
> Did you follow the tuning-math list last year or the year before
> that? This came up a number of times.

Not that closely. I know it's there but it's a bit hard to find.

> > and some kind of explanation would've been
> > nice.
>
> I'll try to include one in part 2 or 3 of the paper. (You can see
> that the paper you have says "Part 1".)

Yep.

> > P.S. In the paper there is a list of 5-prime-limit ratios of
> > Harmonic Distance less than 6. I think the ratios 9/4, 5/2, 15/2
> and
> > 27/1 are missing from that list.
>
> Yikes! I can't believe no one noticed this before! Ouch, ouch,
> ouch . . . how did I let this happen? Well, if you fill in the
> missing lines, the point I'm making about TOP Meantone remains
true,
> right?

Yes.