Planar temperament question for Gene

Can you talk about the planar temperament that results from taking
2025:2048 and 891:896 as unison vectors? (This is the Shrutar
temperament, before some additional unison vector is applied.) Also,
if you'd like to post the result of applying LLL on this, I'd be
interested to see that on

tuning-math@yahoogroups.com

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:

Also,
> if you'd like to post the result of applying LLL on this, I'd be
> interested to see that on

There's no reason not to give the results here, I think: I got a
reduced basis of 176/175 and 9801/9800 from straight LLL, and a
reduced basis of 176/175 and 896/891 after adjusting by multiplying
the values corresponding to a prime p by log2(p).

--- In tuning@y..., genewardsmith@j... wrote:
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
>
> Also,
> > if you'd like to post the result of applying LLL on this, I'd be
> > interested to see that on
>
> There's no reason not to give the results here, I think: I got a
> reduced basis of 176/175 and 9801/9800 from straight LLL, and a
> reduced basis of 176/175 and 896/891 after adjusting by multiplying
> the values corresponding to a prime p by log2(p).

Is 2 considered a prime on equal footing with the others?

Anything else you can say about this temperament, along the lines of
what you've been posting here about other planar temperaments?

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:

> Is 2 considered a prime on equal footing with the others?

As usual. Of course, log2(2)=1 so there isn't much of an adjustment,
but the choice of log2 is arbitrary--I could have used cents, for
instance.

> Anything else you can say about this temperament, along the lines
of
> what you've been posting here about other planar temperaments?

This wasn't the temperament, just a reduced basis. However,
generators for the dual to it are h22,h46,and h80, and reducing this
by LLL either adjusted or nonadjusted gave the same answer. I got a
linear temperament with interval of equivalence sqrt(2); or in other
words, sqrt(2) as a generator, the others being a~11/10 and b~4/3;
the values I got using least squares in the ll-limit being
a=163.1397 cents and b=495.5502 cents. We have 3~2^2 b^(-1), 2.49
cents sharp; 5~2^(3/2) b^2, 4.79 cents sharp; 7~2^(7/2)ab^(-2), 3.21
cents sharp; and 11~2^(5/2)ab^2, 2.92 cents sharp. This is a sort of
modified paultone system, with two chains of fifths a sqrt(2) apart,
modified by having a 5/6-tone generator "a" branching off from each
chain to give 7 and 11 harmonies. It would be interesting to know if
this bears any resemblence to Indian musical theory.

--- In tuning@y..., genewardsmith@j... wrote:

> by LLL either adjusted or nonadjusted gave the same answer. I got a
> linear temperament with interval of equivalence sqrt(2); or in
^^^^^^

Planar--which is to say, 3D. :)

--- In tuning@y..., genewardsmith@j... wrote:
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
>
> > Is 2 considered a prime on equal footing with the others?
>
> As usual. Of course, log2(2)=1 so there isn't much of an
adjustment,
> but the choice of log2 is arbitrary--I could have used cents, for
> instance.

That I see . . . but I meant something different. I'll pursue this on
tuning-math.

> > Anything else you can say about this temperament, along the lines
> of
> > what you've been posting here about other planar temperaments?
>
> This wasn't the temperament, just a reduced basis. However,
> generators for the dual to it are h22,h46,and h80, and reducing
this
> by LLL either adjusted or nonadjusted gave the same answer. I got a
> linear temperament with interval of equivalence sqrt(2); or in
other
> words, sqrt(2) as a generator, the others being a~11/10 and b~4/3;
> the values I got using least squares in the ll-limit being
> a=163.1397 cents and b=495.5502 cents. We have 3~2^2 b^(-1), 2.49
> cents sharp; 5~2^(3/2) b^2, 4.79 cents sharp; 7~2^(7/2)ab^(-2),
3.21
> cents sharp; and 11~2^(5/2)ab^2, 2.92 cents sharp. This is a sort
of
> modified paultone system, with two chains of fifths a sqrt(2)
apart,
> modified by having a 5/6-tone generator "a" branching off from each
> chain to give 7 and 11 harmonies. It would be interesting to know
if
> this bears any resemblence to Indian musical theory.

Well, it's a detempering of the Shrutar linear temperament, whose
relationship to Indian music theory has already been discussed. I'm
unaware of any evidence for attributing 7- or 11-limit ratios to
Indian music theory, but since Erv Wilson called 4374:4375
the "ragisma", he may know something I don't (or have carried his
speculations farther).

Is there a better system for getting 7- and 11-limit harmonies given
that the "center spine" of the tuning will be two chains of fifths a
sqrt(2) apart with 2025:2048 vanishing?

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:

> Is there a better system for getting 7- and 11-limit harmonies
given
> that the "center spine" of the tuning will be two chains of fifths
a
> sqrt(2) apart with 2025:2048 vanishing?

I suspect the answer is "no". There don't seem to be a lot of choices
of linearly independent sets with good ets, and these seem to be
spanning the same group. I tried tossing in three garbage ets (124,
126 and 148) and it turned out to be garbage in, garbage out. This is
not too surprising since this is the situation with linear
temperaments.

--- In tuning@y..., genewardsmith@j... wrote:
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
>
> > Is there a better system for getting 7- and 11-limit harmonies
> given
> > that the "center spine" of the tuning will be two chains of
fifths
> a
> > sqrt(2) apart with 2025:2048 vanishing?
>
> I suspect the answer is "no".

Can you prove it?

> There don't seem to be a lot of choices
> of linearly independent sets with good ets,

What if we didn't care about ets? (I suspect that that doesn't
matter . . . ?)

> and these seem to be
> spanning the same group.

Well that would be encouraging . . . it would justify the Shrutar
tuning based on its premises. I have to wonder what else might be
possible.

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:

> > I suspect the answer is "no".

> Can you prove it?

Maybe. I'll think about it.

> > There don't seem to be a lot of choices
> > of linearly independent sets with good ets,

> What if we didn't care about ets? (I suspect that that doesn't
> matter . . . ?)

Ets are to vals as commas are to intervals; not caring about ets is
like not caring about commas.

--- In tuning@y..., genewardsmith@j... wrote:
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
>
> > > I suspect the answer is "no".
>
> > Can you prove it?
>
> Maybe. I'll think about it.
>
> > > There don't seem to be a lot of choices
> > > of linearly independent sets with good ets,
>
> > What if we didn't care about ets? (I suspect that that doesn't
> > matter . . . ?)
>
> Ets are to vals as commas are to intervals; not caring about ets is
> like not caring about commas.

Well, I guess I meant, "not caring about relatively small ETs"

(Your analogy is a bit mysterious to me, though I think I grasp that
lines are the duals of planes in 3D . . . can you elaborate?)