More on the Shrutar planar temperament

Here's a picture of what it looks like around the half-octave:

16/11 -- 8/5 -- 7/4
| | |
27/14 -- 16/15 -- 7/6
| | |
9/7 -- 45/32 -- 14/9
| | |
12/7 -- 15/8 -- 28/27
| | |
8/7 -- 5/4 -- 11/8

Ets which approximate this temperament are 46,58,68 and 80; the 68 is
perhaps the best choice, though the 11 is flat rather than sharp. The
80 and 58 do well also, and 46 is not too bad.

--- In tuning@y..., genewardsmith@j... wrote:
> Here's a picture of what it looks like around the half-octave:
>
> 16/11 -- 8/5 -- 7/4
> | | |
> 27/14 -- 16/15 -- 7/6
> | | |
> 9/7 -- 45/32 -- 14/9
> | | |
> 12/7 -- 15/8 -- 28/27
> | | |
> 8/7 -- 5/4 -- 11/8
>
> Ets which approximate this temperament are 46,58,68 and 80; the 68
is
> perhaps the best choice, though the 11 is flat rather than sharp.
The
> 80 and 58 do well also, and 46 is not too bad.

Now there is an additional unison vector lurking about, which turns
the system into the Shrutar linear temperament we discussed a bit
ago, and makes 46 look better than all these other alternatives. Can
you identify this unison vector (the answer's not unique, of course,
but there should be an LLL-style "best")?

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

> Now there is an additional unison vector lurking about, which turns
> the system into the Shrutar linear temperament we discussed a bit
> ago, and makes 46 look better than all these other alternatives.

There are always commas in the woodwork, ready to turn the planar
temperament into a variety of linear temperaments. Did we discuss
2401/2400 in this connection?

Can
> you identify this unison vector (the answer's not unique, of
course,
> but there should be an LLL-style "best")?

I dunno about that, it seems to me that the other ets I've discussed
are simply to good to be easily trumped like this. I cooked up a
notation from 31,10,46,58 and 68; dual to this were 2048/2025,
176/175,2401/2400,385/384, and 441/440. The 385/384 of the 58 et and
the 441/440 of the 68 et have their own virtues.

--- In tuning@y..., genewardsmith@j... wrote:
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
>
> > Now there is an additional unison vector lurking about, which
turns
> > the system into the Shrutar linear temperament we discussed a bit
> > ago, and makes 46 look better than all these other alternatives.
>
> There are always commas in the woodwork, ready to turn the planar
> temperament into a variety of linear temperaments. Did we discuss
> 2401/2400 in this connection?

I don't think so . . .
>
> Can
> > you identify this unison vector (the answer's not unique, of
> course,
> > but there should be an LLL-style "best")?
>
> I dunno about that, it seems to me that the other ets I've
discussed
> are simply to good to be easily trumped like this. I cooked up a
> notation from 31,10,46,58 and 68; dual to this were 2048/2025,
> 176/175,2401/2400,385/384, and 441/440. The 385/384 of the 58 et
and
> the 441/440 of the 68 et have their own virtues.

By the latter you mean that these ratios are _not_ unison vectors of
the respective ETs???

Also, I'm pretty sure I mean an LLL-style "best" specifically with
respect to 2025:2048 (non-negotiable) and an additional unison vector
which, when taken with 2025:2048, will form the basis for the same
group we were discussing before (possible bases for which include
2025:2048 & 891/896, or 176/175 & 9801/9800, etc.)?

If that made no sense whatsover, let me ask, what does the 22-tone
Shrutar tuning in 46 become if we use 58 or 68 (or 80?) instead? Is
the answer unambiguous? Are they all omnitetrachordal?

I wrote,

>Are they all omnitetrachordal?

The reason I care is that I'm tuning the open strings of the guitar
to 1/1's and 3/2's, and want the frets to go straight across.

--- In tuning@y..., genewardsmith@j... wrote:
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
>
> > Now there is an additional unison vector lurking about, which
turns
> > the system into the Shrutar linear temperament we discussed a bit
> > ago, and makes 46 look better than all these other alternatives.
>
> There are always commas in the woodwork, ready to turn the planar
> temperament into a variety of linear temperaments.

So can you find the one (or a few) that turn this one into the linear
Shrutar temperament that we talked about, whos optimal tuning is very
close to 46-tET? Just out of curiosity, so that if we find a better
family that is related to 58-, 68-, 80-tET, or perhaps none of these,
we'll know what's changed.

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

> So can you find the one (or a few) that turn this one into the
linear
> Shrutar temperament that we talked about, whos optimal tuning is
very
> close to 46-tET?

I'm not sure what your fishing for, but I did an LLL reduction on
<2048/2025,176/175,385/384,441/440> (which I got from the notation I
mentioned) and got <121/120,2662/2625,126/125,686/675>. We have been
discussing 126/125, and perhaps this is your magic comma?

--- In tuning@y..., genewardsmith@j... wrote:
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
>
> > So can you find the one (or a few) that turn this one into the
> linear
> > Shrutar temperament that we talked about, whos optimal tuning is
> very
> > close to 46-tET?
>
> I'm not sure what your fishing for,

Just one or a few examples of such a UV -- isn't it clear what I mean
(see below for further amplification)?

> but I did an LLL reduction on
> <2048/2025,176/175,385/384,441/440> (which I got from the notation
I
> mentioned) and got <121/120,2662/2625,126/125,686/675>. We have
been
> discussing 126/125, and perhaps this is your magic comma?

So you're saying that 2048/2025, 896/891, and 126/125 together define
the Shrutar linear temperament discussed before whose optimal
generator is close to 52 cents and which has an interval of
repetition of sqrt(2)? Besides 126/125, what else can function in
this role?

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

> > but I did an LLL reduction on
> > <2048/2025,176/175,385/384,441/440> (which I got from the
notation
> I
> > mentioned) and got <121/120,2662/2625,126/125,686/675>. We have
> been
> > discussing 126/125, and perhaps this is your magic comma?

> So you're saying that 2048/2025, 896/891, and 126/125 together
define
> the Shrutar linear temperament discussed before whose optimal
> generator is close to 52 cents and which has an interval of
> repetition of sqrt(2)? Besides 126/125, what else can function in
> this role?

No, this was a reduction of the 11-limit kernel of the 46-et; I
thought we had already determined that aside from 2048/2025 and
896/891 we could use instead 176/175 and 896/891 to define the
Shrutar; it only wants two commas in the 11-limit to define it; there
are five primes, "planar" actually means "3D" and 5-3 = 2 so the
codimension is 2, and so we need two commas.

I was looking at 46,58,68, and 80 with this system. Reducing the
fifth by a half-octave, we get generators

5/29 < 4/23 < 7/40 < 3/17

The 5/29 is of course 58 and 4/23 46, the generator maps being
[1,-2,-8,11] vs [1,-2,-8,-12] we have looked at before. Since
11 = -12 mod 23, we can regard the 58 and 46 as the same, more or
less. Also, from 7/40, the 80-et, we get [1,-2,15,11], and since
-8 = 15 mod 23, 46 is similar here also. On the other hand 68 is
divisible by 4 and we have the generator 3/17 when reduced to lowest
terms, so 68 works differently, with a 2^(1/4) interval of repetition.

--- In tuning@y..., genewardsmith@j... wrote:
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
>
> > > but I did an LLL reduction on
> > > <2048/2025,176/175,385/384,441/440> (which I got from the
> notation
> > I
> > > mentioned) and got <121/120,2662/2625,126/125,686/675>. We have
> > been
> > > discussing 126/125, and perhaps this is your magic comma?
>
> > So you're saying that 2048/2025, 896/891, and 126/125 together
> define
> > the Shrutar linear temperament discussed before whose optimal
> > generator is close to 52 cents and which has an interval of
> > repetition of sqrt(2)? Besides 126/125, what else can function in
> > this role?
>
> No, this was a reduction of the 11-limit kernel of the 46-et; I
> thought we had already determined that aside from 2048/2025 and
> 896/891 we could use instead 176/175 and 896/891 to define the
> Shrutar; it only wants two commas in the 11-limit to define it;
there
> are five primes, "planar" actually means "3D" and 5-3 = 2 so the
> codimension is 2, and so we need two commas.

Arrgh! Once again, (deep breath) what about the Shrutar _linear_
temperament we discussed before, whose optimal generator is close to
52 cents and which has an interval of repetition of sqrt(2)?

> I was looking at 46,58,68, and 80 with this system. Reducing the
> fifth by a half-octave, we get generators
>
> 5/29 < 4/23 < 7/40 < 3/17
>
> The 5/29 is of course 58 and 4/23 46, the generator maps being
> [1,-2,-8,11] vs [1,-2,-8,-12] we have looked at before. Since
> 11 = -12 mod 23, we can regard the 58 and 46 as the same, more or
> less. Also, from 7/40, the 80-et, we get [1,-2,15,11], and since
> -8 = 15 mod 23, 46 is similar here also. On the other hand 68 is
> divisible by 4 and we have the generator 3/17 when reduced to
lowest
> terms, so 68 works differently, with a 2^(1/4) interval of
repetition.

OK, so please tell me, if you take the (10+12) scale in the Shrutar
_linear_ temperament, and detemper the "mystery comma" so that it's
now in a planar temperament, and then put it in 68, what does it look
like?

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> Arrgh! Once again, (deep breath) what about the Shrutar _linear_
> temperament we discussed before, whose optimal generator is close
to
> 52 cents and which has an interval of repetition of sqrt(2)?

If we take 46&68 in the 11-limit, by which I mean the linear
temperament defined by intersecting their kernels, we get a map to
generators of

[ 0 2]
[ 2 3]
[-4 5]
[ 7 5]
[-1 7]

after LLL-reducing the pair. Least squares fit to the 11-limit gives
values of a=52.5135 cents and of course b=600 cents for the second
column. The errors of the primes are 3, 3.07 cents sharp, 5, 3.63
cents sharp, 7, 1.23 cents flat, and 11, 3.83 cents flat. The kernel
is generated by 121/120, 176/175 and 245/243.

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

> The kernel
> is generated by 121/120, 176/175 and 245/243.

So if one started with the UVs 2048/2025 and 896/891, one would have
to add ___________ to generate this kernel?

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

> > The kernel
> > is generated by 121/120, 176/175 and 245/243.

> So if one started with the UVs 2048/2025 and 896/891, one would
have
> to add ___________ to generate this kernel?

The question does not have a unique answer, but either of 121/120 or
245/243 will work; since (2048/2025)/(896/891) = 176/175, this one
*wont* work--it's already in there.

--- In tuning@y..., genewardsmith@j... wrote:
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> > --- In tuning@y..., genewardsmith@j... wrote:
>
> > > The kernel
> > > is generated by 121/120, 176/175 and 245/243.
>
> > So if one started with the UVs 2048/2025 and 896/891, one would
> have
> > to add ___________ to generate this kernel?
>
> The question does not have a unique answer,

As I've stated.

> but either of 121/120 or
> 245/243 will work;

Thank you Gene!

> since (2048/2025)/(896/891) = 176/175, this one
> *wont* work--it's already in there.

Gotcha.