Yahoo Tuning Groups Ultimate Backup tuning-math Is it possible

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul_hjelmstad@a...> wrote:

> Is is possible to find a two 7-limit commas, from two temperaments in
7-
> limit space? I would say no. (I know you can find two temperaments
from
> two commas). Just tell me it can't be done and I'll shut up! (For
> awhile)

From two equal temperaments, you can get a rank two temperament (by
wedging the vals, for one method.) This temperament has a kernel, which
(nonuniqely) has a basis of two commas. If we specify TM reduction, we
can make the basis unique. We don't really need the step of specifying
the rank two temperament, as we can simply find the kernel of the two
vals as a mapping. It's quite possible the software you use can do this.

🔗Paul G Hjelmstad <paul_hjelmstad@allianzlife.com>

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <paul_hjelmstad@a...> wrote:
>
> > Is is possible to find a two 7-limit commas, from two
temperaments in
> 7-
> > limit space? I would say no. (I know you can find two
temperaments
> from
> > two commas). Just tell me it can't be done and I'll shut up! (For
> > awhile)
>
> From two equal temperaments, you can get a rank two temperament (by
> wedging the vals, for one method.) This temperament has a kernel,
which
> (nonuniqely) has a basis of two commas. If we specify TM reduction,
we
> can make the basis unique. We don't really need the step of
specifying
> the rank two temperament, as we can simply find the kernel of the
two
> vals as a mapping. It's quite possible the software you use can do
this.

I just installed Octave (in Cygwin) but I haven't had time to learn
it yet. The reason I thought the above cannot be done in the 7-limit
is that it is like trying to find two planes (7-limit commas) from a
line (two temperaments). Of course you can go the other direction
(two planes that are not parallel determine a line) but I'm glad to
see there is a way to make it work.

Any idea how I would find the kernel of vals using Excel? By vals do
you mean two temperaments like <12 19 28 34| and <31 49 72 87|?

How would you get a rank two temperament from a <<a b c d e f>>
wedgie?

Thanks

Paul Hj

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul_hjelmstad@a...> wrote:

> Any idea how I would find the kernel of vals using Excel?

If it has a kernel or nullspace function, you can use that.
Otherwise you can set up a linear system and solve.

By vals do
> you mean two temperaments like <12 19 28 34| and <31 49 72 87|?

Sort of. I mean anything of that form with integer coefficients,
which serve to map p-limit intervals to integers.

> How would you get a rank two temperament from a <<a b c d e f>>
> wedgie?

That's kind of hard to answer, since my usual point of view is that
*is* a temperament. What's your favored way to represent a 7-limit
rank two temperament?

🔗Paul G Hjelmstad <paul_hjelmstad@allianzlife.com>

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <paul_hjelmstad@a...> wrote:
>
> > Any idea how I would find the kernel of vals using Excel?
>
> If it has a kernel or nullspace function, you can use that.
> Otherwise you can set up a linear system and solve.

Octave should be able to do that...
>
> By vals do
> > you mean two temperaments like <12 19 28 34| and <31 49 72 87|?
>
> Sort of. I mean anything of that form with integer coefficients,
> which serve to map p-limit intervals to integers.

I've been struggling with the meaning of the term val- is this what
you call a 'breed'? So, its not always the obvious values like the
above...

> > How would you get a rank two temperament from a <<a b c d e f>>
> > wedgie?
>
> That's kind of hard to answer, since my usual point of view is
that
> *is* a temperament. What's your favored way to represent a 7-limit
> rank two temperament?

period and generator? I'm going to review your webpage on this, I
know I have asked the same questions repeatedly (in slightly
different ways) Wedgies or "adjutant matrix method", both are fine...

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul_hjelmstad@a...> wrote:

> I've been struggling with the meaning of the term val- is this what
> you call a 'breed'? So, its not always the obvious values like the
> above...

So far as I know it means the same thing. The main point is that the
coefficients are integers; otherwise you get a linear functional, which
could for instance be a tuning map. Certainly they don't need to be
equal temperaments; in fact, a typical example would be the mapping of
primes to generator steps in a linear temperament.

🔗Paul G Hjelmstad <paul_hjelmstad@allianzlife.com>

Paul -Z- Hjelmstad

🔗Paul G Hjelmstad <paul_hjelmstad@allianzlife.com>

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul_hjelmstad@a...> wrote:
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <paul_hjelmstad@a...> wrote:
> > --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> > wrote:
> > > --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> > > <paul_hjelmstad@a...> wrote:
> > >
> > > > Any idea how I would find the kernel of vals using Excel?
> > >
> > > If it has a kernel or nullspace function, you can use that.
> > > Otherwise you can set up a linear system and solve.
> >
> Could I use the adjutant method? Using 1 0 0 0 and ? ? ? ? and 12
19 28
> 34 and 31 49 72 87? I guess I don't know what to use in the second
row.
> Or would it be a simple linear equation? I'll have to learn Octave,
> I don't see how Excel could solve linear equations. Thanks.
>
I'm getting confused. I learned how to use Octave a little, but every
kernel of Ax=b where b=0 is giving me the zero matrix. I'm obviously
doing something wrong here. According to Mathworld, this IS the
solution. (x=0). Must be a Monday...

🔗Paul G Hjelmstad <paul_hjelmstad@allianzlife.com>

Found it in "Octave" - null (a). Could someone give me a good example
of a linear equation to solve? (I'm trying to generate two (nonunique)
commas from vals, such as two temperaments...

🔗Paul G Hjelmstad <paul_hjelmstad@allianzlife.com>

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul_hjelmstad@a...> wrote:
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <paul_hjelmstad@a...> wrote:
> > --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> > <paul_hjelmstad@a...> wrote:
> > > --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> > > <paul_hjelmstad@a...> wrote:
> > > > --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
> > <gwsmith@s...>
> > > > wrote:
> > > > > --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> > > > > <paul_hjelmstad@a...> wrote:
> > > > >
> > > > > > Any idea how I would find the kernel of vals using Excel?
> > > > >
> > > > > If it has a kernel or nullspace function, you can use that.
> > > > > Otherwise you can set up a linear system and solve.
>
> Found it in "Octave" - null (a). Could someone give me a good
example
> of a linear equation to solve? (I'm trying to generate two
(nonunique)
> commas from vals, such as two temperaments...

I'm trying different vals. Taking (12, 19, 28, 34; 19, 30, 44, 54)
I get a kernel with two "commas", but they are reduced to values
below 1. I thought if I find a scaling factor to get "2" to the first
power, I should be able to scale the others and eventually find
integral values, but no such luck. What do I need to do to get the
kernel to integral values?

Thanks

Paul Hj

🔗Paul G Hjelmstad <paul_hjelmstad@allianzlife.com>

> I'm trying different vals. Taking (12, 19, 28, 34; 19, 30, 44, 54)
> I get a kernel with two "commas", but they are reduced to values
> below 1. I thought if I find a scaling factor to get "2" to the first
> power, I should be able to scale the others and eventually find
> integral values, but no such luck. What do I need to do to get the
> kernel to integral values?

Well, I found common denominators with the null spaces of certain vals,
such as the 2 x 4 matrix of [12,19,28,34;43,68,100,121] but the commas
are so absurdly big (like (119, 134, 46, 5) - not very practical. I
take it they need to be TM-reduced? Anyway the kernel function is fun
partly because it doesn't require square matrices:)

> Thanks
>
> Paul Hj

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul_hjelmstad@a...> wrote:

> Well, I found common denominators with the null spaces of certain vals,
> such as the 2 x 4 matrix of [12,19,28,34;43,68,100,121] but the commas
> are so absurdly big (like (119, 134, 46, 5) - not very practical. I
> take it they need to be TM-reduced? Anyway the kernel function is fun
> partly because it doesn't require square matrices:)

I don't know why your program is giving you such large answers, but in
any case you certainly will want to reduce them. Some sort of
reduction algorithm such as LLL or Hermite normal form is a very
useful thing to have available, you might want to check to see what
your program supports.

🔗Paul G Hjelmstad <paul_hjelmstad@allianzlife.com>

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <paul_hjelmstad@a...> wrote:
>
> > Well, I found common denominators with the null spaces of certain
vals,
> > such as the 2 x 4 matrix of [12,19,28,34;43,68,100,121] but the
commas
> > are so absurdly big (like (119, 134, 46, 5) - not very practical. I
> > take it they need to be TM-reduced? Anyway the kernel function is
fun
> > partly because it doesn't require square matrices:)
>
> I don't know why your program is giving you such large answers, but in
> any case you certainly will want to reduce them. Some sort of
> reduction algorithm such as LLL or Hermite normal form is a very
> useful thing to have available, you might want to check to see what
> your program supports.

Thanks. I looked in the Octave manual, and the definition of null (a) is
"Return an orthonormal basis of the null space of a." Usually the
values are integral values divided by the square root of the sum of
squares of something such as sqrt (a^2+b^2+c^2) but not always. The
values are always between -1 and +1. If I could figure out the formula,
then I could really make use of this function...

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul_hjelmstad@a...> wrote:

> Thanks. I looked in the Octave manual, and the definition of null (a) is
> "Return an orthonormal basis of the null space of a." Usually the
> values are integral values divided by the square root of the sum of
> squares of something such as sqrt (a^2+b^2+c^2) but not always. The
> values are always between -1 and +1. If I could figure out the formula,
> then I could really make use of this function...

I don't think you need to figure out the algorithm to use it, as it
tells you what the output is. Clear the denominator of all square
roots, and then clear denominators by muliplying by the lcm. You now
have a basis for at least a subgroup of the kernel. If you wedge it
all together and don't get a common factor, you have the full kernel.
If you do get a common factor, you can try to remove it by performing
elementary row operations, getting a row with a common factor, and
dividing it out. If you know how to do it, you can also divide the
common factor out of the wedge product, and then use that to generate
a full basis.

Of course if you are using wedge products you might as well simply
start off by wedging the rows or columns of the matrix together, and
then using the result.

🔗Paul G Hjelmstad <paul_hjelmstad@allianzlife.com>

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <paul_hjelmstad@a...> wrote:
>
> > Thanks. I looked in the Octave manual, and the definition of null
(a) is
> > "Return an orthonormal basis of the null space of a." Usually the
> > values are integral values divided by the square root of the sum
of
> > squares of something such as sqrt (a^2+b^2+c^2) but not always.
The
> > values are always between -1 and +1. If I could figure out the
formula,
> > then I could really make use of this function...
>
> I don't think you need to figure out the algorithm to use it, as it
> tells you what the output is. Clear the denominator of all square
> roots, and then clear denominators by muliplying by the lcm. You now
> have a basis for at least a subgroup of the kernel. If you wedge it
> all together and don't get a common factor, you have the full
kernel.
> If you do get a common factor, you can try to remove it by
performing
> elementary row operations, getting a row with a common factor, and
> dividing it out. If you know how to do it, you can also divide the
> common factor out of the wedge product, and then use that to
generate
> a full basis.
>
> Of course if you are using wedge products you might as well simply
> start off by wedging the rows or columns of the matrix together, and
> then using the result.

Interesting. I also posted a message on sci.math, and was told that
Octave uses singular value decomposition, and sure enough V^(T)*V=I
like it should. (Mathworld has U as non-square and V as square,
actally I think its the reverse). A=U*D*V', I take it V'=V^-1=V^T.
I wonder how in the world I can implement lcm on numbers like .25566?
Anyway, some fun stuff to try out. Thanks.

🔗Paul G Hjelmstad <paul_hjelmstad@allianzlife.com>

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul_hjelmstad@a...> wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> > <paul_hjelmstad@a...> wrote:
> >
> I don't think you need to figure out the algorithm to use it, as it
> > tells you what the output is. Clear the denominator of all square
> > roots, and then clear denominators by muliplying by the lcm. You
now
> > have a basis for at least a subgroup of the kernel. If you wedge
it
> > all together and don't get a common factor, you have the full
> kernel.
> > If you do get a common factor, you can try to remove it by
> performing
> > elementary row operations, getting a row with a common factor, and
> > dividing it out. If you know how to do it, you can also divide the
> > common factor out of the wedge product, and then use that to
> generate
> > a full basis.
> >
> > Of course if you are using wedge products you might as well simply
> > start off by wedging the rows or columns of the matrix together,
and
> > then using the result.
>
> Interesting. I also posted a message on sci.math, and was told that
> Octave uses singular value decomposition, and sure enough V^(T)*V=I
> like it should. (Mathworld has U as non-square and V as square,
> actally I think its the reverse). A=U*D*V', I take it V'=V^-1=V^T.
> I wonder how in the world I can implement lcm on numbers
like .25566?
> Anyway, some fun stuff to try out. Thanks.

I guess I am still hazy on orthonormal, even though I know its unit
vectors that are orthogonal. Speaking of lcm, how can I figure out
the lcm the numbers I am getting? I've tried squaring the denominator
(to get rid of the square root) inverting, adding it to itself
(summing) using ++ to make it a constant, until I hit a integer, but
that doesn't always work...

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul_hjelmstad@a...> wrote:

> I guess I am still hazy on orthonormal, even though I know its unit
> vectors that are orthogonal. Speaking of lcm, how can I figure out
> the lcm the numbers I am getting? I've tried squaring the denominator
> (to get rid of the square root) inverting, adding it to itself
> (summing) using ++ to make it a constant, until I hit a integer, but
> that doesn't always work...

I wasn't aware when I posted what I did that you didn't know what the
exact form was. You might try dividing through by one of the
coordinates, and then clearing rational denominators, using rational
approximation such as continued fractions if you have them. Better
night be to simply ignore this function and solve the linear system
instead.

🔗Paul G Hjelmstad <paul_hjelmstad@allianzlife.com>

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <paul_hjelmstad@a...> wrote:
>
> > I guess I am still hazy on orthonormal, even though I know its unit
> > vectors that are orthogonal. Speaking of lcm, how can I figure out
> > the lcm the numbers I am getting? I've tried squaring the
denominator
> > (to get rid of the square root) inverting, adding it to itself
> > (summing) using ++ to make it a constant, until I hit a integer,
but
> > that doesn't always work...
>
> I wasn't aware when I posted what I did that you didn't know what the
> exact form was. You might try dividing through by one of the
> coordinates, and then clearing rational denominators, using rational
> approximation such as continued fractions if you have them. Better
> night be to simply ignore this function and solve the linear system
> instead.

Thanks. But if I solve the linear system Ax=b with b equals zero, don't
I get the trivial solution? I've been studying Unitary matrices, SVD,
and so forth on Mathworld, and the Octave manual is helpful to a point.
What I would like to do is the sort of thing I cannot do in Excel,
kernels of non-square matrices, especially getting two commas from two
temperaments in the 7-limit, and vice versa (actually you can do that,
two temperaments from two commas, but only with square matrices). I'm
surprised Excel can't do more than it can, but since I have Octave,
I'll use that. Where I am stuck now is determining values for U, D and
V' in singular value decomposition of the matrix A. (A=UDV', AV=UD, etc)
Does orthogonal merely mean that V^T=V^-1? Or is there more to it than
that? How in the world do they calculate V? I've noticed that simple
vectors like [1,2,3] have the "worst" kernels. The best I can do is:
sqrt(10), sqrt(21), 2 and sqrt(45),sqrt(8),sqrt(17). The funny thing is
that the ones that fare the best are usually things I can calculate in
Excel. (A val from three commas or a comma from three vals; never two
and two).

So I'm working in three main areas now -- Studying the clustering of Z-
relations, using Octave with the nullspace and other functions, and
Steiner systems. Any luck getting M24? It would be fun to calculate
which of the 1,782 set types in ST(22,6) are used by the 77 Steiner
blocks in S(3,6,22) and other stuff. I still need to redo my analysis of
S(5,6,12) with good data.

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul_hjelmstad@a...> wrote:

> Thanks. But if I solve the linear system Ax=b with b equals zero, don't
> I get the trivial solution?

The way to tell is to try it and see, but I would hope Octave could do
better than that.

I've been studying Unitary matrices, SVD,
> and so forth on Mathworld, and the Octave manual is helpful to a point.

How do you propose using that in music theory?

> Does orthogonal merely mean that V^T=V^-1?

For non-square matricies it means the columns (or rows) are orthogonal
vectors.

Any luck getting M24?

Should I email that, and if so, where?

🔗Paul G Hjelmstad <paul_hjelmstad@allianzlife.com>

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <paul_hjelmstad@a...> wrote:
>
> > Thanks. But if I solve the linear system Ax=b with b equals zero,
don't
> > I get the trivial solution?
>
> The way to tell is to try it and see, but I would hope Octave could
do
> better than that.

It doesn't, unfortunately. You type a \ b with b=0 vector and always
get the zero matrix. That's their way of solving for Ax=b
>
> I've been studying Unitary matrices, SVD,
> > and so forth on Mathworld, and the Octave manual is helpful to a
point.
>
> How do you propose using that in music theory?

Not sure. But I definitely want to make use of the kernel function in
Octave. That's what's behind my drive to understand svd.
>
> > Does orthogonal merely mean that V^T=V^-1?
>
> For non-square matricies it means the columns (or rows) are
orthogonal
> vectors.

Right. I checked it out and it works great. I'm going to study the
subtle relationship between orthogonal and unitary matrices...
>
> Any luck getting M24?
>
> Should I email that, and if so, where?

Yes please:
paul.hjelmstad@allianzlife.com

🔗Paul G Hjelmstad <paul_hjelmstad@allianzlife.com>

Paul Hj

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul_hjelmstad@a...> wrote:

> Just wondering, do you ever get orthogonal vectors when you derive
> two commas for say, a 7-limit linear temperament?

An example would be the Pontiac temperament, where the TM basis is
{2401/2400, 65625/65536}.

I looked on you
> website and it doesn't seem so. Would there be any benefit to having
> two commas that are orthogonal so that u.v=0?

There certainly are advantages to Fokker blocks with approximately
orthogonal sides in the breed plane, where one is horizonal and the
other vertical; I've been looking at exactly that in the last few
days. In the 7-limit lattice, rectangular blocks don't seem quite so
interesting. The temperament, unlike the block, does not depend on the
exact choice of commas, so it also seems less likely that it matters
very much to the temperament.

If we pick 2401/2400, 4375/4374 and 65625/65536, then the angle
between 2401/2400 and 4375/4374 is 88.56 degrees, between 4375/4374
and 65625/65536 is 90 degrees, and between 2401/2400 and 65625/65536
is 90 degrees. The block, for 171-et, is almost rectangular.

🔗Paul G Hjelmstad <paul_hjelmstad@allianzlife.com>

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <paul_hjelmstad@a...> wrote:
>
> > Just wondering, do you ever get orthogonal vectors when you
derive
> > two commas for say, a 7-limit linear temperament?
>
> An example would be the Pontiac temperament, where the TM basis is
> {2401/2400, 65625/65536}.
>
> I looked on you
> > website and it doesn't seem so. Would there be any benefit to
having
> > two commas that are orthogonal so that u.v=0?
>
> There certainly are advantages to Fokker blocks with approximately
> orthogonal sides in the breed plane, where one is horizonal and the
> other vertical; I've been looking at exactly that in the last few
> days. In the 7-limit lattice, rectangular blocks don't seem quite so
> interesting. The temperament, unlike the block, does not depend on
the
> exact choice of commas, so it also seems less likely that it matters
> very much to the temperament.
>
> If we pick 2401/2400, 4375/4374 and 65625/65536, then the angle
> between 2401/2400 and 4375/4374 is 88.56 degrees, between 4375/4374
> and 65625/65536 is 90 degrees, and between 2401/2400 and 65625/65536
> is 90 degrees. The block, for 171-et, is almost rectangular.

Interesting. Does this also tie into orthogonal projection, where you
make the comma(s) disappear? (I need to review that anyway)

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul_hjelmstad@a...> wrote:

> > If we pick 2401/2400, 4375/4374 and 65625/65536, then the angle
> > between 2401/2400 and 4375/4374 is 88.56 degrees, between 4375/4374
> > and 65625/65536 is 90 degrees, and between 2401/2400 and 65625/65536
> > is 90 degrees. The block, for 171-et, is almost rectangular.
>
> Interesting. Does this also tie into orthogonal projection, where you
> make the comma(s) disappear? (I need to review that anyway)

It means that 65625/65536 is orthogonal to the plane of note classes
corresponding to the commas of ennealimmal. In the 7-limit, there's
always some interval class which does this, which it might be
interesting to explore. There is a unique ennealimmal pure-octaves
tuning which has the effect that it leaves 65625/65536 fixed and
collapses 2401/2400 and 4375/4374 to the unison. It seems to be a good
choice, too. In general one could define such a tuning for 7-limit
temperaments using the observation on the unique orthogonal note class.

🔗Gene Ward Smith <gwsmith@svpal.org>

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

> There is a unique ennealimmal pure-octaves
> tuning which has the effect that it leaves 65625/65536 fixed and
> collapses 2401/2400 and 4375/4374 to the unison. It seems to be a good
> choice, too. In general one could define such a tuning for 7-limit
> temperaments using the observation on the unique orthogonal note >
class.

The reason it's a good choice turns out to be very simple: it's the
least-squares tuning. This is actually pretty cool; we immediately get
weighted least squares for other possible lattices, including higher
prime limits.

🔗Paul G Hjelmstad <paul_hjelmstad@allianzlife.com>

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul_hjelmstad@a...> wrote:

> > > An example would be the Pontiac temperament, where the TM basis is
> > > {2401/2400, 65625/65536}.

> Would Pontiac be 53&118? Or 171&? (What combinations would work.)

As a 7-limit temperament, either. To extend it to the 11-limit, both
will work, but give different temperaments.

Standard vals giving pontiac are numerous; the list starts out 53, 65,
118, 171, 224, 277, 289 ... . A copop generator for pontiac is
1031/1763, from whose denominator I cooked up the name. This doesn't
give a standard val, but rather <1763 2794 4093 4949|, which may be
preferable. I'm pretty sure we could tune up all maqam music with it. :)

It
> would be great to get a good database of 7-limit temperaments in the
> Database section or Files section. I am happy to see Yahoo has a more
> advanced method of searching now though!

Yes, the searches are working better now. I think the Monzopedia has
an entry on pontiac as a part of the schismatic family.

🔗Paul G Hjelmstad <paul_hjelmstad@allianzlife.com>

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <paul_hjelmstad@a...> wrote:
>
> > > > An example would be the Pontiac temperament, where the TM
basis is
> > > > {2401/2400, 65625/65536}.

Stupid Octave. It can't really do anything I cannot do in Excel. I
was hoping for better. Maybe its just me. I thought if I tried 53&118
etc I would get meaningful orthogonal commas but I don't. Also, going
the other way is hopeless - unless you have three commas, and I can
do that in Excel.

> > Would Pontiac be 53&118? Or 171&? (What combinations would work.)
>
> As a 7-limit temperament, either. To extend it to the 11-limit, both
> will work, but give different temperaments.

> Standard vals giving pontiac are numerous; the list starts out 53,
65,
> 118, 171, 224, 277, 289 ... . A copop generator for pontiac is
> 1031/1763, from whose denominator I cooked up the name. This doesn't
> give a standard val, but rather <1763 2794 4093 4949|, which may be
> preferable. I'm pretty sure we could tune up all maqam music with
it. :)

That's Arabic music, right? Isn't it close to 53-et?
>
> It
> > would be great to get a good database of 7-limit temperaments in
the
> > Database section or Files section. I am happy to see Yahoo has a
more
> > advanced method of searching now though!
>
> Yes, the searches are working better now. I think the Monzopedia has
> an entry on pontiac as a part of the schismatic family.

I can usually find any post I want within a minute!

Now if only the font would return back to normal.

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul_hjelmstad@a...> wrote:

> Stupid Octave. It can't really do anything I cannot do in Excel.

Well, I use Maple, Paul uses Matlab, Graham uses Python, and I think
Carl mostly Mathematica.

> I'm pretty sure we could tune up all maqam music with
> it. :)

> That's Arabic music, right? Isn't it close to 53-et?

Over on the main tuning list, Ozan Yarman is telling use 53 isn't good
enough; in particular it doesn't have a meantone fifth. On the other
hand, three chains of 53, 159edo, may be just the thing.

🔗Ozan Yarman <ozanyarman@superonline.com>

I may be able to cross-check if you will be so kind as to provide the pitches in scala format.
----- Original Message -----
From: Gene Ward Smith
To: tuning-math@yahoogroups.com
Sent: 04 Ekim 2005 Salı 23:49
Subject: [tuning-math] Re: Is it possible

SNIP

This doesn't give a standard val, but rather <1763 2794 4093 4949|, which may be preferable. I'm pretty sure we could tune up all maqam music with it. :)

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Paul G Hjelmstad <paul_hjelmstad@allianzlife.com>

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <paul_hjelmstad@a...> wrote:
>
> > Stupid Octave. It can't really do anything I cannot do in Excel.
>
> Well, I use Maple, Paul uses Matlab, Graham uses Python, and I think
> Carl mostly Mathematica.

Well, Octave is free. I guess so is Python. I'll load that up. Can it
find two commas from two temperaments in the 7-limit using a kernel
function (Graham?)

Thanks

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul_hjelmstad@a...> wrote:

> Well, Octave is free. I guess so is Python. I'll load that up. Can it
> find two commas from two temperaments in the 7-limit using a kernel
> function (Graham?)

Pari is free. I only use it for things Maple doesn't handle well, or
at all, but it could be very useful for other things.

http://pari.math.u-bordeaux.fr/

🔗Graham Breed <gbreed@gmail.com>

Paul G Hjelmstad wrote:

> Well, Octave is free. I guess so is Python. I'll load that up. Can it > find two commas from two temperaments in the 7-limit using a kernel > function (Graham?)

What's a kernel function? Python's Turing complete, so you can do anything. But don't expect to find libraries that do it for you.

My temper library can do the two commas from two temperaments. Erm, hang on

Python 2.3.5 (#1, Mar 20 2005, 20:38:20)
[GCC 3.3 20030304 (Apple Computer, Inc. build 1809)] on darwin
Type "help", "copyright", "credits" or "license" for more information.
>>> import temper
>>> mean = temper.Temperament(12,19,temper.limit7)
>>> map(temper.getRatio, mean.getUnisonVectors())
[(81, 80), (126, 125)]
>>>

It doesn't give good results for more complicated temperaments, though

>>> mystery = temper.Temperament(29,58,temper.limit15)
>>> mystery.getUnisonVectors()
[[46, -29, 0, 0, 0, 0], [-14, 0, -29, 29, 0, 0], [33, 0, 29, 0, -29, 0], [7, 0, 0, 0, 29, -29]]
>>>

That isn't the correct kernel. I've never found any libraries that help much. Numeric (which has a new name now) can do matrices, but only for floating point numbers and so it doesn't have the magic integer "solve" functions Gene uses. I wrote my own wedge products. You can also look at ScientificPython and the GNU Scientific Library. And there's an interface to Matlab, but then it stops being free.

Graham

🔗Graham Breed <gbreed@gmail.com>

Gene Ward Smith wrote:
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <paul_hjelmstad@a...> wrote:
> > >>>>An example would be the Pontiac temperament, where the TM basis is
>>>>{2401/2400, 65625/65536}.
> >>Would Pontiac be 53&118? Or 171&? (What combinations would work.) > > As a 7-limit temperament, either. To extend it to the 11-limit, both
> will work, but give different temperaments.

They're not the same. I get {4375:4374, 32805:32768} for h53&h118 (that is, best 7-limit mappings of 53-equal and 118-equal, which are both 7-limit consistent). 65625:65536 is a comma for this temperament.

> Standard vals giving pontiac are numerous; the list starts out 53, 65,
> 118, 171, 224, 277, 289 ... . A copop generator for pontiac is
> 1031/1763, from whose denominator I cooked up the name. This doesn't
> give a standard val, but rather <1763 2794 4093 4949|, which may be
> preferable. I'm pretty sure we could tune up all maqam music with it. :)

Your "standard val" must be nearest-primes for this to work. 65-equal has a better 7-limit mapping that belongs to a different family. 171 and 277 are also inconsistent in the 7-limit, but the nearest-primes mapping is also the best in the 7-limit, at least the way I calculate it.

Only 171 tempers out 2401:2400 from that list, with mapping [171, 271, 397, 480].

Your mapping of 1763 is optimal in the 7-limit, and so standard in that sense. Also in the 9-limit.

> Yes, the searches are working better now. I think the Monzopedia has
> an entry on pontiac as a part of the schismatic family.

If it's schismatic, it ain't {2401/2400, 65625/65536}.

Graham

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@g...> wrote:
> Gene Ward Smith wrote:

> They're not the same. I get {4375:4374, 32805:32768} for h53&h118
(that
> is, best 7-limit mappings of 53-equal and 118-equal, which are both
> 7-limit consistent). 65625:65536 is a comma for this temperament.

Sorry, I don't know where I got that other answer. 54&118, and
118&171, both give the {4375/4374,32805/32768} temperament, "pontiac".
The {2401/2400, 65635/65536} temperament is tertiaseptal, another
microtemperament. It's supported by neither 53 or 118, is not a
version of schismatic, and might be called 140&171.

> Your "standard val" must be nearest-primes for this to work.

Much to Paul's distress. What would you suggest using instead?

65-equal
> has a better 7-limit mapping that belongs to a different family. 171
> and 277 are also inconsistent in the 7-limit, but the nearest-primes
> mapping is also the best in the 7-limit, at least the way I
calculate it.

Presumably you don't mean to say 171 is 7-inconsistent! It becomes
inconsistent in the 17-limit.

> Only 171 tempers out 2401:2400 from that list, with mapping [171, 271,
> 397, 480].
>
> Your mapping of 1763 is optimal in the 7-limit, and so standard in that
> sense. Also in the 9-limit.

It's p-optimal for both 7 and 9, which is why I picked on it. A 171
MOS has steps of size 10 and 11, which of course is not a heck of a
lot different than 171-et.

🔗Graham Breed <gbreed@gmail.com>

Gene Ward Smith wrote:

>>Your "standard val" must be nearest-primes for this to work. > > Much to Paul's distress. What would you suggest using instead?

I suggest no standard val is defined where there's any chance of ambiguity. With odd limits, that means the temperament should be consistent to have a standard val. As odd limits are out of fashion now, you'll have to make sure any reasonable optimum gives the same result as the nearest-primes.

A document- or thread-specific standard is find, so long as you define it. Optimal optimized mappings tend to agree more with each other than with the nearest-primes-with-pure-octaves rule.

> Presumably you don't mean to say 171 is 7-inconsistent! It becomes
> inconsistent in the 17-limit.

Oh no, I meant what I said, but I was mistaken in my beliefs ;) All down to misreading the decimal point. Hence proving the general rule that an internet post correcting a mistake will itself contain a mistake.

Graham

🔗Paul G Hjelmstad <paul_hjelmstad@allianzlife.com>

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> --- In tuning-math@yahoogroups.com, Graham Breed <gbreed@g...>
wrote:
> > Gene Ward Smith wrote:
>
> > They're not the same. I get {4375:4374, 32805:32768} for h53&h118
> (that
> > is, best 7-limit mappings of 53-equal and 118-equal, which are
both
> > 7-limit consistent). 65625:65536 is a comma for this temperament.
>
> Sorry, I don't know where I got that other answer. 54&118, and
> 118&171, both give the {4375/4374,32805/32768}
temperament, "pontiac".
> The {2401/2400, 65635/65536} temperament is tertiaseptal, another
> microtemperament. It's supported by neither 53 or 118, is not a
> version of schismatic, and might be called 140&171.
>
Leaving aside Hermite reduction, LLL reduction and TM for now, could
someone show me how to calculate two "raw" commas from two
temperaments such as h53, and h118 above? (what's the h for?)

Thanks Paul Hj

🔗monz <monz@tonalsoft.com>

Hi Gene and Graham,

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@g...> wrote:
>
> Gene Ward Smith wrote:

> > <snip> ... I think the Monzopedia has an entry on pontiac
> > as a part of the schismatic family.
>
> If it's schismatic, it ain't {2401/2400, 65625/65536}.

I looked and looked and couldn't find it, and then finally
did a Google search ... the only place in the Tonalsoft
Encylopedia which mentions pontiac is this:

http://tonalsoft.com/enc/h/hahn-metric.aspx

Gene, could you please take a look at the "schismic"
temperament family page and see if pontiac should be
added to that? Thanks.

http://tonalsoft.com/enc/s/schismic.aspx

(Note that i'm calling it "schismic" and not "schismatic",
based on protests from others ... i don't remember who now,
but it's a done deal for the Encyclopedia.)

-monz
http://tonalsoft.com
Tonescape microtonal music software

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning-math@yahoogroups.com, "monz" <monz@t...> wrote:

> (Note that i'm calling it "schismic" and not "schismatic",
> based on protests from others ... i don't remember who now,
> but it's a done deal for the Encyclopedia.)

Wow, there seem to be strong feelings on this issue. I created a
Wikipedia article on "schismic temperament", and someone moved it to
"schismatic temperament", noting that this is the traditional name,
and that "schsmic" is an invention of Graham Breed.

Anyone care to give us the real dope?