Is this a valid mapping

Take the smallest interval representation of 3,5,7
(4/3, 5/4, 8/7). This is 2^(5/12), 2^(1/3), 2^(1/6)
This is (5,4,2) steps in 12t-ET.

Add 4 steps to each (Multiply by 2^(1/4)) and obtain
(9,8,6). Reduce to "absolute" interval size: (3,4,6).

This is 2^(1/4), 2^(1/3), 2^(1/2), which is a decomposition
of 12 based on prime factors, 2 X 2 X 3 (primes and
a power of a prime),

Would this possibly have any significance for mapping
3,5, and 7 to 12-tET?

(Actually, it gets worse... but I'll save that for now...)

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul_hjelmstad@...> wrote:
>
> Take the smallest interval representation of 3,5,7
> (4/3, 5/4, 8/7). This is 2^(5/12), 2^(1/3), 2^(1/6)
> This is (5,4,2) steps in 12t-ET.
>
> Add 4 steps to each (Multiply by 2^(1/4)) and obtain
> (9,8,6). Reduce to "absolute" interval size: (3,4,6).

Of course I meant Multiply by 2^(1/3)

>
> This is 2^(1/4), 2^(1/3), 2^(1/2), which is a decomposition
> of 12 based on prime factors, 2 X 2 X 3 (primes and
> a power of a prime),
>
> Would this possibly have any significance for mapping
> 3,5, and 7 to 12-tET?
>
> (Actually, it gets worse... but I'll save that for now...)
>

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

> This is 2^(1/4), 2^(1/3), 2^(1/2), which is a decomposition
> of 12 based on prime factors, 2 X 2 X 3 (primes and
> a power of a prime),
>
> Would this possibly have any significance for mapping
> 3,5, and 7 to 12-tET?

This is a basis change. If we add 2^1 to the above list, we have that
[2^(1/4), 2^(1/3), 2^(1/2), 2] is a 12-et approximation of
[6/5, 5/4, 7/5, 2]. If we take the corresponding monzos, we get a
4x4 unimodular matrix
M = [<1 1 -1 0|, <-2 0 1 0|, <0 0 -1 1|, <1 0 0 0|].

Now for any 7-limit monzo u, uM^(-1) gives its coordinates in the new
system. Similarly, for any 7-limit val v, Mv are its coordinates in
the new system. If we do this for <12 19 28 34|, we get <3 4 6 12].
Each of these coordinates is a divisor of 12, although the gcd is 1
and we don't have contorsion.

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

> If we take the corresponding monzos, we get a
> 4x4 unimodular matrix
> M = [<1 1 -1 0|, <-2 0 1 0|, <0 0 -1 1|, <1 0 0 0|].

Should be
M = [|1 1 -1 0>, |-2 0 1 0>, |0 0 -1 1>, |1 0 0 0>]
of course.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <paul_hjelmstad@> wrote:
>
> > This is 2^(1/4), 2^(1/3), 2^(1/2), which is a decomposition
> > of 12 based on prime factors, 2 X 2 X 3 (primes and
> > a power of a prime),
> >
> > Would this possibly have any significance for mapping
> > 3,5, and 7 to 12-tET?
>
> This is a basis change. If we add 2^1 to the above list, we have
that
> [2^(1/4), 2^(1/3), 2^(1/2), 2] is a 12-et approximation of
> [6/5, 5/4, 7/5, 2]. If we take the corresponding monzos, we get a
> 4x4 unimodular matrix
> M = [<1 1 -1 0|, <-2 0 1 0|, <0 0 -1 1|, <1 0 0 0|].
>
> Now for any 7-limit monzo u, uM^(-1) gives its coordinates in the
new
> system. Similarly, for any 7-limit val v, Mv are its coordinates in
> the new system. If we do this for <12 19 28 34|, we get <3 4 6 12].
> Each of these coordinates is a divisor of 12, although the gcd is 1
> and we don't have contorsion.
>
You mean the gcd of the whole val is 1, right? What can <3,4,6,12| be
used for?

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul_hjelmstad@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
> <genewardsmith@> wrote:
> >
> > --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> > <paul_hjelmstad@> wrote:
> >
> > > This is 2^(1/4), 2^(1/3), 2^(1/2), which is a decomposition
> > > of 12 based on prime factors, 2 X 2 X 3 (primes and
> > > a power of a prime),
> > >
> > > Would this possibly have any significance for mapping
> > > 3,5, and 7 to 12-tET?
> >
> > This is a basis change. If we add 2^1 to the above list, we have
> that
> > [2^(1/4), 2^(1/3), 2^(1/2), 2] is a 12-et approximation of
> > [6/5, 5/4, 7/5, 2]. If we take the corresponding monzos, we get a
> > 4x4 unimodular matrix
> > M = [<1 1 -1 0|, <-2 0 1 0|, <0 0 -1 1|, <1 0 0 0|].
> >
> > Now for any 7-limit monzo u, uM^(-1) gives its coordinates in the
> new
> > system. Similarly, for any 7-limit val v, Mv are its coordinates
in
> > the new system. If we do this for <12 19 28 34|, we get <3 4 6
12].
> > Each of these coordinates is a divisor of 12, although the gcd is
1
> > and we don't have contorsion.
> >
> You mean the gcd of the whole val is 1, right? What can <3,4,6,12|
be
> used for?
>
Such a "basic" concept - but it's always thrown me for a loop, since
Linear in college. I see that u=|1,1,1,1> transforms to |1,3,3,4>
Disregard my question...

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul_hjelmstad@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <paul_hjelmstad@> wrote:
> >
> > --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
> > <genewardsmith@> wrote:
> > >
> > > --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> > > <paul_hjelmstad@> wrote:
> > >
> > > > This is 2^(1/4), 2^(1/3), 2^(1/2), which is a decomposition
> > > > of 12 based on prime factors, 2 X 2 X 3 (primes and
> > > > a power of a prime),
> > > >
> > > > Would this possibly have any significance for mapping
> > > > 3,5, and 7 to 12-tET?
> > >
> > > This is a basis change. If we add 2^1 to the above list, we
have
> > that
> > > [2^(1/4), 2^(1/3), 2^(1/2), 2] is a 12-et approximation of
> > > [6/5, 5/4, 7/5, 2]. If we take the corresponding monzos, we get
a
> > > 4x4 unimodular matrix
> > > M = [<1 1 -1 0|, <-2 0 1 0|, <0 0 -1 1|, <1 0 0 0|].
> > >
> > > Now for any 7-limit monzo u, uM^(-1) gives its coordinates in
the
> > new
> > > system. Similarly, for any 7-limit val v, Mv are its
coordinates
> in
> > > the new system. If we do this for <12 19 28 34|, we get <3 4 6
> 12].
> > > Each of these coordinates is a divisor of 12, although the gcd
is
> 1
> > > and we don't have contorsion.
> > >
> > You mean the gcd of the whole val is 1, right? What can
<3,4,6,12|
> be
> > used for?
> >
> Such a "basic" concept - but it's always thrown me for a loop, since
> Linear in college. I see that u=|1,1,1,1> transforms to |1,3,3,4>
> Disregard my question...
>
Had my matrix transposed, its |1,3,1,6>. But getting back to my
original post, do you think there is anything worthwhile in this
change of basis? Obviously, I am not mapping 3,5,7 directly, but
using 6/5, 5/4, 7/5, and 2/1. Perhaps that's the best one can do.
Have you done any lattices with this basis?

And now for a real stretch - could interval vector counts in 12-et
(1,1,6,12,28,35,35,35,28,12,6,1,1) have anything to do with such a
structure? Of course, you could just as well be counting necklace
beads with Polya's methods, however, interval vector counts are based
on intervals - which is a tuning consideration...