494

This seems overdue for a deeper look.

The MT reduced basis for the 11-limit h494 is
<3025/3024, 9801/9800, 131072/130977, 234375/234256>.
That immediately suggests the planar temperament defined by
<3025/3024, 9801/9800> as something of interest. The (planar,
not linear!) wedgie is 3025/3024^9801/9800 = h72^h270^h494 =
[2,8,-6,-14,10,-2,-2,5,14,-25], and the Hermite reduced mapping
is

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

This is an excellent microtemperament, but not the one Graham used; which is 3025/3024^825000/823543 = h31^h41^h494 =
[13,-8,-9,-5,22,-17,-10,31,-26,5].
This is both more complex and less accurate than the previous, but it has a very nice feature--it is the detempering of Miralce, which is
h31^h41, by h494. This suggests we look at this proceedure in general--take a linear temperament, and detemper it by wedging with a suitable JI (one which does not cover the orginal temperament.) Then relate it to the original temperament by keeping the two columns of that temperament as the first two columns of the mapping matrix for the new temperament. I'm off to take a look at this.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:

>This suggests we look at this proceedure in general--take a linear >temperament, and detemper it by wedging with a suitable JI (one >which does not cover the orginal temperament.) Then relate it to the >original temperament by keeping the two columns of that temperament >as the first two columns of the mapping matrix for the new >temperament. I'm off to take a look at this.

If we detemper miracle by h270, we get

[1 0 0]
[1 6 0]
[3 -7 4]
[3 -2 2]
[2 15 -3]

This actually doesn't go very well with 270, but it is well covered by
954, which has its generators as 93.005/954 (the secor) and
1.003/954. The temperament is 2401/2400^3025/3024, and is therefore a
detempering of hemiennealimmal, which is 2401/2400^3025/3024^9801/9800. Since it both covers hemiennealimmal and has its own version of miracle, it might be a good "universal" temperament, and in particular a way of notating miracle in hemiennealimmal and vice-versa.

In-Reply-To: <aff3jh+vet5@eGroups.com>
genewardsmith wrote:

> If we detemper miracle by h270, we get
>
> [1 0 0]
> [1 6 0]
> [3 -7 4]
> [3 -2 2]
> [2 15 -3]
>
> This actually doesn't go very well with 270, but it is well covered by
> 954, which has its generators as 93.005/954 (the secor) and
> 1.003/954. The temperament is 2401/2400^3025/3024, and is therefore a
> detempering of hemiennealimmal, which is 2401/2400^3025/3024^9801/9800.
> Since it both covers hemiennealimmal and has its own version of
> miracle, it might be a good "universal" temperament, and in particular
> a way of notating miracle in hemiennealimmal and vice-versa.

Now there's a funny thing. I was looking for the miracle-plus temperament
with 2401:2400^3025:3024 and got the mapping

[1 0 0]
[1 6 2]
[3 -7 -1]
[3 -2 0]
[2 15 4]

It looks like my generator's three times the size of yours. Does it still
work, or is this a torsion problem?

Graham

On Thu, 27 Jun 2002 15:17 +0100 (BST) graham@microtonal.co.uk writes:
> Now there's a funny thing. I was looking for the miracle-plus
> temperament
> with 2401:2400^3025:3024 and got the mapping
>
> [1 0 0]
> [1 6 2]
> [3 -7 -1]
> [3 -2 0]
> [2 15 4]
>
> It looks like my generator's three times the size of yours. Does it
> still
> work, or is this a torsion problem?

The two matricies are equivalent; three times the third column minus the
second column gives you my matrix.