Eigentuning

Here is another excuse to stick "eigen" in front of a word.

1/4-comma meantone, as we know, leaves 2 and 5 fixed. If we take
the 4x4 matrix for [2,5,81/80,126/125] and invert it, we find that we
can project down to an "eigentuning", based just on 2 and 5, by means of
sending a rational q to (a, b) where a is mapped by <1 1 0 -3| and b
is mapped by <0 1/4 1 5/2|. This mapping is therefore one way to do a
tuning map for 1/4-comma, giving a slightly different point of view.
That is, instead of saying 5^(1/4) is a generator, or computing a
projection map with 5 as an eigenmonzo, we simply base it all on 2 and
5. When the intervals involved are more complicated (say, |0 -11 1 15>
for 7-limit rms meantone) then it looks a bit messy but works in the
same way.

At 02:01 PM 5/31/2006, you wrote:
>Here is another excuse to stick "eigen" in front of a word.
>
>1/4-comma meantone, as we know, leaves 2 and 5 fixed. If we take
>the 4x4 matrix for [2,5,81/80,126/125] and invert it, we find that we
>can project down to an "eigentuning", based just on 2 and 5, by means of
>sending a rational q to (a, b) where a is mapped by <1 1 0 -3| and b
>is mapped by <0 1/4 1 5/2|. This mapping is therefore one way to do a
>tuning map for 1/4-comma, giving a slightly different point of view.
>That is, instead of saying 5^(1/4) is a generator, or computing a
>projection map with 5 as an eigenmonzo, we simply base it all on 2 and
>5. When the intervals involved are more complicated (say, |0 -11 1 15>
>for 7-limit rms meantone) then it looks a bit messy but works in the
>same way.

Of use to aural tuners, and perhaps a way to canonical maps?

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:

> Of use to aural tuners, and perhaps a way to canonical maps?

Canonical tuning maps is actually what I was thinking of. If you
represent the abstract temperament group via an interior product
mapping, many tuning maps lead to the same result.

>> Of use to aural tuners, and perhaps a way to canonical maps?
>
>Canonical tuning maps is actually what I was thinking of. If you
>represent the abstract temperament group via an interior product
>mapping, many tuning maps lead to the same result.

This would be a real boon as far as I'm concerned. Maps are
the most accessible representations of temperaments to the
masses, I think. If they can be brought in 1:1 correspondence
with temperaments somehow, it would be fabulous. And if one
can get a bearing plan out of them, so much the better.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:

> This would be a real boon as far as I'm concerned. Maps are
> the most accessible representations of temperaments to the
> masses, I think. If they can be brought in 1:1 correspondence
> with temperaments somehow, it would be fabulous. And if one
> can get a bearing plan out of them, so much the better.

I'm afraid my "canonical mapping" idea is abstract and pretty useless
for the purpose of accessibility. What it's good for is that it does
not depend in any way on a choice of generators, and does not depend
on rank. It represents the temperament group in a more intrinsic way.
Alternatives are Hermite generators and (for rank two) period and
generator, with minimal generators.