more questions about adjoints and mappings

Here's a simple example: Ellis's Duodene

kernel

2 3 5 unison vectors ~cents

[ 1 0 0 ] = 2:1 1200
[ -4 4 -1 ] = 81:80 21.5062896
[ 7 0 -3 ] = 128:125 41.05885841

adjoint

[ -12 0 0 ]
[ -19 -3 1 ]
[ -28 0 4 ]

determinant = | -12 |

"mapping" of UVs

[ 1 0 0 ]
[ 0 1 0 ]
[ 0 0 1 ]

So here, I can see that the h12 mapping does not
temper out the 2:1 ... and I still don't understand
what that means. Is it simply because any "8ve"-based
ET must include 2:1 by definition?

I can also see that the third column of the adjoint
specifies some kind of meantone, which tempers out
the 2:1 and the 81:80, but not the diesis 128:125.
Is there a way to tell what flavor of meantone it is?

And the middle column of the adjoint specifies some
temperament which does not temper out the 81:80.
But can someone explain what kind of tuning this is?

-monz

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monz wrote:

> Here's a simple example: Ellis's Duodene
>
>
> kernel
>
> 2 3 5 unison vectors ~cents
>
> [ 1 0 0 ] = 2:1 1200
> [ -4 4 -1 ] = 81:80 21.5062896
> [ 7 0 -3 ] = 128:125 41.05885841
>
>
> adjoint
>
> [ -12 0 0 ]
> [ -19 -3 1 ]
> [ -28 0 4 ]
>
> determinant = | -12 |
>
>
> "mapping" of UVs
>
> [ 1 0 0 ]
> [ 0 1 0 ]
> [ 0 0 1 ]
>
>

> So here, I can see that the h12 mapping does not
> temper out the 2:1 ... and I still don't understand
> what that means. Is it simply because any "8ve"-based
> ET must include 2:1 by definition?

Tempering out the 2:1 is the same as enforcing octave equivalence. ETs
aren't octave-equivalent. In 12-equal, a unison is 0 steps, an octave is
12 steps and two octaves are 24 steps. These certainly aren't the same.

> I can also see that the third column of the adjoint
> specifies some kind of meantone, which tempers out
> the 2:1 and the 81:80, but not the diesis 128:125.
> Is there a way to tell what flavor of meantone it is?

No, because all you asked is for an octave-equivalent temperament that
tempers out the 81:80. All meantones do that. You could set the
enharmonic diesis to be just, following the method I outline in
<http://x31eq.com/lintemp.htm> and that happens to give
quarter-comma meantone.

> And the middle column of the adjoint specifies some
> temperament which does not temper out the 81:80.
> But can someone explain what kind of tuning this is?

It's an octave-equivalent tuning that tempers out the enharmonic diesis.
That's the difference between three 5:4 major thirds and an octave. So,
if you temper it out, the octave must divide into three equal parts, each
of which can be called a 5:4. The column tells you that by having a GCD
of 3, and a zeros in the 2:1 and 5:1 rows. The 3:1 column doesn't have a
zero in it, so you can choose whatever size for a fifth you like. It's
independent of the major third and octave.

Graham

Thanks, Graham, this helps a lot!

-monz

> From: <graham@microtonal.co.uk>
> To: <tuning-math@yahoogroups.com>
> Sent: Sunday, January 20, 2002 3:28 AM
> Subject: [tuning-math] Re: more questions about adjoints and mappings
>
>
> monz wrote:
>
> > Here's a simple example: Ellis's Duodene
> >
> >
> > kernel
> >
> > 2 3 5 unison vectors ~cents
> >
> > [ 1 0 0 ] = 2:1 1200
> > [ -4 4 -1 ] = 81:80 21.5062896
> > [ 7 0 -3 ] = 128:125 41.05885841
>
> ...
>
> Tempering out the 2:1 is the same as enforcing octave equivalence
> <etc.>

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