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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