Tuning-Math Digests messages 11005 - 11029

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Message: 11005

Date: Wed, 02 Jun 2004 00:50:09

Subject: Re: Family commas

From: Carl Lumma

>> Also, when you say "given that all the previous commas are
>> fixed", does this imply any relation to TM-reduction?
>
>It's a different reduction--sequential reduction or something.
>
>> The words that are coming to mind are, temperament n should
>> be considered an extension of temperament m if m's TM-reduced
>> basis is a subset of n's.  Does that make any sense?
>
>This won't work. It doesn't even work if you replace TM reduction with
>sequential reduction, though that is better for this. The nexial
>approach does it, however.

Noted.

>> >> >> This family stuff looks awesome.  I wish I understood the
>> >> >> half of it.  I'm surprised you're using generator sizes.
>> >> >> How do you standardize the generator representation?  Forgive
>> >> >> me if this is old stuff, I haven't kept up.
>> >> >
>> >> >It's just the TOP tuning for the generators.
>> >> 
>> >> How do you get a unique set of generators out of the TOP
>> >> tuning?
>> >
>> >One way is to apply the TOP tuning to a rational number generator
>> >which works as a reduced generator. For instance, with meantone that
>> >would be 4/3, with miracle 15/14 or 16/15,
>> 
>> This is apparently not giving unique generators...
>
>Sure it does; miracle(15/14) = [0, 1] = miracle(16/15)

Right you are.

-Carl



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Message: 11008

Date: Wed, 02 Jun 2004 00:06:58

Subject: Re: Family commas

From: Carl Lumma

> ... defining a linear temperament in terms of a
> sequence of commas, each at a succesively higher prime limit,
> and each with a minimal Tenney height given that all the
> previous commas are fixed. This sort of whatzit reduction,
> for meantone, would say meantone is the 81/80-temperament,
> dominant sevenths the [81/80, 36/35] temperament, septimal
> meantone the [81/80, 126/125]-temperament, flattone the
> [81/80, 525/512]-temperament. Then 11-limit meantone is
> the [81/80, 126/125, 385/384]-temperament and huygens the
> [81/80, 126/125, 99/98]-temperament. And so forth.

This immediately appeals to me more than generators.

I wonder how it relates to Paul's tratios.  They involve
the LCM... I wonder what good that is.

Also, when you say "given that all the previous commas are
fixed", does this imply any relation to TM-reduction?

The words that are coming to mind are, temperament n should
be considered an extension of temperament m if m's TM-reduced
basis is a subset of n's.  Does that make any sense?

> It should be noted that while this keeps track of the familial
> relationships, we don't necessarily get corresponding
> generators in these family trees, nor do we necessarily get
> rid of contorsion. 7-limit ennealimmal is the [ennealimma,
> breedsma]- temperament, but the wedge product of this has a
> common factor of 4.

Rats.

-Carl


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Message: 11009

Date: Wed, 02 Jun 2004 00:16:28

Subject: Re: The hanson family

From: Carl Lumma

>> >> This family stuff looks awesome.  I wish I understood the
>> >> half of it.  I'm surprised you're using generator sizes.
>> >> How do you standardize the generator representation?  Forgive
>> >> me if this is old stuff, I haven't kept up.
>> >
>> >It's just the TOP tuning for the generators.
>> 
>> How do you get a unique set of generators out of the TOP
>> tuning?
>
>One way is to apply the TOP tuning to a rational number generator
>which works as a reduced generator. For instance, with meantone that
>would be 4/3, with miracle 15/14 or 16/15,

This is apparently not giving unique generators...

-Carl



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