Tuning-Math Digests messages 8886 - 8910

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Message: 8886

Date: Tue, 30 Dec 2003 08:56:52

Subject: Re: Meantone reduced blocks

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> It would be nice to classify 12-note, 5-limit Fokker blocks at least
> up to meantone reduction. While pondering that, I thought I'd see 
how
> an example which does not reduce to Meantone[12] worked out.
> 
> The "thirds" scale, the genus derived from 6/5 and 5/4, can be
> analyzed as a Fokker block using the method I've given as 
> 
> thirds[i] = (25/24)^i (128/125)^round(i/3) (648/625)^round(i/4+1/8)
> 
> In meantone, 25/24 maps to 7, and 128/125 and 648/625 to -12.
>
> The
> meatone reduction therefore is
> 
> 7i - 12(round(i/3) + round(i/4+1/8))

Please express this meantone scale in conventional letter-name-and-
accidental notation.


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Message: 8887

Date: Tue, 30 Dec 2003 16:54:18

Subject: Re: 5-limit, 12-note Fokker blocks (attn Manuel)

From: Manuel Op de Coul

I've paid attention. It will be a useful addition, so I've put it
on the todo list.

Manuel




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Message: 8889

Date: Wed, 31 Dec 2003 22:52:19

Subject: Re: The Two Diadie Scales

From: Carl Lumma

>The two scales using the DIAschisma and the DIEsis of 128/125 are 
>both known, and this seems like a Carl Lumma speciality. They don't 
>reduce to Meantone[12], but 22-et, pajara or orwell seem more to the 
>point. Reduction by 22-et or pajara leads to Pajara[12], but 
>reduction by orwell leads to two interesting new scales. Or at least 
>one is new, reducing the first diadie scale gives us something quite 
>close to lumma.scl, which Carl presented back in 1999.

I did a non-thorough by-hand search for 12-tone 5- and 7-limit
'Fokker blocks' (before I knew the term, and before the subject had
been explored by the list -- I certainly wasn't checking for
epimorphism or monotonicity).  Some of this was done before I joined
the list, on paper with the rectangular lattices I'd learned about
from Doty's JI Primer.

-Carl



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Message: 8895

Date: Thu, 01 Jan 2004 20:54:55

Subject: Re: The Two Diadie Scales

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> Let me start out by saying that 81/80 with either the schisma or 
the 
> pythagorean commas (or those two taken together) give us the 12-
note 
> Pythagorean scale, and that this completes the classification for 
> scales using 81/80, unless you want to go past 0.75 in epimericity.
> 
> The two scales using the DIAschisma and the DIEsis of 128/125 are 
> both known, and this seems like a Carl Lumma speciality. They don't 
> reduce to Meantone[12], but 22-et, pajara or orwell seem more to 
the 
> point. Reduction by 22-et or pajara leads to Pajara[12],

Gene -- you keep saying Pajara but don't you mean Diaschismic?


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Message: 8898

Date: Thu, 01 Jan 2004 21:20:33

Subject: Re: The Two Diadie Scales

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
> wrote:
> 
> > > The two scales using the DIAschisma and the DIEsis of 128/125 
are 
> > > both known, and this seems like a Carl Lumma speciality. They 
> don't 
> > > reduce to Meantone[12], but 22-et, pajara or orwell seem more 
to 
> > the 
> > > point. Reduction by 22-et or pajara leads to Pajara[12],
> > 
> > Gene -- you keep saying Pajara but don't you mean Diaschismic?
> 
> I'm assuming that in 22-equal, it is more correctly called Pajara.

No, Pajara is simply the 7-limit extension of Diaschsimic that you do 
get in 22-equal (and pretty much in no other ET):

GX Networks *

As long as you're talking 5-limit though, there's no reason to bring 
Pajara into it.


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