new 1/6-comma meantone lattice

I've added a new lattice to my "Lattice Diagrams comparing
rational implications of various meantone chains" webpage:

http://www.ixpres.com/interval/monzo/meantone/lattices/lattices.htm

It's about 2/3 of the way down the page: a new lattice
showing a definition of 1/6-comma meantone within a
55-tone periodicity-block... just under the old 1/6-comma
lattice, below this text:

>> And here is a more accurate lattice of the above,
>> showing a closed 55-tone 1/6-comma meantone chain and
>> its implied pitches, all enclosed within a complete
>> periodicity-block defined by the two unison-vectors
>> 81:80 = [-4 4 -1] (the syntonic comma, the shorter
>> boundary extending from south-west to north-east on
>> this diagram) and [-51 19 9] (the long nearly vertical
>> boundary), portrayed here as the white area.
>>
>> For the bounding corners of the periodicity-block, I
>> arbitrarily chose the lattice coordinates [-7.5 -5]
>> for the north-west corner, [-11.5 -4] for north-east,
>> [11.5 4] for south-west, and [7.5 5] for south-east.
>> This produces a 55-tone system centered on n^0.
>>
>> The grey area represents the part of the JI lattice
>> outside the defined periodicity-block (and thus, with
>> each of those pitch-classes in its own periodicity-block),
>> and the lattice should be imagined as extending infinitely
>> in all four directions. The other periodicity-blocks,
>> all identical to this one, can be tiled against it to
>> cover the entire space.

love / peace / harmony ...

-monz
http://www.monz.org
"All roads lead to n^0"

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--- In tuning@y..., "monz" <joemonz@y...> wrote:
> I've added a new lattice to my "Lattice Diagrams comparing
> rational implications of various meantone chains" webpage:
>
> http://www.ixpres.com/interval/monzo/meantone/lattices/lattices.htm
>
>
> It's about 2/3 of the way down the page: a new lattice
> showing a definition of 1/6-comma meantone within a
> 55-tone periodicity-block... just under the old 1/6-comma
> lattice, below this text

[...]

If anyone is interested in why I _don't_ think these lattices depict
the "rational implications" of meantone tunings, you can visit the
discussion at

tuning-math@yahoogroups.com

which is having its most active month since it began earlier in the
year.

Monz,

It looks like your ideas are evolving, and continue
to be interesting.

Whether or not you ideas are "right" or "wrong",
I enjoy seeing them on (this) main tuning list,
in addition to seeing them on "tuning-math".

Thanks for sharing those ideas with a wide array
of folks. Something tells me that there are real
benefits to such wide exposure of your ideas,
which smaller "forums" cannot fully replace...

Sincerely, J Gill

--- In tuning@y..., "monz" <joemonz@y...> wrote:
> I've added a new lattice to my "Lattice Diagrams comparing
> rational implications of various meantone chains" webpage:
>
> http://www.ixpres.com/interval/monzo/meantone/lattices/lattices.htm
>
>
> It's about 2/3 of the way down the page: a new lattice
> showing a definition of 1/6-comma meantone within a
> 55-tone periodicity-block... just under the old 1/6-comma
> lattice, below this text:
>
>
> >> And here is a more accurate lattice of the above,
> >> showing a closed 55-tone 1/6-comma meantone chain and
> >> its implied pitches, all enclosed within a complete
> >> periodicity-block defined by the two unison-vectors
> >> 81:80 = [-4 4 -1] (the syntonic comma, the shorter
> >> boundary extending from south-west to north-east on
> >> this diagram) and [-51 19 9] (the long nearly vertical
> >> boundary), portrayed here as the white area.
> >>
> >> For the bounding corners of the periodicity-block, I
> >> arbitrarily chose the lattice coordinates [-7.5 -5]
> >> for the north-west corner, [-11.5 -4] for north-east,
> >> [11.5 4] for south-west, and [7.5 5] for south-east.
> >> This produces a 55-tone system centered on n^0.
> >>
> >> The grey area represents the part of the JI lattice
> >> outside the defined periodicity-block (and thus, with
> >> each of those pitch-classes in its own periodicity-block),
> >> and the lattice should be imagined as extending infinitely
> >> in all four directions. The other periodicity-blocks,
> >> all identical to this one, can be tiled against it to
> >> cover the entire space.
>
>
>
> love / peace / harmony ...
>
> -monz
> http://www.monz.org
> "All roads lead to n^0"
>
>
>
>
>
> _________________________________________________________
> Do You Yahoo!?
> Get your free @yahoo.com address at http://mail.yahoo.com

> From: paulerlich <paul@stretch-music.com>
> To: <tuning@yahoogroups.com>
> Sent: Thursday, December 27, 2001 1:57 PM
> Subject: [tuning] Re: new 1/6-comma meantone lattice
>
>
> --- In tuning@y..., "monz" <joemonz@y...> wrote:
> > I've added a new lattice to my "Lattice Diagrams comparing
> > rational implications of various meantone chains" webpage:
> >
> > http://www.ixpres.com/interval/monzo/meantone/lattices/lattices.htm
> >
> >
> > It's about 2/3 of the way down the page: a new lattice
> > showing a definition of 1/6-comma meantone within a
> > 55-tone periodicity-block... just under the old 1/6-comma
> > lattice, below this text
>
> [...]
>
> If anyone is interested in why I _don't_ think these lattices depict
> the "rational implications" of meantone tunings, you can visit the
> discussion at
>
> tuning-math@yahoogroups.com
>
> which is having its most active month since it began earlier in the
> year.

Paul, I explain there that the lattice can be tiled with
periodicity-blocks identical to this one. What's the problem?
Perhaps I need to emphasize the equivalence between PBs more?

-monz

_________________________________________________________
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Get your free @yahoo.com address at http://mail.yahoo.com

--- In tuning@y..., "unidala" <JGill99@i...> wrote:
> Monz,
>
> It looks like your ideas are evolving, and continue
> to be interesting.
>
> Whether or not you ideas are "right" or "wrong",
> I enjoy seeing them on (this) main tuning list,
> in addition to seeing them on "tuning-math".
>
> Thanks for sharing those ideas with a wide array
> of folks. Something tells me that there are real
> benefits to such wide exposure of your ideas,
> which smaller "forums" cannot fully replace...

You may not remember, until earlier this year there was no tuning-
math list, and all this kind of stuff was always posted here, by me
and everyone, but there were a few people complaining loudly
about "too much math" or "too off-topic". So, now we have the tuning-
math list for the first category, the metatuning list for the second
category, and these groups are unmoderated and anyone can join
instantly. I didn't want to have a separate tuning-math list, but
certain people really didn't want the stuff here, so I created one.
And I constantly post about it here so that anyone remotely
interested can go over there and join. Now you're accusing me
of "sectarianism" or some such . . . can you see the bind I'm in?

Please reply at metatuning@yahoogroups.com :)

--- In tuning@y..., "monz" <joemonz@y...> wrote:

> Paul, I explain there that the lattice can be tiled with
> periodicity-blocks identical to this one. What's the problem?
> Perhaps I need to emphasize the equivalence between PBs more?

That would be fine if you were talking about JI tunings that
approximated, but failed to function like, the meantones you mention.
But for the meantones themselves, your lattices fail to portray what
to me are musically paramount features. For example, all the fifths
in a meantone tuning are equally consonant, while you portray some of
them as 'direct connections' and others as 'distant relationships'.
This classification has no musical justification whatsoever, for the
meantone tunings themselves, IMHO.