lattices on "TM-reduced lattice basis" updated webpage

hello all,

i've had fun making some nice lattices for my
"TM-reduced lattice basis" Encyclopaedia page,
which quotes several tuning-math posts from Gene
giving TM-reduced lattice bases for several different
prime-limits:

http://tonalsoft.com/enc/tm-reduced-lattice.htm

i started with Gene's simplest examples, from
/tuning-math/message/7391
"some 5-limit TM reduced et bases". they appear
about 1/3 of the way down the page.

i didn't want to make the lattice for the larger ETs
too big, so those are not really legible, but you
can still see the shapes. the 171et toroidal lattices
are really beautiful.

these lattices were made with the copy of the Tonalsoft
Musica software which i have on my computer. it gives
you folks some idea of the lattice-drawing capabilities
of the software.

-monz

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:

> hello all,
>
>
> i've had fun making some nice lattices for my
> "TM-reduced lattice basis" Encyclopaedia page,
> which quotes several tuning-math posts from Gene
> giving TM-reduced lattice bases for several different
> prime-limits:
>
> http://tonalsoft.com/enc/tm-reduced-lattice.htm
>
> <snip>
>
> these lattices were made with the copy of the Tonalsoft
> Musica software which i have on my computer. it gives
> you folks some idea of the lattice-drawing capabilities
> of the software.

i thought it good to add that:

1) size limitations like the one i enforced by using
tiny versions of the 118 and 171 lattices, are not
a problem in the software, because lattices can be
zoomed in or out by any amount;

2) the lattices on the webpage are static 2-D screen-shots
of what are fully 3-D rotatable structures in the software.

in creating the screen-shots of the toroidal lattices,
my criteria were a compromise between: 1) being able to
see the labels on as many lattice-cubes as possible, and
2) getting a nice "top-view" perspective which shows the
donut shape of the torus. the donut shape becomes clearer
as the cardinality of the ETs gets higher ... which is
partly why i think 171 looks so good.

-monz

hi Gene,

this is exactly the kind of thing i was talking
about the other day, regarding adding lots of diagrams
to illustrate your work.

i find it most illuminating ... what used to be
dry lists of numbers are now concrete structures
that i get get my mind around.

-monz

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:

> --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
>
> > hello all,
> >
> >
> > i've had fun making some nice lattices for my
> > "TM-reduced lattice basis" Encyclopaedia page,
> > which quotes several tuning-math posts from Gene
> > giving TM-reduced lattice bases for several different
> > prime-limits:
> >
> > http://tonalsoft.com/enc/tm-reduced-lattice.htm
> >
> > <snip>
> >
> > these lattices were made with the copy of the Tonalsoft
> > Musica software which i have on my computer. it gives
> > you folks some idea of the lattice-drawing capabilities
> > of the software.
>
>
>
> i thought it good to add that:
>
> 1) size limitations like the one i enforced by using
> tiny versions of the 118 and 171 lattices, are not
> a problem in the software, because lattices can be
> zoomed in or out by any amount;
>
> 2) the lattices on the webpage are static 2-D screen-shots
> of what are fully 3-D rotatable structures in the software.
>
> in creating the screen-shots of the toroidal lattices,
> my criteria were a compromise between: 1) being able to
> see the labels on as many lattice-cubes as possible, and
> 2) getting a nice "top-view" perspective which shows the
> donut shape of the torus. the donut shape becomes clearer
> as the cardinality of the ETs gets higher ... which is
> partly why i think 171 looks so good.
>
>
>
> -monz

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:

> i started with Gene's simplest examples, from
> /tuning-math/message/7391
> "some 5-limit TM reduced et bases". they appear
> about 1/3 of the way down the page.

Very cool. I wonder if it would be wise to revise these pages in the
light of our constantly evolving jargon? Given that that seems to be
confusing enough already, it might help.

Also, in case James Tenney ever consults your encyclopedia, I'd fix
the spelling error for his name. :)

> i didn't want to make the lattice for the larger ETs
> too big, so those are not really legible, but you
> can still see the shapes. the 171et toroidal lattices
> are really beautiful.

The 118et lattice is interesting--it is a 9x13 paralleogram with an
added extra note. Keyboard makers take note!

One thing to keep and eye on when looking at these are where the
approximate 8/7 and 7/4 lie on the lattice which derive from
enharmonic bridges. Here is a list of some enharmonic bridges, and
some approximate 8/7 intervals deriving from them:

200/189 27/25
28/27 32/27
36/35 10/9
525/512 75/64
64/63 9/8
875/864 125/108
126/125 144/125
225/224 256/225
5120/5103 729/640
65625/65536 9375/8192
4375/4374 2500/2187

On 31-et, for instance, 126/125 and 225/224, which are commas of
7-limit 31et, lead to 144/125 and 256/225 as approximate 8/7s, which
are represented by 6 steps of 31-equal, the correct note for 8/7 in
31-equal. So we look for the 6 and for differences of 6 on your
lattice diagram; it is a 5-limit interval, but under 31 it maps to
8/7. On your diagram there is a 6 corresponding to 144/125, the
126/125-bridge 8/7, and a 25=31-6 corresponding to 225/128, the
225/224-bridge 7/4. Similarly on the 171et lattice you can spot the 33
which corresponds to 2500/2187, and gives you an 8/7, and also see
that the block is aligned in this direction, so that you'd get a lot
of highly accurate 7-limit intervals by a planar tempering, using
4375/4374, of this block.

> these lattices were made with the copy of the Tonalsoft
> Musica software which i have on my computer. it gives
> you folks some idea of the lattice-drawing capabilities
> of the software.

Leading to the question of when it is to be released. Will we be
seeing these lattice diagrams everywhere when that happens?

hi Gene,

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:

> --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
>
> > i started with Gene's simplest examples, from
> > /tuning-math/message/7391
> > "some 5-limit TM reduced et bases". they appear
> > about 1/3 of the way down the page.
>
> Very cool. I wonder if it would be wise to revise these
> pages in the light of our constantly evolving jargon?
> Given that that seems to be confusing enough already,
> it might help.

i think that's an excellent idea. as you can see, i've
already supplemented the notation in your posts (which
i think is quite cryptic) so that it gives more information
and is arranged more in a tabular format.

the problem with the jargon is ... until just a couple
of weeks ago, i was pretty much on hiatus from the
tuning lists for about the last year, just peeking in
to see if anyone was asking me anything.

the jargon train left me several stations behind.
so i'll need lots of help with that.

anyway, the Tonalsoft Encyclopaedia (like the Tuning
Dictionary before it) is an ongoing, evolving work.
pages which i think are important will always been
getting amended and edited.

> Also, in case James Tenney ever consults your encyclopedia,
> I'd fix the spelling error for his name. :)

thanks for that. i know his name quite well ... it was
just a slipped, and is fixed now.

in fact, Jim had plans on using my book in his course,
but i've been so busy that now i don't even remember if
i ever mailed him a copy! (oops)

> > i didn't want to make the lattice for the larger ETs
> > too big, so those are not really legible, but you
> > can still see the shapes. the 171et toroidal lattices
> > are really beautiful.
>
> The 118et lattice is interesting--it is a 9x13 paralleogram
> with an added extra note. Keyboard makers take note!
>
> One thing to keep and eye on when looking at these are
> where the approximate 8/7 and 7/4 lie on the lattice which
> derive from enharmonic bridges. Here is a list of some
> enharmonic bridges, and some approximate 8/7 intervals
> deriving from them:
>
> <snip>

whoa! "enharmonic bridges"? is that a new term that
i missed? or do you mean "xenharmonic bridges"?

... actually, if the latter, then "enharmonic bridges"
is probably better suited to describe what i mean by
the "xenharmonic bridges" anyway. feedback please.

> > these lattices were made with the copy of the Tonalsoft
> > Musica software which i have on my computer. it gives
> > you folks some idea of the lattice-drawing capabilities
> > of the software.
>
> Leading to the question of when it is to be released.

this is a big project that has been in the works for a
long time. we're already a full year behind our original
hoped-for release date. we're busting our butts now in
hopes of having it out for sale by Christmas.

> Will we be seeing these lattice diagrams everywhere when
> that happens?

everywhere in the Tonalsoft Encyclopaedia, at least.

i wanted to mention before that right now these are only
static 2-D graphics ... but when we make the free Tonalsoft
Lattice Viewer plug-in available, i'll start using the
rotatable virtual-3-D diagrams that Musica creates,
on the webpages. anyone who downloads the Lattice Viewer
will be able to see them (and rotate them with the mouse)
the same way the full-release Musica software does it.

-monz

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:

> whoa! "enharmonic bridges"? is that a new term that
> i missed? or do you mean "xenharmonic bridges"?

The latter; that the xenharmonic bridges are enharmonic bridges makes
it easy to say one in place of the other.

> ... actually, if the latter, then "enharmonic bridges"
> is probably better suited to describe what i mean by
> the "xenharmonic bridges" anyway. feedback please.

OK by me. :)

> this is a big project that has been in the works for a
> long time. we're already a full year behind our original
> hoped-for release date. we're busting our butts now in
> hopes of having it out for sale by Christmas.

Great. Maybe there's still time to work lattices of chords into it.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:

> --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
>
> > whoa! "enharmonic bridges"? is that a new term that
> > i missed? or do you mean "xenharmonic bridges"?
>
> The latter; that the xenharmonic bridges are enharmonic
> bridges makes it easy to say one in place of the other.
>
> > ... actually, if the latter, then "enharmonic bridges"
> > is probably better suited to describe what i mean by
> > the "xenharmonic bridges" anyway. feedback please.
>
> OK by me. :)

if enough others agree, then we should change the term.

> > this is a big project that has been in the works for a
> > long time. we're already a full year behind our original
> > hoped-for release date. we're busting our butts now in
> > hopes of having it out for sale by Christmas.
>
> Great. Maybe there's still time to work lattices of chords
> into it.

extremely doubtful for release 1.0. now we're doing
what's basically "clean-up" work, planning to get the
beta version out by September.

new stuff will have to wait for the next release,
and i suspect that chord-lattices will fall into that
category. there's a lot of stuff i wanted in the
first version that didn't make it yet ... but it's
still killer even as it is.

-monz

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:

> > > i've had fun making some nice lattices for my
> > > "TM-reduced lattice basis" Encyclopaedia page,
> > > which quotes several tuning-math posts from Gene
> > > giving TM-reduced lattice bases for several different
> > > prime-limits:
> > >
> > > http://tonalsoft.com/enc/tm-reduced-lattice.htm
> >
> > in creating the screen-shots of the toroidal lattices,
> > my criteria were a compromise between: 1) being able to
> > see the labels on as many lattice-cubes as possible, and
> > 2) getting a nice "top-view" perspective which shows the
> > donut shape of the torus. the donut shape becomes clearer
> > as the cardinality of the ETs gets higher ... which is
> > partly why i think 171 looks so good.

the page now also contains 7-limit lattices for 31, 41, and
72et.

i've also put in three different perspectives of a 5-limit
612et toroidal lattice. this one looks totally like a donut.

-monz

>i've had fun making some nice lattices for my
>"TM-reduced lattice basis" Encyclopaedia page,
>which quotes several tuning-math posts from Gene
>giving TM-reduced lattice bases for several different
>prime-limits:
>
>http://tonalsoft.com/enc/tm-reduced-lattice.htm

Yo monz!

That's freakin' awesome!

Rock on!

-C.

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >i've had fun making some nice lattices for my
> >"TM-reduced lattice basis" Encyclopaedia page,
> >which quotes several tuning-math posts from Gene
> >giving TM-reduced lattice bases for several different
> >prime-limits:
> >
> >http://tonalsoft.com/enc/tm-reduced-lattice.htm
>
> Yo monz!
>
> That's freakin' awesome!
>
> Rock on!
>
> -C.

thanks a bunch, Carl. i had a feeling that you would
be one of the people who really enjoyed this.

:)

-monz

-- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:

> i've had fun making some nice lattices for my
> "TM-reduced lattice basis" Encyclopaedia page,
> which quotes several tuning-math posts from Gene
> giving TM-reduced lattice bases for several different
> prime-limits:
>
> http://tonalsoft.com/enc/tm-reduced-lattice.htm

the posts from Gene which i quoted on this page
(and one by Paul) were placed in chronological order
by date of posting.

but the order seems very jumbled like this.
i propose to organize this page by prime-limit.

any comments?

-monz

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:

> but the order seems very jumbled like this.
> i propose to organize this page by prime-limit.
>
> any comments?

I think a lot of this stuff could use editing.