lattice diagram "levels" of complexity

is there an accepted method of nomenclature for
describing the "level" of complexity of lattice
diagrams?

i know that "stellation" has something to do
with this, but here i'm talking pretty much about
the algorithm used by Partch to fill out the
Tonality Diamond.

in other words, say we have a 7-limit (3-D) lattice.

(use "Expand Messages" if viewing on Yahoo website
for proper formatting of diagram)

here are the two basic tetrads (otonal and utonal)
which have the 1/1 ratio as their 1-identity:

5:4
/|\
/ | \
/ | \
/ 7:4 \
/. ' ' .\
4:3---------1:1---------3:2
\ '. .' /
\ 8:7 /
\ | /
\ | /
\|/
8:5

this would be "level" 1.

now if we build complete otonal tetrads
on all of the notes in the "basic" utonality
tetrad (i.e., 4/3, 8/5, and 8/7 all become
1-odentities of their respective tetrads), and
complete utonal tetrads on all of the notes in
the "basic" otonality tetrad (i.e, 3/2, 5/4, and
7/4 all become 1-udentities of their respective
tetrads), we get this:

5:3---------5:4
/|\ '. .' /|\
/ | \ 10:7 / | \
/ | \ /|\ / | \
/ 7:6--/-|-\--7:4 \
/. ' '/.\|/.\' ' .\
4:3------/--1:1--\------3:2
\ '. /.' /|\ '.\ .'/
\ 8:7--/-|-\--12:7 /
\ | / | \ | /
\ | / 7:5 \ | /
\|/ .' '. \|/
8:5---------6:5

which would be "level" 2.

is there already an accepted term for what i'm
calling "level"?

and can someone give a very clear and lucid explanation
of how stellation differs from this, if it does?

thanks.

-monz

heya monz,

>is there already an accepted term for what i'm
>calling "level"?

Not to my knowledge, though we often talk about
regions of the lattice within some taxicab
radius. The diamond is r=1, and it sounds like
your levels correspond to successively higher
r values. You might want to check that and tell
me if it's the case.

I said "lattice region". Gene has used the term
"ball", and I think that's a convex hull, plus
everything inside. zthat right, Gene?

>and can someone give a very clear and lucid explanation
>of how stellation differs from this, if it does?

Stellation is Wilson's term. He borrowed it from
geometry, where the term often refers to the process
of adding points above the faces of a polyhedron,
turning *them* into polyhedra. This is indeed what
happens when stellating the hexany -- it's an octahedron
being extended so that each face becomes a tetrahedron.
I'm sure there's a more precise definition on a geometry
website somewhere. Post it here if you are interested
and find it... The one I'm remembering is 'the compound
of a polyhedron and its dual'.

Back when, there was some debate over what stellation
should include when extending other CPSs (the eikosany
was the main inquiry). I said that each existing face
should be completed into a saturated n-limit chord, and
that's it. Others, apparently including Wilson, wanted
to include other points, basically out to the power
set (the compound of all CPSs of a given limit). I was
never clear on why they wanted to do this. Maybe Paul
remembers.

I don't think it's related to your levels, really.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> Others, apparently including Wilson, wanted
> to include other points, basically out to the power
> set (the compound of all CPSs of a given limit).

that isn't the case. look over the old discussions on this list about
stellation. the compound of all CPSs of a given limit can be
constructed in many ways, but most typically (as d'allessandro) as an
euler genus, and this (or any of the other ways) clearly does not
have the symmetry of the original, unstellated CPS, which the
stellation must have by definition.

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
> is there an accepted method of nomenclature for
> describing the "level" of complexity of lattice
> diagrams?
>
> i know that "stellation" has something to do
> with this, but here i'm talking pretty much about
> the algorithm used by Partch to fill out the
> Tonality Diamond.
>
> in other words, say we have a 7-limit (3-D) lattice.
>
> (use "Expand Messages" if viewing on Yahoo website
> for proper formatting of diagram)
>
>
> here are the two basic tetrads (otonal and utonal)
> which have the 1/1 ratio as their 1-identity:
>
>
> 5:4
> /|\
> / | \
> / | \
> / 7:4 \
> /. ' ' .\
> 4:3---------1:1---------3:2
> \ '. .' /
> \ 8:7 /
> \ | /
> \ | /
> \|/
> 8:5
>
> this would be "level" 1.
>
>
> now if we build complete otonal tetrads
> on all of the notes in the "basic" utonality
> tetrad (i.e., 4/3, 8/5, and 8/7 all become
> 1-odentities of their respective tetrads), and
> complete utonal tetrads on all of the notes in
> the "basic" otonality tetrad (i.e, 3/2, 5/4, and
> 7/4 all become 1-udentities of their respective
> tetrads), we get this:
>
> 5:3---------5:4
> /|\ '. .' /|\
> / | \ 10:7 / | \
> / | \ /|\ / | \
> / 7:6--/-|-\--7:4 \
> /. ' '/.\|/.\' ' .\
> 4:3------/--1:1--\------3:2
> \ '. /.' /|\ '.\ .'/
> \ 8:7--/-|-\--12:7 /
> \ | / | \ | /
> \ | / 7:5 \ | /
> \|/ .' '. \|/
> 8:5---------6:5
>
> which would be "level" 2.
>
>
> is there already an accepted term for what i'm
> calling "level"?
>
> and can someone give a very clear and lucid explanation
> of how stellation differs from this, if it does?
>
> thanks.
>
>
>
> -monz

let me suggest a different, cleaner method.

level 1 will be a *single* tetrad (*either* utonal or otonal, it
doesn't matter, but not both).

level 2 results from building a tetrad that is the inverse of the
original one, on each tone of the original tetrad. this is again a
tonality diamond, but is closer to how many people (e.g., john
chalmers) actually think of the diamond -- the cartesian product of a
tetrad with its inverse.

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
> is there an accepted method of nomenclature for
> describing the "level" of complexity of lattice
> diagrams?
>
> i know that "stellation" has something to do
> with this, but here i'm talking pretty much about
> the algorithm used by Partch to fill out the
> Tonality Diamond.
>
> in other words, say we have a 7-limit (3-D) lattice.
>
> (use "Expand Messages" if viewing on Yahoo website
> for proper formatting of diagram)
>
>
> here are the two basic tetrads (otonal and utonal)
> which have the 1/1 ratio as their 1-identity:
>
>
> 5:4
> /|\
> / | \
> / | \
> / 7:4 \
> /. ' ' .\
> 4:3---------1:1---------3:2
> \ '. .' /
> \ 8:7 /
> \ | /
> \ | /
> \|/
> 8:5
>
> this would be "level" 1.
>
>
> now if we build complete otonal tetrads
> on all of the notes in the "basic" utonality
> tetrad (i.e., 4/3, 8/5, and 8/7 all become
> 1-odentities of their respective tetrads), and
> complete utonal tetrads on all of the notes in
> the "basic" otonality tetrad (i.e, 3/2, 5/4, and
> 7/4 all become 1-udentities of their respective
> tetrads), we get this:
>
> 5:3---------5:4
> /|\ '. .' /|\
> / | \ 10:7 / | \
> / | \ /|\ / | \
> / 7:6--/-|-\--7:4 \
> /. ' '/.\|/.\' ' .\
> 4:3------/--1:1--\------3:2
> \ '. /.' /|\ '.\ .'/
> \ 8:7--/-|-\--12:7 /
> \ | / | \ | /
> \ | / 7:5 \ | /
> \|/ .' '. \|/
> 8:5---------6:5
>
> which would be "level" 2.
>
>
> is there already an accepted term for what i'm
> calling "level"?
>
> and can someone give a very clear and lucid explanation
> of how stellation differs from this, if it does?
>
> thanks.
>
>
>
> -monz

i don't really get the appeal of this (what would level 3 be?), but
it's somewhat similar to paul hahn's diameter measure. a *single*
tetrad has a diameter of 1, while *either* a hexany *or* a diamond
would have a diameter of 2, since you'd need no more than 2 consonant
intervals to connect any two pitches . . .

>> Others, apparently including Wilson, wanted
>> to include other points, basically out to the power
>> set (the compound of all CPSs of a given limit).
>
>that isn't the case. look over the old discussions on this list about
>stellation. the compound of all CPSs of a given limit can be
>constructed in many ways, but most typically (as d'allessandro) as an
>euler genus, and this (or any of the other ways) clearly does not
>have the symmetry of the original, unstellated CPS, which the
>stellation must have by definition.

Right, right, you're extending the tones out to a EF Genus in all
directions. A stellated EF Genus is what I called it in that thread.

-Carl

--- In tuning-math@yahoogroups.com, "wallyesterpaulrus
<wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:

> i don't really get the appeal of this (what would level 3 be?), but
> it's somewhat similar to paul hahn's diameter measure. a *single*
> tetrad has a diameter of 1, while *either* a hexany *or* a diamond
> would have a diameter of 2, since you'd need no more than 2
consonant
> intervals to connect any two pitches . . .

This is graph theory language.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> I said "lattice region". Gene has used the term
> "ball", and I think that's a convex hull, plus
> everything inside. zthat right, Gene?

That would be a closed ball in the particular metric that convex
region and origin defined.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> Others, apparently including Wilson, wanted
> >> to include other points, basically out to the power
> >> set (the compound of all CPSs of a given limit).
> >
> >that isn't the case. look over the old discussions on this list
about
> >stellation. the compound of all CPSs of a given limit can be
> >constructed in many ways, but most typically (as d'allessandro) as
an
> >euler genus, and this (or any of the other ways) clearly does not
> >have the symmetry of the original, unstellated CPS, which the
> >stellation must have by definition.
>
> Right, right, you're extending the tones out to a EF Genus in all
> directions.

example?

> A stellated EF Genus is what I called it in that thread.

to my great consternation . . .

> -Carl

>> Right, right, you're extending the tones out to a EF Genus in all
>> directions.
>
>example?

I think that's the way I accounted for a 92-tone structure.

>> A stellated EF Genus is what I called it in that thread.
>
>to my great consternation . . .

What you never showed is how the definition of stellation on
George Hart's site requires the 92-tone, and not the 80-tone
structure.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> Right, right, you're extending the tones out to a EF Genus in all
> >> directions.
> >
> >example?
>
> I think that's the way I accounted for a 92-tone structure.

how about a much simpler example, with much fewer factors? i'm
looking to understand what you mean by "extending the tones out to a
EF Genus in all directions"

> >> A stellated EF Genus is what I called it in that thread.
> >
> >to my great consternation . . .
>
> What you never showed is how the definition of stellation on
> George Hart's site requires the 92-tone, and not the 80-tone
> structure.
>
> -Carl

i think manuel has got that down.

>how about a much simpler example, with much fewer factors?

There really aren't any, since the in the 4-factor case the
"stellated hexany" comes out the same either way. The 5-factor
case is already > 3-D, and doesn't have a symmetrical raw CPS.
The number of tones in a stellated CPS, according to me, is:

> 3 3
> N! (M + (N-M) - N) N!
> ------- + ------------------
> M!(N-M)! (M+1)!(N-M+1)!

I can't remember if this works for 'unsymmetrical' CPSs. In
the case of the dekany, there are 5 triads and 5 tetrads to
complete. That's 15 new notes, 25 notes in all.

Plugging in 2)5 or 3)5 to the above, we get 10 for the first
term and 25 for the second term. So either it doesn't work
on unsymmetrical CPSs, I've missed something in the above
paragraph, or the first term is redundant.

>i'm looking to understand what you mean by "extending the tones
>out to a EF Genus in all directions"

If we imagine completing the triads of a hexany, we see that the
4-factor EF Genus does this for two only two of the triads. If
you look at the mating pattern for "the bottom line" (fig.20
D'Alessandro), and imagine rotating it around the lines of the
eikosany, I believe you get fig.20b.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> >i'm looking to understand what you mean by "extending the tones
> >out to a EF Genus in all directions"
>
> If we imagine completing the triads of a hexany, we see that the
> 4-factor EF Genus does this for two only two of the triads.

right . . .

> If
> you look at the mating pattern for "the bottom line" (fig.20
> D'Alessandro), and imagine rotating it around the lines of the
> eikosany, I believe you get fig.20b.

ok, but the original "EF Genus"ness is irrelevant. there are simpler
structures that, when rotated to all possible positions, will do the
job, and you're kind of mixing apples and oranges in
that "directions" in the EF world would really just refer to the
primes, while in the CPS world there are more operative "directions".

hi paul, Gene, Carl, and others in this thread,

> From: <gwsmith@svpal.org>
> To: <tuning-math@yahoogroups.com>
> Sent: Tuesday, February 18, 2003 10:08 PM
> Subject: [tuning-math] Re: lattice diagram "levels" of complexity
>
>
> --- In tuning-math@yahoogroups.com, "wallyesterpaulrus
> <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:
>
> > i don't really get the appeal of this (what would
> > level 3 be?), but it's somewhat similar to paul hahn's
> > diameter measure. a *single* tetrad has a diameter
> > of 1, while *either* a hexany *or* a diamond would
> > have a diameter of 2, since you'd need no more than
> > 2 consonant intervals to connect any two pitches . . .
>
> This is graph theory language.

well ... since "level 2" results from building
complete otonalities and utonalities on all the
ratios which lie on the outside of the "level 1"
structure ...

i suppose "level 3" would be the structure which
results from building complete otonalities and
utonalities on the ratios which lie on the outside
of the "level 2" structure.

no?

so anyway, is "diameter" more-or-less accepted as
the standard terminology for this kind of thing?

i have my reasons for needing this ... it has to
do with my software project.

-monz

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
> hi paul, Gene, Carl, and others in this thread,
>
>
> > From: <gwsmith@s...>
> > To: <tuning-math@yahoogroups.com>
> > Sent: Tuesday, February 18, 2003 10:08 PM
> > Subject: [tuning-math] Re: lattice diagram "levels" of complexity
> >
> >
> > --- In tuning-math@yahoogroups.com, "wallyesterpaulrus
> > <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:
> >
> > > i don't really get the appeal of this (what would
> > > level 3 be?), but it's somewhat similar to paul hahn's
> > > diameter measure. a *single* tetrad has a diameter
> > > of 1, while *either* a hexany *or* a diamond would
> > > have a diameter of 2, since you'd need no more than
> > > 2 consonant intervals to connect any two pitches . . .
> >
> > This is graph theory language.
>
>
> well ... since "level 2" results from building
> complete otonalities and utonalities on all the
> ratios which lie on the outside of the "level 1"
> structure ...

um . . . not exactly . . . the way you did it was to build
otonalities on *some* of them, and utonalities on *some* others.

> i suppose "level 3" would be the structure which
> results from building complete otonalities and
> utonalities on the ratios which lie on the outside
> of the "level 2" structure.
>
> no?

it doesn't sound like you have a precisely defined concept here. i
hope you'll prove me wrong!

> so anyway, is "diameter" more-or-less accepted as
> the standard terminology for this kind of thing?

yes, but be precise. examples of a diameter-1 structure in the 7-
limit lattice would include the otonal tetrad and the utonal tetrad,
but not the agglomeration of both stuck together. examples of
diameter-2 include the hexany and the diamond. your original "level-
1" structure also has a diameter of 2, since to get from any note in
the utonality to any note in the otonality, assuming neither is 1/1,
you have to pass through 1/1, and thus make 2 steps. there are of
course many other diameter-2 structures possible.

>ok, but the original "EF Genus"ness is irrelevant.

I agree. But I say the same is then true of the 92-tone
structure being a "stellated eikosany". Pending otherwise
via the 'official definition of stellation'. . .

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >ok, but the original "EF Genus"ness is irrelevant.
>
> I agree. But I say the same is then true

how is the same true? the 92-tone structure has the right symmetry to
begin with.

> of the 92-tone
> structure being a "stellated eikosany". Pending otherwise
> via the 'official definition of stellation'. . .
>
> -Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >ok, but the original "EF Genus"ness is irrelevant.
>
> I agree. But I say the same is then true of the 92-tone
> structure being a "stellated eikosany". Pending otherwise
> via the 'official definition of stellation'. . .
>
> -Carl

there is more than one way to stellate something. check this out:

http://www.georgehart.com/virtual-polyhedra/srtc-info.html

>> This is graph theory language.
>
>
>well ... since "level 2" results from building
>complete otonalities and utonalities on all the
>ratios which lie on the outside of the "level 1"
>structure ...

I think Gene meant that "diameter", not "level",
is graph theory lang.

>i suppose "level 3" would be the structure which
>results from building complete otonalities and
>utonalities on the ratios which lie on the outside
>of the "level 2" structure.
>
>no?

That's your call! Have you verified that such a
structure is the collection of lattice points
within a radius of 3 from a given point?

>so anyway, is "diameter" more-or-less accepted as
>the standard terminology for this kind of thing?

No, diameter is diameter...

http://mathworld.wolfram.com/GraphDiameter.html

I don't think it's really related to your levels.

-Carl

>>>ok, but the original "EF Genus"ness is irrelevant.
>>
>>I agree. But I say the same is then true
>
>how is the same true? the 92-tone structure has the
>right symmetry to begin with.

Note that Wilson's 20b is not called "stellated eikosany".
[Oh, by the way, it's obvious that "stellated EF Genus" is awful.
I won't push that anymore.]

Here's some quotes from the old thread...

>a CPS is supposed to be a fancy subset of a tonespace, not a gross
>chunk of it. I find it much more natural to think of stellation as
>simply completing all the chords I had in my original structure.
>Using them all is enough of a challenge, without extras besides.

>As I've been saying, we're running up against adjacent pentadekanies
>here, and completing the pentads to hexads (while ignoring the
>incomplete triads, I might add). So if, as I suggested way back in
>this thread, we add 1 to m at each iteration, stellation terminates
>when m=n. For the hebdomekontany, we must add three tones to each
>chord at the first iteration, then two, and then 1 (again, ignoring
>the "lesser" deficient chords in the adjacent CPSs -- else, three,
>two and six, one and seven).
>
>All this isn't worth the trouble. Might as well just say, "I'm going
>to use the entire lattice". The idea of the CPS as a structure
>providing "special" access to the relations in a tonespace looses all
>meaning.

-Carl

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

> so anyway, is "diameter" more-or-less accepted as
> the standard terminology for this kind of thing?

It's standard math terminology:

http://mathworld.wolfram.com/GraphDiameter.html

http://mathworld.wolfram.com/GraphDistance.html

http://mathworld.wolfram.com/GraphGeodesic.html

Carl and Paul wrote:
>how about a much simpler example, with much fewer factors? i'm
>looking to understand what you mean by "extending the tones out to a
>EF Genus in all directions"

The only way I know to extend a CPS to a EF genus is to add the
remaining combination counts to it. So if you have 2)5 you need to
add 1)5, 3)5, 4)5 and 5)5.

>> What you never showed is how the definition of stellation on
>> George Hart's site requires the 92-tone, and not the 80-tone
>> structure.
>>
>> -Carl

>i think manuel has got that down.

So if I'm not mistaken the 80-tone structure is the
result of the first-order stellation and the 92-tone one of the
complete stellation.
The former is called "stellated" in Scala and Wilson's stellation
is called "superstellated". It's a repeated stellation so that
there are no more unstellated chords anymore.

Manuel