Full thread (20 messages)
From: Gene Ward Smith (2004-02-10)
Subject: Rhombic dodecahedron scale
Here is a scale which arose when I was considering adding to the seven
limit lattices web page. A Voronoi cell for a lattice is every point
at least as close (closer, for an interior point) to a paricular
vertex than to any other vertex. The Voronoi cells for the
face-centered cubic
lattice of 7-limit intervals is the rhombic dodecahedron with the 14
verticies (+-1 0 0), (0 +-1 0), (0 0 +-1), (+-1/2 +-1/2 +-1/2). These
fill the whole space, like a bee's honeycomb. The Delaunay celles of a
lattice are the convex hulls of the lattice points closest to a
Voronoi cell vertex; in this case we get tetrahedra and octahedra,
which are the holes of the lattice, and are tetrads or hexanies. The
six (+-1 0 0) verticies of the Voronoi cell correspond to six
hexanies, and the
eight others to eight tetrads. If we put all of these together, we
obtain the following scale of 19 notes, all of whose intervals are
superparticular ratios:
! rhomb.scl
Union of Delauny cells for the rhombic dodecahedron Voronoi cell
centered at (0 0 0)
19
!
21/20
15/14
8/7
7/6
6/5
5/4
4/3
48/35
7/5
10/7
35/24
3/2
8/5
5/3
12/7
7/4
28/15
40/21
2
Here it is in TOP Marvel:
! rhombmarv.scl
TOP Marvel version of rhomb.scl
19
!
85.229563
115.634597
231.269195
268.545555
316.498758
384.180152
499.814749
547.767953
585.044313
615.449347
652.725707
700.678910
816.313507
883.994902
931.948105
969.224465
1084.859062
1115.264096
1200.493659
From: Carl Lumma (2004-02-10)
Subject: Re: [tuning-math] Rhombic dodecahedron scale
>Here is a scale which arose when I was considering adding to the seven
>limit lattices web page. A Voronoi cell for a lattice is every point
>at least as close (closer, for an interior point) to a paricular
>vertex than to any other vertex. The Voronoi cells for the
>face-centered cubic
>lattice of 7-limit intervals is the rhombic dodecahedron
Something Fuller demonstrated, in his own tongue.
>These
>fill the whole space, like a bee's honeycomb.
Isn't it also the dual to the FCC lattice (hmm, maybe dual isn't
the right word here...)
>The Delaunay celles of a
>lattice are the convex hulls of the lattice points closest to a
>Voronoi cell vertex; in this case we get tetrahedra and octahedra,
Ah, that would be the 'dual' operation I was thinking it above.
I saw a graphic of this on site about Fuller once.
>which are the holes of the lattice, and are tetrads or hexanies. The
>six (+-1 0 0) verticies of the Voronoi cell
*The* Voronoi cell? Which one do you mean?
>correspond to six hexanies, and the eight others to eight tetrads.
>If we put all of these together, we obtain the following scale of
>19 notes, all of whose intervals are superparticular ratios:
Hmm...
-C.
From: Gene Ward Smith (2004-02-10)
Subject: Re: Rhombic dodecahedron scale
--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >Here is a scale which arose when I was considering adding to the seven
> >limit lattices web page. A Voronoi cell for a lattice is every point
> >at least as close (closer, for an interior point) to a paricular
> >vertex than to any other vertex. The Voronoi cells for the
> >face-centered cubic
> >lattice of 7-limit intervals is the rhombic dodecahedron
>
> Something Fuller demonstrated, in his own tongue.
Right. Fuller?
> >These
> >fill the whole space, like a bee's honeycomb.
>
> Isn't it also the dual to the FCC lattice (hmm, maybe dual isn't
> the right word here...)
The dual to the fcc lattice is the bcc lattice (body-centered cubic
lattice.) But we don't seem to be using the same defintion of "dual".
> >The Delaunay celles of a
> >lattice are the convex hulls of the lattice points closest to a
> >Voronoi cell vertex; in this case we get tetrahedra and octahedra,
>
> Ah, that would be the 'dual' operation I was thinking it above.
> I saw a graphic of this on site about Fuller once.
>
> >which are the holes of the lattice, and are tetrads or hexanies. The
> >six (+-1 0 0) verticies of the Voronoi cell
>
> *The* Voronoi cell? Which one do you mean?
The one around the unison, (0 0 0). Others are merely translates.
From: Carl Lumma (2004-02-11)
Subject: Re: [tuning-math] Re: Rhombic dodecahedron scale
>> >A Voronoi cell for a lattice is every point
>> >at least as close (closer, for an interior point) to a paricular
>> >vertex than to any other vertex. The Voronoi cells for the
>> >face-centered cubic lattice of 7-limit intervals is the rhombic
>> >dodecahedron
>>
>> Something Fuller demonstrated, in his own tongue.
>
>Right. Fuller?
Buckminster.
>> >These
>> >fill the whole space, like a bee's honeycomb.
>>
>> Isn't it also the dual to the FCC lattice (hmm, maybe dual isn't
>> the right word here...)
>
>The dual to the fcc lattice is the bcc lattice (body-centered cubic
>lattice.)
Indeed, sorry.
>> >The Delaunay celles of a
>> >lattice are the convex hulls of the lattice points closest to a
>> >Voronoi cell vertex; in this case we get tetrahedra and octahedra,
>> >which are the holes of the lattice, and are tetrads or hexanies.
>> >The six (+-1 0 0) verticies of the Voronoi cell
>>
>> *The* Voronoi cell? Which one do you mean?
>
>The one around the unison, (0 0 0). Others are merely translates.
Ah, yes.
-Carl
From: Paul G Hjelmstad (2004-02-11)
Subject: Re: Rhombic dodecahedron scale
--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> Here is a scale which arose when I was considering adding to the
seven
> limit lattices web page. A Voronoi cell for a lattice is every point
> at least as close (closer, for an interior point) to a paricular
> vertex than to any other vertex. The Voronoi cells for the
> face-centered cubic
> lattice of 7-limit intervals is the rhombic dodecahedron with the 14
> verticies (+-1 0 0), (0 +-1 0), (0 0 +-1), (+-1/2 +-1/2 +-1/2).
These
> fill the whole space, like a bee's honeycomb. The Delaunay celles
of a
> lattice are the convex hulls of the lattice points closest to a
> Voronoi cell vertex; in this case we get tetrahedra and octahedra,
> which are the holes of the lattice, and are tetrads or hexanies. The
> six (+-1 0 0) verticies of the Voronoi cell correspond to six
> hexanies, and the
> eight others to eight tetrads. If we put all of these together, we
> obtain the following scale of 19 notes, all of whose intervals are
> superparticular ratios:
>
I know I'm lagging behind, but I need to ask where the remaining 5
notes come from (14 + 5). Thanks
From: Paul G Hjelmstad (2004-02-11)
Subject: Re: Rhombic dodecahedron scale
--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> wrote:
> > Here is a scale which arose when I was considering adding to the
> seven
> > limit lattices web page. A Voronoi cell for a lattice is every
point
> > at least as close (closer, for an interior point) to a paricular
> > vertex than to any other vertex. The Voronoi cells for the
> > face-centered cubic
> > lattice of 7-limit intervals is the rhombic dodecahedron with the
14
> > verticies (+-1 0 0), (0 +-1 0), (0 0 +-1), (+-1/2 +-1/2 +-1/2).
> These
> > fill the whole space, like a bee's honeycomb. The Delaunay celles
> of a
> > lattice are the convex hulls of the lattice points closest to a
> > Voronoi cell vertex; in this case we get tetrahedra and octahedra,
> > which are the holes of the lattice, and are tetrads or hexanies.
The
> > six (+-1 0 0) verticies of the Voronoi cell correspond to six
> > hexanies, and the
> > eight others to eight tetrads. If we put all of these together, we
> > obtain the following scale of 19 notes, all of whose intervals are
> > superparticular ratios:
> >
> I know I'm lagging behind, but I need to ask where the remaining 5
> notes come from (14 + 5). Thanks
Okay -heres what I know for sure. The 19 tones include 3,5,7,15,21,35
hexany, all divided by 5 and 7. This makes 11 tones, leaving 8. I
can't find any pattern to the 8 remaining however. (Are these the 8
tetrads?). I also discovered that the 19 tones are every combination
of -1, 0 and 1 except for (1,1,1) (-1,-1,-1) triples and every double
of 1,1,0 and -1,-1,0. I guess what I am saying is that I understand
hexanies but don't know what makes a tetrad. Thanks
Paul
From: Paul Erlich (2004-02-11)
Subject: Re: Rhombic dodecahedron scale
--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <paul.hjelmstad@u...> wrote:
> > --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
> <gwsmith@s...>
> > wrote:
> > > Here is a scale which arose when I was considering adding to
the
> > seven
> > > limit lattices web page. A Voronoi cell for a lattice is every
> point
> > > at least as close (closer, for an interior point) to a paricular
> > > vertex than to any other vertex. The Voronoi cells for the
> > > face-centered cubic
> > > lattice of 7-limit intervals is the rhombic dodecahedron with
the
> 14
> > > verticies (+-1 0 0), (0 +-1 0), (0 0 +-1), (+-1/2 +-1/2 +-1/2).
> > These
> > > fill the whole space, like a bee's honeycomb. The Delaunay
celles
> > of a
> > > lattice are the convex hulls of the lattice points closest to a
> > > Voronoi cell vertex; in this case we get tetrahedra and
octahedra,
> > > which are the holes of the lattice, and are tetrads or
hexanies.
> The
> > > six (+-1 0 0) verticies of the Voronoi cell correspond to six
> > > hexanies, and the
> > > eight others to eight tetrads. If we put all of these together,
we
> > > obtain the following scale of 19 notes, all of whose intervals
are
> > > superparticular ratios:
> > >
> > I know I'm lagging behind, but I need to ask where the remaining
5
> > notes come from (14 + 5). Thanks
>
> Okay -heres what I know for sure. The 19 tones include
3,5,7,15,21,35
> hexany, all divided by 5 and 7. This makes 11 tones, leaving 8. I
> can't find any pattern to the 8 remaining however. (Are these the 8
> tetrads?). I also discovered that the 19 tones are every combination
> of -1, 0 and 1 except for (1,1,1) (-1,-1,-1) triples and every
double
> of 1,1,0 and -1,-1,0. I guess what I am saying is that I understand
> hexanies but don't know what makes a tetrad. Thanks
>
> Paul
There are two types of tetrad. 1:3:5:7 is one, and 105:35:21:15 = 1/
(1:3:5:7) is the other.
From: Paul G Hjelmstad (2004-02-11)
Subject: Re: Rhombic dodecahedron scale
--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <paul.hjelmstad@u...> wrote:
> > --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> > <paul.hjelmstad@u...> wrote:
> > > --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
> > <gwsmith@s...>
> > > wrote:
> > > > Here is a scale which arose when I was considering adding to
> the
> > > seven
> > > > limit lattices web page. A Voronoi cell for a lattice is
every
> > point
> > > > at least as close (closer, for an interior point) to a
paricular
> > > > vertex than to any other vertex. The Voronoi cells for the
> > > > face-centered cubic
> > > > lattice of 7-limit intervals is the rhombic dodecahedron with
> the
> > 14
> > > > verticies (+-1 0 0), (0 +-1 0), (0 0 +-1), (+-1/2 +-1/2 +-
1/2).
> > > These
> > > > fill the whole space, like a bee's honeycomb. The Delaunay
> celles
> > > of a
> > > > lattice are the convex hulls of the lattice points closest to
a
> > > > Voronoi cell vertex; in this case we get tetrahedra and
> octahedra,
> > > > which are the holes of the lattice, and are tetrads or
> hexanies.
> > The
> > > > six (+-1 0 0) verticies of the Voronoi cell correspond to six
> > > > hexanies, and the
> > > > eight others to eight tetrads. If we put all of these
together,
> we
> > > > obtain the following scale of 19 notes, all of whose
intervals
> are
> > > > superparticular ratios:
> > > >
> > > I know I'm lagging behind, but I need to ask where the
remaining
> 5
> > > notes come from (14 + 5). Thanks
> >
> > Okay -heres what I know for sure. The 19 tones include
> 3,5,7,15,21,35
> > hexany, all divided by 5 and 7. This makes 11 tones, leaving 8. I
> > can't find any pattern to the 8 remaining however. (Are these the
8
> > tetrads?). I also discovered that the 19 tones are every
combination
> > of -1, 0 and 1 except for (1,1,1) (-1,-1,-1) triples and every
> double
> > of 1,1,0 and -1,-1,0. I guess what I am saying is that I
understand
> > hexanies but don't know what makes a tetrad. Thanks
> >
> > Paul
>
> There are two types of tetrad. 1:3:5:7 is one, and 105:35:21:15 = 1/
> (1:3:5:7) is the other.
I know - but how does this translate to Gene's fractions. Are the
eight tetrads (+-1,+-1,+-1)? But the problem with that is that (1,1,1)
for example doesn't appear in the list (105) and neither do any
(1,1,0) or (-1,0,0) (Notating monzos here, not vertices)
From: Paul Erlich (2004-02-11)
Subject: Re: Rhombic dodecahedron scale
--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> > <paul.hjelmstad@u...> wrote:
> > > --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> > > <paul.hjelmstad@u...> wrote:
> > > > --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
> > > <gwsmith@s...>
> > > > wrote:
> > > > > Here is a scale which arose when I was considering adding
to
> > the
> > > > seven
> > > > > limit lattices web page. A Voronoi cell for a lattice is
> every
> > > point
> > > > > at least as close (closer, for an interior point) to a
> paricular
> > > > > vertex than to any other vertex. The Voronoi cells for the
> > > > > face-centered cubic
> > > > > lattice of 7-limit intervals is the rhombic dodecahedron
with
> > the
> > > 14
> > > > > verticies (+-1 0 0), (0 +-1 0), (0 0 +-1), (+-1/2 +-1/2 +-
> 1/2).
> > > > These
> > > > > fill the whole space, like a bee's honeycomb. The Delaunay
> > celles
> > > > of a
> > > > > lattice are the convex hulls of the lattice points closest
to
> a
> > > > > Voronoi cell vertex; in this case we get tetrahedra and
> > octahedra,
> > > > > which are the holes of the lattice, and are tetrads or
> > hexanies.
> > > The
> > > > > six (+-1 0 0) verticies of the Voronoi cell correspond to
six
> > > > > hexanies, and the
> > > > > eight others to eight tetrads. If we put all of these
> together,
> > we
> > > > > obtain the following scale of 19 notes, all of whose
> intervals
> > are
> > > > > superparticular ratios:
> > > > >
> > > > I know I'm lagging behind, but I need to ask where the
> remaining
> > 5
> > > > notes come from (14 + 5). Thanks
> > >
> > > Okay -heres what I know for sure. The 19 tones include
> > 3,5,7,15,21,35
> > > hexany, all divided by 5 and 7. This makes 11 tones, leaving 8.
I
> > > can't find any pattern to the 8 remaining however. (Are these
the
> 8
> > > tetrads?). I also discovered that the 19 tones are every
> combination
> > > of -1, 0 and 1 except for (1,1,1) (-1,-1,-1) triples and every
> > double
> > > of 1,1,0 and -1,-1,0. I guess what I am saying is that I
> understand
> > > hexanies but don't know what makes a tetrad. Thanks
> > >
> > > Paul
> >
> > There are two types of tetrad. 1:3:5:7 is one, and 105:35:21:15 =
1/
> > (1:3:5:7) is the other.
>
> I know - but how does this translate to Gene's fractions. Are the
> eight tetrads (+-1,+-1,+-1)? But the problem with that is that
(1,1,1)
> for example doesn't appear in the list (105)
These are relative proportions, not absolute figures.
From: Paul G Hjelmstad (2004-02-11)
Subject: Re: Rhombic dodecahedron scale
--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> > <paul.hjelmstad@u...> wrote:
> > > --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> > > <paul.hjelmstad@u...> wrote:
> > > > --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
> > > <gwsmith@s...>
> > > > wrote:
> > > > > Here is a scale which arose when I was considering adding
to
> > the
> > > > seven
> > > > > limit lattices web page. A Voronoi cell for a lattice is
> every
> > > point
> > > > > at least as close (closer, for an interior point) to a
> paricular
> > > > > vertex than to any other vertex. The Voronoi cells for the
> > > > > face-centered cubic
> > > > > lattice of 7-limit intervals is the rhombic dodecahedron
with
> > the
> > > 14
> > > > > verticies (+-1 0 0), (0 +-1 0), (0 0 +-1), (+-1/2 +-1/2 +-
> 1/2).
> > > > These
> > > > > fill the whole space, like a bee's honeycomb. The Delaunay
> > celles
> > > > of a
> > > > > lattice are the convex hulls of the lattice points closest
to
> a
> > > > > Voronoi cell vertex; in this case we get tetrahedra and
> > octahedra,
> > > > > which are the holes of the lattice, and are tetrads or
> > hexanies.
> > > The
> > > > > six (+-1 0 0) verticies of the Voronoi cell correspond to
six
> > > > > hexanies, and the
> > > > > eight others to eight tetrads. If we put all of these
> together,
> > we
> > > > > obtain the following scale of 19 notes, all of whose
> intervals
> > are
> > > > > superparticular ratios:
> > > > >
> > > > I know I'm lagging behind, but I need to ask where the
> remaining
> > 5
> > > > notes come from (14 + 5). Thanks
> > >
> > > Okay -heres what I know for sure. The 19 tones include
> > 3,5,7,15,21,35
> > > hexany, all divided by 5 and 7. This makes 11 tones, leaving 8.
I
> > > can't find any pattern to the 8 remaining however. (Are these
the
> 8
> > > tetrads?). I also discovered that the 19 tones are every
> combination
> > > of -1, 0 and 1 except for (1,1,1) (-1,-1,-1) triples and every
> > double
> > > of 1,1,0 and -1,-1,0. I guess what I am saying is that I
> understand
> > > hexanies but don't know what makes a tetrad. Thanks
> > >
> > > Paul
> >
> > There are two types of tetrad. 1:3:5:7 is one, and 105:35:21:15 =
1/
> > (1:3:5:7) is the other.
>
> I know - but how does this translate to Gene's fractions. Are the
> eight tetrads (+-1,+-1,+-1)? But the problem with that is that
(1,1,1)
> for example doesn't appear in the list (105) and neither do any
> (1,1,0) or (-1,-1,0) (Notating monzos here, not vertices)
Gene? What is the pattern to the remaining 8 fractions: (monzos which
are (0,0,-1),(-1,0,1),(-1,0,0),(1,-1,-1),(0,-1,0),(-1,1,0),(-1,-1,1),
and (-1,1,-1)) Do they represent "the eight tetrads" in someway? Am I
misunderstanding something...Once again, this is after eliminating
fractions based on 3,5,7,15,21,35 /5 and /7. Thanks
From: Paul G Hjelmstad (2004-02-11)
Subject: Re: Rhombic dodecahedron scale
--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> Here is a scale which arose when I was considering adding to the
seven
> limit lattices web page. A Voronoi cell for a lattice is every point
> at least as close (closer, for an interior point) to a paricular
> vertex than to any other vertex. The Voronoi cells for the
> face-centered cubic
> lattice of 7-limit intervals is the rhombic dodecahedron with the 14
> verticies (+-1 0 0), (0 +-1 0), (0 0 +-1), (+-1/2 +-1/2 +-1/2).
These
> fill the whole space, like a bee's honeycomb. The Delaunay celles
of a
> lattice are the convex hulls of the lattice points closest to a
> Voronoi cell vertex; in this case we get tetrahedra and octahedra,
> which are the holes of the lattice, and are tetrads or hexanies. The
> six (+-1 0 0) verticies of the Voronoi cell correspond to six
> hexanies, and the
> eight others to eight tetrads. If we put all of these together, we
> obtain the following scale of 19 notes, all of whose intervals are
> superparticular ratios:
>
> ! rhomb.scl
> Union of Delauny cells for the rhombic dodecahedron Voronoi cell
> centered at (0 0 0)
> 19
> !
> 21/20
> 15/14
> 8/7
> 7/6
> 6/5
> 5/4
> 4/3
> 48/35
> 7/5
> 10/7
> 35/24
> 3/2
> 8/5
> 5/3
> 12/7
> 7/4
> 28/15
> 40/21
> 2
>
I notice a pattern in these fractions: The difference of primes
between num. and den. is 0 or 1, no prime goes beyond p^1. Is there
any significance to the fact that besides (0,0,0) there are 6 scale
members with difference of 0 and 12 with difference of 1? Can someone
put this in geometric terms? Thanks
Paul
From: Gene Ward Smith (2004-02-11)
Subject: Re: Rhombic dodecahedron scale
--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:
> I notice a pattern in these fractions: The difference of primes
> between num. and den. is 0 or 1, no prime goes beyond p^1. Is there
> any significance to the fact that besides (0,0,0) there are 6 scale
> members with difference of 0 and 12 with difference of 1? Can someone
> put this in geometric terms? Thanks
In geometric terms you have symmetrical scales defined by taking
everything inside a ball around a fixed center. If the center is the
unison, you get scales of size 1, 13, 19, 43, ... in that way. In
analytic terms, the generating function for the above problem is a
series (1+2q+2q^4+2q^9+ ... + 2q^n^2 + ...)^3 = 1+12q+6q^2+24q^3+...,
where the coefficient on the q^n term is the number of 7-limit
note-classes at a distance of sqrt(n) from the unison. Similar
generating functions can be defined for distance from the center of
tetrads, hexanies, or the midpoint of the 1-3 interval, with
corresponding scales.
From: Paul G Hjelmstad (2004-02-12)
Subject: Re: Rhombic dodecahedron scale
--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <paul.hjelmstad@u...> wrote:
>
> > I notice a pattern in these fractions: The difference of primes
> > between num. and den. is 0 or 1, no prime goes beyond p^1. Is
there
> > any significance to the fact that besides (0,0,0) there are 6
scale
> > members with difference of 0 and 12 with difference of 1? Can
someone
> > put this in geometric terms? Thanks
>
> In geometric terms you have symmetrical scales defined by taking
> everything inside a ball around a fixed center. If the center is the
> unison, you get scales of size 1, 13, 19, 43, ... in that way. In
> analytic terms, the generating function for the above problem is a
> series (1+2q+2q^4+2q^9+ ... + 2q^n^2 + ...)^3 =
1+12q+6q^2+24q^3+...,
> where the coefficient on the q^n term is the number of 7-limit
> note-classes at a distance of sqrt(n) from the unison. Similar
> generating functions can be defined for distance from the center of
> tetrads, hexanies, or the midpoint of the 1-3 interval, with
> corresponding scales.
Interesting. What's the 1-3 interval?
From: Gene Ward Smith (2004-02-12)
Subject: Re: Rhombic dodecahedron scale
--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:
> Interesting. What's the 1-3 interval?
Twelfth, fifth, etc.
From: Paul G Hjelmstad (2004-02-12)
Subject: Re: Rhombic dodecahedron scale
--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> > <paul.hjelmstad@u...> wrote:
> >
> > > I notice a pattern in these fractions: The difference of primes
> > > between num. and den. is 0 or 1, no prime goes beyond p^1. Is
> there
> > > any significance to the fact that besides (0,0,0) there are 6
> scale
> > > members with difference of 0 and 12 with difference of 1? Can
> someone
> > > put this in geometric terms? Thanks
> >
> > In geometric terms you have symmetrical scales defined by taking
> > everything inside a ball around a fixed center. If the center is
the
> > unison, you get scales of size 1, 13, 19, 43, ... in that way. In
> > analytic terms, the generating function for the above problem is a
> > series (1+2q+2q^4+2q^9+ ... + 2q^n^2 + ...)^3 =
> 1+12q+6q^2+24q^3+...,
*I get 1+6q+12q^2+8q^3+6q^4... am i missing something?*
> > where the coefficient on the q^n term is the number of 7-limit
> > note-classes at a distance of sqrt(n) from the unison. Similar
> > generating functions can be defined for distance from the center
of
> > tetrads, hexanies, or the midpoint of the 1-3 interval, with
> > corresponding scales.
From: Gene Ward Smith (2004-02-12)
Subject: Re: Rhombic dodecahedron scale
--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:
In
> > > analytic terms, the generating function for the above problem is a
> > > series (1+2q+2q^4+2q^9+ ... + 2q^n^2 + ...)^3 =
> > 1+12q+6q^2+24q^3+...,
>
>
> *I get 1+6q+12q^2+8q^3+6q^4... am i missing something?*
Sorry, I wrote down the generating function for the number of tetrads,
not notes. Define
th(q) = 1+2 sum(n=1..infinity) q^n^2 = 1+2q+2q^4+2q^9+...
th01(q) = th(-q) = 1+2 sum((-1)^n q^n^2) = 1-2q+2q^4-2q^9+...
th10(q) = 2 sum q^((n+1/2)^2) = 2 q^(1/4) (1+q^2+q^6+...)
Then the generating function for 7-limit note-classes is
(th(q^(1/2))^3 + th01(q^(1/2))^3)/2
The generating function for balls centered on tetrads is
(th10(q^4)^3)/2 = 4q^3 + 12q^11 + 12q^19 + 16q^27 + ...
For balls centered on hexanies is
(th(q)^3 - th01(q)^3)/2 = 6q + 8q^3 + 24q^5 + 30q^9 + ...
This leads to scales of 4, 16, 30, 46 ... notes for the
tetrad-centered scales, which can be centered on either a minor or a
major tetrad, and 6, 14, 38, 68, ... for the hexany-centered scales.
th(q)
> > > where the coefficient on the q^n term is the number of 7-limit
> > > note-classes at a distance of sqrt(n) from the unison. Similar
> > > generating functions can be defined for distance from the center
> of
> > > tetrads, hexanies, or the midpoint of the 1-3 interval, with
> > > corresponding scales.
From: Paul G Hjelmstad (2004-02-12)
Subject: Re: Rhombic dodecahedron scale
--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <paul.hjelmstad@u...> wrote:
>
> In
> > > > analytic terms, the generating function for the above problem
is a
> > > > series (1+2q+2q^4+2q^9+ ... + 2q^n^2 + ...)^3 =
> > > 1+12q+6q^2+24q^3+...,
> >
> >
> > *I get 1+6q+12q^2+8q^3+6q^4... am i missing something?*
>
> Sorry, I wrote down the generating function for the number of
tetrads,
> not notes. Define
>
> th(q) = 1+2 sum(n=1..infinity) q^n^2 = 1+2q+2q^4+2q^9+...
> th01(q) = th(-q) = 1+2 sum((-1)^n q^n^2) = 1-2q+2q^4-2q^9+...
> th10(q) = 2 sum q^((n+1/2)^2) = 2 q^(1/4) (1+q^2+q^6+...)
>
> Then the generating function for 7-limit note-classes is
>
> (th(q^(1/2))^3 + th01(q^(1/2))^3)/2
>
> The generating function for balls centered on tetrads is
>
> (th10(q^4)^3)/2 = 4q^3 + 12q^11 + 12q^19 + 16q^27 + ...
>
> For balls centered on hexanies is
>
> (th(q)^3 - th01(q)^3)/2 = 6q + 8q^3 + 24q^5 + 30q^9 + ...
>
> This leads to scales of 4, 16, 30, 46 ... notes for the
> tetrad-centered scales, which can be centered on either a minor or a
> major tetrad, and 6, 14, 38, 68, ... for the hexany-centered scales.
>
> th(q)
> > > > where the coefficient on the q^n term is the number of 7-limit
> > > > note-classes at a distance of sqrt(n) from the unison. Similar
> > > > generating functions can be defined for distance from the
center
> > of
> > > > tetrads, hexanies, or the midpoint of the 1-3 interval, with
> > > > corresponding scales.
Thanks Gene. This is neat. It will keep me busy for awhile.
From: Paul G Hjelmstad (2004-02-12)
Subject: Re: Rhombic dodecahedron scale
--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >Here is a scale which arose when I was considering adding to the
seven
> >limit lattices web page. A Voronoi cell for a lattice is every
point
> >at least as close (closer, for an interior point) to a paricular
> >vertex than to any other vertex. The Voronoi cells for the
> >face-centered cubic
> >lattice of 7-limit intervals is the rhombic dodecahedron
>
> Something Fuller demonstrated, in his own tongue.
>
> >These
> >fill the whole space, like a bee's honeycomb.
>
> Isn't it also the dual to the FCC lattice (hmm, maybe dual isn't
> the right word here...)
>
> >The Delaunay celles of a
> >lattice are the convex hulls of the lattice points closest to a
> >Voronoi cell vertex; in this case we get tetrahedra and octahedra,
>
> Ah, that would be the 'dual' operation I was thinking it above.
> I saw a graphic of this on site about Fuller once.
>
> >which are the holes of the lattice, and are tetrads or hexanies.
The
> >six (+-1 0 0) verticies of the Voronoi cell
>
> *The* Voronoi cell? Which one do you mean?
>
> >correspond to six hexanies, and the eight others to eight tetrads.
> >If we put all of these together, we obtain the following scale of
> >19 notes, all of whose intervals are superparticular ratios:
>
Gene, is there a generating function for your scale of 19 notes, or
is it too complex for that? Could you tell me please how this relates
to the 14 points of a rhombic dodecahedron and how that is based on
6 hexanies and 8 tetrads? Thanks. Paul Hj
From: Gene Ward Smith (2004-02-12)
Subject: Re: Rhombic dodecahedron scale
--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> Thanks Gene. This is neat. It will keep me busy for awhile.
If you want to learn more about the Jacobi theta function, you can
read the Wikipedia article "Theta function", which must be a good one
since I wrote it. :)
From: Paul Erlich (2004-02-13)
Subject: Re: Rhombic dodecahedron scale
--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:
> Could you tell me please how this relates
> to the 14 points of a rhombic dodecahedron and how that is based on
> 6 hexanies and 8 tetrads? Thanks. Paul Hj
Paul, did you look at my paper as I suggested? All this stuff is only
3-dimensional, so many of us can best understand it through pictures.