Full thread (19 messages)
From: Gene Ward Smith (2004-02-06)
Subject: Ennealimmal[45] as a chord block
The major chord with root 21/20 is [0,2,0] in the 7-limit chord
lattice, that with root 2401/2400 is [2,-3,-3], and that with
4375/4374 is [6,-6,-3]. If I form the block in the lattice centered at
[0,0,0] and with the inverse matrix coordinates running -1 < coordinat
<= 1, I get a block with 127 chords, consisting of 191 notes. Reducing
this by ennealimmal gives Ennealimmal[45]. Below is a TM reduced JI
scale corresponding to Ennealimmal[45] which people with an aversion
to tempering could use instead, not to mention people who just plain
like the idea. Sticking it in Scala shows 7 of the 18 major tetrads
are very slightly tempered, and the other 11 are pure JI. In the case
of minor tetrads, we have eight tempered ones, and ten untempered. The
maximum error is 2401/2400, or 0.721 cents, in both cases.
We also have 27 supermajor and 18 subminor tetrads, defined as
1--9/7--3/2--9/5 and 1--7/6--3/2--5/3 (or 6:7:9:10 for those who
prefer.) Twelve supermajor and eight subminor tetrads are tempered.
Finding ways of harmonizing things is apparently not a problem.
! enn45.scl
Detempered Ennealimmal[45], TM reduced
45
!
49/48
25/24
21/20
15/14
27/25
54/49
9/8
245/216
81/70
7/6
25/21
175/144
49/40
5/4
63/50
9/7
21/16
250/189
27/20
49/36
25/18
486/343
10/7
35/24
72/49
3/2
49/32
54/35
63/40
100/63
81/50
81/49
5/3
245/144
12/7
7/4
25/14
9/5
90/49
50/27
189/100
27/14
35/18
125/63
2
From: Gene Ward Smith (2004-02-06)
Subject: Re: Ennealimmal[45] as a chord block
--- In [email protected], "Gene Ward Smith" <gwsmith@s...>
wrote:
> The major chord with root 21/20 is [0,2,0] in the 7-limit chord
> lattice, that with root 2401/2400 is [2,-3,-3], and that with
> 4375/4374 is [6,-6,-3]. If I form the block in the lattice centered at
> [0,0,0] and with the inverse matrix coordinates running -1 < coordinat
> <= 1, I get a block with 127 chords, consisting of 191 notes.
Arrgh, as Albert the Alligator once said. Blocks are supposed to run
from -1/2 to 1/2. That gives me something more reasonable, 18 chords,
leading to 48 notes, which reduces to 45 notes after tempering by
2401/2400 and 4375/4374.
The 18 chords are
[[-2, 3, 1], [1, 0, -1], [-1, 0, 1], [1, 1, -1], [-2, 4, 1],
[2, -2, -1], [-1, 4, 0], [-1, 1, 1], [1, -3, 0], [0, 0, 0],
[-2, 0, 2], [0, 1, 0], [1, -2, 0], [2, -3, -1], [-2, 1, 2],
[2, 0, -2], [-1, 3, 0], [2, 1, -2]]
The 48 notes as follows; as a scale this has two highly cheesy
2401/2400 steps, and one even cheesier 4375/4374, all of which
ennealimmal exterminates.
[1, 49/48, 25/24, 21/20, 15/14, 27/25, 54/49, 441/400, 9/8, 567/500,
125/108, 7/6, 25/21, 343/288, 175/144, 49/40, 5/4, 63/50, 9/7, 21/16,
1323/1000,27/20, 49/36, 25/18, 567/400, 343/240, 35/24, 72/49, 3/2,
49/32, 54/35, 63/40,100/63, 81/50, 175/108, 1323/800, 5/3, 245/144,
12/7, 7/4, 343/192, 9/5, 90/49, 50/27, 189/100, 27/14, 35/18, 125/63]
From: Gene Ward Smith (2004-02-07)
Subject: Re: Ennealimmal[45] as a chord block
It seems to work better to forget about 2401/2400 and 4375/4374 to
start out with, and use generators of [0 1 0] and [1 0 -1]. The first
changes major to minor and vice-versa, but two of them together are
the same chord transposed 21/20 up; we have
[0 0 0] ~ {1, 5/4, 3/2, 7/4}
[0 1 0] ~ {21/20, 21/16, 3/2, 7/4}
[0 2 0] ~ {21/20, 21/16, 63/40, 147/80} = (21/20){1, 5/4, 3/2, 7/4}
[1 0 -1] is just transposing up a 7/6. Now we have two ennealimmal
generators, since (7/6)^9 ~ 4 can work in place of (27/25)^9 ~ 2.
If we take [i,j,-i] for i from -4 to 4, j from -1 to 0, we get
Ennealimmal[36] on reducing. Note that [9 0 -9] is the same as [0 0 0]
if we are tempering with ennealimmal. If we take j from -1 to 1, we
get Ennealimmal[45] instead.
We have [9, 0, -9] = 2[2 3 -3] + [5 -6 -3], where the last two are
transposition by 2401/2400 and 4375/4374 respectively, so we may use
the basis [9, 0, -9] and [2 3 -3] for lattice equivalencies. The
determinant of the two generators [0 1 0] and [1 0 -1] together with
[2 3 -3] is one, inverting the unimodular matrix with these rows gives
a transformation from the lattice basis for tetrads I have been using
to one in terms of the generators, plus the 2401/2400 transposition,
given by the matrix with columns [3 1 3], [3 0 2] [-1 0 -1]. We may
therefore use this to change 7-limit tetrads to a basis suitable to
ennealimmal, and drop the last coordinate.
Changing basis in this way we have for instance
Major tetrad on 3/2: [4, 2, -1]
Minor tetrad on 1: [-3, -3, 1]
Major tetrad on 5/4: [6, 5, -2]
and so forth. Dropping the last coodinate tells us where chord should
be placed on the [0 1 0], [1 0 -1] plane, which gets translated as
[1 0 0] and [0, 1, 0] respectively. We can then wrap the plane into a
cylinder by [9 0 -9] ~ [0 0 0], which translates as [0 9 0] ~ [0 0 0].
We then see for instance that the major tetrad on 5/4 could be called
[6, -4] after losing the last coordinate and reducing the second to
the range -4 to 4.
All of which, of course, could be put into Tonalsoft's 1.1 release, in
theory, as a way of doing effective 7-limit JI chord noodling.
From: Gene Ward Smith (2004-02-07)
Subject: Re: Ennealimmal[45] as a chord block
--- In [email protected], "Gene Ward Smith" <gwsmith@s...>
wrote:
> If we take [i,j,-i] for i from -4 to 4, j from -1 to 0, we get
> Ennealimmal[36] on reducing. Note that [9 0 -9] is the same as [0 0 0]
> if we are tempering with ennealimmal.
If we take j from -1 to 1, we
> get Ennealimmal[45] instead.
j from -1 to 2
From: Paul G Hjelmstad (2004-02-09)
Subject: Re: Ennealimmal[45] as a chord block
--- In [email protected], "Gene Ward Smith" <gwsmith@s...>
wrote:
> The major chord with root 21/20 is [0,2,0] in the 7-limit chord
> lattice, that with root 2401/2400 is [2,-3,-3], and that with
> 4375/4374 is [6,-6,-3].
Hate to look dumb, but need to ask how the numbers in brackets are
calculated. [0,1,0] sends major to minor, but how?
Thanx!
Paul
From: Gene Ward Smith (2004-02-09)
Subject: The seven-limit lattices
--- In [email protected], "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:
> Hate to look dumb, but need to ask how the numbers in brackets are
> calculated. [0,1,0] sends major to minor, but how?
Instead of writing a response to this, I made a new web page:
http://66.98.148.43/~xenharmo/sevlat.htm
From: Paul G Hjelmstad (2004-02-09)
Subject: Re: The seven-limit lattices
--- In [email protected], "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In [email protected], "Paul G Hjelmstad"
> <paul.hjelmstad@u...> wrote:
>
> > Hate to look dumb, but need to ask how the numbers in brackets
are
> > calculated. [0,1,0] sends major to minor, but how?
>
> Instead of writing a response to this, I made a new web page:
>
> http://66.98.148.43/~xenharmo/sevlat.htm
Most excellent. Thanks.
From: Paul G Hjelmstad (2004-02-09)
Subject: Re: The seven-limit lattices
--- In [email protected], "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In [email protected], "Paul G Hjelmstad"
> <paul.hjelmstad@u...> wrote:
>
> > Hate to look dumb, but need to ask how the numbers in brackets
are
> > calculated. [0,1,0] sends major to minor, but how?
>
> Instead of writing a response to this, I made a new web page:
>
> http://66.98.148.43/~xenharmo/sevlat.htm
If [a b c] is any triple of integers, then it represents the major
tetrad with root 3^((-a+b+c)/2) 5^((a-b+c)/2) 7^((a+c-c)/2) if a+b+c
7^((a+b-c)/2)?
is even, and the minor tetrad with root 3^((-1-a+b+c)/2) 5^((1+a-
3^((1-a+b+c/2)?
b+c)/2 7^((1+a+b-c)/2) if a+b+c is odd. Each unit cube corresponds to
a stellated hexany, or tetradekany, or dekatesserany, though chord
cube would be less of a mouthful.
How is it that unit cube have 14 tones? Thanks
From: Gene Ward Smith (2004-02-10)
Subject: Re: The seven-limit lattices
--- In [email protected], "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:
> How is it that unit cube have 14 tones? Thanks
The cube is a cube of chords; it has 8 chords but 14 notes.
I've added a bunch, and there is still more that could be said about
all this.
From: Paul G Hjelmstad (2004-02-10)
Subject: Re: The seven-limit lattices
--- In [email protected], "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In [email protected], "Paul G Hjelmstad"
> <paul.hjelmstad@u...> wrote:
>
> > How is it that unit cube have 14 tones? Thanks
>
> The cube is a cube of chords; it has 8 chords but 14 notes.
>
> I've added a bunch, and there is still more that could be said
about
> all this.
8 chords at the vertices (as opposed to notes)? Even after spending
the day looking at the web page, I grasp only parts of it. A few
quick questions:
Para. 1 Holes - are these the areas between sides/vertices?
Para. 2 How are distances scaled up by sqrt(2)? Does the
cubeoctahedron have 12 vertices? In Para. 1 you list the 12 lattice
points in a different form. Why is this new form preferable?
Para. 3 Actually, fine
Para. 4 Dumb question: Are chords or notes at the vertices
Para. 5 Fine
Looking forward to your additions/responses
Paul
From: Paul G Hjelmstad (2004-02-10)
Subject: Re: The seven-limit lattices
--- In [email protected], "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:
> --- In [email protected], "Gene Ward Smith"
<gwsmith@s...>
> wrote:
> > --- In [email protected], "Paul G Hjelmstad"
> > <paul.hjelmstad@u...> wrote:
> >
> > > How is it that unit cube have 14 tones? Thanks
> >
> > The cube is a cube of chords; it has 8 chords but 14 notes.
> >
> > I've added a bunch, and there is still more that could be said
> about
> > all this.
>
> 8 chords at the vertices (as opposed to notes)? Even after spending
> the day looking at the web page, I grasp only parts of it. A few
> quick questions:
>
> Para. 1 Holes - are these the areas between sides/vertices?
> Para. 2 How are distances scaled up by sqrt(2)? Does the
> cubeoctahedron have 12 vertices? In Para. 1 you list the 12 lattice
> points in a different form. Why is this new form preferable?
> Para. 3 Actually, fine
> Para. 4 Dumb question: Are chords or notes at the vertices
> Para. 5 Fine
>
> Looking forward to your additions/responses
>
> Paul
Never mind. I saw your additions, and read them. I think I've figured
out the perpendicular-thing. I also take it that chords are at the
vertices and notes are the faces? (It's funny, once I write down my
own questions, I seem to be better at answering them myself :) )
From: Paul G Hjelmstad (2004-02-10)
Subject: Re: The seven-limit lattices
--- In [email protected], "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In [email protected], "Paul G Hjelmstad"
> <paul.hjelmstad@u...> wrote:
>
> > How is it that unit cube have 14 tones? Thanks
>
> The cube is a cube of chords; it has 8 chords but 14 notes.
>
> I've added a bunch, and there is still more that could be said
about
> all this.
Actually, I have a few questions. I've hopefully thought this through
so that my questions are good ones.
1. The 14 notes of the stellated hexany. Do these correspond to the
sides of the cubeoctohedran? (Or the points on a D3 face centered
lattice)...
2. Where in the cubeoctohedran are the tetrahedra and octahedra?
Could you please give an example of a hexany...
3. I understand the change of of basis in paragraph 2. However
in paragraph 3, "In this new coordinate system" you have (000),(100),
(010) and (001). How are these related to the vertices above?
4. Paragraph 7. Where do the negatives in the (1/2, 1/2, 1/2) terms
come from?
Thanks
From: Paul Erlich (2004-02-10)
Subject: Re: The seven-limit lattices
--- In [email protected], "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:
> --- In [email protected], "Gene Ward Smith"
<gwsmith@s...>
> wrote:
> > --- In [email protected], "Paul G Hjelmstad"
> > <paul.hjelmstad@u...> wrote:
> >
> > > How is it that unit cube have 14 tones? Thanks
> >
> > The cube is a cube of chords; it has 8 chords but 14 notes.
> >
> > I've added a bunch, and there is still more that could be said
> about
> > all this.
>
> Actually, I have a few questions. I've hopefully thought this
through
> so that my questions are good ones.
>
> 1. The 14 notes of the stellated hexany. Do these correspond to the
> sides of the cubeoctohedran? (Or the points on a D3 face centered
> lattice)...
>
> 2. Where in the cubeoctohedran are the tetrahedra and octahedra?
> Could you please give an example of a hexany...
Hi Paul, you may want to read this paper:
http://lumma.org/tuning/erlich/erlich-tFoT.pdf
through the pictures of the hexany and stellated hexany.
-Paul
P.S. Carl -- this used to come up in Google, but no longer does :(
From: Gene Ward Smith (2004-02-10)
Subject: Re: The seven-limit lattices
--- In [email protected], "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:
> 1. The 14 notes of the stellated hexany. Do these correspond to the
> sides of the cubeoctohedran? (Or the points on a D3 face centered
> lattice)...
They correspond to the 14 verticies of a rhombic dodecahedron, which
I'd better add to the page.
> 2. Where in the cubeoctohedran are the tetrahedra and octahedra?
> Could you please give an example of a hexany...
I'm not sure what your question means re the cubeoctahedron; it has 12
verticies corresponding to the 12 intervals nearest the unison, and
forming the 7-limit consonances with it. We've got a 14-note scale of
the rhombic dodecahedron/stellated hexany type, and a 13-note scale of
13 notes in a ball of radius one around the unison.
{3,5,7,15,21,35} is the canonical hexany from Wilson's combinatorial
point of view. Dividing it by 7 and reducing to an octave gives
[1, 15/14, 5/4, 10/7, 3/2, 12/7]; dividing by 5 instead gives
[1, 21/20, 6/5, 7/5, 3/2, 7/4]; these are the sorts of hexanies one
finds (and I found) geometrically, when looking at the ones with the
unison as a lattice point.
> 3. I understand the change of of basis in paragraph 2. However
> in paragraph 3, "In this new coordinate system" you have (000),(100),
> (010) and (001). How are these related to the vertices above?
That's a mistake; I'll correct it.
> 4. Paragraph 7. Where do the negatives in the (1/2, 1/2, 1/2) terms
> come from?
Those are the basis elements for the lattice of mappings.
From: Paul Erlich (2004-02-10)
Subject: Re: The seven-limit lattices
--- In [email protected], "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In [email protected], "Paul G Hjelmstad"
> <paul.hjelmstad@u...> wrote:
>
> > 1. The 14 notes of the stellated hexany. Do these correspond to
the
> > sides of the cubeoctohedran? (Or the points on a D3 face centered
> > lattice)...
>
> They correspond to the 14 verticies of a rhombic dodecahedron, which
> I'd better add to the page.
>
> > 2. Where in the cubeoctohedran are the tetrahedra and octahedra?
> > Could you please give an example of a hexany...
>
> I'm not sure what your question means re the cubeoctahedron; it has
12
> verticies corresponding to the 12 intervals nearest the unison, and
> forming the 7-limit consonances with it. We've got a 14-note scale
of
> the rhombic dodecahedron/stellated hexany type, and a 13-note scale
of
> 13 notes in a ball of radius one around the unison.
The 7-limit diamond -- this, too, is depicted in my paper.
From: Paul G Hjelmstad (2004-02-10)
Subject: Re: The seven-limit lattices
--- In [email protected], "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In [email protected], "Paul G Hjelmstad"
> <paul.hjelmstad@u...> wrote:
>
> > 2. Where in the cubeoctohedran are the tetrahedra and octahedra?
> > Could you please give an example of a hexany...
>
> I'm not sure what your question means re the cubeoctahedron;
I guess I don't understand what "shallow holes" and "deep holes" are
> > 4. Paragraph 7. Where do the negatives in the (1/2, 1/2, 1/2)
terms
> > come from?
>
> Those are the basis elements for the lattice of mappings.
Okay. Thanks. What is the significance of these forming a dot product
with (1,4,10) and equalling "1"?
From: Paul G Hjelmstad (2004-02-10)
Subject: Re: The seven-limit lattices
--- In [email protected], "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:
> --- In [email protected], "Gene Ward Smith"
<gwsmith@s...>
> wrote:
> > --- In [email protected], "Paul G Hjelmstad"
> > <paul.hjelmstad@u...> wrote:
> >
> > > 2. Where in the cubeoctohedran are the tetrahedra and octahedra?
> > > Could you please give an example of a hexany...
> >
> > I'm not sure what your question means re the cubeoctahedron;
>
> I guess I don't understand what "shallow holes" and "deep holes"
are
>
> > > 4. Paragraph 7. Where do the negatives in the (1/2, 1/2, 1/2)
> terms
> > > come from?
> >
> > Those are the basis elements for the lattice of mappings.
>
> Oops. I meant, what is the significance of the dot product of
(13/2 7/2 -5/2) and (0,1,1) equalling 1 "as expected"
From: Gene Ward Smith (2004-02-11)
Subject: Re: The seven-limit lattices
--- In [email protected], "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:
> --- In [email protected], "Gene Ward Smith" <gwsmith@s...>
> wrote:
> > --- In [email protected], "Paul G Hjelmstad"
> > <paul.hjelmstad@u...> wrote:
> >
> > > 2. Where in the cubeoctohedran are the tetrahedra and octahedra?
> > > Could you please give an example of a hexany...
> >
> > I'm not sure what your question means re the cubeoctahedron;
>
> I guess I don't understand what "shallow holes" and "deep holes" are
A hole is centered at a point which is a local maximum for distance
from the lattice; a deep hole is a global maximum.
> > > 4. Paragraph 7. Where do the negatives in the (1/2, 1/2, 1/2)
> terms
> > > come from?
> >
> > Those are the basis elements for the lattice of mappings.
>
> Okay. Thanks. What is the significance of these forming a dot product
> with (1,4,10) and equalling "1"?
(1 4 10) is the meantone mapping to generator steps, the 1 is 1
generator step to get to a fifth.
From: Carl Lumma (2004-02-11)
Subject: Re: [tuning-math] Re: The seven-limit lattices
>P.S. Carl -- this used to come up in Google, but no longer does :(
Dunno why that would be. I'll keep my feelers out on it.
-Carl