Yahoo Tuning Groups Ultimate Backup tuning Re: SCALA scales in FTS

--- In tuning@y..., "Robert Walker" <robertwalker@n...> wrote:

> Hi there,
>
> I've added a button to latest update of the FTS beta preview:

Are you or is anyone thinking of working on retuning software which
will allow a general sort of retuning? One reason I've been asking
about it is that I'd like to create some midi examples of
automorphism groups acting on music to illustrate some things I would
like to discuss on the tuning-math group, and your hexany pieces
would be ideal for some purposes. If you would be interested in what
happens when the group of the octahedron is applied to it (24
elements) or even the full orthogonal group (48 different versions of
each of your hexany pieces) perhaps you could send something in an
ascii version which would allow a perspicuous editing and alteration
of the tuning.

I wrote 3-part hexany canon in the late 70's in order to do this very
thing, but it's long gone and I like what you've done better. :)

🔗Paul Erlich <paul@stretch-music.com>

--- In tuning@y..., genewardsmith@j... wrote:
> --- In tuning@y..., "Robert Walker" <robertwalker@n...> wrote:
>
> > Hi there,
> >
> > I've added a button to latest update of the FTS beta preview:
>
> Are you or is anyone thinking of working on retuning software which
> will allow a general sort of retuning? One reason I've been asking
> about it is that I'd like to create some midi examples of
> automorphism groups acting on music to illustrate some things I
would
> like to discuss on the tuning-math group, and your hexany pieces
> would be ideal for some purposes. If you would be interested in
what
> happens when the group of the octahedron is applied to it (24
> elements)

This I understand.

> or even the full orthogonal group (48 different versions of
> each of your hexany pieces)

This means adding reflections to the process above?

> perhaps you could send something in an
> ascii version which would allow a perspicuous editing and
alteration
> of the tuning.

This sounds like the kind of technique Kraig Grady and Daniel Wolf
have talked about for the Eikosany. A piece in the Eikosany (which is
a 3-out-of-6 Combination Product Set, much like the 2-out-of-4 hexany)
would have 720 variations. Maybe 1440 if you count reflections?
That's one variation a minute for 24 hours.

🔗genewardsmith@juno.com

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:

> > or even the full orthogonal group (48 different versions of
> > each of your hexany pieces)

> This means adding reflections to the process above?

Correct; simply invoke the reciprocal map x |--> 1/x and translate.

> This sounds like the kind of technique Kraig Grady and Daniel Wolf
> have talked about for the Eikosany.

However, in the 7-limit I add the condition that octaves and the 4-et
are invariant. In other words, for the eigenvalue 1, [1 0 0 0] is a
left eigenvector and

[ 4]
[ 6]
[ 9]
[11]

is a right eigenvector. This means octave relationships don't change
and the melody does not change wildly. Modulo 4, [4,6,9,11] becomes
[0,2,1,3] so there is one representative for each prime mod 4 (the
3-et in the 5-limit shares this useful property.) Hence we can make
this work very nicely, and something much like it works in the 5-et.
Further discussion could perhaps migrate to the other list.

🔗Paul Erlich <paul@stretch-music.com>

--- In tuning@y..., genewardsmith@j... wrote:
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
>
> > > or even the full orthogonal group (48 different versions of
> > > each of your hexany pieces)
>
> > This means adding reflections to the process above?
>
> Correct; simply invoke the reciprocal map x |--> 1/x and translate.
>
> > This sounds like the kind of technique Kraig Grady and Daniel
Wolf
> > have talked about for the Eikosany.
>
> However, in the 7-limit I add the condition that octaves and the 4-
et
> are invariant. In other words, for the eigenvalue 1, [1 0 0 0] is a
> left eigenvector and
>
> [ 4]
> [ 6]
> [ 9]
> [11]
>
> is a right eigenvector. This means octave relationships don't
change
> and the melody does not change wildly. Modulo 4, [4,6,9,11] becomes
> [0,2,1,3] so there is one representative for each prime mod 4 (the
> 3-et in the 5-limit shares this useful property.) Hence we can make
> this work very nicely, and something much like it works in the 5-
et.
> Further discussion could perhaps migrate to the other list.

Please do so. I'd like to understand this.

🔗genewardsmith@juno.com

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:

> This sounds like the kind of technique Kraig Grady and Daniel Wolf
> have talked about for the Eikosany. A piece in the Eikosany (which
is
> a 3-out-of-6 Combination Product Set, much like the 2-out-of-4
hexany)
> would have 720 variations. Maybe 1440 if you count reflections?
> That's one variation a minute for 24 hours.

Well, maybe, sort of. What I was talking about was a map not on
scales or pitch classes, but actual pitches; however if you are
considering n primes you end up transforming (n-1)-simplexes
(triangles, tetrahedrons etc.) in (n-1) the (n-1)-dimensional space
of pitch classes before proceeding.

Suppose we start with 1, 3 and 5, and permute them by 1->3->5->1. We
can write this as the matrix

[0 1 0]
[0 0 1]
[1 0 0],

which leaves fixed the vector [1 1 1] corresponding to 15; the action
is therefore reducible. We want 1 to remain fixed, so after permuting
the vertices of the 1-3-5 triangle, we divide by 3; and so get
1->1, 3->5/3, 5->1/3. In matrix form, this is

[1 0 0]
[0 -1 1]
[0 -1 0].

When we do this with all n! permutations in the symmetric group S3 on
n elements, we always get a decomposition of this kind. The
(n-1)x(n-1) matrix so obtained cannot be reduced futher, even over
the complex numbers; it is therefore called an irreducible
representation of Sn. For most purposes we stop here, but for music
we must go onward and lift the irreducible representation so obtained
as best we can to include octaves. It then is no longer irreducible,
but we don't want it to be--we want octaves to be fixed, or else to
invert. We therefore want to do one of two things--either fix 2 and
send 3 to something approximately 3, 5 to an approximation of 5 and
so forth, or we want to send 2 to 1/2, 3 to about 1/3 and so on.
Ideally, we want the matrix that results to be of finite order, but
that is of less significance; we also might not want the harmony to
collpase one of the primes out of existence or do other untoward
things which we can prevent by making it have determinant +-1.

If we apply these principles to the above matrix, we see that sending
2->2, 3->10/3 and 5->16/3 fits the bill; we get the following matrix

[1 0 0]
[1 -1 1]
[4 -1 0].

This is a rotation of the hexagon surrounding 1/1 (3,5,5/3,1/3, 1/5,
3/5) which Paul can no doubt give a name to by 120 degrees. We can
get a 6-cycle by taking the negative of the above matrix,
corresponding to 2->1/2, 3->3/20, 5->3/16, giving us a rotation of 60
degrees. We now want to flip the triangle, which we can do by the
permutation 1->3, 3->1, 5->5. If reduce the matrix as before, we
divide by 3 and find the irreducible representation is

[-1 0]
[-1 1],

which sends 1->1, 3->1/3, 5->5/3, which we can lift to a
transformation including octaves by

[-1 0 0]
[ 0 -1 0]
[-3 -1 1],

which translates to 2->1/2, 3->1/3, 5->5/24. If we take instead minus
this matrix, we get 2->2, 3->3, 5->25/4, the major <--> minor
transformation. We now have a group of order 12 which can be seen as
the dihedral group D6--the isometries of a hexagon. Each element
either preserves or inverts octaves, and correspondingly either keeps
the 3-et

[3]
[5]
[7]

invariant, or inverts it. Note that like all of these groups, it acts
on more than just a single triad or a single hexagon--the group
elements send 5-limit music to 5-limit music, octaves to octaves or
inverted ocatves, and triads to triads (not necesarily in the same
position--that is root, first inversion and second inversion are
permuted.)

If we want to do something corresponding in the 11-limit, we run into
the difficulty that 3^2 = 9 is, after all, smaller than 11. There are
various approaches to this problem, the simplest being to ignore it.
If we take that line, we might began by looking at the 5-cycle
permutation 1->3->5->7->11->1. The element in the irreducible
representation of S5 corresponding to this we may obtain by dividing
everything transformed by the above by 3 (so that as usual 1 is sent
to 1), giving us the matrix

[-1 1 0 0]
[-1 0 1 0]
[-1 0 0 1]
[-1 0 0 0]

We can lift this to include 2 by setting 2->2, 3->10/3, 5->14/3,
7->22/3, 11->32/3; giving us the matrix

[1 0 0 0 0]
[1 -1 1 0 0]
[1 -1 0 1 0]
[1 -1 0 0 1]
[5 -1 0 0 0]

This is of order five, and it preserves the vector

[ 5]
[ 8]
[11]
[14]
[17]

The full group one gets in this way is in general 2.Sn, of order 2n!;
in this case that means a group of order 240. We could certainly work
out all the details of the best liftings and try it out if there was
interest.

🔗Paul Erlich <paul@stretch-music.com>

--- In tuning@y..., genewardsmith@j... wrote:
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
>
> > This sounds like the kind of technique Kraig Grady and Daniel
Wolf
> > have talked about for the Eikosany. A piece in the Eikosany
(which
> is
> > a 3-out-of-6 Combination Product Set, much like the 2-out-of-4
> hexany)
> > would have 720 variations. Maybe 1440 if you count reflections?
> > That's one variation a minute for 24 hours.
>
> Well, maybe, sort of. What I was talking about was a map not on
> scales or pitch classes, but actual pitches; however if you are
> considering n primes

Caution: non-prime odd numbers are usually used in the Eikosany and
similar . . .

> If we want to do something corresponding in the 11-limit, we run
into
> the difficulty that 3^2 = 9 is, after all, smaller than 11. There
are
> various approaches to this problem, the simplest being to ignore
it.
> The full group one gets in this way is in general 2.Sn, of order
2n!;
> in this case that means a group of order 240. We could certainly
work
> out all the details of the best liftings and try it out if there
was
> interest.

Can't absorb it all right now . . . Let's forget about the "lifting"
for the moment, since Kraig Grady and Daniel Wolf weren't interested
in preserving melodic contour in this operation. Do you also get 1440
variations, mapping all simplices to other, similar simplices within
the Eikosany?

Re: SCALA scales in FTS

🔗Robert Walker <robertwalker@ntlworld.com>

🔗genewardsmith@juno.com

🔗Paul Erlich <paul@stretch-music.com>

🔗genewardsmith@juno.com

🔗Paul Erlich <paul@stretch-music.com>

🔗genewardsmith@juno.com

🔗Paul Erlich <paul@stretch-music.com>

🔗genewardsmith@juno.com

🔗Paul Erlich <paul@stretch-music.com>

🔗genewardsmith@juno.com

🔗Paul Erlich <paul@stretch-music.com>

🔗Robert Walker <robertwalker@ntlworld.com>