Comma reduction?

Well, this might seem like a dumb question, but I need to ask it.
When generating commas from a linear temperament, such as 12&19,
you obtain 1 comma in the 5-limit, 2 commas in the 7-limit, 3 commas
in the 11-limit and so on. Gene shew me how to generate commas from
wedgies, to obtain the subgroup commas, but mentioned that these
results are not necc. linearly independent, and something about using
Hermite reduction to simplify all of this..

Are the 2 commas in the 7-limit always linearly independent? How
are they generated, (from wedgies OR matrices)? Also, was told
that the complement of a wedge product in the 5-limit is the same
as the cross-product, how does this work in the 7-limit?

I know, lots of questions. Thanks!

Paul

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:

> Well, this might seem like a dumb question, but I need to ask it.
> When generating commas from a linear temperament, such as 12&19,
> you obtain 1 comma in the 5-limit, 2 commas in the 7-limit, 3
commas
> in the 11-limit and so on. Gene shew me how to generate commas from
> wedgies, to obtain the subgroup commas, but mentioned that these
> results are not necc. linearly independent, and something about
using
> Hermite reduction to simplify all of this..
>
> Are the 2 commas in the 7-limit always linearly independent?

Yes, they are never 'collinear'.

> How
> are they generated, (from wedgies OR matrices)?

You can pick them off the tree. We've been looking at some of
the 'fruits' here.

> Also, was told
> that the complement of a wedge product in the 5-limit is the same
> as the cross-product, how does this work in the 7-limit?

There's a 4-d cross product, but all I know is that when switching
from val (bra vector) basis to monzo (ket vector) basis, the wedge
product reverses order, and some :( of the signs change . . .

>
> I know, lots of questions. Thanks!
>
> Paul

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:

> > Are the 2 commas in the 7-limit always linearly independent?
>
> Yes, they are never 'collinear'.

By definition of a 7-limit linear temperament.

> > How
> > are they generated, (from wedgies OR matrices)?
>
> You can pick them off the tree. We've been looking at some of
> the 'fruits' here.

Trees don't work for me. You can get them from direct comma searches
or extract them out of temperaments, etc.

> > Also, was told
> > that the complement of a wedge product in the 5-limit is the same
> > as the cross-product, how does this work in the 7-limit?

The complement of a 2-val is a 2-monzo, and vice-versa, which just
involves reordering.

~<<||l1 l2 l3 l4 l5 l6|| = <<l6 -l5 l4 l3 -l2 l1||

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
>
> > > Are the 2 commas in the 7-limit always linearly independent?
> >
> > Yes, they are never 'collinear'.
>
> By definition of a 7-limit linear temperament.
>
> > > How
> > > are they generated, (from wedgies OR matrices)?
> >
> > You can pick them off the tree. We've been looking at some of
> > the 'fruits' here.
>
> Trees don't work for me. You can get them from direct comma searches
> or extract them out of temperaments, etc.
>
> > > Also, was told
> > > that the complement of a wedge product in the 5-limit is the
same
> > > as the cross-product, how does this work in the 7-limit?
>
> The complement of a 2-val is a 2-monzo, and vice-versa, which just
> involves reordering.
>
> ~<<||l1 l2 l3 l4 l5 l6|| = <<l6 -l5 l4 l3 -l2 l1||

Shouldn't that be

~||l1 l2 l3 l4 l5 l6>> = <<l6 -l5 l4 l3 -l2 l1||

or something?

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:

> Shouldn't that be
>
> ~||l1 l2 l3 l4 l5 l6>> = <<l6 -l5 l4 l3 -l2 l1||
>
> or something?

Ooops. Yeah, both ways.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
>
> > > Are the 2 commas in the 7-limit always linearly independent?
> >
> > Yes, they are never 'collinear'.
>
> By definition of a 7-limit linear temperament.
>
> > > How
> > > are they generated, (from wedgies OR matrices)?
> >
> > You can pick them off the tree. We've been looking at some of
> > the 'fruits' here.
>
> Trees don't work for me. You can get them from direct comma searches
> or extract them out of temperaments, etc.

I know how to find them using Python. I was kind of interested in the
algorithm used to generate them...

> > > Also, was told
> > > that the complement of a wedge product in the 5-limit is the
same
> > > as the cross-product, how does this work in the 7-limit?
>
> The complement of a 2-val is a 2-monzo, and vice-versa, which just
> involves reordering.
>
> ~<<||l1 l2 l3 l4 l5 l6|| = <<l6 -l5 l4 l3 -l2 l1||

Thanks. Are they called 2-val and 2-monzo because they are "linear"
or is there some other reason?

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:

> Thanks. Are they called 2-val and 2-monzo because they are "linear"
> or is there some other reason?

2-vals are two vals wedged, 2-monzos are two monzos wedged. The former
is linear unless it reduces to the zero wedgie, the latter is linear
only in the 7-limit.

--- 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 Erlich" <perlich@a...>
> wrote:
> >
> > > > Are the 2 commas in the 7-limit always linearly independent?
> > >
> > > Yes, they are never 'collinear'.
> >
> > By definition of a 7-limit linear temperament.
> >
> > > > How
> > > > are they generated, (from wedgies OR matrices)?
> > >
> > > You can pick them off the tree. We've been looking at some of
> > > the 'fruits' here.
> >
> > Trees don't work for me. You can get them from direct comma
searches
> > or extract them out of temperaments, etc.
>
> I know how to find them using Python. I was kind of interested in
the
> algorithm used to generate them...

In Matlab, I simply generate a gigantic n-dimensional lattice in
prime space, and calculate the error and complexity of each point.
This approach cannot be beat for the codimension-1 case, it seems to
me.

> > > > Also, was told
> > > > that the complement of a wedge product in the 5-limit is the
> same
> > > > as the cross-product, how does this work in the 7-limit?
> >
> > The complement of a 2-val is a 2-monzo, and vice-versa, which just
> > involves reordering.
> >
> > ~<<||l1 l2 l3 l4 l5 l6|| = <<l6 -l5 l4 l3 -l2 l1||
>
> Thanks. Are they called 2-val and 2-monzo because they are "linear"
or is there some other reason?

I think the terms bival and bimonzo for these are closer to the
standard "bivector", but since these prefixes can get large and hard
to remember, I guess people here decided to use numbers instead.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <paul.hjelmstad@u...> wrote:
>
> > Thanks. Are they called 2-val and 2-monzo because they
are "linear"
> > or is there some other reason?
>
> 2-vals are two vals wedged, 2-monzos are two monzos wedged. The
former
> is linear unless it reduces to the zero wedgie, the latter is linear
> only in the 7-limit.

Thanks! So the latter is linear in the 7-limit because the 7-limit is
formed from two commas...I see.

--- 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:
> >
> > > Thanks. Are they called 2-val and 2-monzo because they
> are "linear"
> > > or is there some other reason?
> >
> > 2-vals are two vals wedged, 2-monzos are two monzos wedged. The
> former
> > is linear unless it reduces to the zero wedgie, the latter is
linear
> > only in the 7-limit.
>
> Thanks! So the latter is linear in the 7-limit because the 7-limit
is
> formed from two commas...I see.

The 7-limit is 4-dimensional, so if you temper out 2 commas you're
left with a 2-dimensional system, which is what we usually refer to
as "linear". Is that what you meant?

--- 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:
> >
> > > Thanks. Are they called 2-val and 2-monzo because they
> are "linear"
> > > or is there some other reason?
> >
> > 2-vals are two vals wedged, 2-monzos are two monzos wedged. The
> former
> > is linear unless it reduces to the zero wedgie, the latter is
linear
> > only in the 7-limit.
>
> Thanks! So the latter is linear in the 7-limit because the 7-limit
is
> formed from two commas...I see.

I thought I'd also clarify that the dimension of (or number of
entries in) the wedgie can be read off Pascal's triangle -- look at
the row corresponding to the dimension of the prime space (the first
row is row 0), and the entry in that row corresponding to the
dimension of the kernel (the first entry is entry 0).

1
1 1
1 2 1
1 3 3 1
1 4 6 4 1

So 4-dimensional prime space with two independent commas tempered out
requires a 6-dimensional wedgie, as we've seen.

--- 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, "Gene Ward Smith"
> <gwsmith@s...>
> > wrote:
> > > --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> > > <paul.hjelmstad@u...> wrote:
> > >
> > > > Thanks. Are they called 2-val and 2-monzo because they
> > are "linear"
> > > > or is there some other reason?
> > >
> > > 2-vals are two vals wedged, 2-monzos are two monzos wedged. The
> > former
> > > is linear unless it reduces to the zero wedgie, the latter is
> linear
> > > only in the 7-limit.
> >
> > Thanks! So the latter is linear in the 7-limit because the 7-
limit
> is
> > formed from two commas...I see.
>
> I thought I'd also clarify that the dimension of (or number of
> entries in) the wedgie can be read off Pascal's triangle -- look at
> the row corresponding to the dimension of the prime space (the
first
> row is row 0), and the entry in that row corresponding to the
> dimension of the kernel (the first entry is entry 0).
>
> 1
> 1 1
> 1 2 1
> 1 3 3 1
> 1 4 6 4 1
>
> So 4-dimensional prime space with two independent commas tempered
out
> requires a 6-dimensional wedgie, as we've seen.

Thanks Paul and Gene for all the info. This is helping me make better
sense of everthing.

--- 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, "Gene Ward Smith"
> <gwsmith@s...>
> > wrote:
> > > --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> > > <paul.hjelmstad@u...> wrote:
> > >
> > > > Thanks. Are they called 2-val and 2-monzo because they
> > are "linear"
> > > > or is there some other reason?
> > >
> > > 2-vals are two vals wedged, 2-monzos are two monzos wedged. The
> > former
> > > is linear unless it reduces to the zero wedgie, the latter is
> linear
> > > only in the 7-limit.
> >
> > Thanks! So the latter is linear in the 7-limit because the 7-
limit
> is
> > formed from two commas...I see.
>
> The 7-limit is 4-dimensional, so if you temper out 2 commas you're
> left with a 2-dimensional system, which is what we usually refer to
> as "linear". Is that what you meant?

Yes, I guess so. Why does tempering out two commas in a 4-dimensional
system leave a 2-dimensional system?

--- 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, "Gene Ward Smith"
> > <gwsmith@s...>
> > > wrote:
> > > > --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> > > > <paul.hjelmstad@u...> wrote:
> > > >
> > > > > Thanks. Are they called 2-val and 2-monzo because they
> > > are "linear"
> > > > > or is there some other reason?
> > > >
> > > > 2-vals are two vals wedged, 2-monzos are two monzos wedged.
The
> > > former
> > > > is linear unless it reduces to the zero wedgie, the latter is
> > linear
> > > > only in the 7-limit.
> > >
> > > Thanks! So the latter is linear in the 7-limit because the 7-
> limit
> > is
> > > formed from two commas...I see.
> >
> > The 7-limit is 4-dimensional, so if you temper out 2 commas
you're
> > left with a 2-dimensional system, which is what we usually refer
to
> > as "linear". Is that what you meant?
>
> Yes, I guess so. Why does tempering out two commas in a 4-
dimensional
> system leave a 2-dimensional system?

Roughly: the two commas in addition to two other basis vectors will
span the 4-dimensional system (only if the four vectors are linearly
independent). If you temper out the two commas, the remaining two
basis vectors will form a basis for the entire resulting system of
pitches, which we therefore regard as two-dimensional.

--- 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 Erlich" <perlich@a...>
> > wrote:
> > > --- 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:
> > > > >
> > > > > > Thanks. Are they called 2-val and 2-monzo because they
> > > > are "linear"
> > > > > > or is there some other reason?
> > > > >
> > > > > 2-vals are two vals wedged, 2-monzos are two monzos wedged.
> The
> > > > former
> > > > > is linear unless it reduces to the zero wedgie, the latter
is
> > > linear
> > > > > only in the 7-limit.
> > > >
> > > > Thanks! So the latter is linear in the 7-limit because the 7-
> > limit
> > > is
> > > > formed from two commas...I see.
> > >
> > > The 7-limit is 4-dimensional, so if you temper out 2 commas
> you're
> > > left with a 2-dimensional system, which is what we usually
refer
> to
> > > as "linear". Is that what you meant?
> >
> > Yes, I guess so. Why does tempering out two commas in a 4-
> dimensional
> > system leave a 2-dimensional system?
>
> Roughly: the two commas in addition to two other basis vectors will
> span the 4-dimensional system (only if the four vectors are
linearly
> independent). If you temper out the two commas, the remaining two
> basis vectors will form a basis for the entire resulting system of
> pitches, which we therefore regard as two-dimensional.

Got it. How does one find the "remaining two basis vectors?" Is it
with Graham's matrix method?

Thanx

Paul

--- 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 Erlich"
<perlich@a...>
> > > wrote:
> > > > --- 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:
> > > > > >
> > > > > > > Thanks. Are they called 2-val and 2-monzo because they
> > > > > are "linear"
> > > > > > > or is there some other reason?
> > > > > >
> > > > > > 2-vals are two vals wedged, 2-monzos are two monzos
wedged.
> > The
> > > > > former
> > > > > > is linear unless it reduces to the zero wedgie, the
latter
> is
> > > > linear
> > > > > > only in the 7-limit.
> > > > >
> > > > > Thanks! So the latter is linear in the 7-limit because the
7-
> > > limit
> > > > is
> > > > > formed from two commas...I see.
> > > >
> > > > The 7-limit is 4-dimensional, so if you temper out 2 commas
> > you're
> > > > left with a 2-dimensional system, which is what we usually
> refer
> > to
> > > > as "linear". Is that what you meant?
> > >
> > > Yes, I guess so. Why does tempering out two commas in a 4-
> > dimensional
> > > system leave a 2-dimensional system?
> >
> > Roughly: the two commas in addition to two other basis vectors
will
> > span the 4-dimensional system (only if the four vectors are
> linearly
> > independent). If you temper out the two commas, the remaining two
> > basis vectors will form a basis for the entire resulting system
of
> > pitches, which we therefore regard as two-dimensional.
>
> Got it. How does one find the "remaining two basis vectors?" Is it
> with Graham's matrix method?

I suppose, or with Gene's algorithm, in which he uses Hermite
reduction.