Nexials

Suppose p>5 is prime and q is the next largest prime. For any p-limit
linear temperament wedgie, we can project down to a q-limit wedgie by
for instance taking the wedge product of the truncated mapping. Its
elements will be a subset of certain specific p-limit elements, and in
the case which will interest us, we suppose these are all relatively
prime and we have a well-behaved temperament.

If w is a p-limit val belonging to this temperament, and
u = <0 0 ... 1| is p-limit, then both truncated u and tuncated w
belong to the ttemperament, and therefore u^w belongs to the projected
temperament. Adding or subtracting multiples of it will lead to
wedgies which define different p-limit temperaments, but the same
q-limit temperament. We can say they are the same, mod the truncated w.

Given two p-limit wedgies a and b which have the same q-limit
projected temperament, the normalized wedgie for a-b will be of the
u^w form. We can therefore toss the zeros and get a q-limit val, which
I will call the nexus of a and b.

In the 7-limit, the nexus of two wedgies a and b with the same 5-limit
projection is the normalization of <c[3], c[5], c[6]| where
c = a-b. In the 11-limit, this becomes <c[4], c[7], c[9], c[10]|. We
are simply taking the nonzero elements here, which are the ones whose
basis involves a p.

I'm putting this forward in connection with the question of naming and
classifying p-limit temperaments in connection with already named
q-limit temperaments.

>For any p-limit
>linear temperament wedgie, we can project down to a q-limit wedgie

Are p and q reversed here?

-C.

--- In tuning-math@yahoogroups.com, "Carl Lumma" <ekin@l...> wrote:
> >For any p-limit
> >linear temperament wedgie, we can project down to a q-limit wedgie
>
> Are p and q reversed here?

No; q is the next largest prime. If p is 7, q is 5, etc.

>> >For any p-limit
>> >linear temperament wedgie, we can project down to a q-limit
>> >wedgie
>>
>> Are p and q reversed here?
>
>No; q is the next largest prime. If p is 7, q is 5, etc.

Oh, by next largest I took you to mean bigger.

-Carl

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> Suppose p>5 is prime and q is the next largest prime. For any p-
limit
> linear temperament wedgie, we can project down to a q-limit wedgie
by
> for instance taking the wedge product of the truncated mapping. Its
> elements will be a subset of certain specific p-limit elements, and
in
> the case which will interest us, we suppose these are all relatively
> prime and we have a well-behaved temperament.
>
> If w is a p-limit val belonging to this temperament, and
> u = <0 0 ... 1| is p-limit, then both truncated u and tuncated w
> belong to the ttemperament, and therefore u^w belongs to the
projected
> temperament. Adding or subtracting multiples of it will lead to
> wedgies which define different p-limit temperaments, but the same
> q-limit temperament. We can say they are the same, mod the
truncated w.

How is u raised to w? I am confused about w. I know how to raise
u to a bra-ket value, how do you raise a value to a value?

>
> Given two p-limit wedgies a and b which have the same q-limit
> projected temperament, the normalized wedgie for a-b will be of the
> u^w form. We can therefore toss the zeros and get a q-limit val,
which
> I will call the nexus of a and b.
>
> In the 7-limit, the nexus of two wedgies a and b with the same 5-
limit
> projection is the normalization of <c[3], c[5], c[6]| where
> c = a-b. In the 11-limit, this becomes <c[4], c[7], c[9], c[10]|. We
> are simply taking the nonzero elements here, which are the ones
whose
> basis involves a p.

This makes sense - but how does this tranlate into a single value,
like you are using for other posts (53: for example). Is a Nexial
the same as a Nexus???
>
> I'm putting this forward in connection with the question of naming
and
> classifying p-limit temperaments in connection with already named
> q-limit temperaments.

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul.hjelmstad@m...> wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> wrote:
> > Suppose p>5 is prime and q is the next largest prime. For any p-
> limit
> > linear temperament wedgie, we can project down to a q-limit
wedgie
> by
> > for instance taking the wedge product of the truncated mapping.
Its
> > elements will be a subset of certain specific p-limit elements,
and
> in
> > the case which will interest us, we suppose these are all
relatively
> > prime and we have a well-behaved temperament.
> >
> > If w is a p-limit val belonging to this temperament, and
> > u = <0 0 ... 1| is p-limit, then both truncated u and tuncated w
> > belong to the ttemperament, and therefore u^w belongs to the
> projected
> > temperament. Adding or subtracting multiples of it will lead to
> > wedgies which define different p-limit temperaments, but the same
> > q-limit temperament. We can say they are the same, mod the
> truncated w.
>
> How is u raised to w? I am confused about w. I know how to raise
> u to a bra-ket value, how do you raise a value to a value?

I believe Gene was using the symbol "^" to represent a wedge product,
not exponentiation. Better would be to write u/\w, since /\ looks a
lot more like the standard wedge product symbol.

Even better would be if Gene would answer your questions himself.
(and mine)

> > > If w is a p-limit val belonging to this temperament, and
> > > u = <0 0 ... 1| is p-limit, then both truncated u and tuncated w
> > > belong to the ttemperament, and therefore u^w belongs to the
> > projected
> > > temperament. Adding or subtracting multiples of it will lead to
> > > wedgies which define different p-limit temperaments, but the
same
> > > q-limit temperament. We can say they are the same, mod the
> > truncated w.
> >
> > How is u raised to w? I am confused about w. I know how to raise
> > u to a bra-ket value, how do you raise a value to a value?
>
> I believe Gene was using the symbol "^" to represent a wedge
product,
> not exponentiation. Better would be to write u/\w, since /\ looks a
> lot more like the standard wedge product symbol.

Thanx. I can't believe I missed something this obvious. Regarding my
second question: Is the normalization of a wedgie something that
would yield a single value (number?) like 53?
>

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

> Thanx. I can't believe I missed something this obvious. Regarding my
> second question: Is the normalization of a wedgie something that
> would yield a single value (number?) like 53?

Normalization of a bival to a wedgie is accomplished by dividing by
the GCD of the coefficients and making the first nonzero one positive;
this is the convention Graham and I have agreed on.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <paul.hjelmstad@m...> wrote:
>
> > Thanx. I can't believe I missed something this obvious. Regarding
my
> > second question: Is the normalization of a wedgie something that
> > would yield a single value (number?) like 53?
>
> Normalization of a bival to a wedgie is accomplished by dividing by
> the GCD of the coefficients and making the first nonzero one
positive;
> this is the convention Graham and I have agreed on.

So the anwser is no, right? Gene, I think it would be really helpful
if you tried to answer yes/no questions with a "yes" or a "no",
because often I can't tell which it is from your answer, and others
probably have even less idea. I'm sure it always seems obvious to you
but don't forget that you're dealing with a bunch of blockheads like
me.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> > <paul.hjelmstad@m...> wrote:
> >
> > > Thanx. I can't believe I missed something this obvious. Regarding
> my
> > > second question: Is the normalization of a wedgie something that
> > > would yield a single value (number?) like 53?
> >
> > Normalization of a bival to a wedgie is accomplished by dividing by
> > the GCD of the coefficients and making the first nonzero one
> positive;
> > this is the convention Graham and I have agreed on.
>
> So the anwser is no, right?

OK, no. :)

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> > --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> > wrote:
> > > --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> > > <paul.hjelmstad@m...> wrote:
> > >
> > > > Thanx. I can't believe I missed something this obvious.
Regarding
> > my
> > > > second question: Is the normalization of a wedgie something
that
> > > > would yield a single value (number?) like 53?
> > >
> > > Normalization of a bival to a wedgie is accomplished by
dividing by
> > > the GCD of the coefficients and making the first nonzero one
> > positive;
> > > this is the convention Graham and I have agreed on.
> >
> > So the anwser is no, right?
>
> OK, no. :)

So how is that single value (like 53:) generated?

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

> > > So the anwser is no, right?
> >
> > OK, no. :)
>
> So how is that single value (like 53:) generated?

It isn't. The no meant no, it isn't a single value.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <paul.hjelmstad@m...> wrote:
>
> > > > So the anwser is no, right?
> > >
> > > OK, no. :)
> >
> > So how is that single value (like 53:) generated?
>
> It isn't. The no meant no, it isn't a single value.

What I meant is, how is 9: below arrived at. You call it the nexus...

9: <<3 0 3 -7 -4 7|| {15/14, 128/125}
[392.25, 96.15]

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

> What I meant is, how is 9: below arrived at. You call it the nexus...
>
> 9: <<3 0 3 -7 -4 7|| {15/14, 128/125}
> [392.25, 96.15]

The 9 is shorthand for the 9 standard val.

<<3 0 3 -7 -4 7|| = <<3 0 -6 -7 -18 -14|| + <<0 0 9 0 14 21||

In other words, the temperament is augmented plus doubleO 9.