TOP and Tenney space webpage

Gene,

Did you mean (2^n)-1 instead of 2^(n-1)? ("Since we have 2^n corners
to a ball there are 2^(n-1) lines.." etc.)

Paul Hj

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:
> Gene,
>
> Did you mean (2^n)-1 instead of 2^(n-1)? ("Since we have 2^n
corners
> to a ball there are 2^(n-1) lines.." etc.)

The 2^n corners come in 2^(n-1) pairs of opposite corners, and lines
between them pass through the center of the ball, which is the JIP.

What's your take on the acronym JIP? I think it makes sense, since it
is the point corresponding to just intonation, but Paul objects for
reasons not entirely clear to me.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <paul.hjelmstad@u...> wrote:
> > Gene,
> >
> > Did you mean (2^n)-1 instead of 2^(n-1)? ("Since we have 2^n
> corners
> > to a ball there are 2^(n-1) lines.." etc.)
>
> The 2^n corners come in 2^(n-1) pairs of opposite corners, and
lines
> between them pass through the center of the ball, which is the JIP.

Okay. (I had thought JIP was a corner, oops). I had thought 3 and 7
solutions for 5-limit and 7-limit were based on (2^n)-1..

> What's your take on the acronym JIP? I think it makes sense, since
it
> is the point corresponding to just intonation, but Paul objects for
> reasons not entirely clear to me.

Seems fine to me.

--- 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:
> > > Gene,
> > >
> > > Did you mean (2^n)-1 instead of 2^(n-1)? ("Since we have 2^n
> > corners
> > > to a ball there are 2^(n-1) lines.." etc.)
> >
> > The 2^n corners come in 2^(n-1) pairs of opposite corners, and
> lines
> > between them pass through the center of the ball, which is the
JIP.
>
So actually the solutions are 2^(n-1)-1 because you exclude the line
passing through the origin? What is in the opposite corner of the
line that passes through the origin? (Or is that even a valid
question...)
>

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <paul.hjelmstad@u...> wrote:
> > Gene,
> >
> > Did you mean (2^n)-1 instead of 2^(n-1)? ("Since we have 2^n
> corners
> > to a ball there are 2^(n-1) lines.." etc.)
>
> The 2^n corners come in 2^(n-1) pairs of opposite corners, and
lines
> between them pass through the center of the ball, which is the JIP.
>
> What's your take on the acronym JIP? I think it makes sense, since
it
> is the point corresponding to just intonation, but Paul objects for
> reasons not entirely clear to me.

First of all, it measures pitch, something you fail to note at all on
your website, but would be the most comprehensible thing about this
whole business.

Secondly, who discovered the duality between points and linear
functionals? Poincare? That's very recent in the history of
mathematics, and far more advanced than what most musicians could
ever, let alone should be expected to already, comprehend.

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

> > What's your take on the acronym JIP? I think it makes sense,
since
> it
> > is the point corresponding to just intonation, but Paul objects
for
> > reasons not entirely clear to me.
>
> First of all, it measures pitch, something you fail to note at all
on
> your website, but would be the most comprehensible thing about this
> whole business.

It doesn't measure anything. It is the just intonation mapping, just
as other points represent other tunings. Hence, "J I Point".

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> > > What's your take on the acronym JIP? I think it makes sense,
> since
> > it
> > > is the point corresponding to just intonation, but Paul objects
> for
> > > reasons not entirely clear to me.
> >
> > First of all, it measures pitch, something you fail to note at
all
> on
> > your website, but would be the most comprehensible thing about
this
> > whole business.
>
> It doesn't measure anything. It is the just intonation mapping,
just
> as other points represent other tunings. Hence, "J I Point".

I have no idea how to reconcile this with

/tuning-math/message/9797

and with the fact that it's obvious that this linear operator, when
acting on a monzo, is the only one returns its pitch (or interval
size) in cents.

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

> I have no idea how to reconcile this with
>
> /tuning-math/message/9797
>
> and with the fact that it's obvious that this linear operator, when
> acting on a monzo, is the only one returns its pitch (or interval
> size) in cents.

I'm being contrary; it seems to me "measure" isn't really how we want
to look at it, since just intonation is now being viewed as one of an
infinite set of possible tuning maps. It is the just *tuning* point,
but is that a measurement point?

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> > I have no idea how to reconcile this with
> >
> > /tuning-math/message/9797
> >
> > and with the fact that it's obvious that this linear operator,
when
> > acting on a monzo, is the only one returns its pitch (or interval
> > size) in cents.
>
> I'm being contrary; it seems to me "measure" isn't really how we
want
> to look at it, since just intonation is now being viewed as one of
an
> infinite set of possible tuning maps.

True. It only measures pitch (or interval size) in the untempered
case.