What is a "height function?"

The planetmath.com link that appears first on Google is down, and I
can't find anything else on it.

What exactly are the requirements to label a function as a "height"
function? What characteristics must it have? Is it just that it has to
be multilinear?

-Mike

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> The planetmath.com link that appears first on Google is down, and I
> can't find anything else on it.
>
> What exactly are the requirements to label a function as a "height"
> function? What characteristics must it have? Is it just that it has to
> be multilinear?

A number theoretic height function has to create a bound so that there will be a finite number of whatever you are looking at (algebraic numbers, rational points on a variety, or whatever) below that bound. It thereby bounds the informational size of this thing, in terms of digits or whatever kind of information you need to specify this point or number. So Tenney height is a height function for rational numbers: if p/q has Tenney height H, then we know the number of digits in p and q are bounded in terms of H.

--- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:

> A number theoretic height function has to create a bound so that there will be a finite number of whatever you are looking at (algebraic numbers, rational points on a variety, or whatever) below that bound. It thereby bounds the informational size of this thing, in terms of digits or whatever kind of information you need to specify this point or number. So Tenney height is a height function for rational numbers: if p/q has Tenney height H, then we know the number of digits in p and q are bounded in terms of H.
>

Another example of a height function are complexity measures on wedgies. These are heights for rational points on Grassmanian varieties, such that below a given complexity you have only a finite number of points (wedgies), and you can bound how many bytes it takes to store the wedgie in computer mempry by the complexity.

On Thu, Jan 26, 2012 at 8:00 AM, genewardsmith
<genewardsmith@sbcglobal.net> wrote:
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> >
> > The planetmath.com link that appears first on Google is down, and I
> > can't find anything else on it.
> >
> > What exactly are the requirements to label a function as a "height"
> > function? What characteristics must it have? Is it just that it has to
> > be multilinear?
>
> A number theoretic height function has to create a bound so that there will be a finite number of whatever you are looking at (algebraic numbers, rational points on a variety, or whatever) below that bound. It thereby bounds the informational size of this thing, in terms of digits or whatever kind of information you need to specify this point or number.

Ah, that makes sense. So if I define asdfasdf height over R as the
number of base-10 digits that a number takes up, there are only ever a
finite amount of numbers sharing the same asdfasdf height, and it ends
up telling you how complex the number is in a certain sense.

> So Tenney height is a height function for rational numbers: if p/q has Tenney height H, then we know the number of digits in p and q are bounded in terms of H.

OK, so then if I understand correctly, something like p*q for dyads is
a height function, but something like a*b*c*... for an arbitrary chord
a:b:c:... is not. This is because for any arbitrary a*b*c*... product,
there are an infinite amount of chords below that bound. For example,
consider 3/1, with a product of 3: if we consider more than just
dyads, you end up with 1:2,1:1:2, 1:1:1:2, etc all below the bound.
Right?

I wonder what sorts of height functions there are that are sensible in
that they cut across chords, and still work well for concordance.
Maybe a strict height function isn't necessary though.

-Mike

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> OK, so then if I understand correctly, something like p*q for dyads is
> a height function, but something like a*b*c*... for an arbitrary chord
> a:b:c:... is not. This is because for any arbitrary a*b*c*... product,
> there are an infinite amount of chords below that bound. For example,
> consider 3/1, with a product of 3: if we consider more than just
> dyads, you end up with 1:2,1:1:2, 1:1:1:2, etc all below the bound.
> Right?

If you want to count those as separate chords, which seems a little dubious. Of course, just a height on triads, tetrads, etc. separately seems to make more sense anyway.

On Thu, Jan 26, 2012 at 9:21 AM, genewardsmith
<genewardsmith@sbcglobal.net> wrote:
>
> If you want to count those as separate chords, which seems a little dubious. Of course, just a height on triads, tetrads, etc. separately seems to make more sense anyway.

Why do you say this? Much of my thinking recently has been targeting
the exact opposite, which is that I wish we had a way to compare
chords of different cardinalities.

-Mike

--- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@> wrote:
>
> > OK, so then if I understand correctly, something like p*q for dyads is
> > a height function, but something like a*b*c*... for an arbitrary chord
> > a:b:c:... is not. This is because for any arbitrary a*b*c*... product,
> > there are an infinite amount of chords below that bound. For example,
> > consider 3/1, with a product of 3: if we consider more than just
> > dyads, you end up with 1:2,1:1:2, 1:1:1:2, etc all below the bound.
> > Right?
>
> If you want to count those as separate chords, which seems a little dubious. Of course, just a height on triads, tetrads, etc. separately seems to make more sense anyway.

Or you could just consider *sets* of integers/pitches rather than multisets. This makes it a height function, I'm pretty sure. The following are all chords up to generalized Tenney height = 6:

1 (the trivial one-note chord)
1:2
1:3
1:4
1:5
1:6
2:3
1:2:3

Keenan

--- In tuning-math@yahoogroups.com, "Keenan Pepper" <keenanpepper@...> wrote:

> Or you could just consider *sets* of integers/pitches rather than multisets. This makes it a height function, I'm pretty sure.

I wouldn't worry about it; I described what a height function was, I didn't gave a precise definition. I don't know if anyone has attempted to create a general height function theory, which would require such a definition.