Are "Tenney Height", "Benedetti Height", "Kees Height", etc actually height functions?

I know that Paul hates the term "Tenney Height", but that we've been using
it anyway. Tenney Height is log(n*d), and we've also got a Benedetti height
which is n*d, and a Kees height which is the odd-limit of the interval, and
once I finish writing all this stuff up there's soon to be a Farey height
as well, which is either max(n,d) or log(max(n,d)).

I don't care for the notion that we should stop calling log(n*d) Tenney
Height because James Tenney called it "Tenney Harmonic Distance." However,
do these functions actually satisfy the mathematical definition of a
"height function"? If not, then I'd support "officially" calling them
something else, at least when we present this stuff to a mathematical
audience, even if we keep calling it Tenney Height on this board.

-Mike

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

> I don't care for the notion that we should stop calling log(n*d) Tenney
> Height because James Tenney called it "Tenney Harmonic Distance." However,
> do these functions actually satisfy the mathematical definition of a
> "height function"?

They can legitimately be called height functions, though people often want to pack more restrictions into the notion. Check out the "Guiding Principles" in this:

http://www.mathematik.uni-tuebingen.de/ab/arithmetik/h%3Ec/HeightsScript.pdf

On Wed, Aug 15, 2012 at 1:17 PM, genewardsmith <genewardsmith@sbcglobal.net>
wrote:
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...>
> wrote:
>
> > I don't care for the notion that we should stop calling log(n*d) Tenney
> > Height because James Tenney called it "Tenney Harmonic Distance."
> > However,
> > do these functions actually satisfy the mathematical definition of a
> > "height function"?
>
> They can legitimately be called height functions, though people often want
> to pack more restrictions into the notion. Check out the "Guiding
> Principles" in this:
>
> http://www.mathematik.uni-tuebingen.de/ab/arithmetik/h%3Ec/HeightsScript.pdf

It looks like their archetypical example of a function H maps from
some field to [1, \inf). Does that disqualify log(n*d) from being a
height function, since log(1*1) is 0?

Also, what does Q(a): Q mean in the middle of the expression in #2?

-Mike

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

> It looks like their archetypical example of a function H maps from
> some field to [1, \inf). Does that disqualify log(n*d) from being a
> height function, since log(1*1) is 0?

Height functions are arithmetic or logarithmic, depending on whether you take the log or not. Obviously, the two are essentially equivalent.

> Also, what does Q(a): Q mean in the middle of the expression in #2?

It's the degree of the field extension generated by the n algebraic numbers in a over Q. Since we only care about Q, ignore this.

On Wed, Aug 15, 2012 at 3:43 PM, genewardsmith
<genewardsmith@sbcglobal.net> wrote:
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > It looks like their archetypical example of a function H maps from
> > some field to [1, \inf). Does that disqualify log(n*d) from being a
> > height function, since log(1*1) is 0?
>
> Height functions are arithmetic or logarithmic, depending on whether you take the log or not. Obviously, the two are essentially equivalent.

OK. So is there an actual need for us to distinguish between the
arithmetic and logarithmic versions at all, really?

I'm asking because I'm writing something up about Farey norms now, and
now I don't know how to define the phrase "Farey height" - should it
be log(max(n,d)) or max(n,d) or what?

Honestly, I don't see why I'm suddenly supposed to pretend that these
are things that really deserve two different names. Right now it's
totally random whether the normal version or the log version is the
height

1) log(n*d) - Tenney Height
2) n*d - Benedetti Height
3) max(n,d) with powers of 2 removed - Kees Height
4) log(max(n,d)) without powers of 2 removed - Farey height?
5) max(n,d) without powers of 2 removed - Farey height?

Why don't we just say that n*d and log(n*d) are both "Tenney height",
with n*d being "arithmetic Tenney height" and log(n*d) being "log
Tenney height" if we need to get specific? And it can also be on the
wiki page that n*d "is sometimes called Benedetti height", just like
the "cofactors" of a square matrix are sometimes called "adjuncts" or
something.

-Mike

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

> OK. So is there an actual need for us to distinguish between the
> arithmetic and logarithmic versions at all, really?
>
> I'm asking because I'm writing something up about Farey norms now, and
> now I don't know how to define the phrase "Farey height" - should it
> be log(max(n,d)) or max(n,d) or what?

If you don't want both, pick one. But max(|n|, |d|) is the bog-standard height function, so I'm not certain if "Farey height" is a good idea. On the other hand, I don't know of a name for it other than "height".

> Why don't we just say that n*d and log(n*d) are both "Tenney height",
> with n*d being "arithmetic Tenney height" and log(n*d) being "log
> Tenney height" if we need to get specific?

Hell, NO! I changed things at Paul's urging. Enough, already.

On Wed, Aug 15, 2012 at 4:52 PM, genewardsmith <genewardsmith@sbcglobal.net>
wrote:
>
> > Why don't we just say that n*d and log(n*d) are both "Tenney height",
> > with n*d being "arithmetic Tenney height" and log(n*d) being "log
> > Tenney height" if we need to get specific?
>
> Hell, NO! I changed things at Paul's urging. Enough, already.

OK, so any interval n/d can be represented as 2^k * a/b, where a and b
are coprime with one another and each is individually coprime with 2.
Then max(a,b) is the "Kees height" of that interval, as you put on the
wiki. What do I call log(max(a,b))?

-Mike

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

> OK, so any interval n/d can be represented as 2^k * a/b, where a and b
> are coprime with one another and each is individually coprime with 2.
> Then max(a,b) is the "Kees height" of that interval, as you put on the
> wiki. What do I call log(max(a,b))?

The most obvious answer is "Kees expressibility". You could also try "logarithmic Kees height", "Kees distance", "Kees complexity", "Kees harmonic distance" or "Kees harmonic complexity":

On Wed, Aug 15, 2012 at 7:15 PM, genewardsmith <genewardsmith@sbcglobal.net>
wrote:
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...>
> wrote:
>
> > OK, so any interval n/d can be represented as 2^k * a/b, where a and b
> > are coprime with one another and each is individually coprime with 2.
> > Then max(a,b) is the "Kees height" of that interval, as you put on the
> > wiki. What do I call log(max(a,b))?
>
> The most obvious answer is "Kees expressibility". You could also try
> "logarithmic Kees height", "Kees distance", "Kees complexity", "Kees
> harmonic distance" or "Kees harmonic complexity":

OK, and is there a difference between the function on rationals vs the
corresponding function on monzos? For instance, consider log(n*d) vs,
for some monzo |a b c ...> representing the same rational, log2(2)*|a|
+ log2(3)*|b| + log2(5)*|c| + ... . Do both of them get the name
"Tenney height"? And likewise for "Kees expressibility" and so on?

Also, I don't know what Paul has to do with any of this; he hates the
term "Tenney height" in any form and wants log(n*d) to be "Tenney
Harmonic Distance" and "corrects" people if they call it Tenney Height
on Facebook. So are you really that adamant that "logarithmic Kees
height" is OK and "arithmetic Tenney height" is not?

-Mike

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

> OK, and is there a difference between the function on rationals vs the
> corresponding function on monzos?

No, since monzos are a notation.

> Also, I don't know what Paul has to do with any of this; he hates the
> term "Tenney height" in any form and wants log(n*d) to be "Tenney
> Harmonic Distance" and "corrects" people if they call it Tenney Height
> on Facebook.

There was an XA conversation between me, Psul, and who knows who else, but apparently not including you, which led to the decision we'd call the arithmetic height function "Benedetti" and the logarithmic "Tenney", which seemed both to appease Paul and serve a purpose.

So are you really that adamant that "logarithmic Kees
> height" is OK and "arithmetic Tenney height" is not?

Call it what you like, but today I changed "Lp" to "Tp" to appease you and previously I changed "Tenney" to "Benedetti" to appease Paul, and I'm done fiddling with terminology for the moment.

On Wed, Aug 15, 2012 at 9:16 PM, genewardsmith <genewardsmith@sbcglobal.net>
wrote:
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...>
> wrote:
>
> There was an XA conversation between me, Psul, and who knows who else, but
> apparently not including you, which led to the decision we'd call the
> arithmetic height function "Benedetti" and the logarithmic "Tenney", which
> seemed both to appease Paul and serve a purpose.

I remember that conversation, and I'm telling you that, much like the
400-post Multi-MOS conversation we had where Paul and I and everyone
else seemed to reach a compromise, Paul's decided he hates it now.

> Call it what you like, but today I changed "Lp" to "Tp" to appease you and
> previously I changed "Tenney" to "Benedetti" to appease Paul, and I'm done
> fiddling with terminology for the moment.

It would be nice if Paul would stop saying "WRONG; Tenney Harmonic
Distance" then. And it's not like I haven't changed things to appease
you either; I've given up "Lp-TOP" and went with "albitonic" and a
whole bunch of stuff I don't remember. Anyway, I'm going to call both
max(n,d) and log(max(n,d)) "Farey height" for the moment, and if
someone comes up with a better setup later we'll change it.

-Mike