Complexity measures spreadsheet

[Manuel wrote in email:]
>It would be interesting if you added to your spreadsheet
>Paul E.'s log of largest odd factor, Tenney's value
>(log of n*d) and a modified Tenney's with factors of 2 removed
>from n*d, all using the same log base.
>The value of Paul H. and me will be between the first and third one.

Ok. I did it. I also added
Hahn/Op de Coul triangular distance (as you suggested earlier) and
Hahn/Op de Coul with 2's
I'd appreciate it if you'd check a few values for correctness (I haven't
checked). Base 2 logs were used for all.

I've compared these 5 on a separate chart from the others (since they are
log). Scroll down to see it. Also the first chart is too cluttered. Better
to just compare a few at a time. I suggest going to
Chart/Source_Data/Series and removing the ones that don't interest you.

It's still at http://dkeenan.com/Music/HarmonicComplexity.zip 158KB

It's still an Excel 97 spreadsheet. Sorry. It uses features not available
in 5.0/95, namely formulas longer than 255 characters. Which might explain
the problems you had, Manuel, with Euler's functions.

Tenney's looks ok, log of odd-limit is passable (just as useful as
odd-limit of course), Tenney's without 2's is useless (IMHO). Hahn/Op de
Coul with 2's is useless.

For such a complicated algorithm Hahn/Op de Coul triangular distance only
differs from log of odd-limit in a few places (e.g. 10/9, 11/9, 9/7, 9/5)
where it gives a higher complexity. Maybe I've got it wrong?

Regards,
-- Dave Keenan
http://dkeenan.com

>Tenney's looks ok, log of odd-limit is passable (just as useful as
>odd-limit of course), Tenney's without 2's is useless (IMHO).

I would agree with that. Does anyone know if Tenney himself ever did it
this way?

>For such a complicated algorithm Hahn/Op de Coul triangular distance
only
>differs from log of odd-limit in a few places (e.g. 10/9, 11/9, 9/7,
9/5)
>where it gives a higher complexity. Maybe I've got it wrong?

WAIT A MINUTE! Paul Hahn and I have not officially come to a final
agreement on this, but Paul H. was last saying that the vectors in his
algorithm have an entry for each odd number, while Manuel's have an
entry for each prime number. Therefore I don't think it's right to call
the latter a "Hahn/Op de Coul" triangular distance!!!

"Paul H. Erlich" <PErlich@Acadian-Asset.com> wrote

>WAIT A MINUTE! Paul Hahn and I have not officially come to a final
>agreement on this, but Paul H. was last saying that the vectors in his
>algorithm have an entry for each odd number, while Manuel's have an
>entry for each prime number. Therefore I don't think it's right to call
>the latter a "Hahn/Op de Coul" triangular distance!!!

Well it's the one Paul H. posted to the list around 12-March which clearly
assumed primes, and which is the same as Manuel's. Manuel send me his
source code for it so I could double-check, and Manuel referred to it as
his/Hahn's.

If Paul H. wishes to disown it now, he can tell me so and I'll change my
spreadsheet. If Paul H. merely wants to distinguish it from another that
you and he may agree upon in future, I can then call this one Hahn/Op de
Coul *prime* triangular complexity or some such.

Paul E., I'll try to include yours in the spreadsheet too when you tell me
it's settled, (and tell me how to do it).

-- Dave Keenan
http://dkeenan.com

Dave Keenan wrote,

>Well it's the one Paul H. posted to the list around 12-March which clearly
>assumed primes, and which is the same as Manuel's. Manuel send me his
>source code for it so I could double-check, and Manuel referred to it as
>his/Hahn's.

If you'd been following the discussion, you'd see that I too mistook Paul
H's algorithm as assuming primes, in contrast to his stance on odds vs.
primes, but he then claimed that he had been assuming odds all along. Since
the example that accompanied his algorithm only had 3-component vectors
(3,5,7), there was in fact no possible basis for deciding whether the
components were to be primes or odds on the basis of that one post alone.

The problem with Hahn's intended algorithm is that it needs one additional
parameter, the odd limit of the lattice, in order to be unambiguous.

If the lattice is infinite-dimensional, the metric simply yields the log of
the odd-limit of the interval.