Tenney-Euclidean metrics

I wrote this stuff up here:

http://xenharmonic.wikispaces.com/Tenney-Euclidean+metrics

I'm interested to know if

(1) It makes sense, and

(2) If you agree with the presentation

On 31 October 2010 04:42, genewardsmith <genewardsmith@sbcglobal.net> wrote:
> I wrote this stuff up here:
>
> http://xenharmonic.wikispaces.com/Tenney-Euclidean+metrics
>
> I'm interested to know if
>
> (1) It makes sense, and

There should be a link to the first mention of pseudoinverse at least.
I wouldn't expect everybody to know what it means.

> (2) If you agree with the presentation

The point of changing the terminology to TE everything was to simplify
it. Here, your still using "TOP-RMS" and "wedgie complexity" so the
terminology's getting more complicated. I can use temperamental
complexity, though.

Using O as a symbol is widely acknowledged to be a bad idea.

Isn't there a better way of formatting equations on the wiki?

Graham

On 31 October 2010 04:42, genewardsmith <genewardsmith@sbcglobal.net> wrote:
> I wrote this stuff up here:
>
> http://xenharmonic.wikispaces.com/Tenney-Euclidean+metrics
>
> I'm interested to know if
>
> (1) It makes sense, and
>
> (2) If you agree with the presentation

The section on octave equivalent complexity looks wrong as well. It
says "An alternative procedure is to find the normal val list, and
remove the first val from the list, corresponding to the octave or
some fraction thereof, and proceed as in the previous section on
temperamental complexity."

This means, for a rank 2 temperament, taking the weighted RMS of the
octave-orthogonal mapping (octave-equivalent but including the 0).
That gives radically different results for octave-equivalent and
octave-specific complexity. The correct function, as I've explained
at length, is to take the standard deviation instead. You can find a
matrix (and maybe a projection) to generalize this to higher ranks.

This may lead to problems with complexity of intervals. I haven't
worked out the details. But wouldn't 16:15 be ranked much simpler
than 15:14 in 7-limit JI? If it isn't I'd like to know why.

Minimizing the standard deviation of generator steps will still give
you an MOS for the rank 2 case. The only thing is that the tonic is
abstracted out, so an MOS starting on any note gets the same score. I
think that's reasonable, and should be generalizable. The aim is not
to get a cluster of notes close to the origin, but to get a compact
cluster.

Ideally the octave equivalence wouldn't be required. You could ask
for a set of notes, and an algorithm would give you a cluster of
pitches with a reasonably small span. That means you have to score
interval size.

Graham

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:

> The section on octave equivalent complexity looks wrong as well. It
> says "An alternative procedure is to find the normal val list, and
> remove the first val from the list, corresponding to the octave or
> some fraction thereof, and proceed as in the previous section on
> temperamental complexity."

Actually, it's right, but one of the formulas I gave had the wrong weighting. I've corrected that an changed and expanded on the example, showing how the exact same matrix is obtained from two different staring points. As form your subsequent comments, OE must differ in that 2 is projected away. You want to use the matrix with the averages subtracted off used for TE simple badness? Can you give an example showing why that is better, or explain your reasoning?

> This may lead to problems with complexity of intervals. I haven't
> worked out the details. But wouldn't 16:15 be ranked much simpler
> than 15:14 in 7-limit JI? If it isn't I'd like to know why.

Do you mean via 7-limit OE measures?

On 3 November 2010 02:19, genewardsmith <genewardsmith@sbcglobal.net> wrote:

> Actually, it's right, but one of the formulas I gave had the wrong weighting. I've corrected that an changed and expanded on the example, showing how the exact same matrix is obtained from two different staring points. As form your subsequent comments, OE must differ in that 2 is projected away. You want to use the matrix with the averages subtracted off used for TE simple badness? Can you give an example showing why that is better, or explain your reasoning?

I suggested using the standard deviation of the weighted mapping for
the rank 2 case. I thought that suggestion was clear in the paragraph
you cut out. I gave lots of examples in a file called primerr.pdf.
Do I have to search for it again to give you the correct URL? I also
gave metrics for Farey limits in a file called composite.pdf that may
also exist as composite_onecol.pdf. If you want to research
octave-equivalent metrics, you could try working out the metrics that
correspond to odd limits.

>> This may lead to problems with complexity of intervals.  I haven't
>> worked out the details.  But wouldn't 16:15 be ranked much simpler
>> than 15:14 in 7-limit JI?  If it isn't I'd like to know why.
>
> Do you mean via 7-limit OE measures?

Yes. 7-limit JI.

Graham

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:

> I suggested using the standard deviation of the weighted mapping for
> the rank 2 case. I thought that suggestion was clear in the paragraph
> you cut out.

It's completely unclear to me how you can use standard deviation to define an OE norm in any case. If you discussed it or gave examples in primerr.pdf I don't recall it. Did you?

If you want to research
> octave-equivalent metrics, you could try working out the metrics that
> correspond to odd limits.

That's a thought, but it doesn't seem to be a full-fledged proposal. In any case, I already have an OE metric and it is not clear to me why you don't like it.

> >> This may lead to problems with complexity of intervals.  I haven't
> >> worked out the details.  But wouldn't 16:15 be ranked much simpler
> >> than 15:14 in 7-limit JI?

Not so very much simpler: 2.811 for 16/15, and 3.973 for 15/14. Is this bad, and if so, why? After all, odd height gives 15 for 16/15 and 105 for 15/14, so 15/14 loses there too.

"genewardsmith" <genewardsmith@sbcglobal.net> wrote:
>
> --- In tuning-math@yahoogroups.com, Graham Breed
> <gbreed@...> wrote:
>
> > I suggested using the standard deviation of the
> > weighted mapping for the rank 2 case. I thought that
> > suggestion was clear in the paragraph you cut out.
>
> It's completely unclear to me how you can use standard
> deviation to define an OE norm in any case. If you
> discussed it or gave examples in primerr.pdf I don't
> recall it. Did you?

I don't know if it's a norm or not. I'm not a
mathematician.

> If you want to research
> > octave-equivalent metrics, you could try working out
> > the metrics that correspond to odd limits.
>
> That's a thought, but it doesn't seem to be a
> full-fledged proposal. In any case, I already have an OE
> metric and it is not clear to me why you don't like it.

I don't like it because if it 's what I think it is, it
give different results for octave-specific and
octave-equivalent cases.

> > >> This may lead to problems with complexity of
> > >> intervals.  I haven't worked out the details.  But
> > >> wouldn't 16:15 be ranked much simpler than 15:14 in
> > >> 7-limit JI?
>
> Not so very much simpler: 2.811 for 16/15, and 3.973 for
> 15/14. Is this bad, and if so, why? After all, odd height
> gives 15 for 16/15 and 105 for 15/14, so 15/14 loses
> there too.

16:15 has a TE complexity of 7.9 octaves, against 15:14
with 7.7 octaves. They both have the same Kees
expressibility because they both have the same odd identity:
15. Compared to these figures, 2.8 is much simpler than 4.0.

Whatever this "odd height" is, it also looks wrong.

Graham

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:

> > It's completely unclear to me how you can use standard
> > deviation to define an OE norm in any case. If you
> > discussed it or gave examples in primerr.pdf I don't
> > recall it. Did you?
>
> I don't know if it's a norm or not. I'm not a
> mathematician.

So far as I can tell, the problem is it isn't OE. But I'm not sure what you are trying to say.

> > That's a thought, but it doesn't seem to be a
> > full-fledged proposal. In any case, I already have an OE
> > metric and it is not clear to me why you don't like it.
>
> I don't like it because if it 's what I think it is, it
> give different results for octave-specific and
> octave-equivalent cases.

That simply says that you don't like OE measures because they are OE, which isn't a reason at all. A nontrivial measure on equivalence classes under octave equivalence *must*, by definition, give different answers than a measure on the intervals themselves; for one thing, 2 is in the same class as 1, and hence both must, inescapably, have the same measure.

> 16:15 has a TE complexity of 7.9 octaves, against 15:14
> with 7.7 octaves. They both have the same Kees
> expressibility because they both have the same odd identity:
> 15. Compared to these figures, 2.8 is much simpler than 4.0.

Yes, and it should since it is a measure on a class containing 8/15, 4/15, 2/15 and 1/15 as well as 16/15.

> Whatever this "odd height" is, it also looks wrong.

Where "wrong" has the bizarre definition (in a discussion of OE measures) of "valid as an OE measure".