Re: [tuning-math] Digest Number 1590

Hi Gene. You wrote:
> Possible lists are decreasing L1, L2, and L infinity error, everything > below a logflat cutoff figure, such as 1 for L infinity error, > decreasing Pepper ambigutity,

Am I right to think that L1, L2 and L-infinity errors will correspond to what I called "sum of errors" (= mean error), "sum of squared errors" (=mean squared error) and maximum error?

By Pepper ambiguity you mean the ratio between the errors in the closest and second-closest approximation an edo gives for a ratio, right? That leaves undefined how you combine the ambiguities when a temperament is supposed to approximate multiple ratios. In any case I think it will result in the edos being ordered the same way as when the (usual) error is penalised (multiplied) by the cardinality of the scale, don't you think?

> and one that they might find really cute: > nearest integer to increasing size of the absolute value of the integral > of the Z function between successive zeros, with the argument normalized > by log(2)/2 pi.
>
> I wrote a Wikipedia article on the Z function with the intent of
> explaining this stuff on my own web page, and then never did.

I can't see it in the current iteration of the Wikipedia article on the Z function. Have you explained the relationship somewhere I can read about it?

> I like logflat penalty schemes, as they give infinite lists but thin
> them out pretty well.

If I've understood Monz's page on "logflat badness", this would mean (for edos) penalising the error by multiplying it by the cardinality of the scale raised to the exponent 3/2. Is that right? Can you show that that choice of exponent gives the "smallest infinite" sequence, or did you determine that empirically?

--Jon

--- In tuning-math@yahoogroups.com, Jon Wild <wild@...> wrote:

> Am I right to think that L1, L2 and L-infinity errors will
correspond to
> what I called "sum of errors" (= mean error), "sum of squared errors"
> (=mean squared error) and maximum error?

Correct.

> By Pepper ambiguity you mean the ratio between the errors in the
closest
> and second-closest approximation an edo gives for a ratio, right? That
> leaves undefined how you combine the ambiguities when a temperament is
> supposed to approximate multiple ratios.

It's the maximum of these ratios wrt a given diamond.

It seems to me it's a good choice for the integer sequences. It is a
property of the edo only, not of any val, and it generalizes an
integer sequence already in the handbook, A005664. These are the
denominators for the convergents of log2(3), and also the 3-limit list
of decreasing Pepper ambiguity. Here are some more:

5-limit
1, 2, 3, 4, 5, 7, 12, 15, 19, 31, 34, 53, 65, 118, 171, 289, 441, 559,
612, 730, 1171, 1783, 2513, 4296, 6809, 8592

7-limit
1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 19, 22, 27, 31, 41, 68, 72, 99, 130,
140, 171, 202, 270, 342, 441, 612, 1547, 1578, 2019, 3125, 3395, 3566,
5144, 6520, 6691

9-limit
1, 2, 5, 12, 19, 22, 31, 41, 72, 99, 171, 270, 342, 441, 612, 3125, 6691

11-limit
1, 2, 7, 12, 22, 31, 41, 46, 58, 72, 118, 152, 270, 342, 494, 612,
764, 836, 1578, 1848, 6421, 6691, 7927

These look like good replacements for what is there.

> > I wrote a Wikipedia article on the Z function with the intent of
> > explaining this stuff on my own web page, and then never did.
>
> I can't see it in the current iteration of the Wikipedia article on
the Z
> function. Have you explained the relationship somewhere I can read
about
> it?

http://en.wikipedia.org/wiki/Z_function

If you search for "Riemann zeta function" on this list you should find
some old articles.

> > I like logflat penalty schemes, as they give infinite lists but thin
> > them out pretty well.
>
> If I've understood Monz's page on "logflat badness", this would mean
(for
> edos) penalising the error by multiplying it by the cardinality of the
> scale raised to the exponent 3/2. Is that right?

It depends on the prime limit. In the 7 limit that would be right.

Can you show that that
> choice of exponent gives the "smallest infinite" sequence, or did you
> determine that empirically?

It follows from Diophantine approximation theory.

Gene wrote:
[snipped Pepper ambiguity lists]
> These look like good replacements for what is there.

Ok - will you submit them to the OEIS?

>>> I wrote a Wikipedia article on the Z function with the intent
>>> of explaining this stuff on my own web page, and then never
>>> did.
>>
>> I can't see it in the current iteration of the Wikipedia
>> article on the Z function. Have you explained the relationship
>> somewhere I can read about it?
>
> http://en.wikipedia.org/wiki/Z_function

Well of course that's the first place I looked, but there's
absolutely nothing there about the relationship to tunings.

> If you search for "Riemann zeta function" on this list you
> should find some old articles.

Ok, I just did. (Man I hate the search function at yahoogroups.) For
anyone else following along, this is the essence of the relationship,
in Gene's words from 2002:

---
The idea, very briefly, is that when Re(z)>=1 the absolute value of
the Riemann Zeta function will be high when z=s+it is near a scale
division where t = 2 pi n /ln(2), n, being the scale division; this
is because the Diophantine approximation problem for finding a good
division and finding a high value of |Zeta(z)| are essentially the
same. It turns out the relationship extrapolates into the critical
strip.
---

Message 879, also by Gene, lays out a fuller technical description.

By the way I'd recommend the book "The Music of the Primes" by Marcus
du Satoy, all about the Zeta function. It's not very good when it
makes analogies to music, in my opinion, but it was very engaging on
the mathematics and the history of the Riemann hypothesis.

-Jon

--- In tuning-math@yahoogroups.com, "wildatfas" <wild@...> wrote:
>
> Gene wrote:
> [snipped Pepper ambiguity lists]
> > These look like good replacements for what is there.
>
> Ok - will you submit them to the OEIS?

Should I do so as a replacement? What is there now?

Also, I'd like to submit the Z function sequence, which will take some
computing.