# Talk:The Riemann Zeta Function and Tuning/WikispacesArchive

## Contents

# ARCHIVED WIKISPACES DISCUSSION BELOW

**All discussion below is archived from the Wikispaces export in its original unaltered form.**

Please do not add any new discussion to this archive page.

All new discussion should go on Talk:The Riemann Zeta Function and Tuning.

## Two more types of zeta edos?

The lists of zeta edos fascinate me; the most interesting thing to me is that certain very good edos (such as 58 and 94) do not actually appear, while other edos that don't seem to be as good, do.

I was thinking about if there were any other analyses that could be performed on the Z function that might give different lists; in particular, if there are any possible lists of zeta edos that do include 58 or 94, or possibly others.

Two other possible ideas that I thought of, off the top of my head, are:

-Zeta triangle edos: Obtained by measuring the area of the triangle formed "as the crow flies" by two successive zeroes and the peak between them. This area may be less than or greater than than the integral of Z between those zeroes, depending on whether the "sides of the mountain" are convex or concave, respectively. Since it depends on both the size of the gap and the height of the peak, but is independent of the integral, we would expect that the zeta triangle edos would give more weight to the low AND high primes, and less to the middle ones.

-Zeta rectangle edos: Obtained by finding the largest possible rectangle that can fit underneath the Z function, and measuring the area of that rectangle. This is almost the exact opposite of the zeta triangle, since it favors convex sides over concave ones and does not depend on the gap size or peak height. We would expect this to favor the middle primes while giving noticeably less importance to the high and low ones. In other words, it's like zeta integral, but "a lot more so".

I'm not sure how to go about making these calculations (I don't currently have access to Mathematica), but if anyone else on here can, I would LOVE to see some lists of zeta triangle and zeta rectangle edos! Especially if any edos that don't appear on the other lists show up on them.

- **MasonGreen1** April 07, 2016, 08:03:24 PM UTC-0700

## Similar calculations

Some years ago, I did some calculations similar to these, unaware of Gene Ward Smith's work. In 2005 I submitted an article to "Music Theory Online", but it was not accepted. So I put my article online at https://sites.google.com/site/wwwbuch/Zetamusic5.pdf, where it may still be downloaded.

The main statement of my article is that *any* scale that is (approximately) even-spaced and (approximately) harmonic (- not only edos -) will divide the octave in a number of intervals that is related to the Riemann function.

- **peter.buch** December 06, 2012, 07:18:45 AM UTC-0800

Wow! Thanks for the cite, I'll look at your paper.

- **genewardsmith** December 06, 2012, 04:21:02 PM UTC-0800

It says "click here to download" but I can't click.

- **genewardsmith** December 06, 2012, 04:22:42 PM UTC-0800

For me it's the same. Can it be that access is restricted?

When I chose "save target as", I get a file of size 1.8 MB which cannot be opened with Acrobat Reader.

- **xenwolf** December 07, 2012, 12:30:09 AM UTC-0800

I can click th word "here", in Internet Explorer as well as in Firefox. Anyway, I have uploaded the file to this wiki: Zetamusic5.pdf. Hope it works.

Peter

- **peter.buch** December 10, 2012, 05:42:09 AM UTC-0800

Doesn't work. Try this: http://xenharmonic.wikispaces.com/file/view/Zetamusic5.pdf

- **peter.buch** December 10, 2012, 05:53:00 AM UTC-0800

This is ok. I already read a part of it.

- **xenwolf** December 11, 2012, 01:28:46 AM UTC-0800

## Applications

Can someone add a paragraph about the applications of this? Explaining it in plain English?

- **Natebedell** November 03, 2011, 11:23:58 AM UTC-0700