Talk:The Riemann Zeta Function and Tuning/WikispacesArchive

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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