Modification of the Zeta function to work with subgroup temperaments

A subgroup temperament of a certain limit is one that has one or more
"holes" in it, e.g. the 2.3.7.13 subgroup temperament has holes at 5
and 11. This can be extended to the zeta function by taking advantage
of the Euler product formula:

zeta(s) = Prod_p (1/(1-p^(-s)))

If we want to create an infinite product of primes that, say, doesn't
include 5, we can come up with a no-5's zeta formula by simply
multiplying by (1-5^(-s)):

zetano5(s) = Prod_p ((1-5^(-s))/(1-p^(-s)))

One could also ignore 3 as a prime, but treat 9 as a "faux" prime to
work within the 2.9.5.7.11.13.17.... limit:

zetano3with9(s) = (1-3^(-s))/(1-9^(-s)) * Prod_p (1/(1-p^(-s)))

This could also, perhaps, be used to weight the contributions from
different primes differently. (Or, as a side note, perhaps we could
arrive at different prime weightings than sqrt(p) by looking at
zeta^2, etc).

A caveat - the zeta function isn't actually defined as above, but is
defined by an analytic continuation involving the gamma function that
allows it to be defined anywhere except for where s=1. However, if I
remember correctly, this is only a problem if Re{s} > 1, whereas if
we're sticking to the critical line then perhaps the above wouldn't be
a problem. Could this information be used to come up with a zeta
integral sequence for subgroup temperaments? Or, would it require
first coming up with a factorial operator for this new no-3's
9-is-prime number system, and then generalizing that to a new gamma
function?

-Mike

Quick note: If you want to calculate zeta function values for these purposes, do NOT use the GNU Scientific Library implementation (which is the one used by Octave and some other software). It becomes quite inaccurate for values on the critical line with an imaginary part over a few hundred, which corresponds to EDOs of only 30 or 40.

I'm now trying out the Python "mpmath" library which may have a more accurate zeta function.

More detailed post to follow.

Keenan

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

This stuff was going to go into a section of the zeta article I haven't written yet, complete with plots. I'm trying to compose at the moment. The no-3 with 9 stuff had not occurred to me, so I'll think about it.

--- In tuning-math@yahoogroups.com, "Keenan Pepper" <keenanpepper@...> wrote:
>
> Quick note: If you want to calculate zeta function values for these purposes, do NOT use the GNU Scientific Library implementation (which is the one used by Octave and some other software).

The Maple routines are accurate, but very slow. Mathematica seems to do it right.

At 03:00 PM 4/17/2011, you wrote:
>The Maple routines are accurate, but very slow. Mathematica seems to
>do it right.

That seems to be the trend these days.

-Carl

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> A subgroup temperament of a certain limit is one that has one or more
> "holes" in it, e.g. the 2.3.7.13 subgroup temperament has holes at 5
> and 11. This can be extended to the zeta function by taking advantage
> of the Euler product formula:
>
> zeta(s) = Prod_p (1/(1-p^(-s)))
>
> If we want to create an infinite product of primes that, say, doesn't
> include 5, we can come up with a no-5's zeta formula by simply
> multiplying by (1-5^(-s)):
>
> zetano5(s) = Prod_p ((1-5^(-s))/(1-p^(-s)))
>
> One could also ignore 3 as a prime, but treat 9 as a "faux" prime to
> work within the 2.9.5.7.11.13.17.... limit:
>
> zetano3with9(s) = (1-3^(-s))/(1-9^(-s)) * Prod_p (1/(1-p^(-s)))
>
> This could also, perhaps, be used to weight the contributions from
> different primes differently. (Or, as a side note, perhaps we could
> arrive at different prime weightings than sqrt(p) by looking at
> zeta^2, etc).
>
> A caveat - the zeta function isn't actually defined as above, but is
> defined by an analytic continuation involving the gamma function that
> allows it to be defined anywhere except for where s=1. However, if I
> remember correctly, this is only a problem if Re{s} > 1, whereas if
> we're sticking to the critical line then perhaps the above wouldn't be
> a problem.

Actually it's only a problem for Re(s) <= 1. For Re(s) > 1 the series converges, but for Re(s) <= 1 you have to compute the analytic continuation some other way.

> Could this information be used to come up with a zeta
> integral sequence for subgroup temperaments? Or, would it require
> first coming up with a factorial operator for this new no-3's
> 9-is-prime number system, and then generalizing that to a new gamma
> function?

You don't need to do any of that stuff. In spite of the fact that the series doesn't converge for Re(s) <= 1 where we're interested, it's still totally easy as long as you have an implementation of the zeta function.

There are two different things you can do:

If you want to include only a finite set of primes, then you don't even use your zeta function calculator at all. You just plot

Prod_("primes" p to include) 1/(1-p^(-s))

If you only include one prime you always get a periodic function, like this:
https://docs.google.com/viewer?a=v&pid=explorer&chrome=true&srcid=0B9CMyeCjAMQGZGI4Y2QyODQtNjA0Yi00MDUxLTk3MGYtNDk4MDY5OTMxOTY4&hl=en
(Note that all these PDFs are vector plots, so you can zoom in to see more detail)

If you include more than one prime you get a reasonable equal-division-goodness function. For example:

2,3 (near-record maxima at 5, 12, 41, 53, 94) https://docs.google.com/viewer?a=v&pid=explorer&chrome=true&srcid=0B9CMyeCjAMQGMzg0MmQ5MjgtN2E2OC00M2FkLTg4ZmYtZTg4ZGQzZjY4NWNm&hl=en

2,3,5 (near-record maxima at 7, 12, 19, 31, 53) https://docs.google.com/viewer?a=v&pid=explorer&chrome=true&srcid=0B9CMyeCjAMQGNzc1ZTIyODgtMzNlZi00MDdmLWEyMmQtMjNjZTEwZTkxODg5&hl=en

2,3,7 (near-record maxima at 5 and 36) https://docs.google.com/viewer?a=v&pid=explorer&chrome=true&srcid=0B9CMyeCjAMQGMDEwZjY4NzAtNjk4OS00MDY2LTgxMzAtOTM2YzBiMWU3ODZi&hl=en

2,3,5,7 (5, 12, 19, 31, 53, 72, 99) https://docs.google.com/viewer?a=v&pid=explorer&chrome=true&srcid=0B9CMyeCjAMQGNDQ4MDIyZmQtYmFlZi00NTM2LTk5YzQtMjBmNTExOWQ3YmQ2&hl=en

2,3,11 (5, 7, 17, 24) https://docs.google.com/viewer?a=v&pid=explorer&chrome=true&srcid=0B9CMyeCjAMQGMjhhYTNlZjEtNTU2Ny00ZTk4LTk1MTgtYzBmYTZkNTExNzAw&hl=en

Using 4 as a "prime" (so octaves are not included but double-octaves are), we can get for example:
3,4,5 (maxima at 5 and 19 equal divisions of 4/1) https://docs.google.com/viewer?a=v&pid=explorer&chrome=true&srcid=0B9CMyeCjAMQGMTAzMGUzNGMtNGFhNy00YTQ2LThlNTEtYTk0YjQ4MDhhOWQz&hl=en
(The non-octave scale where you take every other note of 19-EDO is pretty mind-bending.)

And for 3,5,7 we get 13 equal divisions of 3/1 (Bohlen-Pierce scale) standing out way above anything else: https://docs.google.com/viewer?a=v&pid=explorer&chrome=true&srcid=0B9CMyeCjAMQGNTY0OWI3ZDgtZTkzZi00NTUwLThkN2UtODY5MWZmMzcyYTk4&hl=en

The second thing we can do is the one you pointed out. If you want to *exclude* only a finite set of primes from the whole zeta function, then you plot

[Prod_(primes p to exclude) (1-p^(-s))] zeta(s)

and you can also add "primes" in that weren't there before.

The reason this works and is well-defined is that the modified zeta function we want is equal to the product of the original zeta function and another meromorphic function for Re(s) > 1, but by analytic continuation the same equation must also hold everywhere none of the functions has a pole.

Some examples are:

Original zeta function (no primes removed) (record maxima at http://oeis.org/A117536 ) https://docs.google.com/viewer?a=v&pid=explorer&chrome=true&srcid=0B9CMyeCjAMQGOGQ1ZmJjNTEtNjg1OC00NDIzLTk0NTgtMTJjNDYyODM5ZTMx&hl=en

2 removed (near-record maxima at these divisions of the 3/1: 2, 4, 7, 9, 13, 15, 17, 26, 32, 39, 56, 69, 71, 75, 88 - note that the B-P scale is still there as a record peak, but much less prominent now that all other primes are included) https://docs.google.com/viewer?a=v&pid=explorer&chrome=true&srcid=0B9CMyeCjAMQGNTI5YzMyNDYtY2ZlNi00ZTU0LTkzYTktNGUyMzU5MTVkNzE0&hl=en

2 removed but 4 added back (These divisions of the 4/1: 5, 15, 19, 49, 67, 87) https://docs.google.com/viewer?a=v&pid=explorer&chrome=true&srcid=0B9CMyeCjAMQGOWRlYzBhYmMtY2VjYi00ZTI5LWFjNmItMDc3NDQ1OTY0YTQ2&hl=en

3 removed (6, 13, 16, 21, 25, 37, 74, 78, 93) https://docs.google.com/viewer?a=v&pid=explorer&chrome=true&srcid=0B9CMyeCjAMQGNjliZmNlMmMtNzUyYS00NmQyLTlmYTItYWIzNzk0MmM2MjNj&hl=en

3 removed but 9 added back (6, 13, 25, 47, 66, 93) https://docs.google.com/viewer?a=v&pid=explorer&chrome=true&srcid=0B9CMyeCjAMQGM2E0MjU0NmQtYTMwMC00OTRlLWJlNDAtODE2YTBmOTVhMGNh&hl=en

5 removed (5, 10, 14, 17, 36, 63, 77, 89, 94) https://docs.google.com/viewer?a=v&pid=explorer&chrome=true&srcid=0B9CMyeCjAMQGYzYxNGE0YjUtZDkxYy00ODI2LWI5ZmMtMDcxOGI3ODlmYTZi&hl=en

One last thing to notice is that a lot of these curves have nonzero local minima, while the original zeta function appears to have none (other than that first one) - which is the Riemann hypothesis. But the weird thing is that whenever you add back some power of all the primes you removed, it sure looks like there are only a finite number of such minima and then they stop. For example, in zeta-no3 the nonzero minima keep going through a little one around 99, but in zeta-no3with9 there's one at 0.15 and another at 0.55... but then they appear to stop. Who knows why?

Keenan

On Mon, Apr 18, 2011 at 3:59 PM, Keenan Pepper <keenanpepper@gmail.com> wrote:
>
> If you only include one prime you always get a periodic function, like this:
> https://docs.google.com/viewer?a=v&pid=explorer&chrome=true&srcid=0B9CMyeCjAMQGZGI4Y2QyODQtNjA0Yi00MDUxLTk3MGYtNDk4MDY5OTMxOTY4&hl=en
> (Note that all these PDFs are vector plots, so you can zoom in to see more detail)

Huh. Why is it periodic?

> If you include more than one prime you get a reasonable equal-division-goodness function. For example:
>
> 2,3 (near-record maxima at 5, 12, 41, 53, 94) https://docs.google.com/viewer?a=v&pid=explorer&chrome=true&srcid=0B9CMyeCjAMQGMzg0MmQ5MjgtN2E2OC00M2FkLTg4ZmYtZTg4ZGQzZjY4NWNm&hl=en
>
> 2,3,5 (near-record maxima at 7, 12, 19, 31, 53) https://docs.google.com/viewer?a=v&pid=explorer&chrome=true&srcid=0B9CMyeCjAMQGNzc1ZTIyODgtMzNlZi00MDdmLWEyMmQtMjNjZTEwZTkxODg5&hl=en
>
> 2,3,7 (near-record maxima at 5 and 36) https://docs.google.com/viewer?a=v&pid=explorer&chrome=true&srcid=0B9CMyeCjAMQGMDEwZjY4NzAtNjk4OS00MDY2LTgxMzAtOTM2YzBiMWU3ODZi&hl=en
>
> 2,3,5,7 (5, 12, 19, 31, 53, 72, 99) https://docs.google.com/viewer?a=v&pid=explorer&chrome=true&srcid=0B9CMyeCjAMQGNDQ4MDIyZmQtYmFlZi00NTM2LTk5YzQtMjBmNTExOWQ3YmQ2&hl=en

Huh! I'm surprised that 19 beat out 22 for the 7-limit.

> Some examples are:
>
> 3 removed but 9 added back (6, 13, 25, 47, 66, 93) https://docs.google.com/viewer?a=v&pid=explorer&chrome=true&srcid=0B9CMyeCjAMQGM2E0MjU0NmQtYTMwMC00OTRlLWJlNDAtODE2YTBmOTVhMGNh&hl=en

13 beats 11?! I'm surprised.

> 5 removed (5, 10, 14, 17, 36, 63, 77, 89, 94) https://docs.google.com/viewer?a=v&pid=explorer&chrome=true&srcid=0B9CMyeCjAMQGYzYxNGE0YjUtZDkxYy00ODI2LWI5ZmMtMDcxOGI3ODlmYTZi&hl=en

Good to see 17 up there.

> One last thing to notice is that a lot of these curves have nonzero local minima, while the original zeta function appears to have none (other than that first one) - which is the Riemann hypothesis. But the weird thing is that whenever you add back some power of all the primes you removed, it sure looks like there are only a finite number of such minima and then they stop. For example, in zeta-no3 the nonzero minima keep going through a little one around 99, but in zeta-no3with9 there's one at 0.15 and another at 0.55... but then they appear to stop. Who knows why?

That is troubling indeed. All I know is that the zeta function is
apparently somehow related to the Mellin transform of the
prime-counting function. Well, I know that the Fourier transform of
the prime-indicator function is going to be periodic, and hence the
Fourier transform of the prime-counting function, which is the
integral of the prime-indicator function, is going to be a decaying
periodic signal. Since the Mellin transform is related to the Fourier
transform, maybe we could figure it out by starting there, except with
a prime-indicator function that has certain numbers removed and such.

-Mike

Also, do you think there's any way to work out a list of the best
linear temperaments? I guess you could always just take the best ETs,
somehow assign an omega-limit val to each one, and wedge them
together, although there has to be a more elegant way to just arrive
at a list of two generators...

I wonder if we'll get a sequence of best, no-limit linear temperaments
that goes something like meantone, pajara, porcupine, miracle, etc.

-Mike

On Mon, Apr 18, 2011 at 6:14 PM, Mike Battaglia <battaglia01@gmail.com> wrote:
> On Mon, Apr 18, 2011 at 3:59 PM, Keenan Pepper <keenanpepper@gmail.com> wrote:
>>
>> If you only include one prime you always get a periodic function, like this:
>> https://docs.google.com/viewer?a=v&pid=explorer&chrome=true&srcid=0B9CMyeCjAMQGZGI4Y2QyODQtNjA0Yi00MDUxLTk3MGYtNDk4MDY5OTMxOTY4&hl=en
>> (Note that all these PDFs are vector plots, so you can zoom in to see more detail)
>
> Huh. Why is it periodic?

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> Huh. Why is it periodic?

One answer is that, since we only told it to care about one prime (for example 2), any equal division of that prime is perfectly good. In other words, all EDOs have equally good octaves.

Another answer is that, since the exponential function is periodic with period 2*pi*i, the function 1/(1-p^-s) = 1/(1-exp(-log(p)*s))
is periodic in s with period 2*pi*i/log(p).

> > 3 removed but 9 added back (6, 13, 25, 47, 66, 93) https://docs.google.com/viewer?a=v&pid=explorer&chrome=true&srcid=0B9CMyeCjAMQGM2E0MjU0NmQtYTMwMC00OTRlLWJlNDAtODE2YTBmOTVhMGNh&hl=en
>
> 13 beats 11?! I'm surprised.

As you can see from the plot, 11 is a pretty high peak, but both 6 and 13 are slightly higher. If you went by peak area instead of height the ordering might be different.

Keenan