A Riemann zeta peek at 75 equal

I've put a jpg of Z(x) for values of x near 75 in the Photos
directory. Here Z is the Riemann zeta function along the critical
line, twisted in the usual way to get a real function of a real
variable, and then rescaled by the factor ln(2)/2 pi in the argument,
so as to make it correspond to divisions of the octave.

The interest of 75 is that there is a zero of the Reimann zeta
function near 75, and two reasonably large peaks, rather than the
usual one, near to it. The highest peak is the flat octaves peak,
which for a range of values around the peak corresponds to the
19-limit val

[75, 119, 174, 211, 260, 278, 307, 319]

which has 7-limit comma basis

[225/224, 1728/1715, 15625/15309]

The other peak is the sharp octaves peak; it has 19-limit val

[75, 119, 174, 210, 259, 277, 306, 318]

and comma basis

[686/675, 875/864, 5120/5103]

The standard val, in case anyone cares, would be

[75, 119, 174, 211, 259, 278, 307, 319]

The three are already distinct in the 11-limit.

considering that it's not so great in the 5-limit, i was surprised to
see it show up as so many of your low-geometric-badness 5-limit
linear temperaments: tetracot, orwell, vavoom, *and* misty!

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> I've put a jpg of Z(x) for values of x near 75 in the Photos
> directory. Here Z is the Riemann zeta function along the critical
> line, twisted in the usual way to get a real function of a real
> variable, and then rescaled by the factor ln(2)/2 pi in the
argument,
> so as to make it correspond to divisions of the octave.
>
> The interest of 75 is that there is a zero of the Reimann zeta
> function near 75, and two reasonably large peaks, rather than the
> usual one, near to it. The highest peak is the flat octaves peak,
> which for a range of values around the peak corresponds to the
> 19-limit val
>
> [75, 119, 174, 211, 260, 278, 307, 319]
>
> which has 7-limit comma basis
>
> [225/224, 1728/1715, 15625/15309]
>
> The other peak is the sharp octaves peak; it has 19-limit val
>
> [75, 119, 174, 210, 259, 277, 306, 318]
>
> and comma basis
>
> [686/675, 875/864, 5120/5103]
>
> The standard val, in case anyone cares, would be
>
> [75, 119, 174, 211, 259, 278, 307, 319]
>
> The three are already distinct in the 11-limit.