50&87 as an 11 and 13 system

Particularly good equal divisions for 11 and 13 taken together are 50
and 87; 87 does 11 and 13, and also an excellent 5. If you take 50&87
as a 13-limit system, you get a temperament with 11/8 as a generator,
such that five generators take us to 64/13, giving 2^21 11^(-5)
13^(-1) as a comma. This makes a heck of a good temperament for anyone
interested only in 11 and 13, and 5 fits in nicely as well. MOS of 5,
7, 11, 13, 24, 37, and 50 are available, and an 11 or 13 note MOS--or
twelve notes, for that matter--would give a scale already containing a
lot of {2,5,11,13} harmonies, nicely in tune.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

> Particularly good equal divisions for 11 and 13 taken
> together are 50 and 87; 87 does 11 and 13, and also an
> excellent 5. If you take 50&87 as a 13-limit system, you
> get a temperament with 11/8 as a generator, such that
> five generators take us to 64/13, giving 2^21 11^(-5)
> 13^(-1) as a comma. This makes a heck of a good temperament
> for anyone interested only in 11 and 13, and 5 fits in
> nicely as well. MOS of 5, 7, 11, 13, 24, 37, and 50 are
> available, and an 11 or 13 note MOS--or twelve notes,
> for that matter--would give a scale already containing a
> lot of {2,5,11,13} harmonies, nicely in tune.

thanks for that, Gene!

i've made two Muzika lattices of it, one open rectangular,
and one closed helical, at tuning_files (delete the line break
in these links):

/tuning-math/files/monz/11-13-
primespace_lattice_rectangular.gif

/tuning-math/files/monz/11-13-
primespace_lattice_closed-helix.gif

(note: these lattices are not of 87-et, but rather of the
r2 (linear) temperament, which is theoretically open at
the top and bottom, and which in my versions has an
arbitrary cutoff point.)

-monz
http://tonalsoft.com
microtonal music software

--- In tuning-math@yahoogroups.com, "monz" <monz@t...> wrote:

> --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
>
> > Particularly good equal divisions for 11 and 13 taken
> > together are 50 and 87; 87 does 11 and 13, and also an
> > excellent 5. If you take 50&87 as a 13-limit system, you
> > get a temperament with 11/8 as a generator, such that
> > five generators take us to 64/13, giving 2^21 11^(-5)
> > 13^(-1) as a comma. This makes a heck of a good temperament
> > for anyone interested only in 11 and 13, and 5 fits in
> > nicely as well. MOS of 5, 7, 11, 13, 24, 37, and 50 are
> > available, and an 11 or 13 note MOS--or twelve notes,
> > for that matter--would give a scale already containing a
> > lot of {2,5,11,13} harmonies, nicely in tune.
>
>
>
> thanks for that, Gene!
>
> i've made two Muzika lattices of it, one open rectangular,
> and one closed helical, at tuning_files (delete the line break
> in these links):

much better to see both geometries at once, side by side:

/tuning-math/files/monz/11-13-
primespace_lattice_rectangular-and-helix.gif

-monz
http://tonalsoft.com
microtonal music software