More TM base postings

Here are some other messages to check:

/tuning-math/message/3189

/tuning-math/message/5053

/tuning-math/message/5761

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> Here are some other messages to check:
>
> /tuning-math/message/3189
>
> /tuning-math/message/5053
>
> /tuning-math/message/5761

the first one assumes the standard val (or something) for
inconsistent ets; i'd at least show the results for other vals.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:

> the first one assumes the standard val (or something) for
> inconsistent ets; i'd at least show the results for other vals.

The 5-limit needs to be what it is in order to include 81/80, and for
some of these, the 7-limit needs to be what it is to avoid torsion.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
>
> > the first one assumes the standard val (or something) for
> > inconsistent ets; i'd at least show the results for other vals.
>
> The 5-limit needs to be what it is in order to include 81/80, and
for
> some of these, the 7-limit needs to be what it is to avoid torsion.

yes, that process should be made explicit though -- i'd rather
explain that 24-equal sometimes doesn't have a reasonable basis due
to torsion, and in those cases is best understood as an equal halving
of each 12-equal step, instead of using hidden rules to provide a
nice-looking answer.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:

> --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
> >
> > > the first one assumes the standard val (or something)
> > > for inconsistent ets; i'd at least show the results
> > > for other vals.
> >
> > The 5-limit needs to be what it is in order to include
> > 81/80, and for some of these, the 7-limit needs to be
> > what it is to avoid torsion.
>
> yes, that process should be made explicit though -- i'd
> rather explain that 24-equal sometimes doesn't have a
> reasonable basis due to torsion, and in those cases is
> best understood as an equal halving of each 12-equal step,
> instead of using hidden rules to provide a nice-looking
> answer.

yes, i agree totally with paul.

in fact, the precise thing that prompted me to write my
original post requesting these TM-reduced bases was that
i tried to create 41-ET with our software, and while it
did give 41edo as the temperament, it gave an 82-tone
periodicity-block for the JI scale.

as soon as i saw that, i suspected that it was due to
torsion, and sure enough, that turned out to be the case.

in the specific case of 24-ET in a [3,5]-prime-space,
it's nice to be able to see how choosing a val of h(5)=8
results in a "double 12-ET", whereas h(5)=7 results in
a true 24-tone periodicity-block ...

... of course, noting that the chromatic semitone disappears
because 2^(7/24) is 350 cents, and is thus a neutral-3rd
which represents both the major-3rd and minor-3rd on this
particular bingo-card, which in turn means that it's not
really doing much in the way of representing 5-limit JI
in the first place.

-monz

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

> ... of course, noting that the chromatic semitone disappears
> because 2^(7/24) is 350 cents, and is thus a neutral-3rd
> which represents both the major-3rd and minor-3rd on this
> particular bingo-card, which in turn means that it's not
> really doing much in the way of representing 5-limit JI
> in the first place.

well, it's still doing so at least in theory, just like all the ETs
on the 'dicot' line on the zoom-1 and zoom-10 charts:

http://www.sonic-arts.org/dict/eqtemp.htm

(this mapping of 24-equal would appear at the intersection of the
aristoxenean and dicot lines; i didn't label it because 24-equal is
consistent in the 5-limit with the same approximations as 12-equal,
so you'll see 24 on the same point as 12 instead.)

and like those with a bingo card where the chromatic semitone
vanishes, for example

/tuning-math/files/perlich/10.gif
/tuning-math/files/Paul/7p.gif