The minerva-marvel-Bigler-orwell complex

225/32 gives us a 5-limit version of 7, and 5625/512 a 5-limit version
of 11. Tempering out both 225/224 and 5625/5632 gives what I've called
"minerva", the 11-limit planar temperament with TM basis {99/98,
176/175}. It has wedgie

<<<1 2 4 -2 -2 4 -5 -9 2 8||

and mapping

[<1 0 0 -5 -9|, <0 1 0 2 2|, <0 0 1 2 4|]

Written as a matrix of five rows and three columns, the bottom two
rows are simply 225/32 and 5625/512, which is typical behavior in
these planar mappings using Hermite reduction.

As often happens with planar temperaments, the optimized tunings can
be rather variable. The minimax tuning is particularly interesting,
since it equates the error of 5 and 7, and hence makes 7/5 pure. In
other words, the minimax tuning for minerva, an extension of 7-limit
marvel, is exactly Bigler 1/2-kleismic marvel. As you may recall, this
is marvel sneaking off in the direction of meantone; 81/80 does not
vanish but it shrinks to a mere 4.47 cents; nor does 385/384 vanish,
but it shrinks also to 1.45 cents. We don't get the huygens version of
11-limit meantone, but we draw close to it--close enough to make a
difference. We end up in a system which can be regarded as either
irregular, inconsistent, or just plain lucky, depending on how you
want to conceptualize it. Whatever you call it, Bigler marvel/minimax
minerva seems like a useful way to temper Fokker blocks and genera.

The rms tuning for minerva is also interesting; the value it gives to
7/6 makes it a good orwell generator; in fact, pretty close to the
orwell of 84+53 = 137-equal. In terms of lattices and genera, the
225/32 approximate 7 lies next to 375/256, an approximate 16/11. We
have 16/11 ~ 35/24 via 385/384, and 35/24 ~ 375/256 via 225/224, so
this is an 11-limit marvel approximation. It isn't as good as 5625/512
for producing complete 11-limit chords, since it is a 16/11 and not an
11/8, but it is excellent at adding in 11-limit intervals. If we
decide we want a 225/224 system in which both 5625/512 is an
approximate 11 *and* 375/256 is an approximate 16/11, we want all of
225/224, 385/384, and 5625/5632 to vanish; this combined
marvel-minerva is 11-limit orwell, 22&31. The rms tuning, and orwell
for that matter, makes the 5-limit more in tune; the rms minerva
tuning has a fifth 1.29 cents flat and a major third only 0.34 cents
sharp. On the other hand, it doesn't shrink 385/384, but actually
makes it larger--8.71 cents. Nor does it do a very good job with
81/80, managing only to cut it down to 16 cents.

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

The minimax tuning is particularly interesting,
> since it equates the error of 5 and 7, and hence makes 7/5 pure. In
> other words, the minimax tuning for minerva, an extension of 7-limit
> marvel, is exactly Bigler 1/2-kleismic marvel.

It's close, but not *exactly* 1/2-kleismic marvel; I forgot that that
required more than 225/224 tempered out and pure 7/5s. Minimax minerva
not only has pure 7/5s (or 7:5s if you prefer) but pure 11/9 (11:9.)
(Gosh this seems silly!) 1/2-kleismic has a fifth flat by
sqrt(225/224); minimax minerva is approximately 8/15-kleisma rather
than 1/2-kleisma. In comma terms, 1/2-kleismic is about 2/11-comma,
and minimax minerva is about 1/5-comma. Not much different; they both
will produce the mutant meantone marvel effect, though minimax minera
works in some pure 11:9 or whatever you want to call them.

This is stupid. I quit. Also, please note that 1/2, 8/15, 2/11 and 1/5
above are *not* musical intervals, and that requiring fractions to
always mean musical intervals would cause this whole group to melt
into a pool of butter.

>> The minimax tuning is particularly interesting,
>> since it equates the error of 5 and 7, and hence makes 7/5 pure. In
>> other words, the minimax tuning for minerva, an extension of 7-limit
>> marvel, is exactly Bigler 1/2-kleismic marvel.
>
>It's close, but not *exactly* 1/2-kleismic marvel; I forgot that that
>required more than 225/224 tempered out and pure 7/5s. Minimax minerva
>not only has pure 7/5s (or 7:5s if you prefer) but pure 11/9 (11:9.)
>(Gosh this seems silly!)

That's the thing about it: we almost *always* mean the interval
around here. Using the less standard colon for the most standard
thing we do seems a bit backward. Anyway, I think we now have
other tools in place to allow correct communication -- other
shit has come to light, as Lebowski says.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> That's the thing about it: we almost *always* mean the interval
> around here.

Except that we don't enforce the idea the interval is undirected that
way. This is actually a good thing.

>> That's the thing about it: we almost *always* mean the interval
>> around here.
>
>Except that we don't enforce the idea the interval is undirected that
>way. This is actually a good thing.

How do you write a directed interval? I've don't remember ever seeing
a compact notation for this from you -- you say it with context.
Similarly, the colon notation does not *enforce* undirectedness, but
rather assumes it unless context says otherwise.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> How do you write a directed interval? I've don't remember ever seeing
> a compact notation for this from you -- you say it with context.

I might, but pretty often I will say "80/81" instead of "a comma
down". for instance.

> Similarly, the colon notation does not *enforce* undirectedness, but
> rather assumes it unless context says otherwise.

It doesn't have a way of saying "3/4" other than 3:2 (or 2:3) down.

>> How do you write a directed interval? I've don't remember ever seeing
>> a compact notation for this from you -- you say it with context.
>
>I might, but pretty often I will say "80/81" instead of "a comma
>down". for instance.

I've never seen you do this, but I might have missed it.

>> Similarly, the colon notation does not *enforce* undirectedness, but
>> rather assumes it unless context says otherwise.
>
>It doesn't have a way of saying "3/4" other than 3:2 (or 2:3) down.

This ordering trick applies as well to the colon as the slash, and
one could argue in favor of adopting it.

-Carl