Lumma's scale lattices and 6:8:11 tempering

[Me, Dave Keenan]
>>Chalmers lattices dont do anything for me I'm afraid.

[Paul Erlich]
>Aren't they the same as the ones we've been using here on the list for the
>7-limit? Beyond that, aren't they like Wilson's lattices except they only
>have one dimension for each prime instead of each odd? What is it that they
>don't do for you?
>
>Are you referring to Monzo's lattices? The unfortunate thing about them is
>that they only show pitch as angle in a few select cases; otherwise
>interpreting angles as pitch doesn't work.

No I was referring to (what might be?) Monzo's new type, drawn for him by
David Chalmers. Sorry I didn't make this clear. Monzo corrected my
oversight in Digest 144. In these, pitch is expressed as radial distance,
not polar angle. See
http://www.ixpres.com/interval/monzo/w-jchalm/1999-3-29/oc-red1.htm

[Erlich]
>That's a nice visualization of Lumma's scale. But what is that mysterious
>connection between d# and ab? An approximate 11:8?

I assume you learned, when you read on, that it was indeed an approximate
11:8. Incidentally the error was 4.5c, not 6.5c as I said in that message.
But that's still not a good enough approximation for such a complex ratio
(particularly in relation to the accuracy of the other ratios in this
scale). Equal weighting of errors *really* fails for ratios of 11.

Paul Erlich
> Otherwise, it is quite
>similar in concept to the way I depicted the decatonic scales in my paper
>(pages 30 and 31 -- instead of note names I have 22-tET degrees) in that
>all 7-limit tetrads are shown as tetrahedra.

Similar in the concept that all 7-limit tetrads are shown as tetrahedra,
yes. But I take this to be a minimum requirement. I thought for a minute
that you were saying there was a decatonic hiding in Lumma's scale.

>Since I was most interested in the
>tetrads, I would eliminate the connections that are not parts of tetrads:
...
>or, since I was also interested in chains of fifths, I might go back to
>the arrangement considered in connection with Fokker's version of this
>scale
> f
> / \
> / \
> a---------e---------b---------f#
> /|\ /|\`. /,'/ \`.\ ,'/
> / | \ / | \ db-/---\--ab /
> / d#--------a# \ | / \ | /
> /,' \`.\ /,'/ `.\|/ \|/
> f------\--c--/------g---------d
> \ | /
> \|/
> f#
>
>so that the chains of fifths are laid out more clearly.
>I might prefer the last one above because chains of fifths are so
>important to harmonic progressions.

Yes. That's a very useful view of it, for the reason you gave. But note
that it fails to show the B and Fm triads, since it completely omits two
consonant intervals. These are in fact the only thing that makes Lumma's
different to Fokker's. So I would feel obliged to extend this to

f---------c
/ \ /
/ \ /
a---------e---------b---------f#
/|\ /|\`. /,'/ \`.\ /,'/
/ | \ / | \ db-/---\--ab /
/ d#--------a# \ | / \ | /
/,'/ \`.\ /,'/ `.\|/ \|/
f--/---\--c--/------g---------d
/ \ | /
/ \|/
b---------f#

even though it duplicates more than the minimum number of notes, or at least

f
/ \/
/ /\ /
a---------e---------b---------f#
/|\ /|\`. /,'/ \`.\ /,'/
/ | \ / | \ db-/---\--ab /
/ d#--------a# \ | / \ | /
/,'/ \`.\ /,'/ `.\|/ \|/
f--/---\--c--/------g---------d
/ \/| /
/\|/
f#

The best understanding is probably gained by looking at mine and yours
together (yours as extended above?). So here's mine again.

a---------e
/|\ /|\
/ | \ / | \
/ d#--------a# \
/,'/:\`.\ /,'/ `.\
f--/-:-\--c--/------g
/|\/ : \/| /
/ |/\ : /\|/
e---------b---------f#
\`. /,'/ \`.\:/,'/
\ db-/---\--ab /
\ | / \ | /
\|/ \|/
g---------d

They are of course maximally opposed; one having the E:G at the periphery
and the other having it at the center. I notice that you lined yours up
vertically with mine; very nice. Why do you favour lowercase for these
diagrams?

At present I'm fooling around with the 6:8:11 (A#:D#:Ab) and its
relationships (in the new temperament that I gave in digest 144), and your
favoured view breaks *them* badly. I didn't show the two 6:11's since it is
too hard in ASCII. The other 6:11 is of course D#:Db, but a 1/11:1/8:1/6
chord is just mud. Mind you, 6:8:11 isn't much better. :-)

I gave the wrong sign for the error in the 6:11 in my new temperament. The
errors are

2:3 4:5 5:6 4:7 5:7 6:7 6:11 8:11
-2.74 -2.74 0 -3.24 -0.5 -0.5 1.47 -1.27 cents

The absolute value of the errors in 6:11 and 8:11 are in the proportion
log(8) to log(6). I figure this gives them equal beats in the 6:8:11 and
hence minimises the roughness of that chord in this scale. It sounds ok to
me, but it wouldn't sound much different if they were made equal in cents.
Notice that the improvement in the 11's is mostly at the expense of the 4:7
which used to be just.

Regards,
-- Dave Keenan
http://dkeenan.com

Dave Keenan wrote,

>Why do you favour lowercase for these
>diagrams?

I had three layers in the third dimension in my diagram, so using case to
distinguish the layers wouldn't work. Plus I didn't want anyone to get the
idea that E was a different note from e, etc.