Looped distorted lattices and Lumma's scale

Hey Monz, glad you've got FTP going again.

I get "not found" for http://www.ixpres.com/interval/td/Erlich/entropy.htm
when I click on the relevant link within the explanation of "harmonic
entropy" in your excellent dictionary.

Chalmers lattices dont do anything for me I'm afraid. Bending to show pitch
radially loses too much other info for my liking.

However, these prompted me to draw (by hand) a similar distorted lattice
for Lumma's scale (the 7-limit 12-tone wafso-just one where the 225/224
disappears). In this case I was not attempting to display pitch, only to
show all linkages without repetition of notes. i.e. Within the one plane I
bent the 7-limit triangular lattice around in a circle and joined it up to
itself (the snake eating its own tail). Alas, the distortion was too great
and (almost) all sense of triangles (triads), terahedra (tetrads) and the
octahedron (hexany) were lost.

It is much easier to understand it as a periodic (i.e. repeating) 3D
lattice. I've drawn a single period below in the way which has minimum
duplication, i.e. I've cut the chain at its weakest link (the E-G minor
third). This puts the hexany at the center too. You could wrap it around to
make a cylinder (e to E and g to G).

a---------e
/|\ /|\
/ | \ / | \
/ D#--------A# \
/,'/:\`.\ /,'/ `.\
f--/-:-\--c--/------g
/|\/ : \/| /
/ |/\ : /\|/
E---------B---------F#
\`. /,'/ \`.\:/,'/
\ db-/---\--ab /
\ | / \ | /
\|/ \|/
G---------D

I think this helps to understand the harmonies and harmonic progressions
available in this tuning better than any of the previous drawings Carl or I
gave. I've used upper and lower case to try to assist in seeing the depth,
i.e. the two 5-limit planes, upper case in front and lower case behind. Not
sure if it really helps.

D#:Ab is 6.5 cents away from being an 8:11 (shown dotted above) and A#:Ab
is x cents away from being a 6:11. This isn't close enough to be useful,
but if they were included in the temper, a 6:8:11 triad (A#:D#:Ab) might be
possible (for whatever that's worth) without damaging the other harmonies
too much.

C or F# or both might also be added to this for some pretty weird chords.
i.e. subsets of

a# C D# F# Ab A# C' D#' F#' Ab' A#'
6 8 11 12 16 22 24
5 7 8 10 14 16
1/16 1/14 1/12 1/10 1/8 1/7 1/6 1/5 1/4

Hey that's even a kind of interesting pentatonic on A#.

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

Dave Keenan wrote,

>Chalmers lattices dont do anything for me I'm afraid.

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?

>Bending to show pitch
>radially loses too much other info for my liking.

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.

> a---------e
> /|\ /|\
> / | \ / | \
> / D#--------A# \
> /,'/:\`.\ /,'/ `.\
> f--/-:-\--c--/------g
> /|\/ : \/| /
> / |/\ : /\|/
>E---------B---------F#
> \`. /,'/ \`.\:/,'/
> \ db-/---\--ab /
> \ | / \ | /
> \|/ \|/
> G---------D

That's a nice visualization of Lumma's scale. But what is that mysterious
connection between d# and ab? An approximate 11:8? 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. Since I was most interested in the
tetrads, I would eliminate the connections that are not parts of tetrads:

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

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 think this helps to understand the harmonies and harmonic progressions
>available in this tuning better than any of the previous drawings Carl or I
>gave.

I might prefer the last one above because chains of fifths are so important
to harmonic progressions.

[Keenan:]
>> Bending to show pitch radially loses too much
>> other info for my liking.

[Erlich:]
> Are you referring to Monzo's lattices?

He's referring, in this sentence at least, to the
*new* lattices Chalmers has made with me:
http://www.ixpres.com/interval/monzo/w-jchalm/1999-3-29/oc-red1.htm

[Erlich:]
> 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.

Huh?

Every point on my lattices is placed according
to an angle derived from its pitch. Are you talking
strictly about its pitch relative to 1/1?

The new lattices are quite different. They 'warp'
the vectors to make them fit concentric 12-eq markings
that serve as the background of the lattice.
That's what Dave doesn't like.

I'm not sure I like them either. It was just an idea.

Joseph L. Monzo....................monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html

|"...I had broken thru the lattice barrier..."|
| - Erv Wilson |

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Joe Monzo wrote:

>[Erlich:]
>> 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.

>Huh?

>Every point on my lattices is placed according
>to an angle derived from its pitch. Are you talking
>strictly about its pitch relative to 1/1?

Yes, or any other way of reckoning pitch. Your lattices do not make it
possible to see the order of pitch of the notes in a scale. So in my view,
the use of angles derived from pitch ends up serving no useful purpose. As I
have told you before, I feel that your lattices would be improved if either
(a) you dropped the whole pitch-angle idea and used angles that made
simple-integer ratios (like 6/5) look closer together than more complex ones
(like 15/8); or (b) you chose angles in accordance with Canright's procedure
so that the pitch order would appear naturally on one axis.