Re: [tuning] seeing is believing [lattices]

Joseph!
try http://www.anaphoria.com/dal01.html this article from xenharmonikon XII has quite a
few different lattices as well as different lattices of the same tuning!

Joseph Pehrson wrote:

> Conversation between Paul Erlich and Kraig Grady, TD 659:
>
> PE:
> > OK Kraig -- points well taken. I just note that in my view, what is truly
> > special about the hexany is the number of 7-limit consonant intervals it
> > contains (12) for the number of notes it contains (6). This feature comes
> > out most clearly on the 3-d triangular lattice, otherwise known as the
> > octahedral-tetrahedral (or oct-tet) lattice, which shows each consonance as
> > a connecting line.
>
> This stuff is really interesting, but where is the best place to go to
> SEE some of these lattices?? Monzo? Archive??
>
> Help.
> Bozo again
> ________________ _________ ___ _
> Joseph Pehrson
>
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-- Kraig Grady
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Joseph Pehrson wrote,

>This stuff is really interesting, but where is the best place to go to
>SEE some of these lattices?? Monzo? Archive??

The 6 7-limit consonant intervals are:

3/2: 5/4: 5/3:
5 5
/ \
2---------3 / \
/ \
/ \
4 3

7/4: 7/5: 7/6:

,7 5 7.
,' | `.
4' | `6
7

where 7 is seen as coming out of the page.

Using 31-tET-consistent note names, the hexany looks like this:

,a#
,'/ \`.
C'-/---\-`G
|\/ \/|
|/\ /\|
f#-------,c#
`.\ /,'
`Eb

An octahedron. Note that each consonant interval occurs twice, for a total
of 12, and that there are only 3 dissonant intervals. By comparison, the
major tetrad

E
/|\
/ | \
/ ,a# \
/,' `.\
C'-------`G

and the minor tetrad

f#-------,c#
\`. ,'/
\ `Eb /
\ | /
\|/
a

are tetrahedra and each have 6 consonant and 0 dissonant intervals. A chunk
of the complete 7-limit lattice looks like this:

C---------G---------D---------A---------E
/|\ /|\ /|\ /|\ /|
/ | \ / | \ / | \ / | \ / |
/ ,F#-------,C#-------,G#-------,D#-------,A#
/,'/|\`.\ /,'/|\`.\ /,'/|\`.\ /,'/|\`.\ /,'/|
Ab-/-|-\-`Eb-/-|-\-`Bb-/-|-\-`F'-/-|-\-`C' / |
| / ,b#-------,fx-------,cx-------,gx-------,dx
|/,'/ \`.\|/,'/ \`.\|/,'/ \`.\|/,'/ \`.\|/,'/
D'-/---\-`A'-/---\-`E'-/---\-`B'-/---\-`F# /
|\/.\ /,\/|\/.\ /,\/|\/.\ /,\/|\/.\ /,\/|\/.\
|/\--Cb-/\|/\--Gb-/\|/\--Db-/\|/\--Ab-/\|/\--Eb
g#-------,d#-------,a#-------,e#-------,b# \ |
\`.\|/,'/ \`.\|/,'/ \`.\|/.'/ \`.\|/,'/ \`.\|
\ `F'-/---\-`C'-/---\-`G'-/---\-`D'-/---\-`A
/,\/|\/.\ /,\/|\/.\ /,\/|\/.\ /,\/|\/.\ /,\/|
Abb/\|/\-Ebb-/\|/\-Bbb-/\|/\--Fb-/\|/\--Cb-/\|
| / ,b.-------,f#-------,c#-------,g#-------,d#
|/,'/ \`.\|/,'/ \`.\|/,'/ \`.\|/,'/ \`.\|/,'/
Db-/---\-`Ab-/---\-`Eb-/---\-`Bb-/---\-`F' /
| / \ | / \ | / \ | / \ | /
|/ \|/ \|/ \|/ \|/
g---------d---------a---------e---------b

The 3-d space is completely filled with tetrahedra (major and minor tetrads)
and octahedra (hexanies). See them?