RE: [tuning] RE: The 3 5 7 cube-continued

Mats wrote,

>i´ve never been able to grasp triangular lattices.

Here, I'll repost a recent explanation:

Consider a fundamental pitch, 1/1, and all the notes that are traditionally
considered consonant with it: 6/5, 5/4, 4/3, 3/2, 8/5, 5/3. The square
lattice you've seen the neigborhood of 1/1 as:

5/3-------5/4------15/8
| | |
| | |
| | |
| | |
| | |
4/3-------1/1-------3/2
| | |
| | |
| | |
| | |
| | |
16/15------8/5-------6/5

According to this diagram, 15/8 and 16/5 are as close to 1/1 as 5/3 and 6/5.
But clearly they are not as consonant with 1/1. The solution is to use a
triangular lattice, where the 6:5 (5:3) relationship is given a line of its
own, equal in length to the 3:2 and 5:4 lines. So the above turns to:

5/3-------5/4------15/8
/ \ / \ /
/ \ / \ /
/ \ / \ /
/ \ / \ /
4/3-------1/1-------3/2
/ \ / \ /
/ \ / \ /
/ \ / \ /
/ \ / \ /
16/15------8/5-------6/5

which better depicts the consonance relationships. Each consonant triad is
shown as a nice little triangle. Really, this is all you need to remember to
understand all the lines in the 5-limit triangular lattice:

5
/ \
/ \
/ \
/ \
1---------3

Similarly, for the 7-limit you may have seen a cubical lattice like this:

35/24-----35/32----105/64
/| /| /|
5/3|------5/4|-----15/8|
/| | /| | /| |
40/21|-|---10/7|-|--15/14| |
| | | | | | | | |
| |7/6-------7/4------21/16
| |/| | |/| | |/|
|4/3|------1/1|------3/2|
|/| | |/| | |/| |
32/21|-|----8/7|-|--12/7 | |
| | | | | | | | |
| 28/15------7/5------21/20
| |/ | |/ | |/
16/15------8/5-------6/5
|/ |/ |/
128/105----64/35----48/35

If not, it's simply produced by defining a new axis to represent
multiplication (in one direction) or division (in the other direction) by 7.
Well, again, this doesn't do a good job of showing the consonances, so we
add new lines to represent the 7:4, 7:5, and 7:6 relationships (plus the 5:3
as before) and redraw the lattice like this:

35/24-----35/32----105/64
,'/ \`. .'/ \`. ,'/
5/3-/---\-5/4-/---\15/8 /
,'/|\/. ,\/|\/. ,\/| /
40/21/-|/\10/7-/\|/\15/14/\|/
/|\/ 7/6-------7/4------21/16
/ |/,'/ \`.\|/.'/ \`.\ /,'/
/ 4/3-/---\-1/1-/---\-3/2 /
/,'/|\/.\ /,\/|\/.\ /,\/| /
32/21/-|/\-8/7-/\|/\12/7 /\|/
/|\/28/15------7/5------21/20
/ |/,' `.\|/,' `.\|/,'
/16/15------8/5-------6/5
/,' `.\ /,' `.\ /.'
128/105----64/35-----48/35

Here, each consonant tetrad is shown as a nice little tetrahedron, but a lot
of the lines are obscured by one another (hey, this is ASCII). But really,
all you need to understand all the lines in all the lattices I've ever drawn
on this list is:

5
/|\
/ | \
/ 7 \
/,' `.\
1---------3

When diagramming scale from ETs, I often put the ET degree numbers instead
of the ratios.

So if you can get from one place in the lattice to another, using only
consonant intervals, and your ending point is the same pitch as the starting
point, you have a "pun". For example, the traditional natural-note diatonic
scale relies on punning the note D:

D---------A---------E---------B
\ / \ / \ / \
\ / \ / \ / \
\ / \ / \ / \
\ / \ / \ / \
F---------C---------G---------D

The two D's can be the same pitch in any meantone (including 12-, 19-, and
31-tET) but not in JI.

My diagrams try to show as many puns as possible.

Let me know if anything is still not clear.