more for Shaun Van Ault

Here's an old message which should help you get a grasp of the lattice
stuff, and some of the issues surrounding it.

> -----Original Message-----
> From: Paul H. Erlich
> Sent: Monday, September 20, 1999 6:35 PM
> To: 'McDougall, Darren Scott - MCDDS001'
> Subject: RE: lattice help
>
> 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.