Decatonic tuning and 7-limit lattices

Sure enough, Paul Erlich does describe decatonic modes in JI,
but it's under the heading "History of 22-tone tunings" and
so I overlooked it. The text implies that lattice diagrams should
be there, but I don't have them. One should look like this:

9
/ \
/ \
8---------4---------0
\ / / \ \ / \
\ 6--/---\--2 / \
\ / 9---------5 \
\ / \ \ / / \
1------\--7--/------3
\ /
\ /
2

The numbers show decatonic steps, hence there are 10 of them(!)
This is the following mode:

0 1 2 3 4 5 6 7 8 9 0
2 2 2 3 2 2 2 3 2 2 =22

It doesn't seem to be any of the named scales, but no matter.
If you bring 6-9-2-4 and 9-2-5-7 together, you get this:


9---------5
/ \ /
/ \ 7 /
8---------4---------0
\ / / \ \ / / \
\ 6--/---\--2 / \
\ / 9 \ / 5 \
\ / \ / \
1---------7---------3

This has some advantages with 5- or 15-limit harmony, but not
really the strict 7-limit decatonic tonality. I could pretend
this is the 3 comma tuning I was thinking of, but actually I lost
the bit of paper I originally worked it out on, and misremembered
the number of commas I got. I decided to post anyway, because I
wanted to start a discussion on this.

For the pentachordal modes, I reckon tuning to 53-equal is an
option. I think the 50/49 is only one step, and so comma sized.
Crucially, 1/53 is smaller than half 1/22, so there's no ambiguity.
The 7-limit chords are tuned significantly better than in 22-equal,
so there is an advantage to offset the presence of the commas.

My point about tuning tritones isn't quite covered in Paul's paper.
To quote: "Since any perfect 6th10 in the scale is likely to
represent 7:5 and 10:7 equally often, the interval achieves its
greatest accuracy at exactly 600 cents." No, at least 1 of the 4
tritones in a pentachordal scale has only one interpretation. My
point is that, as at least two have to be ambiguous, these worst
intervals in 22-equal can't be improved by any tempering. A well
temperament may improve the average tuning, though, and I haven't
looked at this properly yet.

The symmetrical modes require a lot more commas, and so there's
no point in using more than the 10 notes unless you really know
what you're doing.


These 3-dimensional lattices are definitely fcc, aren't they?
I thought they were hexagonal close packing before, but I was
wrong. There is a difference.

In abstract terms, the point of fcc is that, along with hcp, it
has the highest possible packing fraction for spheres. That's why
it's so common in crystals. The musical relevance of the
dodecahedra (which are apparently rhombic) is as follows. Consider
one point on a lattice. That point is directly connected to 12
other points. These correspond to the 12 faces of the
dodecahedron. The lattice looks the same from each point and so
is regular. Furthermore, the 12 points correspond to the 12
7-limit intervals within an octave:

5/3-------5/4
/ \ / \
/ \10/7 / \
O O / 7/6 \ / 7/4 \
| / \ / \
\___/ 4/3-------1/1-------3/2
\ / \ /
\ 8/7 / \12/7 /
\ / 7/5 \ /
\ / \ /
8/5-------6/5

And there's probably a name for that structure, maybe 7-limit
tonality diamond.

I am of the opinion that these lattices are the best way of
representing 7-limit octave invariant scales most of the time.
When you bring the octaves into it, it makes more sense to use
hypercubes. Or, at least, matrices with a rectangular distance
function. I, for one, have difficulty in visualising four
dimensions.

In case people are getting deluged with all this theory, I'll
highlight the point that these lattices are very useful for
constructing 7-limit scales, and taking chords from them. There,
I've done it.


SMTPOriginator: tuning@eartha.mills.edu
From: "Paul H. Erlich"
Subject: RE: Decatonic tuning and 7-limit lattices
PostedDate: 02-02-98 20:17:03
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