The senga

The senga, the 7-limit comma 686/675, forced itself on my attention when I was doing 7-limit comma pumps because of the unusual nature of its most efficient pumps. The otonal tetrad with root 686/675 in the cubic lattice coordinates

http://xenharmonic.wikispaces.com/The+Seven+Limit+Symmetrical+Lattices

is [1 0 -5]. (In general, you can find these starting from any 7-limit monzo |a b c d> as [c+d b+d b+c].) Even if you allow chords to share only a single note, an efficient pump has a lot of utonal tetrads, because the easy way to get to [0 0 -2], which is stepping down by a septimal diatonic semitone of 15/14, is through [0 0 -1], which is utonal: 1-5/4-3/2-7/4 => 1-7/6-7/5-7/4 => 14/15-7/6-7/5-49/30.

The L1 (taxicab) norm of the tetrad [1 0 -5] is 1+0+5 = 6. If you look at a comma basis for 7-limit 46et, you might pick {126/125, 245/243, 1029/1024} because of the low Tenney height of 245/243. However, the corresponding tetrad is [3 -3 -4], with a taxicab size of 10. Instead we could pick 126/125 and 686/675, both 6, and one of the commas with taxicab size of 8: 1029/1024, 6144/6125 or 64827/64000. With any of these, the parallelepiped with the corresponding corners in the lattice of tetrads has volume 92, inclusive of 46 otonal and 46 utonal tetrads. The last comma, 64827/64000, unnamed and unnoticed so far as my knowledge extends, is like 686/675, with a corresponding tetrad of [1 7 0]. I plan to add it to my comma pump examples.

Hope this gives Mike food for thought. :)

On Sat, May 21, 2011 at 2:36 PM, genewardsmith
<genewardsmith@...> wrote:
>
> The senga, the 7-limit comma 686/675, forced itself on my attention when I was doing 7-limit comma pumps because of the unusual nature of its most efficient pumps. The otonal tetrad with root 686/675 in the cubic lattice coordinates
>
> http://xenharmonic.wikispaces.com/The+Seven+Limit+Symmetrical+Lattices

I guess this subspace is going to be a 3d space then, and it's going
to be at a 45 degree angle to the axes. When you say each
octave-equivalency interval class is represented once, you mean that
7/2, 7/4, 7/2, etc all get one representative member? Is this all
basically how to formalize what Petr was doing by just dropping the 2
exponent from the monzo?

Also, for this

"If |-x-y-z z y z> is any element of symmetric interval class space,
then by definition || |-x-y-z x y z> || = sqrt(2)
sqrt(x^2+y^2+z^2+xy+yz+zx) where we may remove the sqrt(2) factor
without changing anything substantial."

Should that first one say |-x-y-z x y z> instead of |-x-y-z z y z>?
Also, is that first thing one whole expression, e.g. |(-x-y-z) x y
z>?

> is [1 0 -5]. (In general, you can find these starting from any 7-limit monzo |a b c d> as [c+d b+d b+c].) Even if you allow chords to share only a single note, an efficient pump has a lot of utonal tetrads, because the easy way to get to [0 0 -2], which is stepping down by a septimal diatonic semitone of 15/14, is through [0 0 -1], which is utonal: 1-5/4-3/2-7/4 => 1-7/6-7/5-7/4 => 14/15-7/6-7/5-49/30.

OK, I think I'm following. In the article, you talk about values for
[2 0 0], [0 2 0] and [0 0 2]. I'll need to take second to work out the
values for [1 0 0], [0 1 0], and [0 0 1] before I can really grasp
this space. Do you have an intuitive way of looking at the coordinates
of this space? The formula

"If [a b c] is any triple of integers, then it represents the major
tetrad with root 3^((-a+b+c)/2) 5^((a-b+c)/2) 7^((a+b-c)/2) if a+b+c
is even, and the minor tetrad with root 3^((-1-a+b+c)/2)
5^((1+a-b+c)/2 7^((1+a+b-c)/2) if a+b+c is odd."

isn't at all intuitive for me to think about, although maybe I'll see
a pattern eventually. BTW, is that supposed to be a -1 in the exponent
for 3? It breaks the pattern...

> The L1 (taxicab) norm of the tetrad [1 0 -5] is 1+0+5 = 6. If you look at a comma basis for 7-limit 46et, you might pick {126/125, 245/243, 1029/1024} because of the low Tenney height of 245/243. However, the corresponding tetrad is [3 -3 -4], with a taxicab size of 10. Instead we could pick 126/125 and 686/675, both 6, and one of the commas with taxicab size of 8: 1029/1024, 6144/6125 or 64827/64000. With any of these, the parallelepiped with the corresponding corners in the lattice of tetrads has volume 92, inclusive of 46 otonal and 46 utonal tetrads. The last comma, 64827/64000, unnamed and unnoticed so far as my knowledge extends, is like 686/675, with a corresponding tetrad of [1 7 0]. I plan to add it to my comma pump examples.
>
> Hope this gives Mike food for thought. :)

Lots of food for thought, though I'm still unsure of how you are using
this space to transform the comma basis for 7-limit 46et into a
tetrad. And I'm also not sure what the L1 norm of this space means -
does it end up meaning chord movements that move by 3, 5, and/or 7? I
note that when you add up (-a+b+c)/2, (a-b+c)/2, and (a+b-c)/2, you
get a+b+c, but I'm still not sure if I have the right concept.

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> I guess this subspace is going to be a 3d space then, and it's going
> to be at a 45 degree angle to the axes.

"This subspace" = ?

When you say each
> octave-equivalency interval class is represented once, you mean that
> 7/2, 7/4, 7/2, etc all get one representative member?

Right: they all are represented by |-1 0 0 1> = 7/2.

Is this all
> basically how to formalize what Petr was doing by just dropping the 2
> exponent from the monzo?

Don't know what ypu mean, sorry.

> Should that first one say |-x-y-z x y z> instead of |-x-y-z z y z>?

Thanks!

> BTW, is that supposed to be a -1 in the exponent
> for 3? It breaks the pattern...

That's because the root of a utonal tetrad is not the note the others are "undertones" of, but the bottom note of the fifth interval; that is, 1 is the root of 1-6/5-3/2-12/7, not 3/2.

> And I'm also not sure what the L1 norm of this space means -
> does it end up meaning chord movements that move by 3, 5, and/or 7?

No, it means chord movements which move via common dyads.