Intro to Hexany page

HI there,

I've just done a page with an introduction to lattices and the
hexany for newbies - complete with midi clips for all the notes
and chords (made with Fractal Tune Smithy).

http://tunesmithy.netfirms.com/tunes/mus_geom/musical_geometry.htm

Robert

--- In tuning@y..., "Robert Walker" <robertwalker@n...> wrote:
> HI there,
>
> I've just done a page with an introduction to lattices and the
> hexany for newbies - complete with midi clips for all the notes
> and chords (made with Fractal Tune Smithy).

Neat! It would be nice to see something on the cubic lattice of 7-limit tetrads sometime; that is a very useful construct.

HI there,

I was wondering, how old is the idea of assigning triads to triangles?

First in 2D for lattice keyboards. Then in 3D - did the idea of studying
geometrical figures come much later? E.g. when Erv Wilson invented the
CPS sets perhaps? At least, studied enough to name them and so forth.

I suppose the 3D and 4D lattices can't have been that long after the 2D ones
- at least if there were any mathematicians in on it :-). Or were they
later too...

Robert

--- In tuning@y..., "Robert Walker" <robertwalker@n...> wrote:
> HI there,
>
> I was wondering, how old is the idea of assigning triads to triangles?

Didn't someone claim a while back it was Hugo Riemann who thought of it?

> First in 2D for lattice keyboards. Then in 3D - did the idea of studying
> geometrical figures come much later? E.g. when Erv Wilson invented the
> CPS sets perhaps? At least, studied enough to name them and so forth.

Wilson discovered the 3D lattice, I think. I rediscovered in in the 70s, and the tuning gang knew about it when I arrived, since someone (Paul?) came up with it also.

> I suppose the 3D and 4D lattices can't have been that long after the 2D ones
> - at least if there were any mathematicians in on it :-). Or were they
> later too...

I'm not sure if anyone had a general lattice definition in N dimensions until I came up with one using quadratic forms.

--- In tuning@y..., "Robert Walker" <robertwalker@n...> wrote:
> HI there,
>
> I was wondering, how old is the idea of assigning triads to
>triangles?

i think daniel wolf said shohe tanaka was the first one to do it. you
should ask him on the spec-mus list.

--- In tuning@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:

> I'm not sure if anyone had a general lattice definition in N
>dimensions until I came up with one using quadratic forms.

isn't it enough to simply state that the N-ad is the N-dimensional
simplex, and that the entire lattice is built from those same
vectors, each of which is found springing from every vertex?

--- In tuning@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:

> I'm not sure if anyone had a general lattice definition in N
>dimensions until I came up with one using quadratic forms.

tenney sure did! his is a rectilinear lattice, where the length of
the vector for the prime p is log(p). simple! this actually works
wonderfully if you don't want to use octave equivalence, and include
2 as one of the primes. otherwise, not so good.

--- In tuning@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:
> --- In tuning@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:
>
> > I'm not sure if anyone had a general lattice definition in N
> >dimensions until I came up with one using quadratic forms.
>
> isn't it enough to simply state that the N-ad is the N-dimensional
> simplex, and that the entire lattice is built from those same
> vectors, each of which is found springing from every vertex?

So long as you define things carefully enough to actually construct the lattice. It's a well-known lattice of Lie type, and can be defined in different ways.

--- In tuning@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:
> --- In tuning@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:
>
> > I'm not sure if anyone had a general lattice definition in N
> >dimensions until I came up with one using quadratic forms.
>
> tenney sure did! his is a rectilinear lattice, where the length of
> the vector for the prime p is log(p).

That's not the same lattice--it isn't even the same metric.

--- In tuning@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:
> --- In tuning@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...>
wrote:
> > --- In tuning@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:
> >
> > > I'm not sure if anyone had a general lattice definition in N
> > >dimensions until I came up with one using quadratic forms.
> >
> > isn't it enough to simply state that the N-ad is the N-
dimensional
> > simplex, and that the entire lattice is built from those same
> > vectors, each of which is found springing from every vertex?
>
> So long as you define things carefully enough to actually construct
> the lattice.

did i do ok? what pitfalls could there be that could prevent it from
being constructible? (maybe reply to tuning-math)

--- In tuning@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:
> --- In tuning@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...>
wrote:
> > --- In tuning@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:
> >
> > > I'm not sure if anyone had a general lattice definition in N
> > >dimensions until I came up with one using quadratic forms.
> >
> > tenney sure did! his is a rectilinear lattice, where the length
of
> > the vector for the prime p is log(p).
>
> That's not the same lattice--it isn't even the same metric.

all you said was "a general lattice definition" -- you didn't say
which one! see above.

--- In tuning@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:

> did i do ok? what pitfalls could there be that could prevent it from
> being constructible? (maybe reply to tuning-math)

It was hand-waving to a degree--not the same thing as defining a basis, for instance.

--- In tuning@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:

> all you said was "a general lattice definition" -- you didn't say
> which one! see above.

I meant A_n, which is what you were talking about.

--- In tuning@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:
> --- In tuning@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...>
wrote:
>
> > did i do ok? what pitfalls could there be that could prevent it
from
> > being constructible? (maybe reply to tuning-math)
>
> It was hand-waving to a degree--not the same thing as defining a
>basis, for instance.

forgive me, i was trained as a physicist, not as a mathematician. i
signed up for math 301 (real analysis) with peter jones and the first
problem set drove me to tears. that was the end of that.

ooh -- i just looked at jones' website and see that he's lost even
more hair than i have (since 1992-3)! :) i feel bettah . . .

waving my hand bye-bye,
paul

--- In tuning@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:

> forgive me, i was trained as a physicist, not as a mathematician. i
> signed up for math 301 (real analysis) with peter jones and the first
> problem set drove me to tears. that was the end of that.

Actually, if someone (Erv?) defined it like this, that would be enough for a gold star, since it could easily be translated back into mathematese. It wouldn't do in a math course, though.