Re: [tuning] Digest Number 3205

Hi Gene,

> > One accessible example of its use is to find out the
> > area or volume of a periodicity block, and so its
> > number of notes.

> I like this, but why not say "basic example" or "simple example"?

Because though mathematically it is maybe simple because
it uses concepts that aren't so very abstract, and
the calculations are often a few lines, the newbie
idea of a simple calculation is one that is easy to do
and calculating the volume of even a 3D periodicity
block by hand isn't particularly easy - I know
nothing compared with proofs that run to many
hundreds of pages or whatever, but it is complex
enough so that one can easily make slips when doing
the calculations by hand - that is newbie complicated.

But anyway, probably a better idea to just say "one example"
and not qualify it in any way at all.

> > The two dimensional case is the simplest - the
> > magnitude of the wedge product of [a b> and [c d>
> > is a*c - b*d

> You need your wild cards in there, don't you? In fact, you could do it
> all with wild cards: det([|* 4 -1>, |* 0 -3>]) = -12,
> det([|-4 * -1>, |7 * -3>]) = 19, det([|-4 4 *>, |7 0 *>]) =-28; which
> are of course minors. This seems to be what you do below, you may as
> well start out with it.

Yes, I need the wild cards - will put them in. Don't want to use Det though
as too techy. I try to avoid using the word determinant here - every use of a
techy math term will stop a newbie in their tracks if they haven't
learnt it yet.

Here is my latest definition, and I've changed it
around again, somewhat back the way it was originally,
and start by explaining what a wedge
product is mathematically but simply.
Then one carefully chosen example from first principles.

=term=
wedge product

=definition=
The wedge product x/\y of two vectors x and y is the area
swept out from one to the other. So it is an
area with a sense of direction and x /\y = - y/\x.
A vector is made up of unit vectors for two or more
directions of travel - similarly a wedge product
of vectors is made up of wedge products of unit vectors.

One example of its use in tuning theory is to find out the
area or volume of a periodicity block. To do that you
write out the monzos in the form of a sum of unit vectors
then multiply them out using two rules: ei/\ei = 0 (area 0)
and because it is a directed area, ei/\ej = - ej/\ei
You can only change the order of the unit vectors using the
second rule.

Example:
tridecimal diesis 26/25 = [* 0, -2 * *, 1> = (-2e2 + e3)
syntonic comma 81/80 = [* 4, -1 * *, 0> = (4e1 - e2)
schisma [* 8, 1> = (8e1 + e2)
(-2e2 + e3) /\ (4e1 - e2) /\ (8e1 + e2)
= (-2e2 + e3) /\ (4e12 -8e21)
= (-2e2 + e3) /\ (-12e21)
= -12 e321
An example would be David Canright's
13 limit twelve tone scale.
You can simplify this calculation
by using the diesis in place of the schisma.

Well the best I can do at present. Don't know how many
hours it took me :-).

Be sure to say if anything there is unclearly
expressed or of course if there is a
typo or something not correct.

Robert

--- In tuning@yahoogroups.com, "Robert Walker" <robertwalker@n...> wrote:

> Example:
> tridecimal diesis 26/25 = [* 0, -2 * *, 1> = (-2e2 + e3)
> syntonic comma 81/80 = [* 4, -1 * *, 0> = (4e1 - e2)
> schisma [* 8, 1> = (8e1 + e2)

Would this be clearer as [* 8, 1 * *, 0>?

> (-2e2 + e3) /\ (4e1 - e2) /\ (8e1 + e2)
> = (-2e2 + e3) /\ (4e12 -8e21)
> = (-2e2 + e3) /\ (-12e21)
> = -12 e321
> An example would be David Canright's
> 13 limit twelve tone scale.

Is Canright's scale a Fokker block using these commas? I'd make that
explicit, and also point out that the "-12 e321" means that it is a 12
note scale.

> You can simplify this calculation
> by using the diesis in place of the schisma.

And get different Fokker blocks too. Another try would be with 65/64.