Re: wedge product

Hi Gene,

I thought perhaps one way to think about newbie
definitions is to list the things you want
the reader to learn from it. Maybe
this could be a useful kind of "how to"
help for making newbie definitions.

So for wedge product ones list may be
simply

1)
learn how to calculate the wedge product
by hand.

In that case Dave Keenan's definition
is just fine. No more is needed.

Not sure what you need in the dictionary
with this approach, probably no more than

wedgie:
useful mathematical tool for tuning
theory - see encyclopedia for method
of calculation.

But maybe you want
1) learn the connection of wedge products with areas
and volumes
2) learn the applications of those to some
tuning situations such as finding the
area of a periodicty block or (... give
more applications)
3) Understand the representation of a wedge
product in terms of wedge products of unit
vectors.
4) learn how to do the calculations by hand.

In that case more is needed. A newbie
reading Dave Keenan's explanation simply
can't be expected to learn anything from
it except 4). I didn't from reading it.
You couldn't possibly read that and
at the end of it know that the
wedge product of the syntonic comma
and the diesis is the number of notes
in a five limit scale with those
as unison vectors - it doesn't make
any connection at all with pitches
and unison vectors. That's not a
criticism at all - it is a post he did
in response to a request to explain
how to calculate the wedge product
so that is what he did, simply
and straighforwardly. But more is needed
that's all.

Then having decided what you want the
newbie to be able to learn, or a
list of things you would like to be
learnt, then try and order them from
the familiar and useful to the less
familiar.

This will seem completely back to
front mathematically, but it is
how it needs to be done if it
is to be user friendly.

so
1) learn one simple application and
show an example of how it works
- best to be a 2D application probably
as simpler to explain, and needs
to be simple but not so simple as to
have no obvious application.

2) explain how it is calculated
for that one application so that
they now already have a useful tool.

3) learn the connection with areas
and volumes for the general case
so that they understand the concept
intuitively - I think this is helpful
so that they have some understanding.
Just saying that it is a directed
area and gives the parallelogram area swept
out is fine.

4) more detailed explanation and
description of how to calculate it.

There 4) and some of 3) are more
suitable for the encyclopedia than
the dictionary.

Now, write out the newbie
style definition a first draft.
Look at it and try to find ways of
making it use natural language more
and symbols less, if possible and not
cumbersome. Also feel free to change
the order again at that point.

The same thing needs to be done
with vals by someone, and the other
techy things presently. But probably
not me, or not yet :-).

Possiby I could do it later though
with all my other commitments
unfortunately I simply don't
have the time right now to
devote much of it to learning
tuning theory. As someone
who is very fond of maths
in all its forms, especially
pure mahts, I'd certainly
enjoy doing it. also bear
in mind that though I have
kept active mathematically
by thinking of things, it
is maybe ten years or more
since I did much serious maths
research so I'm a bit rusty
at reading maths papers, and slow
though that wuold come back
very quickly I'm sure if I had
the time for a few months of
solid maths reasearch.

Robert

Hi Gene,

> This is where the business of "epimorphic" scales comes in; when Scala
> says something is "JI epimorphic" and gives a val it means the val
> mapping order corresponds to the scale order.

Oh right. I just tried it out but it is hard to make
an example, except for things like
1/1 9/8 10/9 5/4 3/2 8/5 2/1
which I wouldn't expect to work.

Are all scales defined as rectangular areas in periodicity blocks
j.i. epimorphic?

> > Gene must have done that, or maybe
> > there is some nice result
> > one can use.

> This goes back to the discussion of Paul's "Hypothesis", which was
indeed a major topic at one point.

I saw that indeed, many posts in the tuning-math
archive. However, not understanding the basic concepts
at the time, I couldn't follow the discussion
and have no idea yet what it is about.

However, probably next time I read it, it will make a lot
more sense.

Robert

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

> Are all scales defined as rectangular areas in periodicity blocks
> j.i. epimorphic?

Nope. Take a block defined by 16/15 and 4374/3125 which goes

25/27, 6/5, 10/9, 36/25, 4/3, 125/108, 3/2, 25/18, 9/5, 5/3, 54/25, 2

Put that in a single octave and order it, and you get

27/25, 10/9, 125/108, 6/5, 4/3, 25/18, 36/25, 3/2, 5/3, 9/5, 50/27, 2

Feed this scale to Scala and you will find it isn't proper, isn't a
constant structure and isn't epimorphic, but we got it from a Fokker
block, and despite its irregularity it has five pure major triads and
four pure minor triads, so you could certainly make music with it.

> However, probably next time I read it, it will make a lot
> more sense.

It took me a while to figure out what exactly Paul wanted to prove,
but once you do that it isn't too hard.

Hi Gene,

Rightio. Thanks for the example of a non j.i. epimorphic
periodicity block - that saves me trying
to puzzle out to prove something that isn't true :-).

> 27/25, 10/9, 125/108, 6/5, 4/3, 25/18, 36/25, 3/2, 5/3, 9/5, 50/27, 2

Here for example, 10/9 is degree 2 and 6/5 is degree 4 so
3/2 should be degree 6 but instead it is degree 8.

So it's "inconsistent", the vals aren't homomorphisms, so it fails.
I suppose the easiest way to check is to run the scale through
Scala - but if the scale has the notes reasonably evenly spaced
then it will work.

I suppose then, one result that one might expect to be able
to show fairly easily is that if each note in the scale
is approximated by a distinct degree in the corresponding et
then it is epimorphic - is that true?

But you probably can get some epimophic scales that aren't
et approximations.

Here - isn't epimorphic just another way of saying
used to be called "consistent", so j.i.
epimorphic just means j.i. consistent?
Or am I missing something?

Robert

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

> I suppose then, one result that one might expect to be able
> to show fairly easily is that if each note in the scale
> is approximated by a distinct degree in the corresponding et
> then it is epimorphic - is that true?

Unfortunately, no, since a single step of an et can be arbitarily
small. The kleisma, amity, comma and monzizma, for instance, are a
single step of 12, as are 2401/2400 and 4375/4374. The wuerschmidt and
semisixth commas are a single step *down*.

> Here - isn't epimorphic just another way of saying
> used to be called "consistent", so j.i.
> epimorphic just means j.i. consistent?
> Or am I missing something?

It's very closely related to "constant structure" scales. I think of
consistency in reference to equal temperaments.