Prime space

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:

/tuning/topicId_55226.html#55241

> i tried to set up "prime-space" as the definition of
> what i think it is you guys mean here ... but both
> Gene and Paul said they didn't like that term.
> i still don't understand why.

I don't like it because it is too vague for me to find it useful. The
problem is that I don't know if there is a way of measuring the
distance between two monzos in prime space, and if there is, what that
measurement is. A defintion of Tenney space would be more useful to me.

hi Gene,

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

> --- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
>
> /tuning/topicId_55226.html#55241
>
> > i tried to set up "prime-space" as the definition of
> > what i think it is you guys mean here ... but both
> > Gene and Paul said they didn't like that term.
> > i still don't understand why.
>
> I don't like it because it is too vague for me to find
> it useful. The problem is that I don't know if there is
> a way of measuring the distance between two monzos in
> prime space, and if there is, what that measurement is.
> A defintion of Tenney space would be more useful to me.

ah, ok, now i see what your objection is.

i've always thought of prime-space in a very general
sense, where the length of a unit step on each
prime-axis can be anything, dependent on the particular
lattice formula used by the theorist drawing the lattice.

where p is a member of the prime-series:

. Tenney-space uses a unit length of log(p),

. Monzo-space (as in the lattices defined on my "lattice" page
http://tonalsoft.com/monzo/lattices/lattices.htm)
uses a unit length of p,

. etc.

the only lattice formula used in release 1.0 of Musica
makes all the unit-lengths the same. future releases
will let the user adjust the unit-length to anything desired.

-monz

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:

> i've always thought of prime-space in a very general
> sense, where the length of a unit step on each
> prime-axis can be anything, dependent on the particular
> lattice formula used by the theorist drawing the lattice.

The way to put that in mathematical language is that prime-space is a
real topological vector space; this should be for some specific prime
limit p and not an infinite-dimensional space, since then this
definition is precise--the vector space "inherits" a topology (product
topology) from the real numbers as a topolgical field. Since a lattice
can be defined in a topological vector space (as a discrete subgroup),
this definition gives a precise, but rather abstract, sense in which
the lattice of monzos for any given prime limit is uniquely defined.
As usual there isn't really a simple web exposition, but there do seem
to be various places you can find out about topolgical vector spaces:

http://en.wikipedia.org/wiki/Topological_vector_space

http://mathworld.wolfram.com/TopologicalVectorSpace.html

http://planetmath.org/encyclopedia/TopologicalVectorSpace.html