Scales

Let's see if I understand the situation. In a p-limit system of
harmony G, containing n primes, we select n-1 unison vectors and the
prime 2, which is a sort of special unison vector. This defines a
kernel K and so congruence classes on G, and a homomorphic image H =
G/K which under the best circumstances is cyclic but which in any
case is a finite abelian group of order h. We now select a set of
coset representatives in G for H by taking elements in one period of
the period lattice defined by the n generators of K.

Now we select out a certain number m+1 of the unison vectors,
including 2, and call them (aside from the 2) chromatic. The
remaining unison vectors now define a temperament whose codimension
is m+1, and hence is an m-dimensional temperament. We now select a
tuning for that temperament, and we have a scale with h tones in an
octave. If m = 0 we have a just scale; if m = 1 we have the scales
with one chromatic vector which people have been talking about.

I'm not sure if this helps anyone but me, but I think I am getting it.

--- In tuning-math@y..., genewardsmith@j... wrote:
> Let's see if I understand the situation. In a p-limit system of
> harmony G, containing n primes, we select n-1 unison vectors and
the
> prime 2, which is a sort of special unison vector. This defines a
> kernel K and so congruence classes on G, and a homomorphic image H
=
> G/K which under the best circumstances is cyclic but which in any
> case is a finite abelian group of order h.

What differentiates the "best circumstances" and just "any case" here?

> We now select a set of
> coset representatives in G for H by taking elements in one period
of
> the period lattice defined by the n generators of K.
>
> Now we select out a certain number m+1 of the unison vectors,
> including 2, and call them (aside from the 2) chromatic. The
> remaining unison vectors now define a temperament whose codimension
> is m+1, and hence is an m-dimensional temperament. We now select a
> tuning for that temperament, and we have a scale with h tones in an
> octave. If m = 0 we have a just scale;

You mean an equal temperament?

> if m = 1 we have the scales
> with one chromatic vector which people have been talking about.

Sounds right.

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning-math@y..., genewardsmith@j... wrote:

> > Let's see if I understand the situation. In a p-limit system of
> > harmony G, containing n primes, we select n-1 unison vectors and
> the
> > prime 2, which is a sort of special unison vector. This defines a
> > kernel K and so congruence classes on G, and a homomorphic image
H
> =
> > G/K which under the best circumstances is cyclic but which in any
> > case is a finite abelian group of order h.

> What differentiates the "best circumstances" and just "any case"
here?

What differentiates them is that in the best case we get a cyclic
group. What differentiates cyclic groups is having only one generator.

If we remove 2 from the generators of the kernel, defining a new
kernel K', the "best case" is still cyclic but now of infinite order,
corresponding to an et. It will fail to be cyclic if and only if
there are torsion elements in G, meaning elements not in the kernel
K' some power of which are in K'.

> You mean an equal temperament?

Sorry. :)