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. :)