I am not yet truly available to post on tuning-math but I have to comment

here.

Dave wrote about CS :

<< Not a very useful scale property >>

Hmmm . . .

Perhaps it might be better to flush the dirty water rather than the baby :)

The following definition of CS found at

<http://www.ixpres.com/interval/dict/constant.htm>

<< A tuning system where each interval occurs always

subtended by the same number of steps.

(THAT IS ALL, NO OTHER RESTRICTIONS) >>

is not, as expressed, a mathematical one but that corresponds, in my

opinion, to the main property of a serious attempt to modelize

mathematically the tonal paradigmatic structures. The temperament questions

don't address that problematic.

The CS property corresponds to the main axiom in the gammoid theory (the

_congruity_ condition) and subtends the _periodicity_ concept in Z-module.

If you flush CS you loss the mathematical sense of _interval class_ (or do

you have a _fuzzy class_ definition to replace it?)

I can't see how you could define mathematically such terms as step, degree,

class, mode . . . in a system S without the existence of a primordial

epimorphism S --> Z which gives its consistence.

Pierre

--- In tuning-math@y..., Pierre Lamothe <plamothe@a...> wrote:

> The CS property corresponds to the main axiom in the gammoid theory

(the

> _congruity_ condition) and subtends the _periodicity_ concept in Z-

module.

> If you flush CS you loss the mathematical sense of _interval class_

(or do

> you have a _fuzzy class_ definition to replace it?)

>

> I can't see how you could define mathematically such terms as step,

degree,

> class, mode . . . in a system S without the existence of a

primordial

> epimorphism S --> Z which gives its consistence.

Hmmm. A Z-module is the same as an abelian group, which I've been

raving about, and an epimorphism to Z sounds like an ET to me. Other

than that I don't know what this is about, but I wonder if Pierre

shows up here from time to time?

--- In tuning-math@y..., genewardsmith@j... wrote:

> but I wonder if Pierre

> shows up here from time to time?

Not often enough, sadly. How's your French?

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

> Not often enough, sadly. How's your French?

Even worse than his English. :(