Carl wrote:

Pierre didn't say that. He said Epimorphic -> CS, which is

exactly what Gene said. Clearly CS /-> Epimorphic. See my

post in this thread.

Sorry Carl, it's not what I said. I justly wrote the first post in the "CS implies EPIMORPHISM" thread

for I saw much ambiguities in many posts about CS and epimorphic.

Gene has defined epimorphic for a scale as

"... there is a val h such that if qn is the nth scale degree, then h(qn) = n "

and a val (in a context of rational tone group) as

"... an homomorphism from the tone group to the integers.

How are interconnected, by definition, scale and tone group ? Subgroup, subset, periodicity block ??

I used simply the well-known mathematical term EPIMORPHISM as surjective morphism and I shown:

If a scale S has the CS property then there exist an epimorphism D applying each interval x,

in the space of all intervals spanned by S, onto an integer corresponding to its scale degree,

not only in S but in any derived scale S' obtained by tonic rotation and/or duality.

Epimorphism don't imply CS

I hope the following counterexample will suffice. It is well-known that the Zarlino scale is both CS and

epimorphic. The unison vectors 81/80 (about 22 cents) and 25/24 (about 70 cents) generate the kernel

of that epimorphism. It is very easy to transform that CS scale in a non-CS scale but respecting that

epimorphism.

The Zarlino degree 6 == 15/8 has approximately 1088 cents. It misses 112 cents to reach the octave.

If you multiply 15/8 by a combination of unison vectors like 81/80 and 25/24, you dont change the class

of the epimorphism, since class 6 + n (class 0) = class 6. If you add at 1088 cents, for instance, two

comma of 22 cents and a chroma of 70 cents, the result is a degree 6 which is about 2 cents over the

octave.

If you don't see immediately that a such scale forcely reordered

0 2 204 386 498 702 884 1200

is non-CS, consider that 204 is subtended by 2 steps between 0 and 204 and only one step between

498 (4/3) and 702 (3/2).

Pierre