Gene's relative connectedness

Gene wrote:
Connected scale

A scale is connected with respect to a set of intervals S if for any
two scale degrees a and b, there is a path a=a_0, a_1, ... a_n = b
such that |a_i - a_{i+1}| is an element of S.
Either this definition use precisely his proper terms
Scale

A discrete set of real numbers, containing 0, and regarded as defining tones
in a logarithmic measure, such as cents or octaves, and such that the distance
between sucessive elements of the scale is bounded both below and above
by positive real numbers. The least upper bound of the intervals between
successive elements of the scale is the maximum scale step, and the greatest
lower bound is the minimum scale step. The element of the scale obtained by
counting up n scale steps is the nth degree, by counting down is the –nth
degree; 0 is the 0th degree. The set of positive real numbers which are the tones
so represented is also regarded as the scale.
or scale and degrees have another sense where it could exist more than one
path between the first and the last degrees of the scale. In that second case
the definition of connected scale would not be precise since scale and degree
would not be defined.

In the first case, using strictly the definitions, on could say
A scale whose steps belong to a set S is said
"connected with respect to a set of intervals S".
In the same way,
A women made pregnant by her lover S could be said
"pregnant with respect to her lover S"
while
"non-pregnant with respect to her husband H".
How to believe that a "relative connectedness" could convey an exact sense, when
it seems that that conveys about nothing? I recall what Gene wrote:
A definition is not supposed to be elegant, it is supposed to be precise. If it
isn't, it does not do the job it needs to do, which is to convey an exact sense
which allows one to understand precisely what is meant, what some is and what
it isn't.
Do a connectedness of a scale relative to S, be a property of the scale or a property
of the relation between the scale and S?

I suppose Gene seeks to adapt, in his simplest way, my concept on contiguity. So,
I would like to say him it does'nt convey any idea of relativity.

A gammier mode is not relatively connected but absolutely connected since the unique
set A for which it has sense to refer the connectedness is strictly determined by the
gammier structure itself. But it is very easy to generalize since I have a propension
to use independant axioms. I recall first my two following definitions with the
correction (k distinct from unison) in contiguity.
Atom definition in an harmoid
a is an atom if
a > u (where u is the unison) and
xy = a has no solution where both (u < x < a) and (u < y < a)
Contiguity axiom is
any interval k is divisible by an atom (k distinct from unison)
or there exist an atom a such that ax = k has a solution
I defined atom in harmoid for it's there it has really sense, but the definition would
remain valid in any finite subset G of rational numbers with standard multiplication,
neutral element 1, and standard order (or a similar additive structure).

In a such subset G respecting the contiguity condition, any element k distinct from the
neutral element is divisible by an atom, say a. Thus, there exist b such that ab = ba = k.
While b is not the neutral element, it remains, like k, divisible by an atom, so there exist
minimally one path, where all steps are atoms, between the neutral element and any k
including sup(G) and min(G), when distinct from the neutral element.

When a such set G is a gammoid, I name these paths modes, reserving the term mode
to denote such (connected) paths when they have the same number of degrees (what
is determined by the congruity condition) in the octave.

To sum up, the contiguity property (maybe the connectedness you seeked to define) is
an essential property of a mode such I conceptualize that, as gammier mode.

Why I would want to introduce infinity, like your simplest way, while my way is so short?

Pierre Lamothe