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

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

> 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.

There can be more than one path according to the definitions as given.
Another way to define it is to define the graph of a scale relative to S, and then it is connected iff the graph is connected, for which see

http://mathworld.wolfram.com/ConnectedGraph.html

> How to believe that a "relative connectedness" could convey an exact sense, when
> it seems that that conveys about nothing?

A scale can for example be connected in the 7-limit but not in the
5-limit, so it conveys something.

> 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?

The scale and S.

> I suppose Gene seeks to adapt, in his simplest way, my concept on contiguity.

No, I seek to apply standard mathematical concepts of path-connectedness in graphs to this particular situation.

> 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.

That could be done with RI scales, but is in general too restrictive for my intended applications.

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

The question of infinity is another one; however the rational numbers are an infinite field, and in fact any ordered field (for which you were giving some of the axioms a while back) is infinite.
In any event, I want to consider more than RI scales.

(PL) (GWS)
> 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?

The scale and S.
So our concepts are not conflictual since the contiguity is a property of a mode in itself.
There can be more than one path according to the definitions as given.
Another way to define it is to define the graph of a scale relative to S, and
then it is connected iff the graph is connected
It's not easy to see that sense from your definition since a graph is defined by a set of nodes and
a set of vertices. It seems the scale would be the set of nodes while S would be the set of vertices.
However in a graph, one vertice link two nodes, but it seems you relie your nodes with a chain of
vertices, implying other nodes, outside the scale, to rely nodes. If I understand the connectedness
in a graph, that corresponds to a possible way between any two nodes, using a unique vertice
between two nodes.
The question of infinity is another one; however the rational numbers are an
infinite field, and in fact any ordered field (for which you were giving some
of the axioms a while back) is infinite
I use only finite sets. In the harmoid frame, I work with the finite classes mod <2> and a set
of intervals representing these classes (first octave or centered octave or pivots). Even if a
relation remains valid at infinity, I never use something requiring the existence of elements
outside the perceptible domain. I can plunge a gammoid structure in a group, for instance,
to show the link with the periodicity block, but I have access at all that from operations in a
finite set.

You could say that using logarithm implies the infinity of the real field. Yes, but I don't use the
operative properties of that field, restricting to Z-module, in which I restrict to classes mod <2>,
and finally I restrict to a finite area well-defined around the unison.

Pierre