Some definitions

Reduced set of short definitions about chordoid and gammier structures permitting to see their relations

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Gammier structure

Gammier structure is
Gammoid structure with
Fertility axiom
Gammoid structure is
Harmoid structure with
Regularity axiom
Contiguity axiom
Congruity axiom
Harmoid structure is
Chordoid structure on
rational numbers with
standard multiplication
standard order
finite chordoid congruence modulo 2
Chordoid structure
See Chordoid structure

It is sufficient to know at this level that any finite set of odd numbers
A = <k1 k2 ... kn>
generates a finite chordoid of classes modulo 2 with the matrix
A\A = [aij]
where the generic element is
aij = kj/ki
and a corresponding harmoid with the set
{2xaij}
where the x are relative integers. Inversely, for any harmoid there exist
a such set of minimal odd values generating it and so called its minimal
harmonic generator.

The minimal genericity is the rank of that minimal generator.

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)
Regularity axiom is
a < 2/a for any atom a
Contiguity axiom is
any interval k is divisible by an atom
or there exist an atom a such that ax = k has a solution
Congruity axiom is
for any interval k there exist a stable number D of atoms
in any variant of a complete atomic decomposition of k
Degree function definition in gammoid
number D(X) of atoms in an interval X
Octave periodicity definition in gammoid
number D(X) where X is the octave
Fertility axiom is
octave periodicity > minimal genericity

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Chordoid structure

Chordoid structure is
Simploid structure with
Right associativity axiom
Commutativity axiom
Chordicity axiom
Simploid structure is
set of elements with
partial binary law
Right simplicity axiom
Right simplicity axiom is
ak = ak' Þ k = k'
Lemme 1 in simploid
ab = c Þ b = a\c

behind
the reverse law \
the interval a\b
the interval domain A\B
which is all x\y where x in A and y in B
Right associativity axiom is
ak = (ab)c Þ k = bc
Commutativity axiom is
k = ab Þ k = ba
Chordicity axiom is
There exist a subset A in E such that E = A\A

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--- In tuning@y..., "Pierre Lamothe" <plamothe@a...> wrote:
> Reduced set of short definitions about chordoid and gammier >
structures permitting to see their relations

Hi Pierre.

Since Gene was asking you about this on tuning-math, and since Gene
axiomatized what "we"'re doing, also on tuning-math (hope you saw
that), perhaps this should be posted there rather than here? It might
make more of an impact that way.