Some definitions

DEFINITIONS

(1) A *tone group* T is a finitely generated subgroup of the group
(R+, *) of the positive real numbers under multiplication. A *tone*
is any element of (R+, *), that is, any positive real number
considered multiplicatively.

Hence, T is generated by n tones {t1, ..., tn}, which have the
property that

t1^e1 * ... * tn^en != 1

so long as the integer exponents e1...en are not all zero. The number
n is the *rank* of T, and any element t \in T is a tone of T. The
tones {t1, ..., tn} are the *generators* of T. The canonical examples
are the p-limit groups, so we also define the *p-limit group* T_p to
be the tone group generated by the primes less than or equal to the
prime p.

(2) A *note group* N is a finitely generated free abelian group which
we may identify with row vectors with integer entries. It is a sort
of generalized musical notation, since the canonical example makes
[a, b] represents "a" octaves and "b" fifths in a meantone system. If
we equate A 440 to [0,0], then any note in ordinary musical notation
may be translated into or out of this system--ordinary musical
notation can be thought of as representing this note group. Here of
course we do *not* equate B# with C, etc. The elements of N we call
notes, or N-notes if we need to be specific.

(3) A *tuning map* or *tuning* for the note group N is a
homomorphism "tune" of N into
(R+, *); it is defined by its values tune([1, 0, .. , 0]) = t1, tune
([0, 1, 0, ...,0]) = t2, ... tune([0, 0, ..., 1]) = tn. The image T =
tune(N) under this map is the *corresponding tone group*; if T is
also of rank n then tune is a *tuning isomorphism* and {t1, ..., tn}
are generators for T.

(4) If there are n primes less than or equal to a prime p, we define
the note group N_p to be the rank n free group, and the just tuning
map to be the tuning

just([1, 0, ..., 0]) = 2, just([0, 1, ... , 0]) = 3, ..., just([0,
0, ..., 1]) = p.

"Just" is therefore a tuning isomorphism from N_p to T_p.

(5) The dual N` to a note group N is the group N` = Hom(N, Z) of
homomorphisms from N into the integers. The elements of N` we call
ets, or N-ets if we need to be specific. If we take N concretely as
consisting of row vectors with integer entries, then we may take N`
to be column vectors with integer entries. If h and g are any two ets
and v is a note, then h+g is an et defined by "h+g"(v) = h(v)+g(v);
the 0-division et "0" which sends all notes v to 0 is the identity;
this defines the group structure on N`. Concretely, it is represented
by adding the two column vectors, just as the group N is defined by
adding row vectors.

If n is any positive integer, we define the et h_n in the dual to the
p-limit group N_p` to be the column vector with entries round(n log_2
(p_i)), so that h_n =

[ n ]
[round(n log_2(3)]
[round(n log_2(5)]
.
.
.
[round(n log_2(p)].

Here "round(x)" is the real number x rounded to the nearest integer,
so that round(x)-1/2<x<=round(x)+1/2 We should note that it is *not*
always the case that h_n + h_m = h_{n+m}.

(6) If M is any subgroup of the note group N, then null(M) is the
subgroup of N` consisting of all elements h \in N such that h(m) = 0
for every m \in M.

(7) If M is any subgroup of the dual to a note group N`, then null(M)
is the subgroup of N consisting of all v \in N such that h(v) = 0 for
every h \in M.

(8) If h \in N` is any nonzero et, then kernel(h) or null(h), the
*kernel* of h is the subgroup of rank n-1 of N defined as null(H),
where H is the group generated by h. In other words, the kernel is
the set of all notes v such that h(v)=0.

(9) A set of n-1 notes {u1, ..., u_{n-1}}in kernel(h) is a
*generating set* if any element u of the kernel is a Z-linear sum u =
j_1 u_1 + ... + j_{n-1}, where the j_i are integers.

(10) A nonzero et h is *reduced* if the coordinates of h are
relatively prime; that is, if no integer greater than one divides all
the v_i where v_i is the number in the ith row of v.

(11) If {u_i} is a generating set for h, then we may divide it
into "a" *commas* and "b" *chromas*, where a+b = n-1. The subset of
kernel(h) generated by the commas is the *commatic kernel* K, and the
quotient group N/K is the *commatic note group*. If all the
generators are commas, the commatic note group is simply Z and if h
is reduced we may identify it with the homomorphism from N to N/K. If
all of the generators are chromas, the commatic kernel is {0} and the
commatic note group is N. In general, the commatic note group is of
rank b.

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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