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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