Definitions of tuning terms

© 1998 by Joseph L. Monzo

All definitions by Joe Monzo unless otherwise cited

Pierre Lamothe's definitions of gammier and chordoid

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

     
    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
     
    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
     

    [from Pierre Lamothe, Yahoo tuning-math message.]

Updated: 2002.1.12

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