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Convexity lemma

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Let L be a lattice of rank k in which is defined an epimorphism

L --> Z having a periodicity p in the octave, and within it,

a set S of intervals modulo 2 .

Any convex segment of S, containing more than 2p-1 elements,

and having an element in each class, contains a unison vector.

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Indeed, any linear sequence in L/<2>Z has a periodicity equal

to 1 or p or any factor of p. A line may contain an infinite

convex sequence of elements without any element of class 0 if

its periodicity is less than p. A convex segment containing an

element of each class has one of it in class 0 and has the

periodicity p.

Since it contains more than 2p-1 elements, it contains forcely

two elements of class 0 due to the periodical return for each

of its classes. One of these elements may be the unison, the

other is forcely a unison vector.

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Plunging the lattice L in an Euclidean space to obtain a linear

temperament with an orthogonal projection on a line, the longest

convex sequence possible, without a hole corresponding to a unison

vector, and containing a mode (so an element in each class), is

then equal to 2p-1, for a good temperament don't permit that an

element of another class, in the same vicinity, might be projected

at the same place than the unison vector.

Pierre