  In this article we solve this ancient problem of perfect tuning in all keys
and present a system were all harmonies are conserved at once. It will become
clear, when we expose our solution, why this solution could not be found in the
way in which earlier on musicians and scientist have been approaching the
problem. We follow indeed a different approach. We first construct a
mathematical representation of the complete harmony by means of a vector space,
where the different tones are represented in complete harmonic way for all keys
at once. One of the essential differences with earlier systems is that tones
will no longer be ordered within an octave, and we find the octave-like
ordering back as a projection of our system. But it is exactly by this
projection procedure that the possibility to create a harmonic system for all
keys at once is lost. So we see why the old way of ordering tones within an
octave could not lead to a solution of the problem. We indicate in which way a
real musical instrument could be built that realizes our harmonic scheme.
Because tones are no longer ordered within an octave such a musical instrument
will be rather unconventional. It is however a physically realizable musical
instrument, at least for the Pythagorean harmony. We also indicate how perfect
harmonies of every dimension could be realized by computers.
