Another interesting property of the Balzano space is that one of its diagonals turns out to be the chromatic scale (1 semitone increase) and the other diagonal appears to be the cycle of fifths (7 semitone increase).  This means you can navigate (move the mouse) easily along a chromatic sequence and along the cycle of fifths simply by just following the diagonals.  Plus there is a useful progression on the horizontal and vertical axes too, in steps of major and minor thirds respectively (4 and 3 semitone increases resp.).

 

 

The Longuet-Higgins space has comparable properties, which we will not describe in detail here.

 

You can find a much more thorough treatment of the theory behind these spaces and a description of an earlier implementation of these principles, called Harmony Space, in the paper by Simon Holland [Holland, 1987]. Note that while tonespace is greatly indebted to the work on Harmony Space, the tonespace implementation deviates in some respects from what is described in this paper, for instance by generalizing spaces to other intervals besides Longuet-Higgins. But the general principles still apply.

 

We will now explain a bit more about scales, keys and chord intervals.

 

 

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