2007/09/05
In 1999, it occurred to me that Archimedean tilings might offer a nice way to look at tonespace. I came up with:
There's apparently enough room here for the 9-limit. Can anyone prove the mapping is complete?
Being a keyboardist, I always thought this would make a nice keyboard. Perhaps even – in the vein of the chord buttons on an accordion – pressing an octagon could cause its neighboring squares to play too.
If you prefer a lattice, you can take the dual of this. Now the short links (squares) represent consonances while the long links (diagonals) represent root motions that preserve a common dyad.
Recently, I thought I'd take a look at other Archimedean tilings. Wikipedia nicely lays out the choices. Considering only those tilings with two different polygons (one for roots and another for non-roots), we rule out 3.4.6.4 and 4.6.12.
For something that packs the greatest number of consonant chords in the least space, we should make progressions which use the highest number of common tones possible as short as possible. ASSs aside, JI admits to at most common-dyad progressions, which occur only between otonal and utonal chords. Therefore, we want a tiling whose dual contains two basic kinds of line segment: one for the common-dyad-preserving progressions and another for the consonances themselves. And there can be no cycles of the progression segment with an odd number of steps, since the quality of the chords must alternate between otonal and utonal.
Given these constraints, I believe we are left with
In the 4.8.8 mapping, the consonances (length 1) are shorter than the modulations (length sqrt(2)). In 3.6.3.6 they are apparently of equal length.
Other ways of mapping tonespace to these are possible. For example, 4.8.8 can be done with primes. Would temperament- or ASS-based mappings allow some of the other tilings to be used?