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Planar temperament lattice diagrams

🔗Gene Ward Smith <genewardsmith@juno.com> <genewardsmith@juno.com>

12/10/2002 9:42:07 PM

The octave equivalence classes of a planar temperament system can be reduced to a plane (hence the name.) Relative complexity gives us a Euclidean metric on that plane, so it is an (honest to goodness, genuine mathematician-style) lattice. If Paul or Monz felt ambitious, lattice diagrams drawn in such a way that this distance was respected might prove illuminating.

To do it for Starling, one might take

6/5 represented by [2.851474223, 0]

3/2 represented by [1.349572175, 4.477746219]

These can of course be rescaled or rotated, but the relative lengths and the angle between them should be kept the same. In Starling, since
126/125~1, we have (6/5)^(-3) 3 ~ 7/4. Hence every 7-limit octave reduced note class can be expressed in terms of the two vectors above.