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Standard ets relative to a temperament

🔗Gene Ward Smith <gwsmith@svpal.org>

5/23/2005 8:36:54 PM

Given an r2 temperament, we can check for vals for n*period equal
divisions of the octave, and see which are minimal for val distance from
n*JIP. Very often there will be only one of them; in fact for most r2
temperaments there can be only one. These can be considered
semistandard vals, but in the typical case where it is unique, the
standard val for edo n*period and given r2 temperament. Of course this
idea applies to other temperament ranks also.

One use for this is in specifying a set of generators for the
temperament, which can be regarded as a "tone space", or abstract
keyboard. For a given temperament, we might call the square grid lattice
with the n-val values along the horizontal axis and the m-val values
along the vertical axis the "temperament(n, m)" lattice for the
temperament. For example the beehive keyboard is based on the
kleismic(1, 3) lattice.

If we pick on meantone for example, the meantone(1, 2) lattice has
whole tones horizontally and fourths vertically. Meantone(-1, 3) has
minor thirds horizontally and fourths vertically, and meantone(2, 3)
has minor thirds horizontally and whole tones vertically.