Equivalence continuum: Difference between revisions

== Geometric interpretation ==

Mathematically, the rank-''k'' '''equivalence continuum''' C(''k'', ''T'', ''S'') associated with a rank-''r'' temperament ''T'' on a rank-''n'' subgroup ''S'' is the space of [[saturation|saturated]] (''n − k'')-dimensional sublattices of the [[kernel]] (set of all intervals tempered out) of ''T'', the rank-(''n − r'') lattice of commas tempered out by ''T''. This is a set of rational points on the Grassmannian '''G''' = '''Gr'''(''n − k'', ''n − r'') of (''n − k'')-dimensional vector subspaces of '''R'''^{''n−r''}, identifying '''R'''^{''n−r''} with the '''R'''-vector space ker(''T'') ⊗ '''R'''.

This has a particularly simple description when ''r'' = 1 (i.e. when ''T'' is an edo), ''n'' = 3 (for example, when ''S'' is the [[5-limit]], 2.3.7 or 2.5.7) and ''k'' = 2 (so that we're considering the equivalence continua of rank-2 temperaments associated with an edo), as then '''G''' = '''Gr'''(1, 2) = '''R'''P¹ (the real projective line), which can be viewed as a circle. Then the continuum corresponds to the set of lines with rational slope passing through the origin on the Cartesian plane '''R'''² where the lattice of ker(''T'') lives. The lattice of ker(''T'') is generated by a [[basis]] of some choice of two commas '''u''' and '''v''' in ''S'' tempered out by the edo; view the plane as having two perpendicular axes corresponding to '''u''' and '''v''' directions. A rational point, i.e. a temperament on the continuum, then corresponds to a rational or infinite ratio ''t'' = ''p''/''q'', where the temperament is defined by the identification ''p'''''u''' ~ ''q'''''v''' (written additively). When ''t'' = 0, this corresponds to the temperament tempering out '''v'''. When ''t'' = (unsigned) infinity, this corresponds to the temperament tempering out '''u'''.

A higher-dimensional example: Say that ''r'' = 1, ''n'' = 4 (e.g. when ''S'' is the [[7-limit]]), and ''k'' = 2, for example the set of rank-2 [[7-limit]] temperaments supported by [[31edo]].  Then our Grassmannian '''G''' becomes '''Gr'''(2, 3). Define a coordinate system (''x'', ''y'', ''z'') for ker(T) using some fixed comma basis '''u'''_''x'', '''u'''_''y'', '''u'''_''z'' for ker(T). Then our Grassmannian can be identified with '''R'''P² (the real projective plane, the 2-dimensional space of lines through the origin in 3-dimensional space) by taking the unique line '''Rv''' perpendicular (according to the dot product given by the given coordinates) to the plane of commas tempered out for each temperament. Note that the real projective plane '''R'''P² can be visualized as a sphere with diametrically opposite points viewed as the same point.

NB: Not all equivalence continua are projective spaces, because not all Grassmannians are; for example, '''Gr'''(2, 4) is not a projective space.

Let's look at where some well-known 7-limit rank-2 temperaments supported by [[31edo]] live in the 2-dimensional equivalence continuum C(2, 31edo, 7-limit). Let '''u'''_''x'', '''u'''_''y'', '''u'''_''z'' = 81/80, 126/125, 1029/1024 be the basis for the kernel of 7-limit [[31edo]]. Then:

* [[septimal meantone]] tempers out 81/80 = '''u'''_''x'' = (1, 0, 0) and 126/125 = '''u'''_''y'' = (0, 1, 0), thus corresponds to the plane ''z'' = 0. This corresponds to '''v''' = (0, 0, 1).