Equivalence continuum: Difference between revisions

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=== 1-dimensional continua ===

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'''P<sup>1</sup> (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'''<sup>2</sup> 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 ''x'' and ''y'' axes corresponding to '''u''' and '''v''' directions. A rational point, i.e. a temperament on the continuum, then corresponds to a line

''y = tx'', of rational or infinite slope ''t'' = ''q''/''p'', 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'''.

with equation ''py = qx'', of rational or infinite slope ''t'' = ''q''/''p'', 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'''.

=== 2-dimensional continua ===

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 === 1-dimensional continua ===
 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'''P<sup>1</sup> (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'''<sup>2</sup> 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 ''x'' and ''y'' axes corresponding to '''u''' and '''v''' directions. A rational point, i.e. a temperament on the continuum, then corresponds to a line
-''y = tx'', of rational or infinite slope ''t'' = ''q''/''p'', 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'''.
+with equation ''py = qx'', of rational or infinite slope ''t'' = ''q''/''p'', 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'''.
 === 2-dimensional continua ===