Generator sequence: Difference between revisions

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Suppose ''u'' and ''v'' are two subwords of ''s'' with Ψ(''u'') = Ψ(''v''). (Ψ(''w'') is the [[Parikh vector]] of ''w'', the interval size subtended by a subword ''w''.) Modulo equaves, either ''u'' is the imperfect generator of the WFGS, in which case ''u'' = ''v'', or we can assume that ''u'' and ''v'' are both stacks of ''n'' resp. ''m'' detempered perfect generators where we take the ''E'' (equave)-complement generator if necessary.

Given a lattice generated by ''E'' and the ~~perfect~~ generators, there exists a map π that identifies all non-imperfect generators. We may choose π so that the MOS's period and generator are linearly independent, and thus ''m'' = ''n''. The images π(''u'') and π(''v'') also satisfy Ψ(π(''u'')) = Ψ(π(''v'')) > 0 and are a stack of ''n'' resp. ''m'' MOS generators (all of which are perfect). Hence Ψ(π(''u'')) = Ψ(π(''v'')) = ''mg + pE''. This expression corresponds to a well-defined number of steps, given the generator ''g'' and the period ''E'' of the MOS, hence ''u'' and ''v'' must subtend the same number of steps.

Given a lattice generated by ''E'' and the generators in the recipe, there exists a map π that identifies all non-imperfect generators. We may choose π so that the MOS's period and generator are linearly independent, and thus ''m'' = ''n''. The images π(''u'') and π(''v'') also satisfy Ψ(π(''u'')) = Ψ(π(''v'')) > 0 and are a stack of ''n'' resp. ''m'' MOS generators (all of which are perfect). Hence Ψ(π(''u'')) = Ψ(π(''v'')) = ''mg + pE''. This expression corresponds to a well-defined number of steps, given the generator ''g'' and the period ''E'' of the MOS, hence ''u'' and ''v'' must subtend the same number of steps.

== JI scales from WFGS series ==

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 Suppose ''u'' and ''v'' are two subwords of ''s'' with Ψ(''u'') = Ψ(''v''). (Ψ(''w'') is the [[Parikh vector]] of ''w'', the interval size subtended by a subword ''w''.) Modulo equaves, either ''u'' is the imperfect generator of the WFGS, in which case ''u'' = ''v'', or we can assume that ''u'' and ''v'' are both stacks of ''n'' resp. ''m'' detempered perfect generators where we take the ''E'' (equave)-complement generator if necessary.
-Given a lattice generated by ''E'' and the perfect generators, there exists a map π that identifies all non-imperfect generators. We may choose π so that the MOS's period and generator are linearly independent, and thus ''m'' = ''n''. The images π(''u'') and π(''v'') also satisfy Ψ(π(''u'')) = Ψ(π(''v'')) > 0 and are a stack of ''n'' resp. ''m'' MOS generators (all of which are perfect). Hence Ψ(π(''u'')) = Ψ(π(''v'')) = ''mg + pE''. This expression corresponds to a well-defined number of steps, given the generator ''g'' and the period ''E'' of the MOS, hence ''u'' and ''v'' must subtend the same number of steps.
+Given a lattice generated by ''E'' and the generators in the recipe, there exists a map π that identifies all non-imperfect generators. We may choose π so that the MOS's period and generator are linearly independent, and thus ''m'' = ''n''. The images π(''u'') and π(''v'') also satisfy Ψ(π(''u'')) = Ψ(π(''v'')) > 0 and are a stack of ''n'' resp. ''m'' MOS generators (all of which are perfect). Hence Ψ(π(''u'')) = Ψ(π(''v'')) = ''mg + pE''. This expression corresponds to a well-defined number of steps, given the generator ''g'' and the period ''E'' of the MOS, hence ''u'' and ''v'' must subtend the same number of steps.
 == JI scales from WFGS series ==