Generator sequence: Difference between revisions

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=== Proof that a WFGS scale is constant structure ===

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. ~~Tempering to~~ a ~~MOS via~~ a map π that identifies all non-imperfect generators, ~~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)~~. We may choose π so that the MOS's period and generator are linearly independent, and thus ''m'' = ''n''~~. 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.

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.

== JI scales from WFGS series ==

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 <!-- todo: Non-WF GSes-->
 === Proof that a WFGS scale is constant structure ===
-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. Tempering to a MOS via a map π that identifies all non-imperfect generators, 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). We may choose π so that the MOS's period and generator are linearly independent, and thus ''m'' = ''n''. 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.
+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.
 == JI scales from WFGS series ==