Generator sequence: Difference between revisions

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Given a choice of equave ''E'' and an AGS ''S'' = AGS(''x''1, ..., ''x''''r''), a ''splitting''{{idiosyncratic}} of ''S'' is a generator sequence AGS(w1, ..., w''r'') where each w''i'' is a sequence of ''k'' = ''k''(''i'') intervals, ''y''''i''1, ..., ''y''''ik'', where ''y''''i''1 + ... + ''y''''ik'' ≡ ''x''''i'' modulo ''E''. If ''k'' does not depend on ''i'', call the splitting ''uniform''{{idiosyncratic}}. For instance, the GS for Zil, AGS(8/7, 7/6, 8/7, 7/6, 8/7, 7/6, 8/7, 189/160, 8/7, 7/6) is a uniform splitting of AGS(4/3, 4/3, 4/3, 27/20, 4/3), which generates Zarlino. Any 2/1-equivalent WFGS with an aggregate generator equal to a voicing of 3/2 is a uniform splitting of AGS(3/2), corresponding to a unique [[pergen]] with a 3/2 period.

=== Proof that a WFGS scale is CS ===

=== 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.

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 Given a choice of equave ''E'' and an AGS ''S'' = AGS(''x''<sub>1</sub>, ..., ''x''<sub>''r''</sub>), a ''splitting''{{idiosyncratic}} of ''S'' is a generator sequence AGS(w<sub>1</sub>, ..., w<sub>''r''</sub>) where each w<sub>''i''</sub> is a sequence of ''k'' = ''k''(''i'') intervals, ''y''<sub>''i''1</sub>, ..., ''y''<sub>''ik''</sub>, where ''y''<sub>''i''1</sub> + ... + ''y''<sub>''ik''</sub> ≡ ''x''<sub>''i''</sub> modulo ''E''. If ''k'' does not depend on ''i'', call the splitting ''uniform''{{idiosyncratic}}. For instance, the GS for Zil, AGS(8/7, 7/6, 8/7, 7/6, 8/7, 7/6, 8/7, 189/160, 8/7, 7/6) is a uniform splitting of AGS(4/3, 4/3, 4/3, 27/20, 4/3), which generates Zarlino. Any 2/1-equivalent WFGS with an aggregate generator equal to a voicing of 3/2 is a uniform splitting of AGS(3/2), corresponding to a unique [[pergen]] with a 3/2 period.
 <!-- todo: Non-WF GSes-->
-=== Proof that a WFGS scale is CS ===
+=== 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.