Generator sequence: Difference between revisions

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== Terminology ==

* Consider a scale whose steps are all positive. Suppose that there exists a positive integer ''k'' such that for every generator ''x''''i'' in the AGS recipe AGS(''x''1, ..., ''x''''r''), every occurrence of ''x''''i'' in the scale [[subtend]]s ''k'' steps.

* This automatically implies that the gap between the next higher equave and the result of stacking len(scale) − 1 of the generators in the recipe, called the ''imperfect generator'' since it is analogous to the imperfect generator in [[MOS]] scales, also subtends this number of steps. Suppose also that the imperfect generator is distinct from all of the generators used in the generator sequence.

* This automatically implies that the gap between the next higher equave and the result of stacking len(scale) − 1 of the generators in the recipe, called the ''imperfect generator'' since it is analogous to the imperfect generator in [[MOS]] scales, also subtends this number of steps. Suppose also that the imperfect generator is distinct from all of the generators used in the generator sequence and occurs only once.

When all of the above hold, this article calls the resulting scale ''well-formed GS'' (WFGS){{idiosyncratic}}. In such a situation, we call the (logarithmic) average of the generators the ''guide generator''. The choice of "well-formed" is informed by the well-formed property of single-period MOS scales: the property that each occurrence of the generator subtends the same number of steps. The reason that WFGSes are studied is that the sequence yields CS scales at sizes of MOS scales generated by the guide generator and with the same period used by the WFGS, as long as the MOS scale in question is not too large. In summary, WFGS scales are made by detempering a MOS's generator chain into a stacked generator sequence, and the MOS sizes of the guide generator can help predict the sizes at which the AGS scale will be CS. This is because WFGS is designed to be exactly the right condition such that when one equates all of the generators of a WFGS chain, one gets a MOS scale which will be CS (this MOS has an abstract generator so there are no concerns about linear independence). As, by assumption, there are no steps in the original scale that are negative relative to the MOS, the original scale will thus be CS as well.

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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 ===

Suppose ''u'' = ''s''[i_1 : i_2] (this notation denotes a slice of the necklace ''s'' from index i_1 to index i_2 - 1) and ''v'' = ''s''[j_1 : j_2] are two subwords of ''s'' with Ψ(''u'') = Ψ(''v''). (Ψ(''w'') is 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 ''p'' resp. ''q'' detempered perfect generators where we take the ''E'' (equave)-complement generator if necessary. Tempering to the MOS with π, the images π(''u'') and π(''v'') also satisfy Ψ(π(u)) = Ψ(π(v)) and are a stack of ''l'' resp. ''M'' MOS generators (all of which are perfect). Since the abstract MOS's period and generator are linearly independent, ''m'' = ''l''. Hence Ψ(π(u)) = Ψ(π(v)) = ''mg + nE''. 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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 == Terminology ==
 * Consider a scale whose steps are all positive. Suppose that there exists a positive integer ''k'' such that for every generator ''x''<sub>''i''</sub> in the AGS recipe AGS(''x''<sub>1</sub>, ..., ''x''<sub>''r''</sub>), every occurrence of ''x''<sub>''i''</sub> in the scale [[subtend]]s ''k'' steps.
-* This automatically implies that the gap between the next higher equave and the result of stacking len(scale) &minus; 1 of the generators in the recipe, called the ''imperfect generator'' since it is analogous to the imperfect generator in [[MOS]] scales, also subtends this number of steps. Suppose also that the imperfect generator is distinct from all of the generators used in the generator sequence.
+* This automatically implies that the gap between the next higher equave and the result of stacking len(scale) &minus; 1 of the generators in the recipe, called the ''imperfect generator'' since it is analogous to the imperfect generator in [[MOS]] scales, also subtends this number of steps. Suppose also that the imperfect generator is distinct from all of the generators used in the generator sequence and occurs only once.
 When all of the above hold, this article calls the resulting scale ''well-formed GS'' (WFGS){{idiosyncratic}}. In such a situation, we call the (logarithmic) average of the generators the ''guide generator''. The choice of "well-formed" is informed by the well-formed property of single-period MOS scales: the property that each occurrence of the generator subtends the same number of steps. The reason that WFGSes are studied is that the sequence yields CS scales at sizes of MOS scales generated by the guide generator and with the same period used by the WFGS, as long as the MOS scale in question is not too large. In summary, WFGS scales are made by detempering a MOS's generator chain into a stacked generator sequence, and the MOS sizes of the guide generator can help predict the sizes at which the AGS scale will be CS. This is because WFGS is designed to be exactly the right condition such that when one equates all of the generators of a WFGS chain, one gets a MOS scale which will be CS (this MOS has an abstract generator so there are no concerns about linear independence). As, by assumption, there are no steps in the original scale that are negative relative to the MOS, the original scale will thus be CS as well.
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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 ===
+Suppose ''u'' = ''s''[i_1 : i_2] (this notation denotes a slice of the necklace ''s'' from index i_1 to index i_2 - 1) and ''v'' = ''s''[j_1 : j_2] are two subwords of ''s'' with Ψ(''u'') = Ψ(''v''). (Ψ(''w'') is 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 ''p'' resp. ''q'' detempered perfect generators where we take the ''E'' (equave)-complement generator if necessary. Tempering to the MOS with π, the images π(''u'') and π(''v'') also satisfy Ψ(π(u)) = Ψ(π(v)) and are a stack of ''l'' resp. ''M'' MOS generators (all of which are perfect). Since the abstract MOS's period and generator are linearly independent, ''m'' = ''l''. Hence Ψ(π(u)) = Ψ(π(v)) = ''mg + nE''. 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 ==