Generator sequence: Difference between revisions

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* This concept was formerly known as '''AGS''' or '''alternating generator sequence''' and was renamed to its current name since ''alternating'' indicates a sequence of 2 generators with period 2.

* One can use [[arity]] terminology for the number of distinct generators in a GS. When we have 2 (resp. 3) distinct generators, the generator sequence is ''binary'' (resp. ''ternary'').

* 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 GS recipe GS(''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 ''closing generator'', or 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 in the scale.

* 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 GS recipe GS(''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 ''closing generator'', or the ''imperfect generator'' since it is analogous to the imperfect generator in [[MOS]] scales, also subtends this number of steps. Suppose also that the closing generator is distinct from all of the generators used in the generator sequence and occurs only once in the scale.

When all of the above hold, this article calls the resulting scale ''well-formed GS'' (WFGS){{idiosyncratic}}. Whereas guided GS is a ''procedure'', WFGS provides a ''stopping condition'' for the procedure of guided GS described above. 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 GS 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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 * This concept was formerly known as '''AGS''' or '''alternating generator sequence''' and was renamed to its current name since ''alternating'' indicates a sequence of 2 generators with period 2.
 * One can use [[arity]] terminology for the number of distinct generators in a GS. When we have 2 (resp. 3) distinct generators, the generator sequence is ''binary'' (resp. ''ternary'').
-* 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 GS recipe GS(''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 ''closing generator'', or 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 in the scale.
+* 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 GS recipe GS(''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 ''closing generator'', or the ''imperfect generator'' since it is analogous to the imperfect generator in [[MOS]] scales, also subtends this number of steps. Suppose also that the closing generator is distinct from all of the generators used in the generator sequence and occurs only once in the scale.
 When all of the above hold, this article calls the resulting scale ''well-formed GS'' (WFGS){{idiosyncratic}}. Whereas guided GS is a ''procedure'', WFGS provides a ''stopping condition'' for the procedure of guided GS described above. 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 GS 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.