Generator sequence: Difference between revisions

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* 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.

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 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.

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.

To exclude the case when the generator is a 1-step or a (len(scale) − 1)-step, the modifier ''non-step''{{idiosyncratic}} can be used.

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, GS(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 GS(4/3, 4/3, 4/3, 27/20, 4/3), which generates Zarlino.

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.

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 * 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.
-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 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.
+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.
 To exclude the case when the generator is a 1-step or a (len(scale) &minus; 1)-step, the modifier ''non-step''{{idiosyncratic}} can be used.
-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, GS(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 GS(4/3, 4/3, 4/3, 27/20, 4/3), which generates Zarlino.
+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.
 <!-- todo: Non-WF GSes-->