Generator sequence: Difference between revisions

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When all generators ''x''i in the AGS recipe AGS(''x''1, ..., ''x''''r'') [[subtend]] the same number of steps (not depending on ''i''), we call the resulting scale ''well-formed GS'' (WFGS){{idiosyncratic}}. This automatically implies that the leftover interval after stacking len(scale) − 1 of the generators in the recipe (analogous to the imperfect generator in [[MOS]] scales) also subtends this number of steps. 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.

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

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 a GS ''S'' = GS(''x''1, ..., ''x''''r''), a ''splitting''{{idiosyncratic}} of ''S'' is a generator sequence GS(w1, ..., wr) 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.

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 When all generators ''x''<sub>i</sub> in the AGS recipe AGS(''x''<sub>1</sub>, ..., ''x''<sub>''r''</sub>) [[subtend]] the same number of steps (not depending on ''i''), we call the resulting scale ''well-formed GS'' (WFGS){{idiosyncratic}}. This automatically implies that the leftover interval after stacking len(scale) &minus; 1 of the generators in the recipe (analogous to the imperfect generator in [[MOS]] scales) also subtends this number of steps. 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.
-To exclude the case when the generator is a 1-step or a (len(scale) &minus; 1)-step, the modifier ''non-step'' can be used.
+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 a GS ''S'' = GS(''x''<sub>1</sub>, ..., ''x''<sub>''r''</sub>), a ''splitting''{{idiosyncratic}} of ''S'' is a generator sequence GS(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.