Generator sequence: Difference between revisions

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Certain [[generator-offset property|generator-offset]] scales are examples. For example, [[diasem]] is AGS(8/7, 7/6) or AGS(7/6, 8/7) depending on [[chirality]]. The trivial case AGS(''x'') is stacking a single generator ''x'' to make a rank-2 scale, such as a [[MOS scale]].

== Terminology ==

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). 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.

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). 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.

Given a choice of equave ''E'' and a GS ''S'' = GS(''x''1, ..., ''x''r), a ''refinement'' of ''S'' is a generator sequence GS(w1, ..., wr) where each wi is a sequence of ''k'' = ''k''(''i'') intervals, ''y''i1, ..., ''y''ik, where ''y''i1 + ... + ''y''ik ≡ ''x''i modulo ''E''. If ''k'' does not depend on ''i'', call the refinement ''uniform''. 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 refinement of GS(4/3, 4/3, 4/3, 27/20, 4/3), which generates Zarlino.

Given a choice of equave ''E'' and a GS ''S'' = GS(''x''1, ..., ''x''''r''), a ''refinement'' 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 refinement ''uniform''. 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 refinement of GS(4/3, 4/3, 4/3, 27/20, 4/3), which generates Zarlino.

== Series arising from well-formed generator sequences ==

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 Certain [[generator-offset property|generator-offset]] scales are examples. For example, [[diasem]] is AGS(8/7, 7/6) or AGS(7/6, 8/7) depending on [[chirality]]. The trivial case AGS(''x'') is stacking a single generator ''x'' to make a rank-2 scale, such as a [[MOS scale]].
 == Terminology ==
-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). 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.
+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). 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.
-Given a choice of equave ''E'' and a GS ''S'' = GS(''x''<sub>1</sub>, ..., ''x''<sub>r</sub>), a ''refinement'' 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>i1</sub>, ..., ''y''<sub>ik</sub>, where ''y''<sub>i1</sub> + ... + ''y''<sub>ik</sub> ≡ ''x''<sub>i</sub> modulo ''E''. If ''k'' does not depend on ''i'', call the refinement ''uniform''. 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 refinement of GS(4/3, 4/3, 4/3, 27/20, 4/3), which generates Zarlino.
+Given a choice of equave ''E'' and a GS ''S'' = GS(''x''<sub>1</sub>, ..., ''x''<sub>''r''</sub>), a ''refinement'' 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 refinement ''uniform''. 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 refinement of GS(4/3, 4/3, 4/3, 27/20, 4/3), which generates Zarlino.
 == Series arising from well-formed generator sequences ==