Generator sequence: Difference between revisions

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'''{{PAGENAME}}''' ('''AGS''') is a scale-building procedure first described by [[Scott Dakota]]. The notation AGS(x1, ..., xr) denotes a scale-building procedure where a ([[Periodic scale|periodic]]) scale is built by stacking x1 first, x2 second, ..., reducing by the scale's [[equave]] when necessary. When xr is stacked, we go back to x1 and start stacking x1 again, then x2, ... Currently, the study of AGSs is dominated by [[constant structure]] AGS scales, which are obtained by stopping the stacking procedure at scale sizes that yield constant-structure scales. The term '''generator sequence''' (GS) may be preferable, as ''alternating'' is usually used for sequences that repeat every two terms.

'''{{PAGENAME}}''' ('''AGS''') is a scale-building procedure first described by [[Scott Dakota]]. The notation AGS(''x''1, ..., ''x''r) denotes a scale-building procedure where a ([[Periodic scale|periodic]]) scale is built by stacking ''x''1 first, ''x''2 second, ..., reducing by the scale's [[equave]] when necessary. When ''x''r is stacked, we go back to ''x''1 and start stacking ''x''1 again, then ''x''2, ... Currently, the study of AGSs is dominated by [[constant structure]] AGS scales, which are obtained by stopping the stacking procedure at scale sizes that yield constant-structure scales. The term '''generator sequence''' (GS) may be preferable, as ''alternating'' is usually used for sequences that repeat every two terms.

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~~ xi ~~in the AGS recipe AGS~~(x1, ..., xr) ~~[[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(s) − 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 WFGS ''S'' = GS(''x''1, ..., ''x''r), a ''refinement'' of ''S'' is a WFGS GS(w1, ..., wr) where each wi is a sequence of ''k'' (''k'' not depending on ''i'') intervals, ''y''i1, ..., ''y''ik, where ''y''i1 + ... + ''y''ik ≡ ''x''i modulo ''E''.

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

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

Only CS sizes at least 5 are listed.

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-'''{{PAGENAME}}''' ('''AGS''') is a scale-building procedure first described by [[Scott Dakota]]. The notation AGS(x<sub>1</sub>, ..., x<sub>r</sub>) denotes a scale-building procedure where a ([[Periodic scale|periodic]]) scale is built by stacking x<sub>1</sub> first, x<sub>2</sub> second, ..., reducing by the scale's [[equave]] when necessary. When x<sub>r</sub> is stacked, we go back to x<sub>1</sub> and start stacking x<sub>1</sub> again, then x<sub>2</sub>, ... Currently, the study of AGSs is dominated by [[constant structure]] AGS scales, which are obtained by stopping the stacking procedure at scale sizes that yield constant-structure scales. The term '''generator sequence''' (GS) may be preferable, as ''alternating'' is usually used for sequences that repeat every two terms.
+'''{{PAGENAME}}''' ('''AGS''') is a scale-building procedure first described by [[Scott Dakota]]. The notation AGS(''x''<sub>1</sub>, ..., ''x''<sub>r</sub>) denotes a scale-building procedure where a ([[Periodic scale|periodic]]) scale is built by stacking ''x''<sub>1</sub> first, ''x''<sub>2</sub> second, ..., reducing by the scale's [[equave]] when necessary. When ''x''<sub>r</sub> is stacked, we go back to ''x''<sub>1</sub> and start stacking ''x''<sub>1</sub> again, then ''x''<sub>2</sub>, ... Currently, the study of AGSs is dominated by [[constant structure]] AGS scales, which are obtained by stopping the stacking procedure at scale sizes that yield constant-structure scales. The term '''generator sequence''' (GS) may be preferable, as ''alternating'' is usually used for sequences that repeat every two terms.
 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(s) &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 WFGS ''S'' = GS(''x''<sub>1</sub>, ..., ''x''<sub>r</sub>), a ''refinement'' of ''S'' is a WFGS GS(w<sub>1</sub>, ..., w<sub>r</sub>) where each w<sub>i</sub> is a sequence of ''k'' (''k'' not depending on ''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''.
-To exclude the case when the generator is a 1-step or a (len(s) &minus; 1)-step, the modifier ''non-step'' can be used.
 == Series arising from well-formed generator sequences ==
 Only CS sizes at least 5 are listed.