Generator sequence: Difference between revisions
Defining refinements of WFGSes. |
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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]]. | 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 == | == 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) − 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) − 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. | ||