Generator sequence: Difference between revisions
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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>, ... This article adopts a convention where an enumerated chord can be used instead for part of whole of the argument, where the chord's steps are generators, e.g. AGS(4:5:6)[7] for [[Zarlino]], which is syntactic sugar for AGS(5/4, 6/5)[7]. | '''{{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>, ... This article adopts a convention where an enumerated chord can be used instead for part of whole of the argument, where the chord's steps are generators, e.g. AGS(4:5:6)[7] for [[Zarlino]], which is syntactic sugar for AGS(5/4, 6/5)[7]. | ||
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 | 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. | ||
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 == | ||
* The term '''generator sequence'''{{idiosyncratic}} (GS) may be preferable, as ''alternating'' is usually used for sequences and series that repeat every two terms in mathematical terminology. | |||
* 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. | * 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) − 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 and occurs only once in the scale. | * 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 and occurs only once in the scale. | ||