Generator sequence: Difference between revisions
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m The finiteness of the scale implies that the generator sequence is finite or cyclic in some way. |
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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. | '''{{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" insinuates that the sequence repeats with period 2. | ||
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]]. | ||
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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, and the leftover interval after stacking ''n'' − 1 of the generators in the recipe (analogous to the imperfect generator in [[MOS]] scales) also subtends this number of steps, we call the resulting scale ''well-formed AGS''. In such a situation, we call the (logarithmic) average of the generators the ''guide generator''. | 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, and the leftover interval after stacking ''n'' − 1 of the generators in the recipe (analogous to the imperfect generator in [[MOS]] scales) also subtends this number of steps, we call the resulting scale ''well-formed AGS''. In such a situation, we call the (logarithmic) average of the generators the ''guide generator''. | ||
== | == Series arising from generator sequences == | ||
Only CS sizes at least 5 are listed. | Only CS sizes at least 5 are listed. | ||
* The Zarlino series, AGS(5/4, 6/5): 5, 7, 10, 17, 24, 41, 65 | * The Zarlino series, AGS(5/4, 6/5): 5, 7, 10, 17, 24, 41, 65 | ||