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]].

When all generators xi in the AGS recipe AGS(x1, ..., xr) [[subtend]] the same number of steps (does not depend on ''i''), ~~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''. 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 (does not

depend on ''i''), we call the resulting scale ''well-formed AGS''. This automatically implies that 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. 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.

== Series arising from 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]].
-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 (does not depend on ''i''), and the leftover interval after stacking ''n'' &minus; 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''. 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 (does not
+depend on ''i''), we call the resulting scale ''well-formed AGS''. This automatically implies that the leftover interval after stacking ''n'' &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.
 == Series arising from generator sequences ==