Generator-offset property: Difference between revisions

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#~~redirect~~ [[Scale ~~properties simplified~~]]

A scale satisfies the '''generator-offset property''' if it satisfies the following properties:

# The scale is generated by two chains of stacked copies of an interval called the ''generator''.

# The two chains are separated by a different interval called the ''offset'' (the difference between the first note of the second chain and the first note of the first chain).

# The lengths of the chains differ by at most one. (1-3 can be restated as: The scale can be built by stacking two alternating generators (called ''alternants'') a1 and a2. Note that a1 does not need to [[subtend]], i.e. occur as, the same number of steps as a2.)

# The generator always occurs as the same number of steps. For example, the generator is never both a 2-step and a 3-step.

[[File:MV3-Labeled.png|thumb|240px|Plot of at-most-decatonic MV3 generator-offset scale regions in the plane.]]

The [[Zarlino]] (3L 2M 2S) JI scale is an example of a generator-offset scale, because it is built by stacking alternating 5/4 and 6/5 generators. The 7-limit version of [[diasem]] (5L 2M 2S) is another example, with generators 7/6 and 8/7.

Generator-offset scales generalize the notion of [[dipentatonic scale|dipentatonic]] and [[diheptatonic scale|diheptatonic]] scales where the pentatonic and heptatonic are [[MOS scales]]. A related but distinct notion is [[alternating generator sequence]]. While scales produced using the generator-offset procedure can be seen as a result of an alternating generator sequence of 2 alternants, the generator-offset perspective views the sum of the two alternants as the "canonical" generator, and the alternants as rather being possible choices of the offset which are effectively equivalent up to chirality. While a well-formed AGS scale requires each alternant in the AGS to subtend the same number of steps, the generator-offset property only requires each (aggregate) generator to subtend the same number of steps.

Note: In Inthar's contribution to [[aberrismic theory]], this term has been superseded by [[guide frame]]s.

== Mathematical definition ==

More formally, a cyclic word ''S'' (representing the steps of a [[periodic scale]]) of size ''n'' is '''generator-offset''' if it satisfies the following properties:

# ''S'' is generated by two chains of stacked generators g separated by a fixed offset δ; either both chains are of size ''n''/2 (implying ''n'' is even), or one chain has size (''n'' + 1)/2 and the second has size (''n'' − 1)/2 (implying ''n'' is odd).

# The scale is ''well-formed'' with respect to g, i.e. all occurrences of the generator g are ''k''-steps for a fixed ''k''.

[[Category:Scale]]

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-#redirect [[Scale properties simplified]]
+A scale satisfies the '''generator-offset property''' if it satisfies the following properties:
+# The scale is generated by two chains of stacked copies of an interval called the ''generator''.
+# The two chains are separated by a different interval called the ''offset'' (the difference between the first note of the second chain and the first note of the first chain).
+# The lengths of the chains differ by at most one. (1-3 can be restated as: The scale can be built by stacking two alternating generators (called ''alternants'') a<sub>1</sub> and a<sub>2</sub>. Note that a<sub>1</sub> does not need to [[subtend]], i.e. occur as, the same number of steps as a<sub>2</sub>.)
+# The generator always occurs as the same number of steps. For example, the generator is never both a 2-step and a 3-step.
+[[File:MV3-Labeled.png|thumb|240px|Plot of at-most-decatonic MV3 generator-offset scale regions in the plane.]]
+The [[Zarlino]] (3L 2M 2S) JI scale is an example of a generator-offset scale, because it is built by stacking alternating 5/4 and 6/5 generators. The 7-limit version of [[diasem]] (5L 2M 2S) is another example, with generators 7/6 and 8/7.
+Generator-offset scales generalize the notion of [[dipentatonic scale|dipentatonic]] and [[diheptatonic scale|diheptatonic]] scales where the pentatonic and heptatonic are [[MOS scales]]. A related but distinct notion is [[alternating generator sequence]]. While scales produced using the generator-offset procedure can be seen as a result of an alternating generator sequence of 2 alternants, the generator-offset perspective views the sum of the two alternants as the "canonical" generator, and the alternants as rather being possible choices of the offset which are effectively equivalent up to chirality. While a well-formed AGS scale requires each alternant in the AGS to subtend the same number of steps, the generator-offset property only requires each (aggregate) generator to subtend the same number of steps.
+Note: In Inthar's contribution to [[aberrismic theory]], this term has been superseded by [[guide frame]]s.
+== Mathematical definition ==
+More formally, a cyclic word ''S'' (representing the steps of a [[periodic scale]]) of size ''n'' is '''generator-offset''' if it satisfies the following properties:
+# ''S'' is generated by two chains of stacked generators g separated by a fixed offset δ; either both chains are of size ''n''/2 (implying ''n'' is even), or one chain has size (''n'' + 1)/2 and the second has size (''n''&nbsp;&minus;&nbsp;1)/2 (implying ''n'' is odd).
+# The scale is ''well-formed'' with respect to g, i.e. all occurrences of the generator g are ''k''-steps for a fixed ''k''.
+[[Category:Scale]]