Generator-offset property: Difference between revisions

Line 9:

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 is a conception developed by [[User:ks26|ground fault]], and generalizes the notion of [[dipentatonic scale|dipentatonic]] and [[diheptatonic scale|diheptatonic]] scales to scales of all sizes. A related notion is [[alternating generator sequence]]. ~~Generator~~-offset ~~scales~~ can be seen as a result of an alternating generator sequence of ~~length~~ 2, ~~but~~ the generator-offset ~~conception~~ views the ~~"aggregrate generator"~~ of the ~~AGS~~ as the "canonical" generator, and the alternants as rather being ~~offsets~~. While a well-formed alternating generator sequence requires each alternant in the AGS and the "imperfect" alternant to subtend the same number of steps, the generator-offset only requires each generator to subtend the same number of steps.

Generator-offset is a conception developed by [[User:ks26|ground fault]], and generalizes the notion of [[dipentatonic scale|dipentatonic]] and [[diheptatonic scale|diheptatonic]] scales to scales of all sizes. A related 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 alternating generator sequence requires each alternant in the AGS and the "imperfect" alternant to subtend the same number of steps, the generator-offset property only requires each generator to subtend the same number of steps.

== Mathematical definition ==

@@ Line 9: / Line 9: @@
 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 is a conception developed by [[User:ks26|ground fault]], and generalizes the notion of [[dipentatonic scale|dipentatonic]] and [[diheptatonic scale|diheptatonic]] scales to scales of all sizes. A related notion is [[alternating generator sequence]]. Generator-offset scales can be seen as a result of an alternating generator sequence of length 2, but the generator-offset conception views the "aggregrate generator" of the AGS as the "canonical" generator, and the alternants as rather being offsets. While a well-formed alternating generator sequence requires each alternant in the AGS and the "imperfect" alternant to subtend the same number of steps, the generator-offset only requires each generator to subtend the same number of steps.
+Generator-offset is a conception developed by [[User:ks26|ground fault]], and generalizes the notion of [[dipentatonic scale|dipentatonic]] and [[diheptatonic scale|diheptatonic]] scales to scales of all sizes. A related 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 alternating generator sequence requires each alternant in the AGS and the "imperfect" alternant to subtend the same number of steps, the generator-offset property only requires each generator to subtend the same number of steps.
 == Mathematical definition ==