User:Ganaram inukshuk/Notes: Difference between revisions

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#* s->sLL...L (n L's)

The chunking operation is contingent on rulesets 5 and 6 since there must be two unique chunks whose string sizes differ by exactly one L or one s. Repeatedly applying these rules as reduction rules on any arbitrary scale of L's and s's reduces the scale to a progenitor scale of either Ls or sL.

The chunking operation is contingent on rulesets 5 and 6 since there must be two unique chunks whose string sizes differ by exactly one L or one s. Repeatedly applying these rules as reduction rules on any arbitrary scale of L's and s's reduces the scale to a progenitor scale of either Ls or sL. The reduction rules can be denoted as such:

However, it may be the case that the reduced scale has only one L or one s, or that the scale started out that way. In either case, rulesets 3 and 4 can be used instead. Using these rules as reduction rules allows for the scale to still be reduced back down to Ls or sL.

* Reduction ruleset 5

** Lss...ss (n+1 s's) -> L

** Lss...s (n s's) -> s

* Reduction ruleset 6

** sLL...LL (n+1 L's) -> L

** sLL...L (n L's) -> s

However, it may be the case that the reduced scale has only one L or one s, or that the scale started out that way. In either case, rulesets 3 and 4 can be used instead:

* Reduction ruleset 3

** Lss...ss (n s's) -> L

** s -> s

* Reduction ruleset 4

** L->sLL...LL (n L's) -> L

** L -> s

For reduction ruleset 3, the entire scale except for one s is replaced with an L. For reduction ruleset 4, all but one L is replaced with an L with the remaining L replaced with an s. This can also be thought of using reduction ruleset 2 (sL -> L and L -> s) followed by reduction ruleset 3.

Using these rules as reduction rules allows for the scale to still be reduced back down to Ls or sL.

Since all MOSses must ultimately come from a pair of generators (represented in the progenitor scale as L and s), then this proves that if an arbitrary scale can be reduced to Ls or sL, then the scale itself must be a MOS.

Note that this only applies to single-period scales; for multi-period scales, such as LLsLsLLsLs, the resulting progenitor scale will be either Ls or sL repeated multiple times, and it cannot be a mix of both Ls and sL.

Note that this only applies to single-period scales; for multi-period scales, such as LLsLsLLsLs, the rules must be applied individually to each period and the resulting progenitor scale will be either Ls or sL repeated multiple times, and it cannot be a mix of both Ls and sL.

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 #* s->sLL...L (n L's)
-The chunking operation is contingent on rulesets 5 and 6 since there must be two unique chunks whose string sizes differ by exactly one L or one s. Repeatedly applying these rules as reduction rules on any arbitrary scale of L's and s's reduces the scale to a progenitor scale of either Ls or sL.
+The chunking operation is contingent on rulesets 5 and 6 since there must be two unique chunks whose string sizes differ by exactly one L or one s. Repeatedly applying these rules as reduction rules on any arbitrary scale of L's and s's reduces the scale to a progenitor scale of either Ls or sL. The reduction rules can be denoted as such:
-However, it may be the case that the reduced scale has only one L or one s, or that the scale started out that way. In either case, rulesets 3 and 4 can be used instead. Using these rules as reduction rules allows for the scale to still be reduced back down to Ls or sL.
+* Reduction ruleset 5
+** Lss...ss (n+1 s's) -> L
+** Lss...s (n s's) -> s
+* Reduction ruleset 6
+** sLL...LL (n+1 L's) -> L
+** sLL...L (n L's) -> s
+However, it may be the case that the reduced scale has only one L or one s, or that the scale started out that way. In either case, rulesets 3 and 4 can be used instead:
+* Reduction ruleset 3
+** Lss...ss (n s's) -> L
+** s -> s
+* Reduction ruleset 4
+** L->sLL...LL (n L's) -> L
+** L -> s
+For reduction ruleset 3, the entire scale except for one s is replaced with an L. For reduction ruleset 4, all but one L is replaced with an L with the remaining L replaced with an s. This can also be thought of using reduction ruleset 2 (sL -> L and L -> s) followed by reduction ruleset 3.
+Using these rules as reduction rules allows for the scale to still be reduced back down to Ls or sL.
 Since all MOSses must ultimately come from a pair of generators (represented in the progenitor scale as L and s), then this proves that if an arbitrary scale can be reduced to Ls or sL, then the scale itself must be a MOS.
-Note that this only applies to single-period scales; for multi-period scales, such as LLsLsLLsLs, the resulting progenitor scale will be either Ls or sL repeated multiple times, and it cannot be a mix of both Ls and sL.
+Note that this only applies to single-period scales; for multi-period scales, such as LLsLsLLsLs, the rules must be applied individually to each period and the resulting progenitor scale will be either Ls or sL repeated multiple times, and it cannot be a mix of both Ls and sL.