User:Ganaram inukshuk/Notes

This page is for miscellaneous xen-related notes that I've written about but don't have an exact place elsewhere on the wiki (yet).

On the Origin of MOS Recursion

MOS recursion describes a set of properties that all moment-of-symmetry scales share that, among other things, allows us to create a few algorithms for determining whether an arbitrary scale of large and small steps has those properties.

The child scale of a MOS follows a distinct pattern in which the large step breaks up into the next large step and the next small step (in some order) and the small step becomes either the next large step or the next small step. As such, we can represent this as two sets of replacement rules:

1. Replacement ruleset 1 (where L - s > s)
   • L -> Ls
   • s -> s
2. Replacement ruleset 2 (where L - s < s)
   • L -> sL
   • s -> L

It should be noted that if the order of L's and s's is reversed (for example, L->sL and s->s for ruleset 1), the rulesets are still valid. The numbering of rulesets is also arbitrary. For explanation purposes, rulesets 1 and 2 and successive pairs of rulesets are denoted as though they were sisters of one another; this sistering process can be described with its own ruleset:

• L->s
• s->L

Applying ruleset 1 to itself n times produces ruleset 3, where L produces an L followed by n s's:

• L->Lss...ss (n s's)
• s->s

As such, applying ruleset 1 to itself n-1 times will result in L producing an L and n-1 s's. If ruleset 2 is applied to this, it creates ruleset 4, where L produces an s followed by n L's, the sister of ruleset 3:

• L->sLL...LL (n L's)
• s->L

Reversing the order of L's and s's of ruleset 2 produces this intermediate ruleset:

• L->Ls
• s->L

Applying ruleset 1 to the reversed form of ruleset 2 n times produces ruleset 5, where L produces an L followed by n+1 s's and s produces an L followed by n s's:

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

Applying ruleset 2 to ruleset 1 n times produces ruleset 6, the sister of ruleset 5 where L produces an s followed by n+1 L's and s produces an s followed by n times:

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

The final rulesets are as follows:

1. Replacement ruleset 1
   • L -> Ls
   • s-> s
2. Replacement ruleset 2
   • L -> sL
   • s -> L
3. Replacement ruleset 3
   • L->Lss...ss (n s's)
   • s->s
4. Replacement ruleset 4
   • L->sLL...LL (n L's)
   • s->L
5. Replacement ruleset 5
   • L->Lss...ss (n+1 s's)
   • s->Lss...s (n s's)
6. Replacement ruleset 6
   • L->sLL...LL (n+1 L's)
   • 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 a valid moment-of-symmetry scale will reduce the scale to a progenitor scale of either Ls or sL. The reduction rules can be denoted as such:

• 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 that the scale started out this way. In either case, rulesets 5 and 6 cannot be used, but 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 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.