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, the rulesets are still valid. The numbering of rulesets is also arbitrary. For simplicity, rulesets 1 and 2 are denoted as though they were sisters of one another (that is, an additional "zeroth" ruleset is applied where L->s and 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

Applying ruleset 1 to itself n times produces a variant of ruleset 3 where L produces 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 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. Ruleset 1
   • L -> Ls
   • s-> s
2. Ruleset 2
   • L -> sL
   • s -> L
3. Ruleset 3
   • L->Lss...ss (n s's)
   • s->s
4. Ruleset 4
   • L->sLL...LL (n L's)
   • s->L
5. Ruleset 5
   • L->Lss...ss (n+1 s's)
   • s->Lss...s (n s's)
6. 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 any arbitrary scale of L's and s's reduces the scale to a progenitor scale of either Ls or sL.

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.

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.