MOS substitution: Difference between revisions

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# Suppose that the perfect generator of ''T'' that we use has ''r''-many X steps and that the imperfect generator has (r + 1)-many X steps. Suppose the sizes for ''r''-steps in ''F'' are ''t''L + ''u''s and (''t'' − 1)L + (''u'' + 1)s. Then the interval class of (''r'' + 1)-steps has either (a) ''t''L + (''u'' + 1)s and (''t'' − 1)''L'' + (''u'' + 2)''s'', or (b) ''t''L + (''u'' + 1)''s'' and (''t'' + 1)L + ''u''s.

#* In case (a), ''S'' becomes a mos after deleting s steps for any ''k'' in {0, ..., ''q'' − 1}.

#* In case (b), ''S'' becomes a mos after deleting s steps for k in {0, ..., ''v''}, where ''v'' is the number of generators stacked to obtain (''t'' + 1)L + ''u''s in the filling MOS F.

#* In case (b), ''S'' becomes a mos after deleting s steps for k in {0, ..., ''v'' − 1}, where ''v'' is the number of generators stacked to obtain (''t'' + 1)L + ''u''s in the filling MOS F.

== Example ==

For 5L2m4s, we exploit gcd(b, c) = 2 and substitute 2m4s into the template MOS 5L6X (LXLXLXLXLXX). Since 2m4s has three distinct modes (ssmssm, smssms, and mssmss) and 5L6X is primitive, we obtain three distinct scales: LsLsLmLsLsm, LsLmLsLsLms, and LmLsLsLmLss. The first two are a chiral pair of billiard scales, and the last is achiral but not deletion-MOS. All three scales admit short generator sequences of 2-steps, respectively GS(L+s, L+s, L+m), GS(L+s, L+m, L+s), and GS(L+m, L+s, L+s), notably representing all 3 possible rotations of (L+s, L+m, L+s).

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 # Suppose that the perfect generator of ''T'' that we use has ''r''-many X steps and that the imperfect generator has (r + 1)-many X steps. Suppose the sizes for ''r''-steps in ''F'' are ''t''L + ''u''s and (''t'' &minus; 1)L + (''u'' + 1)s. Then the interval class of (''r'' + 1)-steps has either (a) ''t''L + (''u'' + 1)s and (''t'' &minus; 1)''L'' + (''u'' + 2)''s'', or (b) ''t''L + (''u'' + 1)''s'' and (''t'' + 1)L + ''u''s.
 #* In case (a), ''S'' becomes a mos after deleting s steps for any ''k'' in {0, ..., ''q'' &minus; 1}.
-#* In case (b), ''S'' becomes a mos after deleting s steps for k in {0, ..., ''v''}, where ''v'' is the number of generators stacked to obtain (''t'' + 1)L + ''u''s in the filling MOS F.
+#* In case (b), ''S'' becomes a mos after deleting s steps for k in {0, ..., ''v'' &minus; 1}, where ''v'' is the number of generators stacked to obtain (''t'' + 1)L + ''u''s in the filling MOS F.
 == Example ==
 For 5L2m4s, we exploit gcd(b, c) = 2 and substitute 2m4s into the template MOS 5L6X (LXLXLXLXLXX). Since 2m4s has three distinct modes (ssmssm, smssms, and mssmss) and 5L6X is primitive, we obtain three distinct scales: LsLsLmLsLsm, LsLmLsLsLms, and LmLsLsLmLss. The first two are a chiral pair of billiard scales, and the last is achiral but not deletion-MOS. All three scales admit short generator sequences of 2-steps, respectively GS(L+s, L+s, L+m), GS(L+s, L+m, L+s), and GS(L+m, L+s, L+s), notably representing all 3 possible rotations of (L+s, L+m, L+s).