MOS substitution: Difference between revisions
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#* If the interval class of (''r'' + 1)-steps has ''t''L + (''u'' + 1)s and (''t'' − 1)''L'' + (''u'' + 2)''s'', ''S'' becomes a mos after deleting s steps for any ''k'' in {0, ..., ''q'' − 1}.<!-- | #* If the interval class of (''r'' + 1)-steps has ''t''L + (''u'' + 1)s and (''t'' − 1)''L'' + (''u'' + 2)''s'', ''S'' becomes a mos after deleting s steps for any ''k'' in {0, ..., ''q'' − 1}.<!-- | ||
#* If the interval class of (''r'' + 1)-steps has ''t''L + (''u'' + 1)''s'' and (''t'' + 1)L + ''u''s, ''S'' becomes a mos after deleting s steps for k in {0, ..., ''q'' − ''v'' − 1}, where ''v'' is the number of occurrences of (''t'' + 1)L + ''u''s in ''F''. | #* If the interval class of (''r'' + 1)-steps has ''t''L + (''u'' + 1)''s'' and (''t'' + 1)L + ''u''s, ''S'' becomes a mos after deleting s steps for k in {0, ..., ''q'' − ''v'' − 1}, where ''v'' is the number of occurrences of (''t'' + 1)L + ''u''s in ''F''. | ||
==Examples== | |||
=== 5L2m4s === | |||
To derive 5L2m4s as <math>\mathsf{mos\_subst\_aberrize}(5, 2, m, 4, k)</math>, 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). | |||
{| class="wikitable" | |||
|+ 5L2m4s as <math>\mathsf{mos\_subst\_aberrize}(5, 2, m, 4, k)</math> | |||
|- | |||
!rowspan=2| ''k'' | |||
!rowspan=2| filling MOS | |||
!rowspan=2| [[UDP]] for filling MOS | |||
!colspan=2| step pattern | |||
!colspan=2| generator sequence | |||
!rowspan=2| MOS for s = 0? | |||
|- | |||
!| template MOS: | |||
|| <code>LXLXLXLXLXX</code> | |||
!| intvl. class of gen.: || 2-steps | |||
|- | |||
| 2 || <code>mssmss</code> || 4|0(2) | |||
|colspan=2 style="text-align:right;"| <code>LmLsLsLmLss</code> | |||
|colspan=2| GS(L+m, L+s, L+s) || yes | |||
|- | |||
| 1 || <code>smssms</code> || 2|2(2) | |||
|colspan=2 style="text-align:right;"| <code>LsLmLsLsLms</code> | |||
|colspan=2| GS(L+s, L+m, L+s) || yes | |||
|- | |||
| 0 || <code>ssmssm</code> || 0|4(2) | |||
|colspan=2 style="text-align:right;"| <code>LsLsLmLsLsm</code> | |||
|colspan=2| GS(L+s, L+s, L+m) || yes | |||
|} | |||
=== 6L7m9s === | |||
{| class="wikitable" | |||
|+ 6L7m9s as <math>\mathsf{mos\_subst\_aberrize}(6, 7, L, 9, k)</math> | |||
|- | |||
!rowspan=2| ''k'' | |||
!rowspan=2| filling MOS (1 period) | |||
!rowspan=2| [[UDP]] for filling MOS | |||
!colspan=2| step pattern | |||
!colspan=2| generator sequence | |||
!rowspan=2| MOS for s = 0? | |||
|- | |||
!| template MOS: | |||
|| <code>mXXmXXmXXmXXmXXmXXmXXX</code> | |||
!| intvl. class of gen.: || 3-steps | |||
|- | |||
| 4 || <code>LsLss</code> || 12|0(3) | |||
|colspan=2 style="text-align:right;"| <code>mLsmLsmsLmsLmssmLsmLss</code> | |||
|colspan=2| GS(L+m+s, L+m+s, L+m+s, L+m+s, m+2s) || yes | |||
|- | |||
| 3 || <code>LssLs</code> || 9|3(3) | |||
|colspan=2 style="text-align:right;"| <code>mLsmsLmsLmssmLsmLsmsLs</code> | |||
|colspan=2| GS(L+m+s, L+m+s, L+m+s, m+2s, L+m+s) || yes | |||
|- | |||
| 2 || <code>sLsLs</code> || 6|6(3) | |||
|colspan=2 style="text-align:right;"| <code>msLmsLmssmLsmLsmsLmsLs</code> | |||
|colspan=2| GS(L+m+s, L+m+s, m+2s, L+m+s, L+m+s) || yes | |||
|- | |||
| 1 || <code>sLssL</code> || 3|9(3) | |||
|colspan=2 style="text-align:right;"| <code>msLmssmLsmLsmsLmsLmssL</code> | |||
|colspan=2| GS(L+m+s, m+2s, L+m+s, L+m+s, L+m+s) || yes | |||
|- | |||
| 0 || <code>ssLsL</code> || 0|12(3) | |||
|colspan=2 style="text-align:right;"| <code>mssmLsmLsmsLmsLmssmLsL</code> | |||
|colspan=2| GS(m+2s, L+m+s, L+m+s, L+m+s, L+m+s) || no | |||
|} | |||