MOS substitution: Difference between revisions

Line 14:

#* 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~~ ==

== Examples ==

~~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).

=== 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

|}

@@ Line 14: / Line 14: @@
 #* 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 ==
+== Examples ==
-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).
+=== 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&#124;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&#124;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&#124;4(2)
+|colspan=2 style="text-align:right;"| <code>LsLsLmLsLsm</code>
+|colspan=2| GS(L+s, L+s, L+m) || yes
+|}