MOS substitution: Difference between revisions

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==Examples==

=== 5L2m4s ===

To derive 5L2m4s as <math>\mathsf{mos\_subst\_aberrize}(5, 2, \mathbf{m}~~, 4~~, k)</math>, we exploit gcd(b, c) = 2 and substitute 2'''m'''4'''s''' into the template MOS 5'''L'''6'''X''' ('''LXLXLXLXLXX'''). Since 2'''m'''4'''s''' has three distinct modes ('''ssmssm''', '''smssms''', and '''mssmss''') and 5'''L'''6'''X''' is primitive, we obtain three distinct scales: '''LsLsLmLsLsm''', '''LsLmLsLsLms''', and '''LmLsLsLmLss'''. 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'''), representing all 3 possible rotations of ('''L'''+'''s''', '''L'''+'''m''', '''L'''+'''s''').

To derive 5L2m4s as <math>\mathsf{mos\_subst\_aberrize}(5, 2, 4, \mathbf{m}, k)</math>, we exploit gcd(b, c) = 2 and substitute 2'''m'''4'''s''' into the template MOS 5'''L'''6'''X''' ('''LXLXLXLXLXX'''). Since 2'''m'''4'''s''' has three distinct modes ('''ssmssm''', '''smssms''', and '''mssmss''') and 5'''L'''6'''X''' is primitive, we obtain three distinct scales: '''LsLsLmLsLsm''', '''LsLmLsLsLms''', and '''LmLsLsLmLss'''. 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'''), representing all 3 possible rotations of ('''L'''+'''s''', '''L'''+'''m''', '''L'''+'''s''').

{| class="wikitable"

|+ 5L2m4s as <math>\mathsf{mos\_subst\_aberrize}(5, 2, \mathbf{m}~~, 4~~, k)</math>

|+ 5L2m4s as <math>\mathsf{mos\_subst\_aberrize}(5, 2, 4, \mathbf{m}, k)</math>

|-

!rowspan=2| ''k''

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=== 6L7m9s ===

{| class="wikitable"

|+ 6L7m9s as <math>\mathsf{mos\_subst\_aberrize}(6, 7, \mathbf{L}~~, 9~~, k)</math>

|+ 6L7m9s as <math>\mathsf{mos\_subst\_aberrize}(6, 7, 9, \mathbf{L}, k)</math>

|-

!rowspan=2| ''k''

@@ Line 18: / Line 18: @@
 ==Examples==
 === 5L2m4s ===
-To derive 5L2m4s as <math>\mathsf{mos\_subst\_aberrize}(5, 2, \mathbf{m}, 4, k)</math>, we exploit gcd(b, c) = 2 and substitute 2'''m'''4'''s''' into the template MOS 5'''L'''6'''X''' ('''LXLXLXLXLXX'''). Since 2'''m'''4'''s''' has three distinct modes ('''ssmssm''', '''smssms''', and '''mssmss''') and 5'''L'''6'''X''' is primitive, we obtain three distinct scales: '''LsLsLmLsLsm''', '''LsLmLsLsLms''', and '''LmLsLsLmLss'''. 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'''), representing all 3 possible rotations of ('''L'''+'''s''', '''L'''+'''m''', '''L'''+'''s''').
+To derive 5L2m4s as <math>\mathsf{mos\_subst\_aberrize}(5, 2, 4, \mathbf{m}, k)</math>, we exploit gcd(b, c) = 2 and substitute 2'''m'''4'''s''' into the template MOS 5'''L'''6'''X''' ('''LXLXLXLXLXX'''). Since 2'''m'''4'''s''' has three distinct modes ('''ssmssm''', '''smssms''', and '''mssmss''') and 5'''L'''6'''X''' is primitive, we obtain three distinct scales: '''LsLsLmLsLsm''', '''LsLmLsLsLms''', and '''LmLsLsLmLss'''. 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'''), representing all 3 possible rotations of ('''L'''+'''s''', '''L'''+'''m''', '''L'''+'''s''').
 {| class="wikitable"
-|+ 5L2m4s as <math>\mathsf{mos\_subst\_aberrize}(5, 2, \mathbf{m}, 4, k)</math>
+|+ 5L2m4s as <math>\mathsf{mos\_subst\_aberrize}(5, 2, 4, \mathbf{m}, k)</math>
 |-
 !rowspan=2| ''k''
@@ Line 49: / Line 49: @@
 === 6L7m9s ===
 {| class="wikitable"
-|+ 6L7m9s as <math>\mathsf{mos\_subst\_aberrize}(6, 7, \mathbf{L}, 9, k)</math>
+|+ 6L7m9s as <math>\mathsf{mos\_subst\_aberrize}(6, 7, 9, \mathbf{L}, k)</math>
 |-
 !rowspan=2| ''k''