Ternary scale theorems: Difference between revisions

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=== Theorem 6.2 (Classification of MV3 scales) ===

# A primitive MV3 scale is either

A primitive MV3 scale is either

## '''balanced''' (classified by the previous theorem),

# '''balanced''' (classified by the previous theorem),

## '''sporadic non-balanced''': equivalent to '''XYZYX''', or

# '''sporadic non-balanced''': equivalent to '''XYZYX''', or

## '''twisted''': equivalent to a word constructed as follows:

# '''twisted''': equivalent to a word constructed as follows:

##* Start with the brightest multiMOS word ''ka'''''X'''''kb'''''Z''' with ''c'' being an even number.

#* Start with the brightest multiMOS word ''ka'''''X'''''kb'''''Z''' with ''c'' being an even number.

##* Interchange a '''Z''' and an '''X''' at some (possibly more than one) of the boundaries of these copies of the MOS word ''w''. Here, the boundary of two consecutive copies of ''w'' is the last letter of the first word and the first letter of the second word. (At the ends of the whole multiMOS word, the boundaries are just the first and last letters of the word.) For example, let ''w'' be the multiMOS word 8'''X'''6'''Z''', '''XXZXZXZXXZXZXZ'''. Then the border between the copies of the MOS subword '''XXZXZXZ''' are ''w''[7]''w''[8] and ''w''[14]''w''[1] (using one-based numbering).

#* Interchange a '''Z''' and an '''X''' at some (possibly more than one) of the boundaries of these copies of the MOS word ''w''. Here, the boundary of two consecutive copies of ''w'' is the last letter of the first word and the first letter of the second word. (At the ends of the whole multiMOS word, the boundaries are just the first and last letters of the word.) For example, let ''w'' be the multiMOS word 8'''X'''6'''Z''', '''XXZXZXZXXZXZXZ'''. Then the border between the copies of the MOS subword '''XXZXZXZ''' are ''w''[7]''w''[8] and ''w''[14]''w''[1] (using one-based numbering).

##* Replace every other '''X''' with '''Y''' in ''w''.

#* Replace every other '''X''' with '''Y''' in ''w''.

=== Proof ===

@@ Line 255: / Line 255: @@
 === Theorem 6.2 (Classification of MV3 scales) ===
-# A primitive MV3 scale is either
+A primitive MV3 scale is either
-## '''balanced''' (classified by the previous theorem),
+# '''balanced''' (classified by the previous theorem),
-## '''sporadic non-balanced''': equivalent to '''XYZYX''', or
+# '''sporadic non-balanced''': equivalent to '''XYZYX''', or
-## '''twisted''': equivalent to a word constructed as follows:
+# '''twisted''': equivalent to a word constructed as follows:
-##* Start with the brightest multiMOS word ''ka'''''X'''''kb'''''Z''' with ''c'' being an even number.
+#* Start with the brightest multiMOS word ''ka'''''X'''''kb'''''Z''' with ''c'' being an even number.
-##* Interchange a '''Z''' and an '''X''' at some (possibly more than one) of the boundaries of these copies of the MOS word ''w''. Here, the boundary of two consecutive copies of ''w'' is the last letter of the first word and the first letter of the second word. (At the ends of the whole multiMOS word, the boundaries are just the first and last letters of the word.) For example, let ''w'' be the multiMOS word 8'''X'''6'''Z''', '''XXZXZXZXXZXZXZ'''. Then the border between the copies of the MOS subword '''XXZXZXZ''' are ''w''[7]''w''[8] and ''w''[14]''w''[1] (using one-based numbering).
+#* Interchange a '''Z''' and an '''X''' at some (possibly more than one) of the boundaries of these copies of the MOS word ''w''. Here, the boundary of two consecutive copies of ''w'' is the last letter of the first word and the first letter of the second word. (At the ends of the whole multiMOS word, the boundaries are just the first and last letters of the word.) For example, let ''w'' be the multiMOS word 8'''X'''6'''Z''', '''XXZXZXZXXZXZXZ'''. Then the border between the copies of the MOS subword '''XXZXZXZ''' are ''w''[7]''w''[8] and ''w''[14]''w''[1] (using one-based numbering).
-##* Replace every other '''X''' with '''Y''' in ''w''.
+#* Replace every other '''X''' with '''Y''' in ''w''.
 === Proof ===