Ternary scale theorems: Difference between revisions

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# A primitive MV3 scale is either (1) balanced, (2) equivalent to XYZYX, or (3) a "twisted" word constructed as follows:

## Start with a multimos word <math>\mathsf{Christoffel}_{a,b}(\mathbf{X}, \mathbf{Z})^k</math> such that ''a'' is even.

## Interchange some of the '''Z'''s and '''X'''s at some of the boundaries of these copies of the mos word ''w''. Here, if ''w'' is a word and ''w'' = u'uvv', the ''boundary'' between ''u'' and ''v'' is ''u''[len(''u'')]''v''[1] within '''w'''. (If w is a circular word and ''w'' = [w'] where ''w''' is a linear word, replace ''w'' with w' in the previous sentence.) 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].

## Interchange some of the '''Z'''s and '''X'''s at some of the boundaries of these copies of the mos word ''w''. Here, if ''w'' is a word and ''w'' = u'uvv', the ''boundary'' between ''u'' and ''v'' is ''u''[len(''u'')]''v''[1] within '''w'''. (If w is a circular word and ''w'' = [w'] where w' is a linear word, replace ''w'' with w' in the previous sentence.) 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].

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

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 # A primitive MV3 scale is either (1) balanced, (2) equivalent to XYZYX, or (3) a "twisted" word constructed as follows:
 ## Start with a multimos word <math>\mathsf{Christoffel}_{a,b}(\mathbf{X}, \mathbf{Z})^k</math> such that ''a'' is even.
-## Interchange some of the '''Z'''s and '''X'''s at some of the boundaries of these copies of the mos word ''w''. Here, if ''w'' is a word and ''w'' = <i>u'uvv'</i>, the ''boundary'' between ''u'' and ''v'' is ''u''[len(''u'')]''v''[1] within '''w'''. (If w is a circular word and ''w'' = [<i>w'</i>] where ''w''' is a linear word, replace ''w'' with <i>w'</i> in the previous sentence.) 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].
+## Interchange some of the '''Z'''s and '''X'''s at some of the boundaries of these copies of the mos word ''w''. Here, if ''w'' is a word and ''w'' = <i>u'uvv'</i>, the ''boundary'' between ''u'' and ''v'' is ''u''[len(''u'')]''v''[1] within '''w'''. (If w is a circular word and ''w'' = [<i>w'</i>] where <i>w'</i> is a linear word, replace ''w'' with <i>w'</i> in the previous sentence.) 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].
 ## Replace every other '''X''' with '''Y''' in ''w''.