Operations on MOSes: Difference between revisions

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== Sistering ==

'''Sistering''' is the operation of taking a MOS pattern xL ys and reversing the roles of large and small steps, thus creating a yL xs pattern, called the ''sister'' of xL ys. It is called thus because a MOS pattern and its sister share the same MOS as a subset (for example, [[5L 2s]] and [[2L 5s]] both have [[2L 3s]] subsets), thus they share the same parent on the tree of MOS patterns (which corresponds to the [[scale tree]], via taking generator ranges). (More generally, given an ''r''-step scale pattern ''a1X1 ... arXr'' with ''r'' step sizes ''X''1 > ... > ''Xr'', we call the set of ~~all~~ patterns ''aπ(1)X1 ... aπ(r)Xr'' over all permutations π on {1, ..., ''r''} the ''sisterhood'' of ''a1 X1 ... arXr''.)

'''Sistering''' is the operation of taking a MOS pattern xL ys and reversing the roles of large and small steps, thus creating a yL xs pattern, called the ''sister'' of xL ys. It is called thus because a MOS pattern and its sister share the same MOS as a subset (for example, [[5L 2s]] and [[2L 5s]] both have [[2L 3s]] subsets), thus they share the same parent on the tree of MOS patterns (which corresponds to the [[scale tree]], via taking generator ranges). (More generally, given an ''r''-step scale pattern a1X1 ... arXr with r step sizes ''X''1 > ... > ''Xr'', we call the set of patterns aπ(1)X1 ... aπ(r)Xr over all permutations π on {1, ..., r} the ''sisterhood'' of ''a1 X1 ... arXr''.)

If xL ys has a generator range between a\x and b\(x+y) (it always holds that a < b), then its sister yL xs has a generator range between b\(x+y) and (b-a)\y.

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 == Sistering ==
-'''Sistering''' is the operation of taking a MOS pattern xL ys and reversing the roles of large and small steps, thus creating a yL xs pattern, called the ''sister'' of xL ys. It is called thus because a MOS pattern and its sister share the same MOS as a subset (for example, [[5L 2s]] and [[2L 5s]] both have [[2L 3s]] subsets), thus they share the same parent on the tree of MOS patterns (which corresponds to the [[scale tree]], via taking generator ranges). (More generally, given an ''r''-step scale pattern ''a<sub>1</sub>X<sub>1</sub> ... a<sub>r</sub>X<sub>r</sub>'' with ''r'' step sizes ''X''<sub>1</sub> > ... > ''X<sub>r</sub>'', we call the set of all patterns ''a<sub>π(1)</sub>X<sub>1</sub> ... a<sub>π(r)</sub>X<sub>r</sub>'' over all permutations π on {1, ..., ''r''} the ''sisterhood'' of ''a<sub>1</sub> X<sub>1</sub> ... a<sub>r</sub>X<sub>r</sub>''.)
+'''Sistering''' is the operation of taking a MOS pattern xL ys and reversing the roles of large and small steps, thus creating a yL xs pattern, called the ''sister'' of xL ys. It is called thus because a MOS pattern and its sister share the same MOS as a subset (for example, [[5L 2s]] and [[2L 5s]] both have [[2L 3s]] subsets), thus they share the same parent on the tree of MOS patterns (which corresponds to the [[scale tree]], via taking generator ranges). (More generally, given an ''r''-step scale pattern a<sub>1</sub>X<sub>1</sub> ... a<sub>r</sub>X<sub>r</sub> with r step sizes ''X''<sub>1</sub> > ... > ''X<sub>r</sub>'', we call the set of patterns a<sub>π(1)</sub>X<sub>1</sub> ... a<sub>π(r)</sub>X<sub>r</sub> over all permutations π on {1, ..., r} the ''sisterhood'' of ''a<sub>1</sub> X<sub>1</sub> ... a<sub>r</sub>X<sub>r</sub>''.)
 If xL ys has a generator range between a\x and b\(x+y) (it always holds that a < b), then its sister yL xs has a generator range between b\(x+y) and (b-a)\y.