MOS substitution: Difference between revisions
→Open questions: Add open question about block balance of MOS substitution scales |
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The converse is false, as the scale in 5 letters [9/8 28/27 9/8 64/63 9/8 28/27 243/224 28/27 64/63 567/512 64/63] is almost a parallelogram. | The converse is false, as the scale in 5 letters [9/8 28/27 9/8 64/63 9/8 28/27 243/224 28/27 64/63 567/512 64/63] is almost a parallelogram. | ||
=== MOS substitution scales have block balance at most 2 === | |||
Consider a MOS substitution scale a'''X''' (b'''Y''' c'''Z'''). It is obvious that '''X''' has block balance 1, since we can replace the MOS substitution scale with the MOS scale a'''X''' (b + c)'''W''' to make this argument. '''Y''' and '''Z''' have block balance at most 2, since we can consider windows of the MOS scale of size ''k'' or ''k'' + 1, and the number of times '''Y''' (and also '''Z''') differs by at most 2. This is proved below for '''Y''', but it's exactly the same argument for '''Z''': | |||
Case 1: one of ''k'' and ''k + 1'' equals (''b'' + ''c'') and '''Y''' occurs exactly ''b'' times or ''b'' plus or minus 1 in this case. | |||
Case 2: neither of ''k'' and ''k + 1'' equals (''b'' + ''c''). Here, if '''Y''' occurs ''j'' or ''j'' + 1 times in a window of size ''k'', then ''Y'' occurs ''j'' + 1 or ''j'' + 2 times in a window of size ''k'' + 2. | |||
== Open questions == | == Open questions == | ||