Neutral and interordinal intervals in MOS scales: Difference between revisions
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Tags: Mobile edit Mobile web edit |
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For positive integers n and k and for real x, floor((n+k)x) - floor(nx) >= floor(kx). | For positive integers n and k and for real x, floor((n+k)x) - floor(nx) >= floor(kx). | ||
==== Proof ==== | ==== Proof ==== | ||
This is trivially true for integer x, so assume | This is trivially true for integer x, so assume kx is not an integer. Then: | ||
floor((n+k)x) - floor(nx) = ceil((n+k)x) + ceil(-nx) | floor((n+k)x) - floor(nx) = -1 + ceil((n+k)x) + ceil(-nx) >= ceil((n+k)x - nx) - 1 = ceil(kx) - 1 = floor(kx). | ||
=== Discretizing Lemma === | === Discretizing Lemma === | ||