Neutral and interordinal intervals in MOS scales: Difference between revisions
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Tags: Mobile edit Mobile web edit |
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=== Lemma 1 === | === Lemma 1 === | ||
Let n and k be positive integers and x be a real number such that kx is not an integer. Then floor((n+k)x) - floor(nx) >= floor(kx). | Let n and k be positive integers, and let x be a real number such that kx is not an integer. Then floor((n+k)x) - floor(nx) >= floor(kx). | ||
==== Proof ==== | ==== Proof ==== | ||
floor((n+k)x) - floor(nx) = -1 + ceil((n+k)x) + ceil(-nx) >= ceil((n+k)x - nx) - 1 = ceil(kx) - 1 = floor(kx). | floor((n+k)x) - floor(nx) = -1 + ceil((n+k)x) + ceil(-nx) >= ceil((n+k)x - nx) - 1 = ceil(kx) - 1 = floor(kx). | ||