Rank-3 scale theorems: Difference between revisions

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===== MV2 is equivalent to LQ in 2-step scales (WIP) =====

Every mos is slope-LQ. This follows from the fact that every k-step is of the form pL qs or (p+1)L (q-1)s, where p/q <= a/b <= (p+1)/(q-1). This in turn follows from the fact that a mos has a generator, since every k-step is an octave reduction of either k perfect generators or k-1 perfect generators + 1 imperfect generator, and the perfect generator and the imperfect generator differ by changing one L to an s. If both the perfect generator and the imperfect generator had #L/#s > b/a (it must definitely be smaller than ceil(b/a) + 1, since the perfect generator is a collection of L...Ls chunks, and the size of such a chunk is <= ceil(b/a)), this would imply that n-1 perfect generators + 1 imperfect generator would not close at a period multiple.

Every mos is slope-LQ. This follows from the fact that every k-step is of the form pL qs or (p+1)L (q-1)s, where p/q <= a/b <= (p+1)/(q-1). This in turn follows from the fact that a mos has a generator, since every k-step is an octave reduction of either k perfect generators or k-1 perfect generators + 1 imperfect generator, and the perfect generator and the imperfect generator differ by changing one L to an s. If both the perfect generator and the imperfect generator had #L/#s > b/a (it must definitely be smaller than ceil(b/a) + 1, since the perfect generator is a collection of L...Ls chunks, and the size of such a chunk is <= ceil(b/a)), this would imply that n-1 perfect generators + 1 imperfect generator would not close at a period multiple. It is obvious that slope-LQ implies MV2.

~~There should be some Calc 1 stuff that I can do to show that mosses are~~ pointwise-LQ.

Slope-LQ => pointwise-LQ:

Pointwise-LQ => slope LQ:

==== MV3 Theorem 1 (WIP) ====

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 ===== MV2 is equivalent to LQ in 2-step scales (WIP) =====
-Every mos is slope-LQ. This follows from the fact that every k-step is of the form pL qs or (p+1)L (q-1)s, where p/q <= a/b <= (p+1)/(q-1). This in turn follows from the fact that a mos has a generator, since every k-step is an octave reduction of either k perfect generators or k-1 perfect generators + 1 imperfect generator, and the perfect generator and the imperfect generator differ by changing one L to an s. If both the perfect generator and the imperfect generator had #L/#s > b/a (it must definitely be smaller than ceil(b/a) + 1, since the perfect generator is a collection of L...Ls chunks, and the size of such a chunk is <= ceil(b/a)), this would imply that n-1 perfect generators + 1 imperfect generator would not close at a period multiple.
+Every mos is slope-LQ. This follows from the fact that every k-step is of the form pL qs or (p+1)L (q-1)s, where p/q <= a/b <= (p+1)/(q-1). This in turn follows from the fact that a mos has a generator, since every k-step is an octave reduction of either k perfect generators or k-1 perfect generators + 1 imperfect generator, and the perfect generator and the imperfect generator differ by changing one L to an s. If both the perfect generator and the imperfect generator had #L/#s > b/a (it must definitely be smaller than ceil(b/a) + 1, since the perfect generator is a collection of L...Ls chunks, and the size of such a chunk is <= ceil(b/a)), this would imply that n-1 perfect generators + 1 imperfect generator would not close at a period multiple. It is obvious that slope-LQ implies MV2.
-There should be some Calc 1 stuff that I can do to show that mosses are pointwise-LQ.
+Slope-LQ => pointwise-LQ:
+Pointwise-LQ => slope LQ:
 ==== MV3 Theorem 1 (WIP) ====