Interleaving: Difference between revisions
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(c) is similar to (b). | (c) is similar to (b). | ||
=== | === Attempted proof of Conjecture === | ||
Proof of Conjecture: Suppose the scale is made of two interleaved subsets offset by the abstract interval '''δ'''. | Proof of Conjecture: Suppose the scale is made of two interleaved subsets offset by the abstract interval '''δ'''. | ||
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Case 3: gcd(2(''a'' + ''b''), ''k'') > 1. ''k'' being even contradicts the interleaving property, hence ''k'' = ''a'' + ''b'' which must be odd. | Case 3: gcd(2(''a'' + ''b''), ''k'') > 1. ''k'' being even contradicts the interleaving property, hence ''k'' = ''a'' + ''b'' which must be odd. | ||
Scoot the (''a'' + ''b'')-letter subword across ''s''. On ''a'' + ''b'' consecutive notes, the interval subtended by this word is '''δ''', and on the other ''a'' + ''b'' notes, the interval subtended by this word is ‖''s''‖ - '''δ'''. | Scoot the (''a'' + ''b'')-letter subword across ''s''. On ''a'' + ''b'' consecutive notes, the interval subtended by this word is '''δ''', and on the other ''a'' + ''b'' notes, the interval subtended by this word is ‖''s''‖ - '''δ'''. We get a linear relation probably, but I haven't checked. | ||
== Generalizations == | == Generalizations == | ||