Recursive structure of MOS scales: Difference between revisions
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Suppose ''S'' is a MOS aLbs (a+b=n; WLOG equave = octave) and gcd(a, b) = d > 1. | Suppose ''S'' is a MOS aLbs (a+b=n; WLOG equave = octave) and gcd(a, b) = d > 1. | ||
Assume (n/d)-steps came in 2 sizes, pa+qb and ra+sb. Then at least one size, say pa+qb, must differ from (a/d)L + (b/d)s. WLOG p > a/d and pa+qb occurs on degree 0. Since equave = aL + bs, on a degree k(n/d) for some integer k > 0, there must be another (n/d)-step with fewer L's than a/d. This involves more than two changes from L to s. Scooting an (n/d)-step one step at a time changes its size one substitution at a time, showing that intermediate (n/d)-steps also exist. This violates the MOS property. | Assume (n/d)-steps came in 2 sizes, pa+qb and ra+sb. Then at least one size, say pa+qb, must differ from (a/d)L + (b/d)s. WLOG p > a/d and pa+qb occurs on degree 0. Since equave = aL + bs, on a degree k(n/d) for some integer k > 0, there must be another (n/d)-step with fewer L's than a/d. This involves more than two changes from L to s. Scooting an (n/d)-step one step at a time from degree 0 to k(n/d) changes its size one substitution at a time, showing that intermediate (n/d)-steps also exist. This violates the MOS property. | ||
=== Preservation of generators === | === Preservation of generators === | ||