Recursive structure of MOS scales: Difference between revisions
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for appropriate integers ''c, d, e, f'', where |''c''| < ''n'' and |''e''| < ''n''. The following two steps depend only on "all sizes {0, ''g'', ..., (''n'' − 1)''g''} are distinct": | for appropriate integers ''c, d, e, f'', where |''c''| < ''n'' and |''e''| < ''n''. The following two steps depend only on "all sizes {0, ''g'', ..., (''n'' − 1)''g''} are distinct": | ||
# By assumption ''a''L + ''b''s = (''ac'' + ''be'')''g'' + (''ad'' + ''bf'')''p'' = ''p''. Since ''a''L + ''b''s occurs on the "brightest" mode, from generatedness we have ''ac'' + ''be'' ∈ {0, ..., ''n'' − 1}. Hence we must have ''ac'' + ''be'' = 0 | # By assumption ''a''L + ''b''s = (''ac'' + ''be'')''g'' + (''ad'' + ''bf'')''p'' = ''p''. Since ''a''L + ''b''s occurs on the "brightest" mode, from generatedness we have ''ac'' + ''be'' ∈ {0, ..., ''n'' − 1}. Hence we must have ''ac'' + ''be'' = 0, and thus ''c'' = ±''b'' and ''e'' = ∓''a''. | ||
# We hence must have gcd(''a'', ''b'') = 1, since otherwise ''g'' cannot occur in the scale. | # We hence must have gcd(''a'', ''b'') = 1, since otherwise ''g'' cannot occur in the scale. | ||