Ternary scale theorems: Difference between revisions
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(b) Assume that the projection ''p''<sub>'''YZ'''</sub>(''s'') identifying '''Y''' and '''Z''' of a primitive PMOS scale ''s'' with signature ''ar'''''X''' ''b'''''Y''' ''c'''''Z''' is an ''r''-period MOS with step signature ''ar'''''X''' ''dr'''''W'''. We claim that neither ''b'' nor ''c'' is divisible by ''r''. Since ''p''<sub>'''XY'''</sub>(''s'') is the MOS (''ar'' + ''b'')'''W''' ''c'''''Z''' and ''p''<sub>'''XZ'''</sub>(''s'') is the MOS (''ar'' + ''c'')'''W''' ''b'''''Y''', if either ''b'' and ''c'' is divisible by ''r'', then the distribution of the third letter also has ''r'' periods, meaning that ''s'' itself has ''r'' periods. It suffices to show that ''r'' = 2. If ''r'' > 2, then... | (b) Assume that the projection ''p''<sub>'''YZ'''</sub>(''s'') identifying '''Y''' and '''Z''' of a primitive PMOS scale ''s'' with signature ''ar'''''X''' ''b'''''Y''' ''c'''''Z''' is an ''r''-period MOS with step signature ''ar'''''X''' ''dr'''''W'''. We claim that neither ''b'' nor ''c'' is divisible by ''r''. Since ''p''<sub>'''XY'''</sub>(''s'') is the MOS (''ar'' + ''b'')'''W''' ''c'''''Z''' and ''p''<sub>'''XZ'''</sub>(''s'') is the MOS (''ar'' + ''c'')'''W''' ''b'''''Y''', if either ''b'' and ''c'' is divisible by ''r'', then the distribution of the third letter also has ''r'' periods, meaning that ''s'' itself has ''r'' periods. It suffices to show that ''r'' = 2. If ''r'' > 2, then... | ||
For 7.1.2: Suppose ''s'' is balanced and has at least three sizes for ''k''-steps, ''a''<sub>''i''</sub>'''X''' + ''b''<sub>''i''</sub>'''Y''' + ''c''<sub>''i''</sub>'''Z''' = (''a''<sub>i</sub>, ''b''<sub>i</sub>, ''c''<sub>i</sub>) for ''i'' = 1, 2, 3. We may assume (''a''<sub>2</sub>, ''b''<sub>2</sub>, ''c''<sub>2</sub>) = (''a''<sub>1</sub>, ''b''<sub>1</sub> + 1, ''c''<sub>1</sub> - 1). Then either (''a''<sub>3</sub>, ''b''<sub>3</sub>, ''c''<sub>3</sub>) = (''a''<sub>1</sub> + 1, ''b''<sub>1</sub>, ''c''<sub>1</sub> - 1) or (''a''<sub>3</sub>, ''b''<sub>3</sub>, ''c''<sub>3</sub>) = (''a''<sub>1</sub> - 1, ''b''<sub>1</sub> + 1, ''c''<sub>1</sub>). In both cases, by balancedness applied to subwords of length ''k'', the three vectors represent the only possible interval sizes. | For 7.1.2: Suppose ''s'' is balanced and has at least three sizes for ''k''-steps, ''a''<sub>''i''</sub>'''X''' + ''b''<sub>''i''</sub>'''Y''' + ''c''<sub>''i''</sub>'''Z''' = (''a''<sub>''i''</sub>, ''b''<sub>''i''</sub>, ''c''<sub>''i''</sub>) for ''i'' = 1, 2, 3. We may assume (''a''<sub>2</sub>, ''b''<sub>2</sub>, ''c''<sub>2</sub>) = (''a''<sub>1</sub>, ''b''<sub>1</sub> + 1, ''c''<sub>1</sub> - 1). Then either (''a''<sub>3</sub>, ''b''<sub>3</sub>, ''c''<sub>3</sub>) = (''a''<sub>1</sub> + 1, ''b''<sub>1</sub>, ''c''<sub>1</sub> - 1) or (''a''<sub>3</sub>, ''b''<sub>3</sub>, ''c''<sub>3</sub>) = (''a''<sub>1</sub> - 1, ''b''<sub>1</sub> + 1, ''c''<sub>1</sub>). In both cases, by balancedness applied to subwords of length ''k'', the three vectors represent the only possible interval sizes. | ||
For 7.1.3: The ternary Fraenkel word may be verified as SV3 by inspection, and we have already shown in Theorem 1 that odd-regular balanced scales are SV3. To show that even-regular balanced scales are ''not'' SV3, observe that (''a''+''c'')-steps come in only 2 sizes in such a scale ''s'': floor(''a''/2)'''X''' + ceil(''a''/2)'''Y''' + ''c'''''Z''' and ceil(''a''/2)'''X''' + floor(''a''/2)'''Y''' + ''c'''''Z''', since the underlying MOS 2''a'''''X'''2''c'''''Y''' only has the (''a''+''c'')-step ''a'''''X''' + ''c'''''Z'''. The construction replaces the '''X'''s in these subwords with alternating '''X'''s and '''Y'''s; either of '''X''' or '''Y''' may occur first, corresponding to the two possible sizes, since ''a'' is odd and thus the (''a''+''c'')-step subword ''s''[''k'' : ''k''+''a''+''c''] becomes the subword ''s''[''k''+''a''+''c'' : ''k''+2''a''+2''c''] via interchanging '''X''' and '''Y'''. | For 7.1.3: The ternary Fraenkel word may be verified as SV3 by inspection, and we have already shown in Theorem 1 that odd-regular balanced scales are SV3. To show that even-regular balanced scales are ''not'' SV3, observe that (''a''+''c'')-steps come in only 2 sizes in such a scale ''s'': floor(''a''/2)'''X''' + ceil(''a''/2)'''Y''' + ''c'''''Z''' and ceil(''a''/2)'''X''' + floor(''a''/2)'''Y''' + ''c'''''Z''', since the underlying MOS 2''a'''''X'''2''c'''''Y''' only has the (''a''+''c'')-step ''a'''''X''' + ''c'''''Z'''. The construction replaces the '''X'''s in these subwords with alternating '''X'''s and '''Y'''s; either of '''X''' or '''Y''' may occur first, corresponding to the two possible sizes, since ''a'' is odd and thus the (''a''+''c'')-step subword ''s''[''k'' : ''k''+''a''+''c''] becomes the subword ''s''[''k''+''a''+''c'' : ''k''+2''a''+2''c''] via interchanging '''X''' and '''Y'''. | ||