Ternary scale theorems: Difference between revisions
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For 7.1.2: Suppose ''s'' is balanced and has 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. (''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 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'''. | ||