Ternary scale theorems: Difference between revisions

Line 272:

==== Proof ====

For 7.1.1: It suffices to show that ternary balanced words are pairwise-MOS and that non-PWF pairwise-MOS scales are even-regular.

For 7.1.1: It suffices to show that (a) ternary balanced words are pairwise-MOS and that (b) non-PWF pairwise-MOS scales are even-regular.

Let ''s'' be a ternary balanced word; then for any given letter '''y''' the number of '''y''''s in a subword of any given length ''L'' varies by at most 1. Thus the same is true when we count all non-'''y''' letters in any subword of length ''L''; thus when we equate '''x''' and '''z''', the count of the resulting letter in any subword of length ''L'' differs by 1. Being a binary balanced word is one characterization of the MOS property.

(a) Let ''s'' be a ternary balanced word; then for any given letter '''y''' the number of '''y'''s in a subword of any given length ''L'' varies by at most 1. Thus the same is true when we count all non-'''y''' letters in any subword of length ''L''; thus when we equate '''x''' and '''z''', the count of the resulting letter in any subword of length ''L'' differs by 1. Being a binary balanced word is one characterization of the MOS property.

For 7.1.2: Suppose ''s'' is balanced and has three sizes for ''k''-steps, ''a''''i'''''X''' + ''b''''i'''''Y''' + ''c''''i'''''Z''' = (''a''i, ''b''i, ''c''i) for ''i'' = 1, 2, 3. We may assume (''a''2, ''b''2, ''c''2) = (''a''1, ''b''1 + 1, ''c''1 - 1). Then either (''a''3, ''b''3, ''c''3) = (''a''1 + 1, ''b''1, ''c''1 - 1) or (''a''3, ''b''3, ''c''3) = (''a''1 - 1, ''b''1 + 1, ''c''1). In both cases, by balancedness applied to subwords of length ''k'', the three vectors represent the only possible interval sizes.

(b) Assume that the projection ''p'''''YZ'''(''s'') identifying '''Y''' and '''Z''' of a PMOS scale ''s'' is an ''r''-period MOS with step signature ''ar'''''X''' ''br'''''W'''...

For 7.1.2: Suppose ''s'' is balanced and has at least three sizes for ''k''-steps, ''a''''i'''''X''' + ''b''''i'''''Y''' + ''c''''i'''''Z''' = (''a''i, ''b''i, ''c''i) for ''i'' = 1, 2, 3. We may assume (''a''2, ''b''2, ''c''2) = (''a''1, ''b''1 + 1, ''c''1 - 1). Then either (''a''3, ''b''3, ''c''3) = (''a''1 + 1, ''b''1, ''c''1 - 1) or (''a''3, ''b''3, ''c''3) = (''a''1 - 1, ''b''1 + 1, ''c''1). 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'''.

@@ Line 272: / Line 272: @@
 ==== Proof ====
-For 7.1.1: It suffices to show that ternary balanced words are pairwise-MOS and that non-PWF pairwise-MOS scales are even-regular.
+For 7.1.1: It suffices to show that (a) ternary balanced words are pairwise-MOS and that (b) non-PWF pairwise-MOS scales are even-regular.
-Let ''s'' be a ternary balanced word; then for any given letter '''y''' the number of '''y''''s in a subword of any given length ''L'' varies by at most 1. Thus the same is true when we count all non-'''y''' letters in any subword of length ''L''; thus when we equate '''x''' and '''z''', the count of the resulting letter in any subword of length ''L'' differs by 1. Being a binary balanced word is one characterization of the MOS property.
+(a) Let ''s'' be a ternary balanced word; then for any given letter '''y''' the number of '''y'''s in a subword of any given length ''L'' varies by at most 1. Thus the same is true when we count all non-'''y''' letters in any subword of length ''L''; thus when we equate '''x''' and '''z''', the count of the resulting letter in any subword of length ''L'' differs by 1. Being a binary balanced word is one characterization of the MOS property.
-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.
+(b) Assume that the projection ''p''<sub>'''YZ'''</sub>(''s'') identifying '''Y''' and '''Z''' of a PMOS scale ''s'' is an ''r''-period MOS with step signature ''ar'''''X''' ''br'''''W'''...
+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'''.

Ternary scale theorems: Difference between revisions

Inthar (talk | contribs)