Ternary scale theorems: Difference between revisions

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== Theorem 7 (Classification of MV3 scales) ==

In the following, ''equivalent'' means "is the same circular word after permuting '''X''', '''Y''', and '''Z'''." This means that '''XYXZXYX''' is equivalent to '''YZYXYZY''', or '''XZXYXZX''', and so on.

=== Theorem 7.1 (Classification of ternary balanced scales) ===

# A primitive [[balanced]] MV3 scale ''s'' satisfies one of the following:

## '''sporadic balanced''': ''s'' is equivalent to '''XYXZXYX''', the ternary [[Fraenkel word]], with step signature 4'''X'''2'''Y'''1'''Z'''.

## '''odd-regular''': len(''s'') is odd, and ''s'' is equivalent to a word constructed from taking the brightest mode of the MOS ''c'''''X'''''b'''''Z''' with ''c'' even and gcd(''c'', ''b'') {{=}} 1, and replacing every other '''X''' with '''Y'''. We assume '''X''' > '''Z''' when constructing the MOS. In particular, ''s'' has [[step signature]] ''a'''''X'''''a'''''Y'''''b'''''Z''' where ''b'' is odd (with ''a'' {{=}} ''c''/2).

## '''odd-regular''': len(''s'') is odd, and ''s'' is equivalent to a word constructed from taking the brightest mode of the MOS ''c'''''X'''''b'''''Z''' with ''c'' even and {{nowrap|gcd(''c'', ''b'') {{=}} 1}}, and replacing every other '''X''' with '''Y'''. We assume {{nowrap|'''X''' > '''Z'''}} when constructing the MOS. In particular, ''s'' has [[step signature]] ''a'''''X'''''a'''''Y'''''b'''''Z''' where ''b'' is odd (with {{nowrap|''a'' {{=}} ''c''/2}}).

## '''even-regular''': len(''s'') is even, and ''s'' is equivalent to a word constructed from taking the brightest mode of the MOS 2''a'''''X'''2''c'''''Z''' with ''a'' odd and gcd(''a'', ''c'') {{=}} 1, and replacing every other '''X''' with '''Y'''. In particular, ''s'' has [[step signature]] ''a'''''X'''''a'''''Y'''''b'''''Z''' with ''a'' odd and ''b'' even.

## '''even-regular''': len(''s'') is even, and ''s'' is equivalent to a word constructed from taking the brightest mode of the MOS 2''a'''''X'''2''c'''''Z''' with ''a'' odd and {{nowrap|gcd(''a'', ''c'') {{=}} 1}}, and replacing every other '''X''' with '''Y'''. In particular, ''s'' has [[step signature]] ''a'''''X'''''a'''''Y'''''b'''''Z''' with ''a'' odd and ''b'' even.

# All primitive balanced ternary scales are MV3.

# A balanced primitive ternary scale is SV3 if and only if it is not even-regular.

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(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.

(b) Assume that the projection ''p'''''YZ'''(''s'') identifying '''Y''' and '''Z''' of a primitive PMOS scale ''s'' with signature ''ar'''''X''' ''b'''''Y''' ''c'''''Z''' is an ''r''-period MOS, ''r'' > 1, with step signature ''ar'''''X''' ''dr'''''W'''. We claim that neither ''b'' nor ''c'' is divisible by ''r''. Since ''p'''''XY'''(''s'') is the MOS (''ar'' + ''b'')'''W''' ''c'''''Z''' and ''p'''''XZ'''(''s'') is the MOS (''ar'' + ''c'')'''W''' ''b'''''Y''', if either ''b'' and ''c'' is divisible by ''r'', then the distributions of two of the letters have ''r'' periods. Then the distribution of the third letter also has ''r'' periods, meaning that ''s'' itself has ''r'' periods, a contradiction. It suffices to show that ''r'' {{=}} 2. If ''r'' > 2, then...

(b) Assume that the projection ''p'''''YZ'''(''s'') identifying '''Y''' and '''Z''' of a primitive PMOS scale ''s'' with signature ''ar'''''X''' ''b'''''Y''' ''c'''''Z''' is an ''r''-period MOS, {{nowrap|''r'' > 1}}, with step signature ''ar'''''X''' ''dr'''''W'''. We claim that neither ''b'' nor ''c'' is divisible by ''r''. Since ''p'''''XY'''(''s'') is the MOS {{nowrap|(''ar'' + ''b'')'''W''' ''c'''''Z'''}} and ''p'''''XZ'''(''s'') is the MOS {{nowrap|(''ar'' + ''c'')'''W''' ''b'''''Y'''}}, if either ''b'' and ''c'' is divisible by ''r'', then the distributions of two of the letters have ''r'' periods. Then the distribution of the third letter also has ''r'' periods, meaning that ''s'' itself has ''r'' periods, a contradiction. It suffices to show that {{nowrap|''r'' {{=}} 2}}. If {{nowrap|''r'' > 2}}, then...

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

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 {{nowrap|(''a'' + ''c'')}}-steps come in only 2 sizes in such a scale ''s'': {{nowrap|{{floor|''a''/2}}'''X''' + {{ceil|''a''/2}}'''Y''' + ''c'''''Z'''}} and {{nowrap|{{ceil|''a''/2}}'''X''' + {{floor|''a''/2}}'''Y''' + ''c'''''Z'''}}, since the underlying MOS 2''a'''''X'''2''c'''''Y''' only has the {{nowrap|(''a'' + ''c'')}}-step {{nowrap|''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 {{nowrap|(''a'' + ''c'')}}-step subword {{nowrap|''s''[''k'' : ''k'' + ''a'' + ''c'']}} becomes the subword {{nowrap|''s''[''k'' + ''a'' + ''c'' : ''k'' + 2''a'' + 2''c'']}} via interchanging '''X''' and '''Y'''.

Claim 7.1.4 can be verified by noting that such scales are PWF and using Theorem 4.

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 == Theorem 7 (Classification of MV3 scales) ==
 In the following, ''equivalent'' means "is the same circular word after permuting '''X''', '''Y''', and '''Z'''." This means that '''XYXZXYX''' is equivalent to '''YZYXYZY''', or '''XZXYXZX''', and so on.
 === Theorem 7.1 (Classification of ternary balanced scales) ===
 # A primitive [[balanced]] MV3 scale ''s'' satisfies one of the following:
 ## '''sporadic balanced''': ''s'' is equivalent to '''XYXZXYX''', the ternary [[Fraenkel word]], with step signature 4'''X'''2'''Y'''1'''Z'''.
-## '''odd-regular''': len(''s'') is odd, and ''s'' is equivalent to a word constructed from taking the brightest mode of the MOS ''c'''''X'''''b'''''Z''' with ''c'' even and gcd(''c'', ''b'') {{=}} 1, and replacing every other '''X''' with '''Y'''. We assume '''X''' &gt; '''Z''' when constructing the MOS. In particular, ''s'' has [[step signature]] ''a'''''X'''''a'''''Y'''''b'''''Z''' where ''b'' is odd (with ''a'' {{=}} ''c''/2).
+## '''odd-regular''': len(''s'') is odd, and ''s'' is equivalent to a word constructed from taking the brightest mode of the MOS ''c'''''X'''''b'''''Z''' with ''c'' even and {{nowrap|gcd(''c'', ''b'') {{=}} 1}}, and replacing every other '''X''' with '''Y'''. We assume {{nowrap|'''X''' &gt; '''Z'''}} when constructing the MOS. In particular, ''s'' has [[step signature]] ''a'''''X'''''a'''''Y'''''b'''''Z''' where ''b'' is odd (with {{nowrap|''a'' {{=}} ''c''/2}}).
-## '''even-regular''': len(''s'') is even, and ''s'' is equivalent to a word constructed from taking the brightest mode of the MOS 2''a'''''X'''2''c'''''Z''' with ''a'' odd and gcd(''a'', ''c'') {{=}} 1, and replacing every other '''X''' with '''Y'''. In particular,  ''s'' has [[step signature]] ''a'''''X'''''a'''''Y'''''b'''''Z''' with ''a'' odd and ''b'' even.
+## '''even-regular''': len(''s'') is even, and ''s'' is equivalent to a word constructed from taking the brightest mode of the MOS 2''a'''''X'''2''c'''''Z''' with ''a'' odd and {{nowrap|gcd(''a'', ''c'') {{=}} 1}}, and replacing every other '''X''' with '''Y'''. In particular,  ''s'' has [[step signature]] ''a'''''X'''''a'''''Y'''''b'''''Z''' with ''a'' odd and ''b'' even.
 # All primitive balanced ternary scales are MV3.
 # A balanced primitive ternary scale is SV3 if and only if it is not even-regular.
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 (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.
-(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, ''r'' &gt; 1, 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 distributions of two of the letters have ''r'' periods. Then the distribution of the third letter also has ''r'' periods, meaning that ''s'' itself has ''r'' periods, a contradiction. It suffices to show that ''r'' {{=}} 2. If ''r'' &gt; 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, {{nowrap|''r'' &gt; 1}}, 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 {{nowrap|(''ar'' + ''b'')'''W''' ''c'''''Z'''}} and  ''p''<sub>'''XZ'''</sub>(''s'') is the MOS {{nowrap|(''ar'' + ''c'')'''W''' ''b'''''Y'''}}, if either ''b'' and ''c'' is divisible by ''r'', then the distributions of two of the letters have ''r'' periods. Then the distribution of the third letter also has ''r'' periods, meaning that ''s'' itself has ''r'' periods, a contradiction. It suffices to show that {{nowrap|''r'' {{=}} 2}}. If {{nowrap|''r'' &gt; 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, {{nowrap|''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 {{nowrap|''i'' ∈ {{(}}1, 2, 3{{)}}}}. We may assume {{nowrap|(''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 {{nowrap|(''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 {{nowrap|(''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 {{nowrap|(''a'' + ''c'')}}-steps come in only 2 sizes in such a scale ''s'': {{nowrap|{{floor|''a''/2}}'''X''' + {{ceil|''a''/2}}'''Y''' + ''c'''''Z'''}} and {{nowrap|{{ceil|''a''/2}}'''X''' + {{floor|''a''/2}}'''Y''' + ''c'''''Z'''}}, since the underlying MOS 2''a'''''X'''2''c'''''Y''' only has the {{nowrap|(''a'' + ''c'')}}-step {{nowrap|''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 {{nowrap|(''a'' + ''c'')}}-step subword {{nowrap|''s''[''k'' : ''k'' + ''a'' + ''c'']}} becomes the subword {{nowrap|''s''[''k'' + ''a'' + ''c'' : ''k'' + 2''a'' + 2''c'']}} via interchanging '''X''' and '''Y'''.
 Claim 7.1.4 can be verified by noting that such scales are PWF and using Theorem 4. <!--{{Qed}}-->