Ternary scale theorems: Difference between revisions

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# If ''n'' is odd, ''s'' is of the form ''a'''''x''' ''b'''''y''' ''b'''''z''' for some permutation {{nowrap|('''x''', '''y''', '''z''')}} of {{nowrap|('''L''', '''M''', '''s''')}}.

# If ''n'' is odd, ''s'' is abstractly SV3 (i.e. SV3 for almost all tunings).

# If ''n'' is odd, ''s'' is pairwise-MOS. That is, the following operations each result in a [[MOS]]: setting {{nowrap|'''L''' {{=}} '''M'''}}, setting {{nowrap|'''L''' {{=}} '''s'''}}, and setting {{nowrap|'''M''' {{=}} '''s'''}}.

# If ''n'' is odd, {{nowrap|''s'' {{=}} ''a'''''X''' ''b'''''Y''' ''b'''''Z'''}} is obtained from some mode of the (primitive) MOS ''a'''''X''' 2''b'''''W''' by replacing all the '''W'''s successively with alternating '''Y'''s and '''Z'''s (or alternating '''Z'''s and '''Y'''s for the other chirality, fixing the mode of ''a'''''X''' 2''b'''''W'''). The two alternants differ by replacing one '''Y''' with a '''Z'''.

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==== Statement (4) ====

~~Odd-numbered AGS scales are [[Fokker block]]s (in the 2-dimensional lattice generated by the generator and the offset). To see this, consider the following lattice depiction of such a scale:~~

~~x x x ... x~~

~~x x x ... x x~~

and use the vectors (−1, 2) and ({{ceil|n/2}}, 1) as the Fokker block chromas. A rank-3 Fokker block has the property that tempering out by each of the chromas gives two MOSes. These correspond to two of the temperings {{nowrap|'''X''' {{=}} '''Y'''|'''Y''' {{=}} '''Z'''}}, and {{nowrap|'''X''' {{=}} '''Z'''}}. The third tempering follows by symmetry (by taking the other chirality).

~~==== Statement (5) ====~~

By part (2), we have that ''s'' has step signature {{nowrap|''a'''''X''' ''b'''''Y''' ''b'''''Z'''}}, ''a'' odd. By part (4), we have that {{nowrap|''T''('''X''', '''W''') {{=}} ''s''('''X''', '''W''', '''W''')}} is a MOS scale ''a'''''X'''2''b'''''W'''. If {{nowrap|''b'' {{=}} 1}}, there's nothing to prove, so assume {{nowrap|''b'' > 1}}.

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 # If ''n'' is odd, ''s'' is of the form ''a'''''x''' ''b'''''y''' ''b'''''z''' for some permutation {{nowrap|('''x''', '''y''', '''z''')}} of {{nowrap|('''L''', '''M''', '''s''')}}.
 # If ''n'' is odd, ''s'' is abstractly SV3 (i.e. SV3 for almost all tunings).
-# If ''n'' is odd, ''s'' is pairwise-MOS. That is, the following operations each result in a [[MOS]]: setting {{nowrap|'''L''' {{=}} '''M'''}}, setting {{nowrap|'''L''' {{=}} '''s'''}}, and setting {{nowrap|'''M''' {{=}} '''s'''}}.
 # If ''n'' is odd, {{nowrap|''s'' {{=}} ''a'''''X''' ''b'''''Y''' ''b'''''Z'''}} is obtained from some mode of the (primitive) MOS ''a'''''X''' 2''b'''''W''' by replacing all the '''W'''s successively with alternating '''Y'''s and '''Z'''s (or alternating '''Z'''s and '''Y'''s for the other chirality, fixing the mode of ''a'''''X''' 2''b'''''W'''). The two alternants differ by replacing one '''Y''' with a '''Z'''.
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 ==== Statement (4) ====
-Odd-numbered AGS scales are [[Fokker block]]s (in the 2-dimensional lattice generated by the generator and the offset). To see this, consider the following lattice depiction of such a scale:
- x x x ... x
- x x x ... x x
-and use the vectors (−1,&nbsp;2) and ({{ceil|n/2}},&nbsp;1) as the Fokker block chromas. A rank-3 Fokker block has the property that tempering out by each of the chromas gives two MOSes. These correspond to two of the temperings {{nowrap|'''X''' {{=}} '''Y'''|'''Y''' {{=}} '''Z'''}}, and {{nowrap|'''X''' {{=}} '''Z'''}}. The third tempering follows by symmetry (by taking the other chirality).
-==== Statement (5) ====
 By part (2), we have that ''s'' has step signature {{nowrap|''a'''''X''' ''b'''''Y''' ''b'''''Z'''}}, ''a'' odd. By part (4), we have that {{nowrap|''T''('''X''', '''W''') {{=}} ''s''('''X''', '''W''', '''W''')}} is a MOS scale ''a'''''X'''2''b'''''W'''. If {{nowrap|''b'' {{=}} 1}}, there's nothing to prove, so assume {{nowrap|''b'' &gt; 1}}.