Ternary scale theorems: Difference between revisions

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== Theorem 3 (Properties of even generator-offset ternary scales) ==

A primitive generator-offset ternary scale ''s'' of even size 6 or greater, where the generator '''g''' is an even-step, has the following properties:

# ''s'' is a union of two copies of a primitive MOS ''M'' of size ''n''/2 generated by g; thus it is a [[flought scale]] obtained by taking two offset copies of said primitive MOS.

# ''s'' is a union of two copies of a primitive MOS ''M'' of size {{sfrac|''n''|2}} generated by '''g'''; thus it is a [[flought scale]] obtained by taking two offset copies of said primitive MOS.

# ''s'' is ''not'' SV3.

# ''s'' is ''not'' chiral.

# If ''M'' {{=}} ''M''('''y''', '''z''') is the primitive MOS necklace above, then ''s'' {{=}} ''M''('''XY''', '''XZ''') for some assignment of variable names '''X''', '''Y''', and '''Z''' to the three letters of ''s''.

# If {{nowrap|''M'' {{=}} ''M''('''y''', '''z''')}} is the primitive MOS necklace above, then {{nowrap|''s'' {{=}} ''M''('''XY''', '''XZ''')}} for some assignment of variable names '''X''', '''Y''', and '''Z''' to the three letters of ''s''.

=== Proof ===

(1) and (2) were proved in the proof of Proposition 1 (the part that we appeal to, from "all multiples of the generator '''g''' must be even-steps ..." to "These are all distinct by ℤ-linear independence", does not rely on ''s'' having the SGA property). (3) and (4) are easy to check using (1). {{qed}}

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 == Theorem 3 (Properties of even generator-offset ternary scales) ==
 A primitive generator-offset ternary scale ''s'' of even size 6 or greater, where the generator '''g''' is an even-step, has the following properties:
-# ''s'' is a union of two copies of a primitive MOS ''M'' of size ''n''/2 generated by g; thus it is a [[flought scale]] obtained by taking two offset copies of said primitive MOS.
+# ''s'' is a union of two copies of a primitive MOS ''M'' of size {{sfrac|''n''|2}} generated by '''g'''; thus it is a [[flought scale]] obtained by taking two offset copies of said primitive MOS.
 # ''s'' is ''not'' SV3.
 # ''s'' is ''not'' chiral.
-# If ''M'' {{=}} ''M''('''y''', '''z''') is the primitive MOS necklace above, then ''s'' {{=}} ''M''('''XY''', '''XZ''') for some assignment of variable names '''X''', '''Y''', and '''Z''' to the three letters of ''s''.
+# If {{nowrap|''M'' {{=}} ''M''('''y''', '''z''')}} is the primitive MOS necklace above, then {{nowrap|''s'' {{=}} ''M''('''XY''', '''XZ''')}} for some assignment of variable names '''X''', '''Y''', and '''Z''' to the three letters of ''s''.
 === Proof ===
 (1) and (2) were proved in the proof of Proposition 1 (the part that we appeal to, from "all multiples of the generator '''g''' must be even-steps ..." to "These are all distinct by ℤ-linear independence", does not rely on ''s'' having the SGA property). (3) and (4) are easy to check using (1). {{qed}}