Ternary scale theorems: Difference between revisions

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* In the singly even case, since there are evenly many slot letters in both ''s''1 and ''s''2, there are oddly many non-slot letters in both. Since ''s''1 and ''s''2 differ by interchanging '''y''' and '''z''', they have "opposite" filling letters, '''x''' + '''y''' being the opposite of '''x''' + '''z'''. This makes ''s''1 and ''s''2 opposite chiralities of an odd-regular MV3 scale.

* In the doubly even case, the number of non-slot letters in ''s''1 and ''s''2 is even, and we have a filling MOS of period 2. Since ''s''1 and ''s''2 are both primitive, they are both even-regular scales. {{Qed}}

== Theorem 7 (Ternary parallelogram scales are MOS substitution) ==

:''Main article: [[Ternary parallelogram scales are MOS substitution]]''

Ternary parallelogram scale words are [[MOS substitution]] scale words, where the period count of the template MOS is the number of rows of the parallelogram parallel to the unique step size parallel to a side of the parallelogram.

== Open problems ==

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 * In the singly even case, since there are evenly many slot letters in both ''s''<sub>1</sub> and ''s''<sub>2</sub>, there are oddly many non-slot letters in both. Since ''s''<sub>1</sub> and ''s''<sub>2</sub> differ by interchanging '''y''' and '''z''', they have "opposite" filling letters, '''x''' + '''y''' being the opposite of '''x''' + '''z'''. This makes ''s''<sub>1</sub> and ''s''<sub>2</sub> opposite chiralities of an odd-regular MV3 scale.
 * In the doubly even case, the number of non-slot letters in ''s''<sub>1</sub> and ''s''<sub>2</sub> is even, and we have a filling MOS of period 2. Since ''s''<sub>1</sub> and ''s''<sub>2</sub> are both primitive, they are both even-regular scales. {{Qed}}
+== Theorem 7 (Ternary parallelogram scales are MOS substitution) ==
+:''Main article: [[Ternary parallelogram scales are MOS substitution]]''
+Ternary parallelogram scale words are [[MOS substitution]] scale words, where the period count of the template MOS is the number of rows of the parallelogram parallel to the unique step size parallel to a side of the parallelogram.
 == Open problems ==