Ternary scale theorems: Difference between revisions

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== Theorem 8 (Even-regular scales as (contra)interleavings) ==

Let ''s'' be a primitive even-regular scale of [[MOS substitution]] type ''a'''''x'''(''k'''''y''' ''k'''''z''') where ''a'' is even and gcd(''a'', ''k'') = 1. Let ''n'' = |''s''| = ''a'' + 2''k''.

# If ''n'' is ~~oddly~~ even, then ''s'' is a [[interleaving|contrainterleaving]] of the two opposite chiralities of an odd-regular scale.

# If ''n'' is singly even, then ''s'' is a [[interleaving|contrainterleaving]] of the two opposite chiralities of an odd-regular scale.

# If ''n'' is ~~evenly~~ even and > 4, then ''s'' is an [[interleaving]] of two copies of a smaller even-regular scale.

# If ''n'' is doubly even and > 4, then ''s'' is an [[interleaving]] of two copies of a smaller even-regular scale.

# If ''n'' = 4, then ''s'' is an interleaving of a 2-note MOS.

=== Proof ===

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Now we count the letters that occur in these MOS substitution words of 2-steps. We wish to show that ''s'' altogether has doubly evenly many 2-steps that are slot letters. Consider the chunk boundaries of the template MOS. For every boundary between chunks, there are two slot letters in the template MOS word of 2-steps. So it suffices that we have evenly many boundaries between (nonempty) chunks. Equivalently, we have to prove that there are evenly many steps in the template MOS ''a'''''x''' 2''k'''''X''', which is true by assumption (''a'' is even, 2''k'' is even). 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'''.

* In the ~~oddly~~ 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. This makes ''s''1 and ''s''2 opposite chiralities of an odd-regular MV3 scale.

* 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. This makes ''s''1 and ''s''2 opposite chiralities of an odd-regular MV3 scale.

* In the ~~evenly~~ 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.

* 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.

== Open problems ==

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 == Theorem 8 (Even-regular scales as (contra)interleavings) ==
 Let ''s'' be a primitive even-regular scale of [[MOS substitution]] type ''a'''''x'''(''k'''''y''' ''k'''''z''') where ''a'' is even and gcd(''a'', ''k'') = 1. Let ''n'' = |''s''| = ''a'' + 2''k''.
-# If ''n'' is oddly even, then ''s'' is a [[interleaving|contrainterleaving]] of the two opposite chiralities of an odd-regular scale.
+# If ''n'' is singly even, then ''s'' is a [[interleaving|contrainterleaving]] of the two opposite chiralities of an odd-regular scale.
-# If ''n'' is evenly even and > 4, then ''s'' is an [[interleaving]] of two copies of a smaller even-regular scale.
+# If ''n'' is doubly even and > 4, then ''s'' is an [[interleaving]] of two copies of a smaller even-regular scale.
 # If ''n'' = 4, then ''s'' is an interleaving of a 2-note MOS.
 === Proof ===
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 Now we count the letters that occur in these MOS substitution words of 2-steps. We wish to show that ''s'' altogether has doubly evenly many 2-steps that are slot letters. Consider the chunk boundaries of the template MOS. For every boundary between chunks, there are two slot letters in the template MOS word of 2-steps. So it suffices that we have evenly many boundaries between (nonempty) chunks. Equivalently, we have to prove that there are evenly many steps in the template MOS ''a'''''x''' 2''k'''''X''', which is true by assumption (''a'' is even, 2''k'' is even). 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'''.
-* In the oddly 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. This makes ''s''<sub>1</sub> and ''s''<sub>2</sub> opposite chiralities of an odd-regular MV3 scale.
+* 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. This makes ''s''<sub>1</sub> and ''s''<sub>2</sub> opposite chiralities of an odd-regular MV3 scale.
-* In the evenly 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.
+* 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.
 == Open problems ==