User:Inthar/Epimorphic temperament: Difference between revisions

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=== Definition: constant structure (CS) ===

Given a periodic scale ''S'', let <math>C_k</math> be the set of ''k''-steps of ''S''. Then ''S'' is ''constant structure'' (CS) if for any <math>i, j \in \mathbb{Z}</math> we have <math>C_i \cap C_j = \varnothing.</math>

=== Epimorphic scales are CS ===

{{proof|contents=

Let <math>x \in C_j</math> where without loss of generality ''j'' > 0. Then there exists <math>i > 0</math> such that <math>S[i+j]/S[i] = x.</math> Suppose by way of contradiction there exist <math>k > 0, k \neq j</math> and <math>i > 0</math> such that <math>S[i+k]/S[i] = x.</math>

Then <math>v(x) = v(S[i+j]/S[i]) = v(S[i+j]) - v(S[i]) = i+k - i = j,</math> but also <math>v(x) = v(S[i^\prime+k]/S[i^\prime]) = v(S[i^\prime+k]) - v(S[i^\prime]) = k,</math> a contradiction.

}}

=== If the steps of a CS scale are linearly independent, then the scale is epimorphic ===

Theorem: Suppose ''S'' is a 2/1-equivalent increasing constant structure JI scale of length ''n''. Let <math>C_1</math> be the set of 1-steps of ''S'', and suppose that <math>C_1</math> is a ''basis'' for the JI group ''A'' generated by it. Then there exists an epimorphic val <math> v: A \to \mathbb{Z}</math> which is a val of ''n''-edo (and a similar statement holds for other equaves).

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That <math>v(2) = n</math> is also automatic.

}}

~~=== Epimorphic scales are CS ===~~

~~{{proof|contents=~~

Let <math>x \in C_j</math> where without loss of generality ''j'' > 0. Then there exists <math>i > 0</math> such that <math>S[i+j]/S[i] = x.</math> Suppose by way of contradiction there exist <math>k > 0, k \neq j</math> and <math>i > 0</math> such that <math>S[i+k]/S[i] = x.</math>

~~Then <math>v(x) = v(S[i+j]/S[i]) = v(S[i+j]) - v(S[i]) = i+k - i = j,</math> but also <math>v(x) = v(S[i^\prime+k]/S[i^\prime]) = v(S[i^\prime+k]) - v(S[i^\prime]) = k,</math> a contradiction.~~

}}

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 === Definition: constant structure (CS) ===
 Given a periodic scale ''S'', let <math>C_k</math> be the set of ''k''-steps of ''S''. Then ''S'' is ''constant structure'' (CS) if for any <math>i, j \in \mathbb{Z}</math> we have <math>C_i \cap C_j = \varnothing.</math>
+=== Epimorphic scales are CS ===
+{{proof|contents=
+Let <math>x \in C_j</math> where without loss of generality ''j'' > 0. Then there exists <math>i > 0</math> such that <math>S[i+j]/S[i] = x.</math> Suppose by way of contradiction there exist <math>k > 0, k \neq j</math> and <math>i > 0</math> such that <math>S[i+k]/S[i] = x.</math>
+Then <math>v(x) = v(S[i+j]/S[i]) = v(S[i+j]) - v(S[i]) = i+k - i = j,</math> but also <math>v(x) = v(S[i^\prime+k]/S[i^\prime]) = v(S[i^\prime+k]) - v(S[i^\prime]) = k,</math> a contradiction.
+}}
 === If the steps of a CS scale are linearly independent, then the scale is epimorphic ===
 Theorem: Suppose ''S'' is a 2/1-equivalent increasing constant structure JI scale of length ''n''. Let <math>C_1</math> be the set of 1-steps of ''S'', and suppose that <math>C_1</math> is a ''basis'' for the JI group ''A'' generated by it. Then there exists an epimorphic val <math> v: A \to \mathbb{Z}</math> which is a val of ''n''-edo (and a similar statement holds for other equaves).
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 That <math>v(2) = n</math> is also automatic.
-}}
-=== Epimorphic scales are CS ===
-{{proof|contents=
-Let <math>x \in C_j</math> where without loss of generality ''j'' > 0. Then there exists <math>i > 0</math> such that <math>S[i+j]/S[i] = x.</math> Suppose by way of contradiction there exist <math>k > 0, k \neq j</math> and <math>i > 0</math> such that <math>S[i+k]/S[i] = x.</math>
-Then <math>v(x) = v(S[i+j]/S[i]) = v(S[i+j]) - v(S[i]) = i+k - i = j,</math> but also <math>v(x) = v(S[i^\prime+k]/S[i^\prime]) = v(S[i^\prime+k]) - v(S[i^\prime]) = k,</math> a contradiction.
 }}