Epimorphic scale: Difference between revisions

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A JI scale ''S'' is '''epimorphic''' if on the JI subgroup ''A'' generated by the intervals of ''S'', there exists a linear map, called an '''epimorphism''', ''v'': ''A'' → ℤ such that ''v''(''S''[''i'']) = ''i''.

A JI scale ''S'' is '''epimorphic''' if on the JI subgroup ''A'' generated by the intervals of ''S'', there exists a linear map, called an '''epimorphism''', ''v'': ''A'' → ℤ such that ''v''(''S''[''i'']) = ''i'' for all ''i'' ∈ ℤ.

An '''epimorphic temperament''' of an epimorphic scale ''S'' on a JI group ''A'' is a temperament supported by its epimorphism on ''A''. Some [[exotemperament]]s (including [[val]]s for small edos) can be used as epimorphic temperaments for small epimorphic scales:

* The 2.3.5 temperament [[dicot]] supports [[nicetone]] (3L2M2s), [[blackdye]] (5L2M3s) and superzarlino (a 17-form) scale structures.

* The 2.3.7 temperament [[semaphore]] supports [[archylino]] (2L3M2s), [[diasem]] (5L2M2s), and other scales in the [[Tas series]].

== Example ==

Consider the Ptolemaic diatonic scale, {9/8, 5/4, 4/3, 3/2, 5/3, 15/8, 2/1}, which is nicetone with L = 9/8, M = 10/9, and s = 16/15. This scale is epimorphic because we can apply ⟨7 11 16], the [[7edo]] [[patent val]], to map the intervals into the number of scale steps:

<math>

\left(\begin{array} {rrr}

7 & 11 & 16

\end{array} \right)

\left(\begin{array}{rrrrrrr}

-3 & -2 & 2 & -1 & 0 & -3 & 1 \\

2 & 0 & -1 & 1 & -1 & 1 & 0 \\

0 & 1 & 0 & 0 & 1 & 1 & 0

\end{array}\right)

=

\left(\begin{array}{rrrrrrr}

1 & 2 & 3 & 4 & 5 & 6 & 7

\end{array}\right)

</math>

where the columns of the 3×7 matrix are the scale intervals written in [[monzo]] form. Hence, 7edo (equipped with its patent val) is an epimorphic temperament of the Ptolemaic diatonic scale.

== Facts ==

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

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-A JI scale ''S'' is '''epimorphic''' if on the JI subgroup ''A'' generated by the intervals of ''S'', there exists a linear map, called an '''epimorphism''', ''v'': ''A'' → ℤ such that ''v''(''S''[''i'']) = ''i''.
+A JI scale ''S'' is '''epimorphic''' if on the JI subgroup ''A'' generated by the intervals of ''S'', there exists a linear map, called an '''epimorphism''', ''v'': ''A'' → ℤ such that ''v''(''S''[''i'']) = ''i'' for all ''i'' ∈ ℤ.
 An '''epimorphic temperament''' of an epimorphic scale ''S'' on a JI group ''A'' is a temperament supported by its epimorphism on ''A''. Some [[exotemperament]]s (including [[val]]s for small edos) can be used as epimorphic temperaments for small epimorphic scales:
 * The 2.3.5 temperament [[dicot]] supports [[nicetone]] (3L2M2s), [[blackdye]] (5L2M3s) and superzarlino (a 17-form) scale structures.
 * The 2.3.7 temperament [[semaphore]] supports [[archylino]] (2L3M2s), [[diasem]] (5L2M2s), and other scales in the [[Tas series]].
+== Example ==
+Consider the Ptolemaic diatonic scale, {9/8, 5/4, 4/3, 3/2, 5/3, 15/8, 2/1}, which is nicetone with L = 9/8, M = 10/9, and s = 16/15. This scale is epimorphic because we can apply ⟨7 11 16], the [[7edo]] [[patent val]], to map the intervals into the number of scale steps:
+<math>
+\left(\begin{array} {rrr}
+& 11 & 16
+\end{array} \right)
+\left(\begin{array}{rrrrrrr}
+-3 & -2 & 2 & -1 & 0 & -3 & 1 \\
+& 0 & -1 & 1 & -1 & 1 & 0 \\
+& 1 & 0 & 0 & 1 & 1 & 0
+\end{array}\right)
+=
+\left(\begin{array}{rrrrrrr}
+& 2 & 3 & 4 & 5 & 6 & 7
+\end{array}\right)
+</math>
+where the columns of the 3×7 matrix are the scale intervals written in [[monzo]] form. Hence, 7edo (equipped with its patent val) is an epimorphic temperament of the Ptolemaic diatonic scale.
 == Facts ==
 === Definition: constant structure (CS) ===