Elf: Difference between revisions

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An '''elf''' is a scale in a [[regular temperament]] which is tempered from a [[just intonation]] (JI) scale in the group of the temperament which is [[Periodic scale#Epimorphism|epimorphic]] via a val V which may not be, and characteristically is not, a val supporting the temperament. This allows the elf to have more freedom in scale size and structure but still to possess the coherence induced by the epimorphic mapping.

To construct an elf, take the intervals in the JI group of the temperament which lie within an octave and keep only the least complex (in terms of [[Benedetti height]]) representative for each corresponding interval of the temperament. Order the remaining JI intervals by increasing temperamental complexity, breaking ties by increasing Benedetti complexity. For each integer value 1 ≤ i ≤ V(2), set the ith element of a [[transversal]] for the scale to be the first interval c in the listing such that V(c) = i; which is to say, the interval of least temperamental complexity with ties broken by Benedetti height. The tempering of this transversal by a tuning map for the temperament is the elf.

To construct an elf, take the intervals in the JI group of the temperament which lie within an octave and keep only the least complex (in terms of [[Benedetti height]]) representative for each corresponding interval of the temperament. Order the remaining JI intervals by increasing temperamental complexity, breaking ties by increasing Benedetti complexity. For each integer value 1 ≤ i ≤ V(2), set the ith element of a [[transversal]] for the scale to be the first interval c in the listing such that V(c) = i; which is to say, the interval of least temperamental complexity with ties broken by Benedetti height. The tempering of this transversal by a [[tuning map]] for the temperament is the elf.

== Rank two examples ==

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 An '''elf''' is a scale in a [[regular temperament]] which is tempered from a [[just intonation]] (JI) scale in the group of the temperament which is [[Periodic scale#Epimorphism|epimorphic]] via a val V which may not be, and characteristically is not, a val supporting the temperament. This allows the elf to have more freedom in scale size and structure but still to possess the coherence induced by the epimorphic mapping.
-To construct an elf, take the intervals in the JI group of the temperament which lie within an octave and keep only the least complex (in terms of [[Benedetti height]]) representative for each corresponding interval of the temperament. Order the remaining JI intervals by increasing temperamental complexity, breaking ties by increasing Benedetti complexity. For each integer value 1 ≤ i ≤ V(2), set the ith element of a [[transversal]] for the scale to be the first interval c in the listing such that V(c) = i; which is to say, the interval of least temperamental complexity with ties broken by Benedetti height. The tempering of this transversal by a tuning map for the temperament is the elf.
+To construct an elf, take the intervals in the JI group of the temperament which lie within an octave and keep only the least complex (in terms of [[Benedetti height]]) representative for each corresponding interval of the temperament. Order the remaining JI intervals by increasing temperamental complexity, breaking ties by increasing Benedetti complexity. For each integer value 1 ≤ i ≤ V(2), set the ith element of a [[transversal]] for the scale to be the first interval c in the listing such that V(c) = i; which is to say, the interval of least temperamental complexity with ties broken by Benedetti height. The tempering of this transversal by a [[tuning map]] for the temperament is the elf.
 == Rank two examples ==