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.

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, the following steps are used:

# Take all intervals in the JI group of the temperament which lie within an octave.

# For each interval of a temperament, keep only the least complex (in terms of [[Benedetti height]]) JI interpretation of that interval.

# Order the remaining JI intervals by increasing temperamental complexity, breaking ties by increasing Benedetti complexity.

# For each integer value 1 ≤ i ≤ n = V(2) = scale size, set the ith degree of the scale to be the least (according to the ordering in step 3) 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.

# Temper this detempering of n-edo using a [[tuning map]] for the temperament.

The result is an 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.
+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, the following steps are used:
+# Take all intervals in the JI group of the temperament which lie within an octave.
+# For each interval of a temperament, keep only the least complex (in terms of [[Benedetti height]]) JI interpretation of that interval.
+# Order the remaining JI intervals by increasing temperamental complexity, breaking ties by increasing Benedetti complexity.
+# For each integer value 1 ≤ i ≤ n = V(2) = scale size, set the ith degree of the scale to be the least (according to the ordering in step 3) 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.
+# Temper this detempering of n-edo using a [[tuning map]] for the temperament.
+The result is an elf.
 == Rank two examples ==