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 a one-to-one [[detempering]] of ''n''-edo via a [[val]] ''V'' which may not, and characteristically does not, support 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 a one-to-one [[detempering]] of ''n''-edo via a [[val]] ''V'' which may not, and characteristically does not, support the temperament. This allows the elf to have more freedom in scale size and structure but still to possess the coherence induced by the one-to-one detempering.

To construct an elf given an ''n''-edo val ''V'' and a temperament, the following steps are used:

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

# For each interval of the 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.~~

# Construct a detempering of ''n''-edo as follows: For each integer value {{nowrap|1 ≤ ''i'' ≤ ''n''}}, select the ''i''th degree of the scale from the JI preimage of ''i''\''n'' under ''V'' to be the least temperamentally-complex interval (with ties broken by Benedetti height) among the JI intervals that remain after step 2.

# Construct a detempering of n-edo as follows: For each integer value {{nowrap|1 ≤ ''i'' ≤ ''n''}}, ~~set~~ the ''i''th degree of the scale ~~to be~~ the ~~least (according to the ordering in step 3) interval~~ ''c'' ~~in the listing such that {{nowrap|~~''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. This step is done so that it becomes a scale of the given temperament. The result is an elf.

# 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 a one-to-one [[detempering]] of ''n''-edo via a [[val]] ''V'' which may not, and characteristically does not, support 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 a one-to-one [[detempering]] of ''n''-edo via a [[val]] ''V'' which may not, and characteristically does not, support the temperament. This allows the elf to have more freedom in scale size and structure but still to possess the coherence induced by the one-to-one detempering.
 To construct an elf given an ''n''-edo val ''V'' and a temperament, the following steps are used:
 # Take all intervals in the JI group of the temperament which lie within an octave.
 # For each interval of the 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.
+# Construct a detempering of ''n''-edo as follows: For each integer value {{nowrap|1 &le; ''i'' &le; ''n''}}, select the ''i''th degree of the scale from the JI preimage of ''i''\''n'' under ''V'' to be the least temperamentally-complex interval (with ties broken by Benedetti height) among the JI intervals that remain after step 2.
-# Construct a detempering of n-edo as follows: For each integer value {{nowrap|1 &le; ''i'' &le; ''n''}}, set the ''i''th degree of the scale to be the least (according to the ordering in step 3) interval ''c'' in the listing such that {{nowrap|''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. This step is done so that it becomes a scale of the given temperament. The result is an elf.
-# Temper this detempering of n-edo using a [[tuning map]] for the temperament.
-The result is an elf.
 == Rank two examples ==