Elf: Difference between revisions

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# 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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 # 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 ==