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.

# 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 temperamentally-complex interval ''c'' in~~ the preimage of ''i''\''n'' ~~such that {{nowrap|~~''V''~~(''c'') {{=}} ''i''}}; which is~~ to ~~say,~~ the ~~interval of~~ least ~~[[Complexity#Complexity of an~~ interval ~~in a temperament|temperamental complexity]]~~ with ties broken by Benedetti height.

# 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.

# Temper this detempering of ''n''-edo using a [[tuning map]] for the temperament. The result is an elf.

# 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.

== 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.
-# 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 temperamentally-complex interval ''c'' in the preimage of ''i''\''n'' such that {{nowrap|''V''(''c'') {{=}} ''i''}}; which is to say, the interval of least [[Complexity#Complexity of an interval in a temperament|temperamental complexity]] with ties broken by Benedetti height.
+# 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.
-# Temper this detempering of ''n''-edo using a [[tuning map]] for the temperament. The result is an elf.
+# 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.
 == Rank two examples ==