Elf: Difference between revisions
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# Take all intervals in the JI group of the temperament which lie within an octave. | # 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. | # 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''}}, select the ''i''th degree of the scale from the preimage of ''i''\''n'' under ''V'' to be the least temperamentally-complex interval (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 preimage of ''i''\''n'' under ''V'' to be the least temperamentally-complex interval (with ties broken by Benedetti height) in the portion of the JI preimage of ''i''\''n'' that remains after step 2. | ||
# 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. This step is done so that it becomes a scale of the given temperament. The result is an elf. | ||
Revision as of 19:39, 17 May 2026
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.
- Construct a detempering of n-edo as follows: For each integer value 1 ≤ i ≤ n, select the ith degree of the scale from the preimage of i\n under V to be the least temperamentally-complex interval (with ties broken by Benedetti height) in the portion of the JI preimage of i\n that remains after step 2.
- 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
13-limit leapday
11-limit magic
11-limit miracle
13-limit myna
13-limit octacot
13-limit qilin
13-limit sensus
11-limit valentine
Rank three examples
11-limit jove
13-limit madagascar
- elfmadagascar7
- elfmadagascar8d
- elfmadagascar9
- elfmadagascar10
- elfmadagascar12f
- elfmadagascar14c
- elfmadagascar15
11-limit portent
11-limit thrush
11-limit zeus
Rank four examples
Keenanismic
- elfkeenanismic7
- elfkeenanismic8d
- elfkeenanismic9
- elfkeenanismic10
- elfkeenanismic11c
- elfkeenanismic12
- elfkeenanismic19